T T Thai

P Tuan-Anh

R Pello

I Goovaerts
email: ilias.goovaerts@lam.fr

J Richard

A Claeyssens

G Mahler

D J Lagattuta

G De La Vieuville

E Salvador- Solé

T Garel

F E Bauer

R Goovaerts

D Pello

T T Burgarella

J Thai

A Richard

P Claeyssens

R C Tuan-Anh

Arango-Toro

Submitted

E Salvador-Solé

A Jeanneau

B Clément

J Matthee

E Carinos

. Probing the faint-end luminosity function of Lyman-alpha emitters at 3 < z < 7 behind 17 MUSE lensing clusters

Keywords: gravitational lensing, strong -galaxies, luminosity function, mass function -galaxies, high-redshift -Galaxy, general

Brightest pixel.

Participation aux conférences et écoles d'été au cours de la période de thèse :

Abstract

At the end of the Dark Ages, when the first structures of the Universe were formed, radiation from these structures ionized the neutral hydrogen atom surrounding its environment. This crucial phase is known as cosmic re-ionization. Today, various hypotheses exist concerning the main contributors to this process, such as Active Galactic Nuclei (AGN), star-forming galaxies, etc. In this thesis, I will investigate the contribution of the star-forming galaxies to cosmic reionization by studying the evolution of the luminosity function with redshift and estimating their star formation rate density at different redshift ranges.

We have used data of Lyman alpha emitters collected behind 17 lensing clusters observed with Multi Unit Spectroscopic Explorer (MUSE/VLT). Due to the gravitational lensing effect, the signals from background sources are magnified by a factor of ∼10, and distorted, allowing us to observe multiple images of the background galaxy in the most extreme cases. The advantage of employing gravitational lensing lies in the amplified signal, which enables us to detect sources at fainter luminosity compared to the observations conducted in the blank fields. However, the data analyses become complex due to the requirements of projection between source and image planes. To save computational time and avoid redundancy, we perform a selection process for each multiple-imaged system. We choose one representative image based on criteria such as high signal-to-noise ratio,accuracy of magnification, and isolation from other images of the same multiple-imaged system, particularly concerning the presence of bright cluster galaxies. Finally, we retain 600 images of Lyman alpha emitters with redshift ranging from 2.9 to 6.7. Their luminosity is distributed within the range (39 < logL[erg s -1 ] < 43).

After introducing the subject, I present the method used in the thesis to investigate the Luminosity Function (LF), called V max method. V max of a source is the volume of the survey in which that individual source could be detected. Inverting this parameter is the contribution of the source to the galaxy number density. This procedure was developed by de la Vieuville et al., 2019 (hereafter DLV 2019) and has been applied in the present work with four times larger data sample, along with improvements of the lensing and quantity measuring the probability for the source to be detected, which is called completeness, which is described in Chapter4 of the thesis.

The data collected from the gravitational lensing is highly efficient for studying the LF at the fainter limits. However, this data becomes less sensitive when investigating the function surrounding the characteristic luminosity value of the Schechter function.

To address this, we utilized the average LF point values obtained from literature in the same luminosity and redshift bins as a constraint at the bright end. The final results revealed a slight evolution of the slope value with four redshift ranges 2.9 < z < 4.0, 4.0 < z < 5.0, 5.0 < z < 6.7, and 2.9 < z < 6.7. The characteristic value L * of the Schechter function is well measured and consistent with the previous literature, whereas the normalization factor Φ * shows a strong dependence on the literature value used as a constraint.

Some LF points allocated at the faintest luminosity bins are not included in the fitting due to high magnification and too low completeness. They tend to show a turnover at the faintest bins and at the highest redshift intervals. To address this, we have introduced a modified Schechter function by adding an exponential part representing the turnover. This shape has been discussed in previous literature. We compared our results in the highest redshift bin 5.0 < z < 6.7 to the theoretical/numerical predictions by Garel et al. (2021), which correct for the attenuation of IGM, as well as those of Salvador-Solé et al. (2022), who study single and double reionization scenarios. This comparison shows good consistency between our results and the predictions, without any re-normalization, but it does not allow us to distinguish between different scenarios. This thesis also discusses the impact of source selection, completeness threshold, and flux measurement on the final shape of the LF at the global redshift range. The best-fit results of the Schechter function at different redshift ranges allow us to determine the luminosity density and convert it to the star formation rate density. These results, compared with those of the critical value for the star formation density (i.e., the one which would be required to reionize the Universe at a given redshift), suggest that galaxies selected by their Lyα emission could be responsible for reionization assuming a Lyα photon escape fraction of 8%, with a typical clumping factor of ∼ 3. In any case, they have a substantial contribution to the reionization process.

After my thesis, I plan to continue this work on the LF of LAEs using data collected behind 25 lensing clusters belonging to the MUSE GTO program, or already available on the ESO archives. I also worked during my thesis on the construction of the catalogs for the fields used in this work, by determining the redshifts of all the background galaxies from signatures other than Lyα, such as the lines in the Balmer's series or OII-3727 at lower redshift. I also plan to work on JWST and Euclid data in a higher redshift range for the detection of LAEs. The determination of the LF remains fundamental in order to study the global evolution of ionizing sources as a function of redshift. The excellent combination between JWST and MUSE also offers the possibility to analyze the production of ionizing photons with high efficiency.

Keywords: gravitational lensing clusters, faint luminosity function, star formation rate density.

Résumé

A la fin des âges des sombres, lorsque les premières structures de l'Univers ont été formées, le rayonnement de ces structures a ionisé les atomes d'hydrogène neutre entourant leur environnement. C'est une phase cruciale de l'histoire de l'Univers connue comme réionisation cosmique. De nos jours, il existe un certain nombre d'hypothèses concernant les principaux contributeurs à ce processus, tels que les AGN (Active Galactic Nuclei ou Noyaux Actifs de Galaxies), ou les galaxies formant des étoiles, pour ne citer que les principaux contributeurs. Dans cette thèse, j'étudierai la contribution des galaxies formant des étoiles à la réionisation cosmique, en étudiant l'évolution de la fonction de luminosité avec le décalage spectral vers le rouge (noté "redshift" par la suite), et de leur densité du taux de formation d'étoiles à différentes plages de redshift.

Nous avons utilisé des données correspondantes à des émetteurs Lyman alpha, recueillies derrière 17 amas de galaxies connus en tant que lentilles gravitationnelles, observés avec le Multi Unit Spectroscopic Explorer (MUSE/VLT). En raison de l'effet de lentille gravitationnelle, les signaux des sources d'arrière-plan sont amplifiés d'un facteur ∼ 10 et déformés, ce qui nous permet d'observer plusieurs images du même objet d'arrière-plan dans les cas les plus extrêmes. L'avantage d'employer des lentilles gravitationnelles réside dans l'amplification du signal, qui permet de détecter des sources à la luminosité plus faible par rapport aux observations effectuées dans des champ vides. La limitation de la méthode est que le traitement à effectuer pour exploiter les données devient très complexe en raison des exigences de projection entre les plans source et image, à l'aide d'un modèle de lentille bien adapté à chacun des amas. Pour gagner du temps de calcul et éviter des calculs redondants, pour chaque système à images multiples nous choisissons une seule image représentative de la galaxie originale, en fonction de critères bien définis, tels que le rapport signal sur bruit élevé, la fiabilité du grossissement, l'isolement de l'image choisie par rapport aux autres images du même système multiple, par rapport des galaxies de l'amas. Au final, l'échantillon contient 600 images d'émetteurs Lyman alpha, avec des décalage vers le rouge allant de 2.9 à 6.7, et avec une luminosité dans la gamme (39 < logL[erg s -1 ] < 43).

Après avoir introduit le sujet, je présente la méthode qui a été utilisée dans cette thèse pour étudier la fonction de luminosité, appelée méthode V max . V max est le volume sondé dans l'étude dans lequel chaque source individuelle aurait pu être détectée, compte tenu de sa luminosité, et de la plage de décalage vers le rouge sondée. L'inversion de cette valeur, affectée d'un poids décrit dans la thèse, représente la contribution de chaque galaxie à la densité numérique des galaxies, pour un bin de luminosité et dans une plage de redshift. Cette méthode a été développée à l'origine par de la Vieuville et al., 2019 (ci-après DLV 2019), et a été appliquée dans le présent travail sur un échantillon contenant quatre fois plus de données, après avoir introduit des améliorations substantielles par rapport à DLV 2019 concernant la prise en compte de la valeur du grossissement et la complétude de la source dans le cube de données.

La collecte de données à partir de lentilles gravitationnelles est assez efficace pour étudier la fonction de luminosité aux limites les plus faibles. Cependant, ces données deviennent moins sensibles lors de l'étude de la fonction entourant la valeur de luminosité caractéristique de la fonction de Schechter. Pour résoudre ce problème, nous avons incorporé des valeurs moyennes de la fonction de luminosité issues de la littérature pour compléter notre fonction de luminosité vers les luminosités les plus brillantes, pour les mêmes plages de redshift, ce qui permet un ajustement global de la Fonction de Luminosité. Le résultat final a révélé une légère évolution de la valeur de la pente avec quatre plages de décalage vers le rouge : 2.9 < z < 4.0, 4.0 < z < 5.0, 5.0 < z < 6.7, et 2.9 < z < 6.7. La valeur caractéristique L * de la fonction de Schechter est conforme à la littérature précédente, alors que le facteur de normalisation Φ * affiche une forte dépendance à la valeur de la littérature utilisée comme contrainte.

Lors de l'étude de la Fonction de Luminosité, certains points dans la région des plus faibles luminosités n'ont pas été pris en compte dans les ajustements, en raison en raison d'un grossissement élevé (avec une très grosse incertitude), d'une complétude trop faible et de données jugées statistiquement insuffisantes. Malgré ces précautions, la Fonction de Luminosité semble montrer un aplatissement, voir une inflexion, aux tranches les plus faibles, pour les plages de redshift les plus élevées. Pour bien appréhender ce comportement, nous avons introduit une fonction de Schechter modifiée en ajoutant une partie exponentielle représentant la configuration observée. Cette forme a été discutée dans la thèse et est conforme à la littérature. Nous avons comparé nos résultats dans le bin de redshift le plus élevé 5.0 < z < 6.7 aux prédictions théoriques/numériques par Garel et al. (2021), qui corrigent de l'atténuation d'IGM, ainsi qu'à celles de Salvador-Solé et al. (2022), qui étudient des scénarios de simple et double réionisation. Cette comparaison montre une grande cohérence entre nos résultats et les prédictions, sans aucune ré-normalisation, sans pour autant pouvoir trancher entre des scénarios différents. Cette thèse discute également sur l'impact de la sélection des sources et le seuil de complétude sur la forme finale de la fonction de luminosité, dans la plage globale de redshift. Les meilleurs résultats d'ajustement de la fonction de Schechter à différentes plages redshift nous permettent de déterminer la densité de luminosité et de la convertir en densité de formation d'étoiles. Ces résultats, comparés à ceux de la valeur critique de la densité de formation d'étoiles (c'est à dire, celui qui serait nécessaire pour réioniser l'univers à un redshift donné) suggèrent que les galaxies sélectionnées par leur émission Lyα pourraient être responsables de la réionisation si on suppose une fraction d'échappement des photon Lyα de 8%, avec un facteur d'agglutination (clumping factor) typique de ∼ 3. Dans tous les cas, ils ont une contribution substantielle au processus de réionisation.

Par la suite, j'espère continuer ce travail sur la fonction de luminosité des émetteurs Lyman alpha en utilisant les données collectées derrière 25 amas-lentilles appartenant au programme MUSE GTO, ou déjà, disponibles sur les archives de l'ESO. J'ai aussi travaillé pendant ma thèse à la constitution des catalogues pour les champs sur lesquels j'ai travaillé, en déterminant les redshifts de toutes les galaxies d'arrièreplan à partir d'autres signatures que Lyα, telles que les raies de la série de Balmer ou OII3727 à plus bas redshift. J'ai prévu également de travailler sur les données JWST et Euclid dans une plage de redshift plus élevé pour les émetteurs Lyman alpha. La détermination de la fonction Luminosité reste fondamentale afin d'étudier l'évolution globale des sources ionisantes en fonction du redshift. L'excellente combinaison entre JWST et MUSE offre également la possibilité d'analyser la production de photons ionisants avec efficacité.

Tóm tắt

Ở giai đoạn cuối của thời kì tăm tối, khi các cấu trúc đầu tiên của Vũ trụ được hình thành, phát xạ từ chúng ra bên ngoài môi trường làm ion hóa nguyên tử hydro trung hòa ở xung quanh. Quá trình này được gọi là tái ion hóa, đánh dấu chuyển đổi pha của vật chất trong Vũ trụ. Cho đến nay, có rất nhiều giả thuyết liên quan đến đối tượng chính đóng góp vào quá trình này: tâm thiên hà hoạt động (AGN), thiên hà có hoạt động hình thành sao... Trong luận án này, tôi trình bày đóng góp của các thiên hà có hoạt động hình thành sao đối với quá trình tái ion hóa Vũ trụ bằng cách nghiên cứu sự tiến triển của hàm độ trưng theo giá trị dịch chuyển đỏ và ước lượng mật độ hình thành sao trong những khoảng giá trị dịch chuyển đỏ khác nhau.

Dữ liệu sử dụng trong luận án là các nguồn phát xạ vạch Lyman alpha đằng sau 17 cụm thiên hà thấu kính hấp dẫn quan sát bởi máy quang phổ đa đối tượng Multi Unit Spectroscopic Explorer (MUSE/VLT). Do hiệu ứng thấu kính hấp dẫn, tín hiệu từ các nguồn này được phóng đại (trong một số trường hợp, tín hiệu phóng đạicó thể gấp 10 lần), đường đi của tín hiệu bị bẻ cong, do đó ta có thể quan sát thất nhiều hình ảnh khác nhau của cùng một nguồn. Ưu điểm này cho phép chúng ta tìm kiếm được nhiều nguồn trong những khoảng độ trưng mờ nhất so với các quan sát được thực hiện trong trường trống. Tuy nhiên, quá trình tính toán và xử lí số liệu sẽ trở nên phức tạp hơn do yêu cầu về việc tạo ảnh và giải ảnh thấu kính qua lại giữa mặt phẳng nguồn và mặt phẳng ảnh. Để rút ngắn quá trình tính toán và tránh trùng lặp, chúng tôi chọn ra một ảnh đại diện cho mỗi nguồn phát xạ vạch Lyman alpha dựa trên các tiêu chí về tỉ lệ tín hiệu trên nhiễu, độ chính xác của giá trị khuếch đại, tính cô lập của ảnh được chọn so với những ảnh khác của cùng một nguồn... Cuối cùng, 600 ảnh đại diện cho các nguồn phát xạ vạch Lyman alpha được lựa chọn, giá trị dịch chuyển đỏ nằm trong khoảng 2.9 đến 6.7, phân bố trong khoảng độ trưng (39 < logL[erg s -1 ] < 43).

Sau khi giới thiệu về chủ đề nghiên cứu, tôi trình bày về phương pháp V max được sử dụng trong luận án để nghiên cứu hàm độ trưng (LF -Luminosity function). V max là thể tích mà một nguồn riêng lẻ được phát hiện trong khảo sát. Nghịch đảo của đại lượng này xác định đóng góp của nguồn đó vào mật độ thiên hà. Phương pháp này được phát triển bởi de la Vieuville et al., 2019 (sau đây gọi tắt là DLV 2019), và được chúng tôi áp dụng với số lượng mẫu nhiều hơn bốn lần, kết hợp với một số cải tiến liên quan đến việc hiệu chỉnh độ khuếch đại, giá trị bổ chính liên quan đến khả năng tìm kiếm nguồn thực tế trong khối dữ liệu MUSE.

Dữ liệu thu được từ thiên hà thấu kính hấp dẫn tỏ ra hiệu quả để nghiên cứu hàm độ trưng trong những khoảng giới hạn mờ nhất. Tuy nhiên, dữ liệu này trở nên kém nhạy hơn khi nghiên cứu ở những vùng có giá trị độ trưng lớn, đặc biệt ở vị trí "điểm uốn"L * của hàm Schechter. Để giải quyết vấn đề này, chúng tôi sử dụng giá trị trung bình của hàm độ trưng từ các kết quả nghiên cứu trước đó được thực hiện trong cùng khoảng độ trưng và dịch chuyển đỏ để ràng buộc ở vùng giá trị độ trưng lớn. Kết quả cuối cùng chỉ ra sự tiến triển nhỏ giá trị độ dốc của hàm độ trưng theo bốn khoảng dịch chuyển đỏ 2.9 < z < 4.0, 4.0 < z < 5.0, 5.0 < z < 6.7, và 2.9 < z < 6.7. Khớp hàm Schechter với các điểm LF, "điểm uốn"L * của hàm Schechter hoàn toàn phù hợp với các kết quả nghiên cứu trước đó, trong khi giá trị chuẩn hóa Φ * phụ thuộc nhiều vào giá trị ràng buộc ở phần độ trưng lớn.

Trong quá trình khớp hàm, có một vài điểm LF ở những phần giới hạn độ trưng mờ nhất không được tính đến do hầu hết các nguồn phân bố trong khoảng này được có giá trị phóng đại vô cùng lớn, giá trị hoàn chỉnh (completeness) nhỏ... Giá trị hàm độ trưng trong những khoảng này gợi ý xu hướng quay đầu (turnover) ở những khoảng dịch chuyển đỏ lớn nhất. Đặc điểm này đã được thảo luận trong nhiều nghiên cứu trước đây. Ở khoảng dịch chuyển đỏ lớn nhất 5.0 < z < 6.7, chúng tôi so sánh kết quả của luận án với các tiên đoán lí thuyết của Garel et al. (2021) về sự phụ thuộc hàm độ trưng với dịch chuyển đỏ khi tính đến ảnh hưởng của IGM, hay nghiên cứu của Salvador-Solé et al. (2022) với mô hình tái ion hóa đơn, tái ion hóa kép. Tiên đoán từ những mô hình này phù hợp với kết quả chúng tôi mà không cần tới bất kì hệ số chuẩn hoá thêm nào. Tuy vậy kết quả hiện tại của chúng tôi chưa cho phép phân biệt chất lượng các mô hình khác nhau. Luận án cũng thảo luận về ảnh hưởng của việc lựa chọn nguồn, cắt ngưỡng giá trị hoàn chỉnh, phương pháp đo thông lượng nguồn, đến hình dáng của hàm độ trưng khi nghiên cứu sự tiến triển ở khoảng dịch chuyển đỏ 2.9 < z < 6.7. Kết quả khớp hàm Schechter trong những khoảng dịch chuyển đỏ khác nhau cho phép xác định mật độ độ trưng và mật độ tốc độ hình thành sao. So sánh kết quả này với giá trị tới hạn về mật độ hình thành sao (cụ thể là mật độ cần thiết để tái ion hóa Vũ trụ tại một dịch chuyển đỏ nhất định) cho thấy rằng, các thiên hà phát xạ vạch Lyman alpha đóng vai trò quan trọng trong quá trình tái ion hóa, với giả sử tỉ lệ thoát Lyman alpha photon là 8% và hệ số cụm (clump) điển hình là 3.

Sau luận án này, tôi dự định tiếp tục công việc này bằng cách sử dụng dữ liệu thu thập sau 25 cụm thiên hà thấu kính hấp dẫn thuộc chương trình MUSE GTO, hoặc có sẵn trên kho lưu trữ của ESO. Trong quá trình thực hiện luận án, bên cạnh việc xác định giá trị dịch chuyển đỏ của các nguồn phát xạ vạch Lyman alpha, dạnh mục các nguồn phát xạ vạch Balmer, OII-3727 cũng đã được xây dựng. Do đó tôi sẽ nghiên cứu sự tiến triển hàm độ trưng của các nguồn phát xạ những vạch này ở khoảng dịch chuyển đỏ thấp hơn. Tôi cũng lên kế hoạch làm việc với dữ liệu của JWST và Euclid để tìm kiếm các nguồn phát xạ vạch Lyman alpha trong những khoảng dịch chuyển đỏ lớn hơn. Những kết quả nghiên cứu này cho phép xác định sự tiến triển toàn cục của các nguồn tái ion hóa đối với dịch chuyển đỏ. Sự kết hợp tuyệt vời giữa JWST và MUSE cũng mang đến khả năng phân tích quá trình tạo ra các photon ion hóa với hiệu quả cao.

Substantial summary

The success in operating astronomical instruments in recent years, such as Multi-Unit Spectroscopic Explorer (MUSE) at the Very Large Telescope (VLT), James Webb Space Telescope (JWST), etc., has driven research in the early Universe to become more dynamic than ever before. The ultimate goal of scientists working in this field is to study the sources of cosmic re-ionization and estimate their contribution to the re-ionization budget.

The re-ionization phase holds significant importance in the evolution of matter in the Universe. It began shortly after the formation of the first structures when their emissions started ionizing the neutral hydrogen atoms surrounding them. To this day, the question of the main contributors to cosmic reionization remains unclear. It could be the Active Galactic Nuclei (AGN) owning to their brightness and high escape fraction of ionizing photons. It could also be the star-forming galaxies, powered by young massive stars inside them. In the context of the thesis, I have focused on studying star-forming galaxies by investigating the evolution of the luminosity function of these sources. This will provide valuable insights into the density of the galaxies per cosmic co-moving volume per luminosity and its contribution to cosmic reionization. Previous research has predicted that the faint galaxies with low mass would contribute significantly to this process. However, observations conducted in blank fields have not been able to reach the faintest luminosity limits at the higher redshift ranges. Observations via gravitational lensing have considerably improved the sample size in the regime, as the signal has been magnified. This thesis will approach the study of the star-forming galaxies that emit the Lyman alpha line behind 17 lensing clusters observed by MUSE/VLT.

The first chapter introduces the research topic, the current understanding of the first structures in the Universe, the ionization and recombination process. It also provides an overview of the spectral properties of star-forming galaxies, including characteristic features and methods to identify them from a given data cube.

The second chapter briefly presents information about the Multi-Unit Spectroscopic Explorer (MUSE/VLT), its scientific goals, and the recent impressive results. In the third part of the chapter, I present the general features of 17 lensing clusters based on previous results and mass models of such clusters.

Because of the lensing effect, the signals from background galaxies are magnified, and their images are distorted, allowing us to observe multiple images of the same source. To reduce computational redundancy and save time, this study selects one representative image for each source based on criteria, such as high signal-to-noise, magnification uncertainty, and isolation compared to the other images of the same system. The chapter concludes with general information of 600 images of Lyman alpha emitters behind 17 gravitational lensing clusters within a redshift range of 2.9 < z < 6.7. These images are distributed in a luminosity interval of 39 < log L [erg s -1 ] < 43. A comparison is made between the present data sample and the data from a previous study conducted by DLV 2019. The data in the DLV 2019 were collected behind only four gravitational lensing clusters, with 128 images of Lyman alpha emitters retained for the LF computation. It is clear that the present data sample is four times larger than the previous one. At the lower luminosity limit (log L[erg s -1 ] ∼ 40), the present number of sources is substantially higher in comparison. Consequently, the sample size used for the thesis significantly improves the information about the luminosity distribution and also its statistical significance in the faintest regions. A comparison between the present sample and the previous one used in the DLV 2019 is shown in Figure 1.

Chapter 3 describes the process of computing the V max value for each source within our survey. This thesis includes data collected in 18 regions (MACS0416 was observed in two regions); therefore, the V max of individual sources must be calculated for all regions, including those where the source is absent. The pipeline was initially developed by DLV 2019 to study the luminosity function of 128 Lyman alpha emitters behind only four lensing clusters. In this chapter, I summarize the main steps, with some new improvements that have been taken into account the for calculation procedure, and made a comparison of the obtained result on the volume of the survey to the previous work. It is worth mentioning that the present results indicate a total co-moving volume of approximately 50000 Mpc 3 , which is three times larger than that in DLV 2019. However, this value is significantly smaller than the one found in the works using data observed in the blank fields, due to the constraints imposed by gravitational lensing effects, which reduce the effective total field of view. Chapter 4 presents the main results of the thesis on the luminosity function of 600 Lyman alpha emitters in four redshift intervals. By definition, the luminosity function is defined as the number of galaxies per unit of co-moving volume (number density), per luminosity interval. To compute the luminosity function, in addition to determining the V max of a source as described above, one must account for the fact that not all sources in the data sample have a 100% probability of being detected. Hence, there is a need to correct for completeness, which gives the true probability of finding a source in the cube. The procedure was carried out for each source by reconstructing the profile of the galaxy in the image plane and randomly injecting this model into different places in the cube, to capture the real effect of the noise. The 1: Total co-moving volume of 17 gravitational lensing clusters in the present work within a redshift range of 2.9 < z < 6.7.

Figure 3: Source distribution in the redshift interval 2.9 < z < 6.7 for a luminosity bin width of 0.25 in logarithmic scale. The blue and orange lines are the distribution before and after correcting for completeness. considered to be detected when the maximum deviation between the injected position and the detected position of the source is less than 0.8". The completeness is determined directly by the ratio between the number of successful detections, under the same conditions as the original extraction, and the number of objects injected.

There are several differences between this work and that of DLV 2019 in consideration of completeness, as well as in the detailed discussion of selection effects.

-The impact of image size on the final result of completeness value has been discussed. The size has to be large enough to capture the noise in the source region of the slice where the Lyman alpha emission line reaches the peak. In the thesis, I increase the size from 30" × 30" to 80" × 80".

-With the goal of maximizing the number of sources used for the luminosity function computation and subsequent advancements, we have excluded sources with completeness values below 1%. Compared to the work by DLV 2019, this value was set at 10%.

-The dependence of detected flux, noise level, and completeness on the redshift of the individual sources is shown in Figure 2. The number of sources used for the luminosity function in the whole redshift interval, for different luminosity intervals, has been presented in Figure 3, before and after completeness correction. Most weak sources have an extremely low completeness value. However, when the correction of this parameter is taken into account, the number of sources in these bins increases significantly. On the other hand, for sources allocated in the bright parts, the probability of detection is already high, so whether we apply the completeness correction or not does not markedly change the source distribution. In the luminosity range 42.25 < log L [erg s -1 ] < 42.5 of Figure 3, the appearance of a spike can be explained by the Luminosity function for 2.9 < z < 6.7

Blanc 2011(1.9 < z < 3.8) Drake 2017(2.9 < z < 6.7) Herenz 2019(2.9 < z < 6.7) Sobral 2018(2.5 < z < 6) de La Vieuville 2019(2.9 < z < 6. Luminosity function for 2.9 < z < 4.0 Figure 5: From left to right, top to bottom shows the luminosity function in different redshift intervals: 2.9 < z < 6.7, 2.9 < z < 4.0, 4.0 < z < 5.0, and 5.0 < z < 6.7.

The red crosses represent the results of this work, while other colors in the bright end indicate data from the literature. The blue ones are results from (DLV 2019) used for comparison purposes. The solid brown reddish line is the best-fit Schechter function. The gray crosses obtained from the present work are not included in the fit. The reddish dashed lines correspond to the best fit with a Schechter function modified to take into account the turnover observed in the data points towards the faint end. Redshift Φ * [10 -4 Mpc -3 ] log L * [erg s -1 ] α 2.9 < z < 6. 7 7.41 +2.70 existence of low completeness sources distributed within the given luminosity range.

The luminosity function point within a given redshift bin and a given luminosity range follows the function below:

Φ(L i ) = 1 ∆logL i j 1 C j V max,j (0.1)
where C j ,V max,j is the completeness and V max of a given source, ∆l og L i is the i t h luminosity bin width. The luminosity function points obtained from the present sample are fitted using the Schechter function:

Φ(L)d L = Φ * L L * α exp - L L * d L L * (0.2)
where α is the slope of the Schechter function at the faint end, L * is the characteristic luminosity when the Schechter function transitions from exponential law at the bright end to the power law at the faint end, Φ * is the normalization factor. This luminosity function is investigated in four redshift intervals: 2.9 < z < 6.7, 2.9 < z < 4.0, 4.0 < z < 5.0 and 5.0 < z < 6.7. Using data collected from gravitational lensing clusters has proven to be effective in studying the luminosity function at the faint end. However, such data becomes less effective at the nearby and higher characteristic luminosity values around L * . This problem can be addressed by including the results from previous research on the luminosity function conducted within the same redshift and luminosity range as a constraint factor for the bright end of the luminosity function.

The results of the Schechter function fitting are shown in Figure 5, with gray crosses excluded from the fit due to low completeness and highly magnified sources residing in these bins.

The best-fit values of the three free parameters of the Schechter function are presented in Table 2. The slope values α of the Schechter function display a slight evolution with redshift. Within the uncertainties, the L * and Φ * are consistent with the previous results.

At higher redshift intervals and fainter luminosity limits, several studies have discussed the possibility of a turnover in the shape of the luminosity function, such as Atek 2018; Bouwens et al. 2022. To gain a better understanding of this behavior, we have introduced a modified Schechter function by adding an exponential part to represent the observed configuration: where L T is the value where the luminosity function starts showing a turnover, it is also the value in which d Φ/d L = 0, m is the curvature value orienting the shape (upward if m < 0 or downward if m > 0) of the turnover part. This is presented in the lower panels of Figure 5. In the thesis, we discuss the effects of the criterion adopted by DLV 2019 to ignore sources that have completeness values below 10% and study the evolution of luminosity function with redshift. Finally, whether sources with completeness value below 1% or 10% are removed, the turnover in shape can still be seen in the vicinity of log L [erg s -1 ] = 41.

The thesis also presents the effect of source selection on the shape of the luminosity function versus redshift. It is worth noting that, in the present work we are using sources with a secure redshift (zcon f = 2, 3) to analyze the evolution of the luminosity function and ignore sources having a lower secure zcon f = 1 due to low signal to noise or displaying an ambiguous line shape. For this reason, we have conducted two tests by including half of zcon f = 1 and all the zcon f = 1 to the present data sample. In addition to source selection effects, we also discuss various factors contributing to systematic uncertainties in the faint end slope, including different completeness thresholds, different flux measurements, and varying fitting models. The final results are reported in Table 3. We compared our results in the highest redshift bin 5.0 < z < 6.7 to the theoretical/numerical predictions by Garel et al. 2021, which correct for the attenuation of IGM, as well as those of Salvador-Solé et al. 2022, who study single and double reionization scenarios. This comparison shows good consistency between our results and the predictions, without requiring any re-normalization. However, it does not enable us to distinguish between different scenarios, as shown in Figure 6.

Based on the obtained results on the luminosity function, Chapter 5 focuses on luminosity density and the conversion of this parameter into star formation rate density by integrating the luminosity function with respect to luminosity. While the shape of the luminosity function at the bright end remains unaffected by the completeness threshold value and source selection and is entirely consistent with the previous studies conducted in the blank fields, the faintest part is constrained by both these mentioned factors. For this reason, the result of the integration is not significantly influenced by the upper limitation, whereas the lower limitation is certainly an impacting factor. I applied two lower limitation values: one is log L [erg s -1 ] = 39.5 to cover all the data samples, and the other is log L [erg s -1 ] = 41 to include only the most secure determinations of the LF. These results, when compared to the critical value for the star formation density (i.e., the amount required to reionize the Universe at a given redshift), suggest that galaxies selected by their Lyα emission could be responsible for reionization assuming a Lyα photon escape fraction of 8%, with a typical clumping factor of ∼ 3. In any case, they appear to play a substantial role in the reionization process. Figure 8: Evolution of star formation rate density with redshift. The red and brownreddish crosses represent the results obtained when applying lower integration limits of log L [erg s -1 ] = 39.5 and log L [erg s -1 ] = 41.0, respectively. The blue, orange, and green solid lines indicate the critical values obtained with escape fraction values of 3%, 8% and 25%, combined with a typical clumping factor of 3.

Chapter 6 presents a general discussion of the final results and future prospects. Subsequently, I hope to continue this work on the Luminosity Function of Lyman alpha emitters using data collected behind 25 lens clusters belonging to the MUSE GTO program, or already available on the ESO archives. I also worked during my thesis on the construction of the catalogs for the fields on which I worked, by determining the redshifts of all the background galaxies from signatures other than Lyα, such as the lines of Balmer's series or OII3727 at lower redshift. I also plan to work on JWST data and Euclid in a higher redshift range for Lyman alpha emitters. The determination of the Luminosity function remains fundamental in order to study the global evolution of ionizing sources according to the redshift. The excellent combination between JWST and MUSE also offers the possibility to analyze the production of ionizing photons with high efficiency.

Résumé substantiel

Le succès dans l'exploitation d'instruments astronomiques au cours des dernières années, tels que l'Explorateur Spectroscopique Multi-Unités (MUSE) sur le Very Large Telescope (VLT) de l'ESO, ou le télescope spatial James Webb (JWST), a poussé la recherche dans l'Univers primordial à devenir plus dynamique qu'il ne l'avait jamais été auparavant. Le but ultime des scientifiques travaillant dans ce domaine est d'étudier les sources de la réionisation cosmique et d'estimer leur contribution relative à ce processus.

La phase de réionisation revêt une importance significative dans l'évolution de la matière dans l'Univers. Cela a commencé peu de temps après la formation des premières structures lorsque les émissions des premières étoiles et galaxies ont commencé à ioniser les atomes d'hydrogène neutre qui les entouraient. A ce jour, la question des principaux contributeurs à la réionisation cosmique reste encore floue. Il peut s'agir des noyaux galactiques actifs (AGN) possédant, à cause de leur luminosité, une fraction d'échappement élevée de photons ionisants. Il peut également s'agir des galaxies à formation d'étoiles, avec un flux ionisant provenant des jeunes étoiles massives à l'intérieur de ces galaxies. Dans le cadre de la thèse, je me suis concentrée sur l'étude des galaxies à formation stellaire, en étudiant l'évolution de la fonction de luminosité de ces sources avec le redshift. L'idée ici est de fournir des informations précieuses sur la densité des galaxies format des étoiles par unité de volume et de luminosité, et de déterminer ainsi leur contribution à la réionisation cosmique. Des recherches antérieures ont prédit que les galaxies faibles à faible masse contribueraient de manière significative à ce processus. Cependant, les observations effectuées dans des champ vides n'ont pas pu atteindre les limites les plus faibles en luminosité qui sont nécessaires pour voir ces sources directement, dans les plages de redshift les plus élevées, tandis que les observations dans les champs des lentilles gravitationnelles ont considérablement amélioré la taille de l'échantillon grâce à l'effet d'amplification du signal. Cette thèse abordera l'étude des galaxies formant des étoiles qui émettent la raie Lyman alpha de derrière 17 amas de lentilles observés par MUSE/VLT.

Le premier chapitre présente le sujet de recherche, la compréhension actuelle des premières structures de l'Univers, l'ionisation et le processus de recombinaison. Il fournit également un aperçu des propriétés spectrales des galaxies formant des étoiles à travers certaines caractéristiques et méthodes pour les identifier à partir d'un cube de données donné.

Le deuxième chapitre présente brièvement des informations sur le Multi-Unit Spectroscopic Explorer (MUSE/VLT), ses objectifs scientifiques et ses récents résultats impressionnants. Dans la troisième partie du chapitre, j'ai présenté les caractéristiques générales des 17 amas-lentilles basées sur des résultats antérieurs, et surtout des modèles de distribution masse de ces amas, utilisés dans la thèse. En raison de l'effet de lentille, le signal des galaxies d'arrière-plan a été amplifié, les images sont déformées nous permettant d'observer parfois plusieurs images de la même source. Pour réduire la redondance de calcul et gagner du temps, cette étude sélectionne une image représentative dans chaque système multiple en fonction de critères spécifiques, tels qu'un rapport signal/bruit élevé, une valeur de l'amplification bien connu, et la source sélectionnée est toujours la plus isolée par rapport aux autres images du même système par rapport aux galaxies brillantes de l'amas. Le chapitre se termine par des informations générales sur les 600 images d'émetteurs Lyman alpha derrière 17 amas de lentilles gravitationnelles utilisées dans ce travail, dans une plage de décalage vers le rouge comprise dans l'intervalle 2.9 < z < 6.7 , et distribués dans un intervalle de luminosité 39 < log L [erg s -1 ] < 43. Une comparaison est faite entre le présent échantillon de données et les données d'une étude précédente menée par DLV 2019. Le chapitre 3 décrit le processus de calcul de la valeur du V max pour chaque source dans le volume étudié par notre relevé. La thèse inclut les données recueillies dans 18 régions ; par conséquent, la valeur de V max pour les sources individuelles doit être calculée en considérant toutes les régions, y compris les régions dans lesquelles la source n'est pas présente. Le pipeline a été développé par DLV 2019 pour étudier la fonction de luminosité de 128 émetteurs Lyman alpha derrière seulement quatre amas de lentilles. Par conséquent, dans ce chapitre, j'ai résumé les principales étapes en présentant quelques nouvelles améliorations qui ont été introduites pour la procédure de calcul, et fait une comparaison critique entre les résultats actuels et les anciens. Il convient de mentionner que la présente étude s'étend sur un volume comobile total Le chapitre 4 présente les principaux résultats de la thèse sur la fonction de luminosité à partir des 600 émetteurs Lyman alpha dans quatre intervalles de redshift. Par définition, la fonction de luminosité est une grandeur décrivant le nombre de galaxies présentes par unité de volume comobile, par bin de luminosité. Cependant, toutes les sources de l'échantillon de données n'ont pas une probabilité de 100% d'être détectées, d'où la nécessité de corriger de la complétude, qui donne la probabilité réelle de trouver une source dans le cube. La procédure a été effectuée en reconstruisant le profil de la galaxie dans le plan image, et en l'injectant aléatoirement ce modèle dans différents endroits du cube, pour capturer l'effet réel du bruit. La source est considérée comme détectée lorsque l'écart maximal entre la position injectée et la position détectée de la source est de 0.8" d'arc. La complétude est donnée directement par le rapport entre le nombre de détections réussites, avec les mêmes conditions que pour l'extraction originale, et le nombre d'objets injectés. Il y a plusieurs différences entre ce travail et celui de DLV 2019 dans la prise en compte de la complétude, ainsi que dans la discussion détaillée des effets de sélection.

log(L) -6 -5 -4 -3 -2 -1 0 1 log(Φ [(∆ log L = 1) -1 .M pc -3 ])
Luminosity function for 2.9 < z < 6.7

log(L) -6 -5 -4 -3 -2 -1 0 1 log(Φ [(∆ log L = 1) -1 .M pc -3 ])
Luminosity function for 2.9 < z < 4.0

-1 0 1 log(Φ [(∆ log L = 1) -1 .M pc -3 ]) 6 -5 -4 -3 -2
Luminosity function for 5.0 < z < 6.7 FIGURE 5 : De gauche à droite, et de haut en bas, l'évolution de la fonction de luminosité avec différents intervalles de redshift : 2.9 < z < 6.7, 2.9 < z < 4.0, 4.0 < z < 5.0 et 5.0 < z < 6.7. Les croix rouges sont le résultat de ce travail ; les autres couleurs dans la partie claire sont issues de la littérature. Les symbols bleus sont les résultats de (DLV 2019) utilisés à des fins de comparaison. La ligne continue rougeâtre montre la meilleure solution pour l'ajustement avec une fonction de Schechter. Les croix grises obtenues à partir du présent travail n'ont pas été incluses dans l'ajustement. Les lignes pointillées rougeâtres correspondent à l'ajustement avec une fonction de Schechter modifiée afin de prendre en compte l'aplatissement observé dans les données vers les faibles luminosités.

-L'effet de la taille de la zone de recherche où les sources sont injectées a été discuté. La taille doit être suffisamment grande pour présenter le bruit de la tranche dans laquelle la raie d'émission Lyman alpha atteint le pic. Dans la thèse, j'ai augmenté la taille de 30" × 30" à 80" × 80".

-Dans le but de maximiser le nombre de sources utilisées pour le calcul de la fonction de luminosité et les avancées ultérieures, nous avons exclu les sources avec des valeurs de complétude inférieures à 1% par rapport aux travaux de (DLV 2019) où cette valeur a été fixée à 10% -La corrélation entre le flux détecté, le décalage vers le rouge, le niveau de bruit et la valeur de complétude des sources individuelles est illustrée à la Figure 2. Le nombre de sources utilisées pour la fonction de luminosité dans l'ensemble de l'intervalle en redshift, pour différents intervalles de luminosité, a été présenté sur la Figure 3, avant et après correction de complétude. La plupart des sources faibles ont une valeur de complétude extrêmement faible. Cependant, lorsque la correction de ce paramètre est prise en compte, le nombre de sources dans ces bins augmente significativement. D'autre part, pour les sources réparties dans les parties lumineuses, la probabilité de détection est déjà élevée, donc que nous appliquions ou non la correction de complétude ne change pas sensiblement la distribution des sources. Dans la gamme de luminosité 42.25 < log L [erg s -1 ] < 42.5 de la Figure 3, l'apparition d'un pic peut s'expliquer par l'existence de sources avec faible complétude réparties dans la gamme de luminosité donnée.

Les points de la fonction de luminosité dans un intervalle de redshift donné et pour une plage de luminosité donnée suit la fonction ci-dessous :

Φ(L i ) = 1 ∆logL i j 1 C j V max,j (0.4)
où C j , V max,j est la complétude et V max d'une source donnée, log L i est la largeur de ième bin de luminosité. Ensuite l'évolution de la fonction de luminosité avec le redshift sera ajustée par la fonction de Schechter :

Φ(L)d L = Φ * L L * α exp - L L * d L L * (0.5)
où α est la valeur de la pente de la fonction de Schechter à la partie faible, L * est la luminosité caractéristique lorsque la fonction de Schechter passe de la loi exponentielle à la partie en loi de puissances, vers la partie faible, et Φ * est le facteur de normalisation. Cette évolution est étudiée dans quatre intervalles de redshift 2.9 < z < 6.7, 2.9 < z < 4.0, 4.0 < z < 5.0 et 5.0 < z < 6.7. Le fait d'utiliser des données collectées derrière les amas-lentilles gravitationnelles s'est avéré efficace pour étudier la fonction de luminosité vers sa partie faible. Cependant, ces données deviennent moins efficaces à la valeur de luminosité caractéristique proche et supérieure à L * . Ce problème peut être résolu en incluant les résultats de recherches antérieures sur la fonction de luminosité, menées dans la même plage de décalage vers le rouge et de luminosité. Ces données seront utilisées comme des contraintes vers l'extrémité TABLEAU 2 : Les meilleures valeurs d'ajustement de la fonction de Schechter dans différents intervalles de redshift.

Redshift

Φ * [10 -4 Mpc -3 ] log L * [erg s -1 ] α 2.9 < z < 6. 7 7.41 +2.70 brillante de la fonction de luminosité. Les résultats de l'ajustement de la fonction de Schechter sont présentés sur la Figure 5. Les croix grises ne sont pas incluses dans l'ajustement en raison de la faible complétude dans ces intervalles dominés par des galaxies fortement amplifiées. La meilleure valeur d'ajustement de trois paramètres libres de la fonction de Schechter est presentée dans le Tableau 2. Les valeurs de la pente de la fonction de Schechter affichent une légère évolution avec le décalage vers le rouge. Les valeurs de L * et de Φ * se montrent en bon accord avec les résultats précédents, une fois qu'on prends en compte les barres d'erreur.

Pour les intervalles de décalage vers le rouge les plus élevés et vers les plus faibles limites en luminosité, plusieurs études ont discuté de la possibilité d'avoir une fonction de luminosité présentant un aplatissement, voir une inflexion, comme ATEK 2018 ; BOUWENS et al. 2022. Pour bien appréhender ce comportement, nous avons introduit une fonction de Schechter modifiée en ajoutant une partie exponentielle représentant la configuration observée :

Φ(L)exp(-L T /L) m = Φ * L * L L * α exp(-L/L * )exp((-L T /L) m ) (0.6)
où L T est la valeur où d Φ/d L = 0, dit d'une autre manière, c'est la valeur où la fonction de luminosité commence à montrer un retournement de forme, m est la valeur de courbure orientant la forme (vers le haut si m < 0 ou vers le bas si m > 0) dans la partie où la courbe s'inflexit. Ceci est présenté dans la partie inférieure de la Figure 5.

Dans thèse nous discutons des effets du critère adopté par DLV 2019 pour ignorer les sources qui ont des valeurs de complétude inférieures à 10% pour étudier l'évolution de la fonction de luminosité avec le décalage vers le rouge. Enfin, que les sources avec une valeur de complétude inférieure à 1% ou à 10% soient supprimées, l'effet du retournement sur la forme de la fonction de luminosité est toujours visible aux alentours du log L [erg s -1 ] = 41.

La thèse présente également l'effet de la sélection des sources sur la forme de la fonction de luminosité en fonction du décalage vers le rouge. Il convient de noter que, dans le présent travail, nous utilisons des sources avec un redshift sécurisé (zcon f = 2, 3) pour analyser l'évolution de la fonction de luminosité et nous ignorons les sources ayant un moindre valeur, soit zcon f = 1, en raison d'un faible signal sur 6.

Sur la base des résultats obtenus sur l'évolution de la fonction de luminosité, le chapitre 5 présente la densité de luminosité et convertit ce paramètre en densité de taux de formation d'étoiles en intégrant la fonction de luminosité à la luminosité. La forme de la fonction de luminosité au niveau de la partie lumineuse n'est pas affectée pas la valeur du seuil de complétude. Dans ce régime, la sélection de nos sources est entièrement cohérente avec les études précédentes menées dans des champ vides. Par contre, la précision dans le calcul de la fonction de luminosité dans la partie la plus faible est limitée par les deux facteurs mentionnés. Pour cette raison, le résultat de l'intégration ne sera pas significativement influencé par la limite supérieure alors que la limite inférieure a un impact. J'ai appliqué deux valeurs pour la limitate inférieure, l'une est log L [erg s -1 ] = 39.5 pour couvrir entièrement le domaine exploré par notre échantillon, et l'autre est log L [erg s -1 ] = 41 visant à inclure uniquement les déterminations les plus sûres de la LF. Ces résultats, comparés à ceux de la valeur critique de la densité de formation d'étoiles (c'est à dire, celui qui serait nécessaire pour réioniser l'univers à un redshift donné) suggèrent que les galaxies sélectionnées par leur émission Lyman alpha pourraient être responsables de la réionisation si on suppose une fraction d'échappement des photon Lya de 8%, avec un facteur d'agglutination (clumping factor) typique de ∼ 3. Dans tous les cas, ils ont une contribution substantielle au processus de réionisation. Le chapitre 6 présente une discussion générale des résultats finaux et des perspectives d'avenir. Par la suite, j'espère continuer ce travail sur la fonction de luminosité des émetteurs Lyman alpha en utilisant les données collectées derrière 25 amas-lentilles appartenant au programme MUSE GTO, ou déjà, disponibles sur les archives de l'ESO. J'ai aussi travaillé pendant ma thèse à la constitution des catalogues pour les champs sur lesquels j'ai travaillé, en déterminant les redshifts de toutes les galaxies d'arrièreplan à partir d'autres signatures que Lya, telles que les raies de la série de Balmer ou OII3727 à plus bas redshift. J'ai prévu également de travailler sur les données JWST et Euclid dans une plage de redshift plus élevé pour les émetteurs Lyman alpha. La détermination de la fonction Luminosité reste fondamentale afin d'étudier l'évolution globale des sources ionisantes en fonction du redshift. L'excellente combinaison entre JWST et MUSE offre également la possibilité d'analyser la production de photons ionisants avec efficacité.

Tóm tắt

Thành công trong việc đưa vào khai thác các hệ kính thiên văn trong những năm gần đây như JWST, MUSE/VLT, VIMOS Ultra Deep Survey đã thúc đẩy hướng nghiên cứu các thiên hà thời kì đầu Vũ trụ trở nên năng động hơn bao giờ hết. Mục tiêu khoa học trong lĩnh vực này đó là nghiên cứu các thiên hà trong thời kì tái ion hóa và ước lượng đóng góp các nguồn này đối với quá trình tái ion hóa Vũ trụ.

Quá trình tái ion hóa là giai đoạn chuyển đổi trạng thái vật chất quan trọng của Vũ trụ. Quá trình này bắt đầu ngay sau khi những cấu trúc đầu tiên trong Vũ trụ hình thành, bức xạ từ chúng ra ngoài môi trường đã làm ion hóa nguyên tử Hydro trung hòa. Cho đến nay, hiểu biết về các nguồn chịu trách nhiệm cho quá trình tái ion hóa vẫn chưa rõ ràng, có thể là thiên hà có tâm hoạt động (Active Galactic Nuclei AGN) do những tính chất nổi bật như độ sáng lớn, tỉ lệ thoát của các photon ion hóa cao..., nhưng cũng có thể là thiên hà hình thành sao (star forming galaxies) với nguồn cung cấp photon ion hóa từ các sao trẻ và nặng. Trong luận án này, tôi tập trung nghiên cứu các thiên hà hình thành sao bằng việc tìm hiểu sự tiến triển giá trị hàm độ trưng theo dịch chuyển đỏ, nhờ đó cung cấp những thông tin hữu ích liên quan đến phân bố mật độ thiên hà trong một đơn vị độ trưng, và đóng góp của chúng cho quá trình tái ion hóa. Các nghiên cứu trước đó đã dự đoán các thiên hà mờ có khối lượng thấp đóng góp đáng kể cho quá trình tái ion hóa này. Tuy nhiên những quan sát hiện tại được thực hiện trong trường trống chưa thể đạt đến giới hạn mờ nhất của độ trưng trong khi các quan sát dựa trên hiện tượng thấu kính hấp dẫn lại cải thiện đáng kể số lượng mẫu ở vùng có độ trưng mờ nhất, do tín hiệu được phóng đại lên nhiều lần. Luận án này nghiên cứu hàm độ trưng của các thiên hà hình thành sao phát xạ vạch Lyman alpha ở sau các thiên hà thấu kính hấp dẫn quan sát bởi MUSE/VLT.

Chương 1 của luận án giới thiệu về chủ đề nghiên cứu, hiểu biết hiện tại về những cấu trúc đầu tiên trong Vũ trụ, trạng thái ion hóa và tái kết hợp diễn ra ngay sau đó; giới thiệu về tính chất phổ của các thiên hà hình thành sao thông qua một số vạch đặc trưng; và cách xác định các thiên hà này từ khối dữ liệu dạng datacube.

Chương 2 của luận án giới thiệu ngắn gọn các thông số về máy quang phổ đa đối tượng MUSE/VLT, mục tiêu khoa học của hệ thiết bị, những kết quả quan sát ấn tượng gần đây. Ở phần thứ 3 của chương này, tôi trình bày một số kết quả nghiên cứu trước đó về 17 cụm thiên hà thấu kính hấp dẫn, thông tin về mô hình thấu kính hấp dẫn được sử dụng trong luận án. Do ảnh hưởng của hiệu ứng thấu kính hấp dẫn nên tín hiệu từ các thiên hà ở xa được phóng đại nhiều lần, hình ảnh bị bẻ cong cho phép quan sát thấy nhiều ảnh của cùng một nguồn. Để hạn chế quá trình tính toán trùng lặp và tiết kiệm thời gian, trong nghiên cứu này chúng tôi chọn ra 1 ảnh đại diện đặc trưng cho mỗi hệ dựa trên các tiêu chí về tỉ lệ tín hiệu trên nhiễu, giá trị độ phóng đại, ảnh được chọn thường nằm riêng lẻ so với những ảnh khác của cùng một hệ. Chương 2 kết thúc với thông tin chung về 600 nguồn phát xạ Lyman alpha được tìm thấy phía sau 17 cụm thiên hà thấu kính hấp dẫn có dịch chuyển đỏ nằm trong khoảng 2.9 < z < 6.7, phân bố trong bốn bậc độ trưng 39 < logL[erg/s] < 43 và so sánh thông tin của 600 nguồn này với dữ liệu trong nghiên cứu trước đó được thực hiện bởi DLV 2019. Chương 3 mô tả quy trình tính toán đại lượng V max đặc trưng cho thể tích khảo sát mà từng nguồn riêng lẻ có thể được tìm kiếm. Nghiên cứu hiện tại được thực hiện trong 18 vùng quan sát khác nhau, do đó giá trị V max của từng nguồn riêng lẻ phải tính đến trong tất cả các vùng này, bao gồm cả những vùng nguồn không xuất hiện. Quy trình này được phát triển trước đó bởi DLV 2019 khi nghiên cứu hàm độ trưng của 128 nguồn phát xạ Lyman alpha phía sau bốn cụm thiên hà thấu kính hấp dẫn. Do đó, ở chương này chúng tôi tóm tắt những bước chính và trình bày một số cải tiến mới trong quy trình tính toán, so sánh thể tích khảo sát thu được từ nghiên cứu này với những kết quả nghiên cứu trước đó. Kết quả nghiên cứu hiện tại đã chỉ ra tổng thể tích khảo sát đã hiệu chỉnh cho quá trình giãn nở Vũ trụ khoảng 50 000 Mpc cứu được thực hiện trong trường trống, do nghiên cứu hiện tại chịu sự ràng buộc về giá trị độ phóng đại gây ra bởi thấu kính hấp dẫn, làm giảm hiệu quả trường nhìn. Chương 4 của luận án trình bày những kết quả chính nghiên cứu hàm độ trưng của 600 nguồn phát xạ vạch Lyman alpha trong bốn khoảng dịch chuyển đỏ. Theo định nghĩa, hàm độ trưng là đại lượng mô tả số lượng thiên hà ứng với một đơn vị độ trưng và một đơn vị thể tích khảo sát khi đã tính đến sự giãn nở Vũ trụ. Tuy nhiên, không phải tất cả các nguồn trong mẫu đều có xác suất được phát hiện 100%, do đó cần phải hiệu chỉnh cho xác suất thực sự tìm thấy nguồn trong khối dữ liệu (completeness). Quy trình này được thực hiện cho từng nguồn bằng cách sử dụng phân bố độ sáng của thiên hà trong mặt phẳng ảnh và đưa nó một cách ngẫu nhiên vào các vị trí khác nhau trong khối datacube. Nguồn được coi là được phát hiện khi độ lệch tối đa giữa vị trí thêm vào ngẫu nhiên và vị trí được phát hiện lại là 0.8 giây cung. Hệ số hoàn chỉnh được tính bởi tỷ lệ giữa số lần phát hiện nguồn thành công, với các điều kiện tương tự như đối với nguồn ban đầu, trên tổng số lần thử. Về điểm này, luận án có một số khác biệt so với DLV 2019 liên quan tới việc chọn ngưỡng cắt của hệ số hoàn chỉnh cũng như ảnh hưởng của việc chọn nguồn.

-Chúng tôi chỉ ra kích thước của ảnh (chứa nguồn) ảnh hưởng tới hệ số hoàn Hình 3: Phân bố của nguồn có dịch chuyển đỏ trong khoảng 2.9 < z < 6.7 trong mỗi khoảng độ trưng có độ rộng 0.25 theo thang logarit. Đường màu xanh hiển thị số lượng nguồn trước khi tính đến hệ số hoàn chỉnh (completeness). Đường màu cam hiển thị phân bố số lượng nguồn khi việc hiệu chỉnh được tính đến.

log(L) -6 -5 -4 -3 -2 -1 0 1 log(Φ [(∆ log L = 1) -1 .M pc -3 ])
Luminosity function for 2.9 < z < 6.7

log(L) -6 -5 -4 -3 -2 -1 0 1 log(Φ [(∆ log L = 1) -1 .M pc -3 ])
Luminosity function for 2.9 < z < 4.0

-1 0 1 log(Φ [(∆ log L = 1) -1 .M pc -3 ]) 6 -5 -4 -3 -2
Luminosity function for 5.0 < z < 6.7 Hình 5: Từ trên xuống dưới, từ trái sang phải, Kết quả nghiên cứu sự tiến triển của hàm độ trưng theo dịch chuyển đỏ 2.9 < z < 6.7, 2.9 < z < 4.0, 4.0 < z < 5.0 và 5.0 < z < 6. -Với mục tiêu tận dụng tối đa số lượng nguồn trong mẫu ban đầu cho quá trình tính toán hàm độ trưng và nghiên cứu sự tiến triển ngay sau đó, chúng tôi loại bỏ những nguồn có giá trị hệ số hoàn chỉnh nhỏ hơn 1%, trong khi nghiên cứu của DLV 2019 đã loại bỏ những nguồn có giá trị hoàn chỉnh nhỏ hơn 10%.

-Mối quan hệ giữa độ sáng đo được, độ dịch chuyển đỏ, tỉ lệ tín hiệu trên nhiễu và giá trị hoàn chỉnh của từng nguồn riêng lẻ được trình bày ở Hình 2. Hình 3 chỉ ra số lượng nguồn được sử dụng cho hàm độ trưng trong toàn bộ khoảng dịch chuyển đỏ, trước và sau khi hiệu chỉnh cho hệ số hoàn chỉnh. Hầu hết số lượng nguồn có độ trưng yếu đều rất thấp. Tuy nhiên, khi tính đến việc hiệu chỉnh tham số này, số lượng nguồn trong các khoảng độ trưng này tăng lên đáng kể. Ở khoảng 42.25 < logL[ergs -1 ] < 42.5 (Hình 3) có sự xuất hiện bất thường của một đỉnh trong phân bố là do sự tồn tại của một nguồn có giá trị hệ số hoàn chỉnh nhỏ được nằm trong khoảng độ trưng đã cho.

Giá trị hàm độ trưng trong một khoảng độ trưng và dịch chuyển đỏ cho trước được tính bởi phương trình:

Φ(L i ) = 1 ∆logL i j 1 C j V max,j (0.7)
trong đó, C j ,V j là hệ số hoàn chỉnh và thể tích khảo sát của nguồn thứ j , ∆logL i là độ rộng của độ trưng ở bin i . Các điểm LF thu được từ mẫu đang khảo sát được khớp hàm Schechter dạng:

Φ(logL)d (logL) = L log e Φ * L * L L * α exp - L L * d (logL) (0.8)
trong đó α là giá trị độ dốc của hàm Schechter về phía độ trưng mờ, L * là giá trị đặc trưng của hàm Schechter khi chuyển từ hàm exponential (phía có độ trưng lớn) sang hàm mũ (phía có độ trưng nhỏ), Φ * là hệ số chuẩn hóa. Sự tiến triển này được nghiên cứu trong 4 khoảng dịch chuyển đỏ 2.9 < z < 6.7, 2.9 < z < 4.0, 4.0 < z < 5.0 và 5.0 < z < 6. Nghiên cứu trước đó thực hiện bởi DLV 2019, đã loại bỏ các nguồn có giá trị hoàn chỉnh nhỏ hơn 10% trong quá trình tính toán hàm độ trưng. Trong luận án, chúng tôi thảo luận về tác động của tiêu chí lựa chọn giá trị hoàn chỉnh thực hiện bởi DLV 2019 đối với kết quả nghiên cứu hàm độ trưng. Cuối cùng, dù loại bỏ các nguồn có giá trị hoàn chỉnh 1% hay 10% , sự thay đổi "quay đầu" của hàm độ trưng vẫn được quan sát lân cận giá trị logL[ergs -1 ] ∼ 41.

Luận án cũng trình bày ảnh hưởng của phương pháp chọn nguồn đối với kết quả nghiên cứu của hàm độ trưng theo dịch chuyển đỏ. Cần lưu ý rằng, trong nghiên cứu hiện tại chúng tôi chỉ sử dụng những nguồn với độ tin cậy zcon f = 2, 3 trong quá trình xác định và đánh giá độ dịch chuyển đỏ. Chúng tôi loại bỏ những nguồn được đánh giá zcon f = 1 bởi tỉ lệ tín hiệu trên nhiễu thấp, hình dáng của phổ phát xạ và các vạch đi kèm không thực sự rõ ràng để nhận biết. Do đó, chúng tôi tiến hành hai phép thử (thêm 50% và thêm 100%) số lượng nguồn zcon f = 1 để kiểm tra ảnh hưởng của chúng vào mẫu hiện tại. Kết quả cho thấy các nguồn zcon f = 1 không ảnh hưởng đáng kể tới hình dạng của hàm độ trưng. Bên cạnh hiệu ứng lựa chọn nguồn zcon f =1, chúng tôi cũng thảo luận một số nhân tố khác ảnh hưởng tới độ dốc α, bao gồm việc lựa chọn các cắt khác nhau với hệ số hoàn chỉnh để loại bỏ nguồn yếu, việc xác định thông lượng đo đạc của nguồn LAEs dựa theo hai phương pháp khác nhau, việc lựa chọn mô hình khác nhau để khớp hàm. Các kết quả này được trình bày trong Bảng 3 và được dùng để ước tính sai số hệ thống liên quan tới độ dốc α ở vùng độ trưng mờ.

Chúng tôi đã so sánh kết quả của mình trong khoảng dịch chuyển đỏ cao nhất 5.0 < z < 6.7 với tính toán của Garel et al. 2021 nghiên cứu sự tiến triển của hàm độ trưng khi tính đến ảnh hưởng của IGM, cũng như của Salvador-Solé et al. 2022 khi nghiên cứu quá trình tái ion hóa đơn và tái ion hóa kép. So sánh này cho thấy sự phù hợp giữa kết quả của chúng tôi và các tính toán lí thuyết nói trên, mà không cần thêm bất cứ hệ số chuẩn hoá nào. Tuy vậy kết quả từ dữ liệu hiện tại vẫn chưa cho phép chúng tôi phân biệt chất lượng của các mô hình khác nhau. Điều này được thể hiện trong Hình 6.

Ở chương 5 của luận án, dựa trên các kết quả nghiên cứu đạt được về hàm độ trưng, chúng tôi lấy tích phân đường của hàm độ trưng theo L để nghiên cứu mật độ độ trưng và chuyển đổi đại lượng này sang mật độ hình thành sao theo hàm của dịch chuyển đỏ. Hình dáng hàm độ trưng ở phía có độ trưng lớn hoàn toàn phù hợp với các nghiên cứu trước đó và không bị ảnh hưởng bởi các yếu tố về hệ số hoàn chỉnh cũng như hiệu ứng chọn nguồn. Do đó, cận trên của tích phân đường không gây ra bất cứ ảnh hưởng nghiêm trọng nào đến kết quả về mật độ độ trưng và mật độ hình thành sao. Tuy nhiên, trong những khoảng độ trưng mờ nhất, hình dáng hàm độ trưng bị ảnh hưởng nhiều bởi các yếu tố chọn lựa và giới hạn tìm kiếm các nguồn nên cận dưới của tích phân là yếu tố gây ảnh hưởng nhiều đến kết quả cuối cùng. Trong phần này, chúng tôi lựa chọn hai cận dưới của tích phân, logL[ergs -1 ] = 39.5 nhằm mục đích bao phủ toàn bộ khoảng độ trưng của dữ liệu hiện tại, và logL[ergs -1 ] = 41 nhằm đảm bảo độ tin cậy cao của các nguồn được dùng cho xây dựng hàm độ trưng. Dựa trên những giả thuyết về mật độ photon ion hóa tối thiểu đủ để ion hoá Vũ trụ sơ khai, để chuyển đổi sang mật độ hình thành sao tối thiểu, luận án ước tính tỉ lệ thoát photon ion hóa trung bình trong các thiên hà ∼ 8% và số lượng cụm phổ biến (clumpy) bằng 3. After the Big Bang, the temperature of the Universe was so hot that protons and electrons could not combine together to form neutral hydrogen atoms. As the Universe expanded, its temperature cooled down. At ∼ 3000 K, it is cool enough for the formation of neutral hydrogen, marking the beginning of the Dark Age. The Universe became transparent and the radiation from the Big Bang referred as Cosmic Microwave Background (CMB), could freely travel. This happened roughly half a billion years before the first stars and first galaxies started lighting up our Universe. In this stage, the dark matter particles collapse to form halos. As time passes, these halos become more massive, and the thermal pressure inside tends to prevent the gas from collapsing into the halo. However, the gas can continue collapsing if the dark matter halos which have been formed reach a threshold where the gravity force overcomes the thermal pressure. This threshold can be treated as compensation for the thermal pressure of the gas called the Jeans mass and is defined as follow:

Conclusions and perspective futures

List of Tables

M J = π 2.5 c 3 s 6(G 3 ρ) 0.5 (1.1)
in which ρ is the density of the gas, G is the gravitational constant, c s is the sound speed identified by c s =

γk b T µm p 0.5
(γ is the adiabatic index, T is the gas temperature, µ is the mean molecular weight, m p is the mass of the proton, and k b is the Boltz-1 Introductionmann constant). During this stage, the mean gas density can be computed using the following form:

ρ b = ρ c Ω b (1 + z) 3 (1.2)
where ρ c is the critical density of the Universe, the baryon density parameter and redshift are Ω b , and z, respectively. Peebles 1993 have shown that at redshift z t = 136, the temperature can be estimated as T = 2.73(1+ z t ) = 374 K. At this point, the thermal coupling of CMB blackbody and other objects is no longer significantly affected by the Compton scattering, so the gas will undergo adiabatic cooling, and its temperature will decrease by a factor of a -2 , where a is expanding factor. The minimum halo mass needed for the gas to collapse, when ignoring streaming velocities, can be estimated at redshift z = 30, M J ∼ 3.2 ×10 4 M ⊙ and its temperature decreases as a -1.5 [START_REF] Barkana | Constraints on Warm Dark Matter from Cosmological Reionization[END_REF]). Note that streaming velocity is the relative velocity of the gas that was already coupled to radiation before the recombination process happened [START_REF] Tseliakhovich | Relative velocity of dark matter and baryonic fluids and the formation of the first structures[END_REF]. This calculation allows us to evaluate the spatial fluctuations of the lower mass limit for the halos that are able to bind gas.

The second major phase of transition happened when the first objects inside hydrogen clouds in the massive dark matter halos collapsed under gravity to form the first stars and galaxies. Properties of galaxies in the early Universe have been identified and quantified up to a few hundred million years after the Big Bang. These intense star-forming galaxies (SFGs) often host very massive and short lifetimes stars (O, B types). Their intense stellar UV emissions easily ionize the neutral hydrogen atoms, marking the end of the Dark Age. This important process happened at redshift z ∼

The first structures

The primordial stars are formed mostly from hydrogen and helium atoms, and then the heavier elements are formed by nucleosynthesis. The gas formed from metal-poor stars is less effective than from metal-rich stars for cooling environment, which leads to the formation of more massive stars than nowadays [START_REF] Bromm | The fragmentation of pre-enriched primordial objects[END_REF][START_REF] Abel | The Formation of the First Star in the Universe[END_REF]. The atomic cooling mechanism allows the gas temperature to decrease to 10 K. For this reason, the halos with a mass over 10 8 M ⊙ will have their thermal pressure compensated by gravity. However, for the halos with a mass below 10 8 M ⊙ , the stars formed in this region will require another mechanism to cool down the gas temperature below 10 4 K. This mechanism is called molecular cooling caused by the formation of H 2 molecules:

H + e -- → H -+ γ H + H -- → H 2 + e -
Considering feedback of H 2 formation combined with simulations until the end of reionization, Wise et al. 2012 found a constant rate of metal-free stars at 3 × 10 -5 M ⊙ yr -1 Mpc -3 . Such metal-free stars are believed more massive than nowadays stars [START_REF] Bromm | Forming the First Stars in the Universe: The Fragmentation of Primordial Gas[END_REF][START_REF] Abel | The Formation of the First Star in the Universe[END_REF][START_REF] Hirano | ONE HUNDRED FIRST STARS: PROTOSTELLAR EVOLUTION AND THE FINAL MASSES[END_REF]) and they efficiently create radiation during their time (3-10 Myr) (Schaerer 2002). At the end of their life, these stars explode as supernovae, outflowing a large quantity of chemical elements into their surrounding environment on a scale of 10-100 ckpc (co-moving kpc). This process also expels most of the gas from their host halo, partially halting the formation of other stars. Finally, some metal elements are included in the stars, forming a different type of star.

For the first galaxies, the dark matter halos keep increasing their mass, where stars and larger structures can be formed. Such stars gather into clusters and eventually form proto-galaxies that are efficient for atomic line cooling. However, as mentioned above, the supernovae from stars at the end of their life deplete the gas in their host halos, delaying the star formation process. This can be continuous at the order of sSF R = Ṁ * M * ∼ 10 -8 yr -1 if halo mass is higher than the atomic cooling limit (Kimm and Cen 2014; [START_REF] Kimm | Feedback-regulated star formation and escape of LyC photons from mini-haloes during reionization[END_REF]. Of course, the value of special star formation rate can vary from one halo to another depending on the feedback from supernovae explosions. Some recent research has shown that the first galaxies can contain a range of 10 6-7 M ⊙ of metal-poor stars and their host halos can contain a dark matter at a mass of 10 9 M ⊙ [START_REF] Xu | Lens galaxies in the Illustris simulation: power-law models and the bias of the Hubble constant from time delays[END_REF]).

The galaxies reside in the halos of dark matter formed from both dark matter and baryons (gas, dust, and stars). This makes galaxies sensitive to physical processes that do not affect the cold dark matter. For example, stellar feedback or black hole physics can play a role in the formation and evolution of galaxies. Galaxies are composed of several components of gas, dust, and dark matter. For this reason, their morphology, mass, and size can vary widely. The most massive galaxy contains 10 12 stars while the smallest one has only 10 7 stars. Depending on their morphology, galaxies are classified into four types:

1 Introduction -1.2 Circum galactic medium -Spiral galaxies are the most common type of galaxy. They have a spiral shape, with a central bulge and a disk of stars, dust, and gas. The spiral arms are made up of denser gas, which makes them more likely to form stars. The stars in the spiral arms are usually young and hot, while the stars in the bulge are older and cooler.

-Elliptical galaxies have a smooth, elliptical shape and a very old stellar population. They usually include very little or even no stars due to the very low amount of gas and dust. As a result, the populations of stars are therefore older.

-Lenticular galaxies are the intermediate galaxies between spiral and elliptical phases with a large massive bulge, around which revolves a disk that is not quite active in the sense of stellar formation and in which we could not discern a spiral structure.

-Irregular galaxies with a complicated morphology. They often show chaotic shapes, including young stars and large amounts of gas.

Circum galactic medium

In the mid-1950s, while observing spectra of hot stars at high galactic latitudes, Guido Munch observed absorption of neutral sodium (NaI) and single ionized calcium (CaII). The appearance of these lines has been integrated by Spitzer 1956 as evidence for diffuse, extraplanar hot gas (T ∼ 10 6 K). In 1956, Lyman Spitzer introduced the concept of the "galactic corona", an encompassing region that surrounds galaxies for up to CGM as a region surrounding a galaxy within a radius of 300 kpc. Nowadays, we have investigated four big issues implicated in the evolution of the CGM: the difference in formation and evolution of galaxies may relate to the different massive dark matter halos; there is a small fraction of baryons and metals in the galaxies; the inflow and outflow of the regular gas passed into/through the CGM.

In the following paragraphs, I will present five approaches that have been employed to investigate this intriguing phenomenon.

-Transverse absorption line studies. Observing the CGM in absorption against a luminous background source, such as a quasar has three advantages over other methods. i) it can be used on the extremely low column densities N ∼ 10 12 cm -2 due to high sensitivity ii) one can access a wide density range iii) it is not affected by the redshift or luminosity of the host galaxy. However, there is one drawback to using transverse absorption lines, we can only measure gas surface density and are usually limited to one sightline per galaxy due to lacking background quasars. At the scale level of the local Universe (a few Mpc), it is possible to use multiple sightlines (Lehner et al. 2015). The method can be extended to the high redshift galaxies by using multiple lensed images of background quasars to constrain the size of absorbed sources [START_REF] Rauch | Faint resonantly scattered Lyα emission from the absorption troughs of damped Lyα systems at z∼ 3[END_REF][START_REF]DISSECTING THE GASEOUS HALOS OF z 2 DAMPED Lyα SYSTEMS WITH CLOSE QUASAR PAIRS[END_REF]. In general, the CGM map made from absorption lines detected from many galaxies is obtained from statistical sampling of gas. For the massive optical spectroscopic surveys, the sample in low ions MgII and CaII have grown to hundreds or thousands [START_REF] Zhu | The JHU-SDSS Metal Absorption Line Catalog: Redshift Evolution and Properties of Mg II Absorbers[END_REF]). This sample may extend out to z ∼ 4 for quasar/galaxy pairings (Turner et al. 2014). It's also interesting to study HI column density regimes using different approaches. For example, lines up to log N ∼ 15 could be probed with Voigt profile fitting or Equivalent Width (EW). This value is quite low for the CGM but is a bit high for the Lyα forest within a region of 100 kpc in which HI is not seen (Tumlinson 2013; S. D. Johnson et al. 2014). At log L ∼ 16 the absorption becomes a major factor, and robust column densities have to be obtained from profile fitting or higher Lyα series lines if the system is redshifted enough. This value is quite close to the Lyman limit system N H I > 10 16 cm -2 . If the limitation is covered (z > 0.24 for Hubble), the decreasing of flux at λ = 912(1 + z) Å allows us to measure precisely log N H I and metallicity properties. At the higher level, i.e. log N H I ∼ 18, where N H I is the HI column density, the Lyman limitation is completely opaque, the Lyman series is saturated so the true column density has to be computed by fitting the Lyman profile for the Lyman limit systems and damped Lyα series.

-Stacking analyses: A novel method for the investigation of halo gas could be conducted by massive spectroscopic surveys. The faint signal of the absorption line 1 Introduction -1.2 Circum galactic medium dataset could be extracted by stacking, which requires a catalog of redshifts for either absorbers or foreground galaxies. For this reason, spectra of background sources could be shifted to their rest frame and be continuum-normalized and then co-added together. The addition reduces statistical noise, allowing low absorption signals to be measured at the cost of averaging over the individual absorption profiles. The stack can be performed to examine the variation of mean profiles of input parameters such as mass, Star formation Rate (SFR), color, orientations, and radius, which are included in the subsets of the data (York 2006; [START_REF] Bordoloi | [END_REF]). Due to the appearance of a large amount of dust in the CGM of galaxies [START_REF] Ménard | Measuring the galaxy-mass and galaxy-dust correlations through magnification and reddening[END_REF][START_REF] Peek | Dust in the Circumgalactic Medium of Low-redshift Galaxies[END_REF], the stacking may relate to the reddening of quasars. In addition, the stacking method can help exploit more faint sources by stacking spectra of background galaxies as the one was conducted in Steidel et al. 2010 to characterize the CGM of galaxies at redshift z ∼ 3.

-Down-the-barrel is another method that is efficient for studying the inflow and outflow of galaxies (Henry et al. 2015;[START_REF] Heckman | THE SYSTEMATIC PROPERTIES OF THE WARM PHASE OF STARBURST-DRIVEN GALACTIC WINDS[END_REF], as well as accretion (Rubin 2012) by using data collected from spectroscopy of star-forming galaxies. The main idea of the method is to use the starlight of our own galaxy as a background source for detecting absorption lines, such as CaII, NaI, MgII, and FeII. The downthe-barrel measurements are important for understanding the CGM because we can directly trace the outflow gas at galactocentric radii that are inefficiently covered by background sources. However, one disadvantage of this method is that it does not constrain the galactocentric radius of any detected absorption which will be anywhere along the light of sight.

-Emission line map: An emission line map is a method that aims to search for the photons emitted from the CGM gas. However, this method is still a challenge due to the measurement being scaled by a factor of n 2 , where n is the number density of the gas and the CGM has n H ∼ 10 -2 , which makes it very difficult to detect emission lines from the gas. At the UV/optical wavelength range, M. [START_REF] Hayes | [END_REF] detected an extended O, VI halo with a radius of 20 kpc surrounding a low redshift starburst galaxy. An extended Lyα emission has been observed out to ∼ 100 kpc way from galaxies and Quasi-Stellar Object (QSO)s at redshift z ∼ 2.5 (Cantalupo et al. 2014). The emission lines map can constrain the morphology, density profile, and physical extent of gas more than aggregated pencil beam sightlines (Corlies et al. 2016).

-Hydrodynamic simulations: Besides using data from observations to examine CGM, one can build models providing information on the environments, physical properties such as histories, and the future of gas. One can simulate the evolution of the cosmic web and galaxies by including the effects of dark matter, gravity, and hydrodynamics. These

Epoch of re-ionization

When the first structures of the Universe were formed, its radiation would ionize their surrounding environment. This stage starts between 0.1 and 0.2 Gyrs after the Big Bang. The ionized photons from such structures can travel freely in the neutral intergalactic medium (IGM) before being absorbed. This distance is called the mean free path, and expressed in the form:

l = 1 σ(λ)n H (z) (1.3)
where σ(λ) is the hydrogen cross-section for a photo-ionizing photon at a given redshift z, n H (z) is the average hydrogen number density with respect to a given redshift. Planck Collaboration: et al. 2013 estimated the mean free path in the IGM at redshift z = 20 was about l ∼0.2 ckpc at redshift z = 20 and l =1.7 ckpc at redshift z = 10. Such results are much shorter than the normal size of the halo (100 ckpc), suggesting that the ionizing photon can not significantly travel in the neutral regions and will be absorbed by the neutral hydrogen atom at the boundary of the regions that have already ionized. For this reason, most of the ionized regions have a spherical shape which, is called of ionization bubble. Such a bubble always starts at a source and keeps

1 Introduction -1.3 Epoch of re-ionization growing with a sharp ionization front beyond. The ionizing process has been shown to have a negative feedback effect. Once the ionization happens, the gas inside the halo can not cool down as efficiently as before because the cooling is done by the atomic cooling mechanism. When the neutral hydrogen content decreases, the probability for the collision excitation also decreases significantly (Wiersma et al. 2009). In addition, this ionizing process heats the gas up to 10 4 K, efficiently photo-evaporating proto-halos of virial temperature below 10 4 K. The sources responsible for cosmic re-ionization are still under debate. Many candidates have been invoked to explain this major phase transition, but the evidence is still inconclusive. Massive population III stars were first thought to be the main contributors, but their short lifetimes prevented them from contributing much to the process (Ricotti et al. 2004 Trebitsch et al. 2021). Until now, the question related to the contribution of AGN to the reionization of the Universe has not been answered yet and needs to be debated.

Quasars were once thought to be of importance, but it soon turned out that they are responsible mostly for the ionization of helium, not hydrogen atoms. The recent research (Matsuoka et al. 2018) shows that the ionizing photons created from quasars is only ṅion = 10 48.8±0.1 s -1 Mpc -3 . This is only 10% less than the critical rate needed to keep the IGM ionized, but it suggests that quasars are not the main contributor to cosmic reionization.

Recently, there have been several observational results that suggest that star-forming galaxies may be the best candidates for the contribution of the ionization photons to the EoR (Robertson, Furlanetto, et 2023). These galaxies have low masses but high densities, which makes them likely to become dominant in driving the reionization process. However, the lack of observational data, in particular in the faint luminosity regime, makes the role of this population still uncertain. The luminosity function (LF), defined as the number of galaxies per given cosmic comoving volume, is of importance in helping to quantify their role in the ionization process.

1 Introduction -1.4 Recombination

Recombination

In addition to studying the ionization of neutral hydrogen, there is another process that needs to be considered to model the reionization process: recombination. Recombination occurs when an electron in an excited state decays to the ground state, emitting a photon. If this photon is absorbed by another hydrogen atom, it can ionize that atom. The probability of recombination is determined by the number density of electrons and protons, as well as the temperature of the gas. The recombination coefficient α A is a measure of the probability of recombination per unit volume per second. At a temperature of T ∼ 10 4 K, the recombination coefficient for hydrogen is: α A (H 0 , 10 4 K ) = 4.2 × 10 -13 cm 3 s -1 . This value is much larger than the typical value of 10 -14 cm 3 s -1 because of the Coulomb focusing of the incoming electron. It is significant at the lower temperature environment in which the recombination coefficient will be proportional to T -0.7 . There are two main channels for recombination: direct recombination, in which the electron decays to the ground state, and indirect recombination, in which the electron decays to a higher excited state before decaying to the ground state. The recombination coefficient being discussed includes both channels. However, in the case of direct recombination, the photon that is emitted can be reabsorbed by another hydrogen atom, ionizing that atom. This means that the effective recombination coefficient is actually lower than the value mentioned above. A correction factor has been calculated for this effect, and the resulting recombination coefficient is α B (H 0 , 10 4 K ) = 2.6 × 10 -13 cm 3 s -1 .

There are two main cases of recombination: case A, in which the hydrogen atom can move to many different excited states before decaying to the ground state, and case B, in which the atom decays directly from the first excited state to the ground state. The recombination time can be estimated from the recombination coefficient α as follows: t r ec (T, z) = (α(T )n H (z)) -1 , where α is the recombination coefficient found from one of the two cases above at a given temperature. The recombination time for case B is about half that of case A, so it is often used in numerical simulations to reduce the computational time. However, this is only valid for the mean density of the IGM. In reality, the IGM is clumpy, with small neutral clumps that have a higher number density than the mean. These clumps have a high recombination rate, which can slow down the ionization process [START_REF] Mcquinn | ON LYMAN-LIMIT SYSTEMS AND THE EVO-LUTION OF THE INTERGALACTIC IONIZING BACKGROUND[END_REF][START_REF] Haardt | Radiative Transfer in a Clumpy Universe. IV. New Synthesis Models of the Cosmic UV/X-Ray Background[END_REF]). The clumping factor is a measure of the deviation from the mean density, and it can be used to estimate the effective recombination rate. The effective recombination rate is then used to calculate the recombination time. The effect of clumping on the SFRD will be discussed later in Chapter 5 of the thesis.

1 Introduction -1.5 Spectral features of star-forming galaxies 1.5 Spectral features of star-forming galaxies

The Lyman alpha line

The Lyα line is a spectral line of hydrogen atoms that is emitted when an electron transits from the excited state (n=2) to the ground state (n=1), where n is the quantum principal number. Due to the spin-orbit interaction, the line is split into a fine structure having two components, referred to as the Lyα doublet. Three states 2S 1/2 , 2P 1/2 , and 2P 3/2 are associated with n=2 but only two transitions between 2P and 1S states can produce Lyman-alpha photons because of the quantum selection rule, which states that the total angular momentum of the initial and final states must be the same (Fig. 1.4). Lyman-alpha doublet (right, credit: https://en.wikipedia.org/wiki/Lyman-alpha_line).

Lyα emission may come from various sources. Young massive stars and AGN in the central regions of Interstellar Medium (ISM) are for example the sources of ionized hydrogen atoms. At larger scales, in CGM, the outflows from galaxies cause the excitation of HI region via shock heating. The infalling gas towards the galaxy center in cold accretion mode releases a large amount of gravitational energy via collisional excitation. The fluorescence light from hydrogen in the CGM and IGM photoionized by UV radiation background from QSOs may also contribute to ionized hydrogen clouds. Lyα line is one of the important probes to study early star-forming galaxies. Neutral hydrogen is the most abundant element in the Universe, making up about 70% of the mass, and exists at temperature ranges of 3000 -7000 K. Most of the data observed in the SEDs of starburst galaxies show that the Lyα emission lines are formed through the combination of hydrogen atom inside the numerous HII region surrounding the most massive and hottest stars of these galaxies (O & B stars). At ∼ 10,000K or so, stars emit a large amount of ionizing photons blueward the Lyα limit, allowing them to ionize the surrounding ISM neutral hydrogen gas and to emit Lyα emission lines. The wavelength of the Lyα photon at the rest frame is 121.6 nm. The expansion of 1 Introduction -1.5 Spectral features of star-forming galaxies the Universe makes the light from high-z star-forming galaxies redshifted and can be observed with a ground-based telescope in the visible and near-infrared ranges. This is one of the brightest emission lines of high-z galaxies and therefore is an important tool to probe the early Universe. In the environment of CGM, one can study the diffused Lyα emission extended over 10 kpc, or even more. The intrinsic Lyα luminosity in HII regions can be computed, based on the number of ionizing photons emitted from stars (D. Schaerer 2002):

L(Lyα) = 2/3(1 -f esc )hν 0 Q H erg s -1
, where Q H is total flux of ionizing photons, f esc is fraction of Lyα photons which do not ionize hydrogen atoms, ν 0 is frequency of Lyα photons in the rest frame. Under the optical thick assumption, which is typical of HII regions, all Lyα photons emitted by recombination of hydrogen atoms are likely to be re-absorbed. All three states 2S 1/2 , 2P 1/2 and 2P 3/2 of neutral hydrogen atoms are equally populated, namely 2/3 of the recombination in HII regions converts to Lyman-alpha photons [START_REF] Osterbrock | Astrophysics of gaseous nebulae and active galactic nuclei[END_REF]).

The Lyα line is one of the brightest lines from star-forming galaxies (Partridge et al. 1967;Charlot et al. 1993;[START_REF] Malhotra | [END_REF], and it can be observed in the far-UV range. Galaxies that are identified by using the Lyman-alpha emission line are called Lyman Alpha Emitters (LAEs). They are in the early stage of their evolution, having masses ranging from 10 8 to 10 9 M ⊙ , more compact in physical size with radii of ∼1 kpc, and less dusty than other types of high-z star forming LBGs (Taniguchi et al. 2009). Thanks to both the prominent strength of the Lyman-alpha emission and the advent of modern spectrographs such as MUSE/VLT, properties of host galaxies can be studied in some detail such as their Lyman-alpha luminosity, their EW, star formation rate by Lyα photons, the time scale of the Lyα emission.

The intrinsic Lyα line is very bright, and its contribution is up to 7% of the total bolometric luminosity of a star-forming galaxy (Partridge et al. 1967). However, as a resonant line, the radiative transfer of the Lyα photon is very complex. As these photons travel through the ISM, they are subject to being absorbed and/or scattered by neutral hydrogen atoms. The excited atoms then re-emit Lyα photons with slightly different frequencies in random directions, which is referred to as the resonant radiation process. In high-density regions of neutral hydrogen in the ISM of a star-forming galaxy, the resonant scattering increases the path lengths of the Lyα photons and hence their probability to be absorbed by dust grain. As a result, the transportation of the Lyα photon depends very much on the mass of neutral hydrogen clouds, its dynamics, and the dust content of the IGM. These interactions leave strong imprints on the measured features of the Lyα line such as its line strength, its shape, its EW, etc. All the factors mentioned above must be taken into account when interpreting the observed Lyα profile and the intrinsic properties of high-z star-forming galaxies.

1 Introduction -1.5 Spectral features of star-forming galaxies In addition to the high brightness of the Lyα line, the other feature of Lyman EW has been also studied. The EW is the ratio between the integrated luminosity of the line and the monochromatic luminosity of the continuum: EW = L line /L continuum [START_REF] Duval | Lyman-α radiative transfer in Star-forming galaxies[END_REF]. L line , L continuum and EW are in units of erg s -1 , erg s -1 Å -1 , and Å, respectively. The EW of LAEs could reach 240 Å for normal stellar metallicities and 360 Å for metalpoor stars (Charlot et al. 1993). The EW declines with stellar populations aging and depends also on star formation activity. Its evolution is displayed in Fig. 1.7 (left). The lower limit of their EW is ∼20 Å obtained from the samples using narrowbands 1 Introduction -1.5 Spectral features of star-forming galaxies (NBs) having a transmission width ∼ 1% of the central wavelength. Fig. 1.7 (right) displays the average fraction of Lyα emitting galaxy having EW > 25 Å for faint galaxies (M UV >-20.25) and bright ones (M UV < -20.25). This fraction increases with redshift between 4 and 6, then decreases at higher redshift, implying that the opacity of Lyα and the cosmic neutral hydrogen fraction do increase beyond 6. It implies also that the detection of LAEs at the middle and early epoch of re-ionization becomes more challenging. 1 Introduction -1.5 Spectral features of star-forming galaxies

In the Lyα analysis, such halos may be extended by a factor of 5 to 15 compared to the UV emission. By dividing Lyα emission into two components, one is close to the size of UV emission, and the other is more extended, they were able to investigate the properties of the central emission from those of the extended halo. The Lyα emission distribution may follow an exponential law with a characteristic length of a few kiloparsecs and most of them come from the extended component of the halo. One year later, Leclercq et al. 2017 studied a larger sample of LAEs with 145 Lyα halos and found a correlation between extended Lyα emission component and UV properties of galaxies in both size and magnitude. 1 Introduction -1.5 Spectral features of star-forming galaxies

The Lyman Break galaxies

Lyman Break galaxies (LBGs) are star-forming galaxies detected by using the traditional technique. They are identified by comparing their fluxes in different broad-band filters. Surrounding neutral gas both in the interstellar environment ISM and IGM absorb completely radiation photons having energy higher than the Lyman limit of 912 Å, making galaxies "drop-out" bluewards with respect to this limit. The method has been extensively used and has become the traditional way to look for high-z galaxies in the early universe. The properties of these LBGs depend strongly on the host galaxies (UV luminosity, neutral hydrogen mass fraction, etc.). The less massive galaxies tend to have fainter UV luminosities and lower HI masses. Such properties make them difficult to be detected in the UV continuum. LBGs are often found in more massive galaxies having larger HI masses and are associated with active star-forming galaxies. As a result, LBGs emit a bright UV continuum making them easy to be identified. In addition, this effect also can be seen from the observed profile of the Lyα line. The UV continuum flux in the blue part of the Lyα line is lower than that of the red one. This can be explained as the foreground neutral IGM hydrogen clouds absorb significant UV radiation making the observed continuum level in the blue part of the Lyman-alpha line reduced accordingly. For galaxies at the early EoR, the observed UV continuum flux toward the blue of Lyα lines is expected to be zero, namely, they are completely absorbed. This is the so-called Gunn-Peterson effect [START_REF] Gunn | On the Density of Neutral Hydrogen in Intergalactic Space[END_REF].

Many thousands of LBGs have been found. Their properties are summarized as follows, in comparison with those of LAEs. They often have a high star formation rate from 10 to 100 M ⊙ yr -1 at the redshift z ∼ 3 [START_REF] Porciani | The Clustering Properties of Lyman Break Galaxies at Redshift z 3[END_REF]. They are a bit older than LAEs, typically ∼300 Myrs at redshift z ∼ 3 (Shapley 2001), more massive, 10

Lyα forest

Spectra of high-z galaxies in the early Universe often display a series of absorption lines that are related to the Lyα line of the neutral hydrogen atom, which is at 121.6 nm in the UV range. The absorption feature is caused by the presence of neutral clouds in the IGM along the line of sight. The appearance of the Lyα series is treated as a post-reionization version of the Gunn-Peterson trough so it can not be used directly to study the EoR. They are usually used to quantify the inhomogeneous properties (i.e.: the gas cloud's physical and chemical states) of neutral fraction after EoR with a lower limit of neutral fraction n H I > 0.94 at redshift z = 5.9 (Mesinger 2010; Greig et al. 2017). The lower column density region N H I ∼ 10 12-16 cm -2 is an ideal environment to observe Lyα forest as these clouds are very sensitive to photo-ionization and photoheating from UV background. Based on these properties one can estimate ionization rate and temperature after EoR, those help us have a deeper understanding of the behaviors during the reionization. 1 Introduction -1.5 Spectral features of star-forming galaxies

The very first confirmation of the presence of Lyα series from the mostly absorption of HI Lyα was done in Lynds 1970. After that, hundreds of individual absorption lines of quasars at redshift z > 2 could be resolved using a higher resolution spectrograph, its appearance gave raising to the label of the Lyα forest [START_REF] Ray | Absorption Lines in the Spectra of Quasistellar Objects[END_REF]. However, there are some missing lines in the forest coming from UV transitions of common metal or heavy element ions such as the ionization stages of Fe, Si, Mg, Al, C, and O which are usually seen in the spectral observations. Such lines always appear with a strong Lyα line.

The Gunn -Peterson effect

It is already known that the Lyα line is formed from a transition between the ground state and the first excited state at a wavelength of 1215 Å. This only happens to the neutral hydrogen which can undergo such a transition (absorption and emission properties), so it's a clue for the existence of neutral hydrogen along the line of sight. In addition, the quasars are known to have a continuum emission of the same order magnitude as the Lyα line. In some observations of quasars at redshift z > 6, one usually sees a strong absorption pattern blueward of the Lyα line or redward of the Lyman beta line (Fan et In general, from such a spectrum, one can access the position of neutral hydrogen clouds and find the local neutral fraction along the line of sight as well. However, there is a problem of this reconstruction method related to the optical depth at the Lyα wavelength. The increase of neutral hydrogen gas of the Universe at the higher redshift 1 Introduction -1.5 Spectral features of star-forming galaxies was confirmed in many previous results, so when the neutral fraction x H I > 10 -4 exists in the environment of IGM, this region will become optically thick to the Lyα photon leading a total absorption and limitation the reconstruction process to the very end of EoR. However, there are still some other affect a depth observation of the EoR. Firstly, using the radiations from QSO shows that the hydrogen vicinity of AGN regions seems to be more ionized than the normal IGM. It means that the radiation towards the blueward of Lyα is more weakly absorbed suggesting that one can study further on the ionized regions surrounding the quasars. In addition, for a given width of the Lyα line, the photons can be absorbed at the wing of the line with an even lower probability.

Even at the lowest level, when the neutral hydrogen column density may reach up to N H I = 10 20 cm -2 , the absorption property can be observed in the wing.

LBG technique and photometric redshifts

The typical spectrum of a high redshift star-forming galaxy (Fig. 1.13) leads to the LBG selection method. An emitted photon having a wavelength less than 912 Å is completely absorbed by neutral hydrogen gas in the ISM and IGM to make the Lyman break in the observed spectrum. At redshift z > 3, the Lyman break is shifted to the visible range making high-z galaxies become accessible to ground-based optical telescopes. To identify a high-z LBG, three traditional broadband filters (U, G, R) have been used to look for variations of colors of a galaxy. Nearby galaxies show their appearance in all three filters but if sources disappear in the bluestone, they are high-z

1 Introduction -1.6 LBG technique and photometric redshifts candidates. This is illustrated in Fig. 1.14. This technique has been proven to be very efficient in detecting high-z star-forming galaxies, even up to z ∼ 12 -13 (Pacucci et al. 2022). Following up on spectroscopy observations of these sources, which is time-consuming, will help to confirm their redshifts. Many thousands of LBGs have been identified using this selection method.

Figure 1.13: Typical spectrum of a high redshift galaxy. The dotted colored lines denote the transmission of photometry filters. In some cases, galaxies can be selected as LAE and LBG. At high redshift z > 6, we may observe the Gunn Peterson trough of galaxies due to a large amount of HI surrounding them. Credit: Vieuville 2019.

In addition to this color-color selection, which is relatively fast and simple to implement, a more advanced technique using SED-fitting has been used to estimate more precise photometric redshift (Bolzonella et al. 2000). The fit procedure makes use of photometric measurements and template spectra having a wide range of parameters such as star formation, metallicity, initial mass functions, etc. An example of the SED fitting technique is illustrated in Fig. 1.15. The technique allows us to study a large number of objects that could not be accessed by spectroscopic observations or that would have required a long time for the available instruments. This tool allows users to estimate the redshift of galaxies, quasars, AGN, etc. using the input values as magnitude/flux measured from long/medium band photometry and identify the spectral break as well. This method provides less accurate results than the one obtained from spectroscopic owing to being dependent on the filters.

To measure the photometric redshift of sources, there is a very popular method is being applied named SED fitting method [START_REF] Kenneth | Star-forming galaxies at very high redshifts[END_REF]Fernández-Soto et al. 1999). It is a combination of the observed values obtained from filters at the range of 912-4000 Å and several reference spectra obtained from theoretical research aiming to find a minimum difference χ 2 which is performed as the one proposed by Bolzonella et al. 2000:

χ 2 (z) = N f i l t er s i =1 F obs,i -b × F t emp,i (z) σ i 2 (1.4)
where F obs,i , F t emp,i and σ i is the observed flux/AB magnitude, template flux, and its uncertainty with respect to the filter i and b is a normalization constant.

The most advantage of this method is easy to use and does not require any spectral sample. The user can be flexible to modify some parameters/choose several models to get the best fitting values. However, this is also the disadvantage of the method as we have to choose a few representative models valid for all objects in the present sample.

Selection of LAEs: narrow-band technique

In the rest frame, the wavelength of the Lyα line emission is 1215 Å. When emitted by high-redshift galaxies, the line is shifted to the optical and near IR ranges. With the advent of modern optical telescopes and filters, one now uses the Narrow Band (NB) technique to detect high redshift star-forming galaxies, using both a NB filter displaying a flux excess on the line and broadband filters overlapping with it or redder. Fig. 1.16 shows an example of the detection of an LAE at z = 6.96 using the Lyα emission line [START_REF] Iye | A galaxy at a redshift z = 6.96[END_REF]. A measure of the quality of identification of a Lyα line is the so-called rest-frame Equivalent Width (EW) defined as the ratio between the line flux and the continuum spectral flux density; one usually retains lines having EW in excess of 20 Å.

Thousands of LAEs at various redshifts, up to z = 8, have been identified using large telescopes such as KECK, VLT, and Subaru, equipped with very sensitive Charge Coupled Device (CCD) cameras, which started operation in the late 1990s (Hu et al. 1998;Masami Ouchi et al. 2010). With the advent of MUSE/VLT, an integral field spectrograph, associated with high performance Adaptive Optic (AO), major progress has been achieved in the search for LAEs having redshifts in the range from 2.9 to 6.7 (see Chapter 2).

The availability of a large sample of high-redshift detected LAEs made it possible

Selection of LAEs: Integral Field Units

A normal spectrograph usually captures data from a line through an image and provides spectral data over one spatial dimension. However, the new technique known as Integral Field Units allows a spectrograph with a slit opening to collect data across a two-dimensional field. The signal in each pixel is fed into a spectrograph to generate a spectrum for that pixel. After this stage, the result will be arranged into a datacube with 2D entire FoV and 1D drawn from the spectrograph. This technique is associated in the second generation of VLT instruments aiming at producing high-quality images with a large Field of View (1 ′ ×1 ′ in Wide Field Mode (WFM) and 7"×7" in Narrow Field Mode (NFM)). It has excellent spectroscopic capabilities, covering a large wavelength range (475 -935 nm) with spectral resolution R ∼ 3000. It is equipped with powerful AO to correct for atmospheric turbulence and is well suited for spectroscopy studies of crowded fields such as lensing clusters. The AO-corrected field of view is split into 24 sub-fields, each fed into a spectrograph called Integral Field Unit (IFU). An image slicer in front of each IFU serves as entrance slit, thus producing a spatially resolved spectrum of the whole sub-field. Wide-field narrow-band surveys using the MUSE pencil beam provide a direct measure of the redshift and the practical procedure to select LAEs is presented in Section 2.4.

1 Introduction -1.9 The galaxy luminosity function

The galaxy luminosity function

The Luminosity Function (LF) is defined as the dependence on luminosity of the number of galaxies detectable in a given co-volume, called number-density. It is measured in units of (erg s -1 Mpc 3 ) -1 . It is one of the most fundamental quantities used in the study of the evolution of galaxies in the early Universe. The LF usually displays a transition between high values toward its faint-end at low luminosities and low values toward its bright end from a power-law to an exponential decline. One refers to it as the turnover or knee. The position of the knee and the values taken by the slope are characteristic parameters of the LF. In the present work, we limit the study to luminosities smaller than the knee in order to secure the reliability of the evaluation of the LF. In the study of the evolution of LAEs, one usually uses a parameterization of

φ(L Lyα )d L Lyα = φ * L Lyα L Lyα * α exp - L Lyα L Lyα * d L Lyα L Lyα * (1.5)
where L Lyα is the LAE luminosity, φ * a normalization factor, L Lyα * is the value taken by the LF at the knee, α the value of the slope at the faint end. The Schechter function may be re-written in logarithmic form:

φ(logL Lyα ) = (ln10)φ * 10 (α+1)(logL Lyα -logL Lyα * ) exp(-10 (logL Lyα -logL Lyα * ) ) (1.6)
In order to reveal the evolution of galaxies, one needs to study the dependence on redshift of the LF. Two decades ago, Loveday 2004, using data from the SDSS Data Release 1, and found a significant evolution of galaxies in r-band, corresponding to a range of redshifts z = 0 -0.3. It could be interpreted in terms of an evolution of the density, of the luminosity, of an intrinsic change of shape, or of any combination of these. Recent studies have given evidence for an increase of Lyα LF at low redshifts (z = 0 - 3-45.5 , slope value of the Schechter function of more than 18000 galaxies and AGN was found at α = -1.70 +0. 13 -0.14 by Zhang et al. 2021. Konno et al. 2018 examined 1266 LAEs at redshift z = 5.7 and z = 6.6 using data of Subaru/Hyper Suprime Cam Survey with luminosity in the range of log L = 10 42.8-43.8 erg s -1 measured α = -2.6 +0. 6 -0.4 and α = -2.5 +0.5 -0.5 , respectively. Itoh et al. 2018 compared LAEs with other galaxies at higher redshift (z ∼ 7.0) in a field of view ∼ 3.1 deg 2 observed with Subaru Telescope and measured α = -2.5. The same value was found by Akira Konno 2014 for sources at redshift z ∼ 7.3. These previous studies suffer from a clear lack of deep enough observations of Lyα emissions. To cope with it, the fits were made with a fixed value of α = -2.5 at the faint end, providing a satisfactory picture of the overall evolution of the LF of LAEs. Figure 1.17 shows an evolution of LF at different redshifts of these works. Yet, some questions are still unanswered, as concerning the contribution to Lyα emission from low mass dark matter halos surrounding SFGs. The best fit Schechter function, providing an estimate of the LAE luminosity density, is an essential input for understanding the evolution of galaxies during the epoch of cosmic reionization, including evaluations of the SFRD and of the fraction of hydrogen ionizing photons that escape from galaxies into the IGM. It is therefore important to obtain as precise as possible a shape of the LF. In particular, this requires a good understanding of the impact of having fixed the slope at the faint end, where a turnover [START_REF] Liu | Dark-ages reionization and galaxy formation simulation -IV. UV luminosity functions of high-redshift galaxies[END_REF]) is expected to be caused by the efficiency of gas cooling in SFG halos [START_REF] Atek | The extreme faint end of the UV luminosity function at z 6 through gravitational telescopes: a comprehensive assessment of strong lensing uncertainties[END_REF]Bouwens et al. 2022). ). Gravitational lensing enables the detection of sources which could not be seen previously, but the complex relation between the source and its images implies intricate and computer-time-expensive calculations for the evaluation of the LF. The data used by DLV 2019 include 156 LAEs reaching faint luminosities, log L = 10 39-43 erg s -1 . The evolution of the LF is studied in four redshift ranges, 2.9 < z < 6.7, 2.9 < z < 4.0, 4.0 < z < 5.0, and 5.0 < z < 6.7; the measured values of the slope are -1.69, -1.58, -1.72, and -1.87, respectively.

Overview of this work

Taking advantage of the availability of new MUSE Guarantee Time of Observation (GTO) observations of regions of the sky covering seventeen lensing clusters, the present thesis studies LAEs detected behind these in the range of redshifts 2.9 < z < 6.7 and measures their LF. Much attention is paid to the evaluation of the dependence of the LF on redshift, in particular toward faint luminosities. The results give an 1 Introduction -1.10 Overview of this work important contribution to the current progress in this field, which keeps asking for improved statistics. We proceed as follows: The first step is the selection of detected LAE images behind the lensing clusters. We need to calculate the LF in the source plane, implying a measurement of the number of sources detected in a given covolume as a function of luminosity. This requires transforming from the image plane to the source plane and evaluating for each source its probability of having been detected, its luminosity distribution corrected for various instrumental effects and the volume in which it could have been detected. These evaluations are not independent and require to proceed by iterations in a complex sequence of calculations. In addition to the complication resulting from the production of multiple images from a single source by gravitational lensing and the need to know the values taken by the associated magnifications, we need to account for the effects of noise and of seeing.

The selection of LAEs images in each of the 17 MUSE datacubes uses standard procedures: looking for line emissions in the spectral distribution (integrated over the whole field of view), subtracting continuum contribution, evaluating for each line a region of the sky containing most of the line emission. In addition, we associate each obtained image with UV HST images in the same region of the sky, providing a much more precise knowledge of the position of the image and telling us which LAE source (listed in an available catalog) it is the image of. The ∼ 1400 images detected at this stage, are in fact associated with only ∼ 600 different sources, because some images are associated with a same source. The next step consists, for each image, in evaluating its probability of having been detected and the volume to be used in evaluating the luminosity density. To start with, we select, for each source, one of its detected images (when there is more than one) as considered the cleanest based on reasonable criteria. For each of the 600 selected images, we evaluate, in each of the 17 MUSE intensity maps, what is the map of the probability for this particular image to have been detected above noise (we call it a mask). The masks obtained in the image planes need to be de-projected onto the source planes. This procedure is heavy in terms of computer time and requires particular care in dealing with faint sources. Accounting for noise level leads to the introduction of the concept of completeness, measuring the probability of successfully detecting the source. Accounting for seeing leads to the concept of convolution/deconvolution using the PSF associated with it.

The selection of LAEs behind the 17 lensing clusters, with MUSE directly providing the associated datacube, differs from the method adopted in blank field observations, where the telescope is made to point toward a previously observed source. The main difficulty is to account for the lensing effect in the best possible way. This requires a proper evaluation of the magnification associated with each image of a given source as well as of the dependence on redshift of the effective surveyed volume. We start with the identification of multiple-image systems observed in the cluster cores, for which we select the best images of a given source. Lensing effects are carefully taken into account when computing the effective volume associated with each source and evaluating its incompleteness. This is presented in Chapters 3, and 4.

In order to evaluate the LF of a given source we use an approach originally developed by DLV 2019and referred to as the V max method. Here, V max is a measure of 1 Introduction -1.10 Overview of this work the maximum volume of the survey in which an individual source could have been detected. One needs to estimate the incompleteness affecting each individual source in order for its statistical weight in the LF to be properly obtained. We have introduced some improvements to the original V max method in order to better account for the effect of lensing. These improvements made it possible, in order to include as many sources as possible in the final sample, to lower the completeness threshold down to 1%. Together with the enlarged sample of lensing clusters (17 compared to 4 in DLV 2019) they make this work a unique study of the faintest end of the Lya LF.

Having applied the improved V max method and corrected for completeness, we are in a position to calculate the value of LF in each luminosity bin of in a given redshift range and fit the result to a Schechter function. In practice, we ignored particularly high magnification contributions to very faint luminosity bins, associated with large uncertainties, and used a modified Schechter function to account for the presence of a turnover in the faintest luminosities at the highest redshifts. The best-fit values of the parameters of the Schechter function measured in each redshift interval are then used to compute the SFRD as a function of redshift and estimate the escape fraction of Lyα photons. These results are presented and discussed in Chapter 5. They have been published in three refereed articles and presented at international conferences.

General

VLT stands for Very Large Telescope located on the Paranal of the Atacama Desert in the Northern part of Chile, at 2635 m above sea level. This system consists of 4 telescopes, 8.2-metre in diameter, mostly working in optical and infrared modes (see Fig. 2.1). It can be operated for individual observations, in visitor mode, or remotely. Sometimes, when the weather is good enough, the system is used for interferometry for a limited number of nights each year. The auxiliary telescopes of the system having a mirror of 1.8 metre can be operated every night with a maximum baseline in interferometer mode up to 140-metre. The four names of the main telescopes are Antu, Kueyen, Melipal, and Yepun. MUSE stands for Multi Unit Spectroscopic Explorer (Bacon 2010). It is the second generation VLT panoramic integral-field spectrograph developed by the European Southern Observatory (ESO) for the VLT. This system is associated with AO to correct for atmospheric turbulence on the surface of the Earth. There are seven big institutes in the MUSE consortium: AIP Potsdam, CRAL Lyon, ESO, ETH Zurich, IRAP Toulouse, Leiden, and IAG Gottingen. The data used in this work come from the GTO granted to the consortium by ESO as a reward for building MUSE. The information presented in the following sections is provided by the ESO MUSE User Manual. This project has been started since 2000 and the instrument was commissioned in 2014 by its current PI, R. Bacon. MUSE is an IFU, a spectrograph providing one spectrum per spatial element with a field of view of 1 ′ × 1 ′ sampled at 0.2" × 0.2" in WFM and 7.5" × 7.5" in NFM. A MUSE datacube contains 2048 × 2048 spatial pixels and 4096 wavelength channels spanning a range from 4650 Å to 9300 Å with a spectral resolution varying between R=2000 to R=4000. A simulation for the MUSE cube is shown in Fig. 2.2. When the light from an astronomical object reaches the Earth's atmosphere, the turbulence of the atmosphere may make the image distort and move in various ways. MUSE overcomes this disadvantage by using AO to compensate for the blurring caused by the Earth's atmosphere. The MUSE project went through long planning, construction, and testing before its installation at the VLT. After a decade of preparation, the first light came to MUSE, capturing high quality images of planetary and galaxy in January 2014. MUSE has been coupled with the powerful AO system between 2017 and 2018, using four 22-watt lasers that shine into the sky to make sodium atoms in the upper atmosphere glow. They produce spots of light in the sky that mimic stars (artificial guiding stars). The Sensor of the AO module Ground Atmospheric Layer Adaptive Corrector for Spectroscopic Imaging (GALACSI) uses these signals as an artificial guide star to capture the changes in the atmospheric conditions. The signal is sent to the AO a thousand times per second to calculate the correction and then move the shape of the telescope's deformable secondary mirror to compensate for blurring caused by the Earth's atmosphere. The combination of an optimal site for the VLT facilities and the use of AO makes MUSE images sharper, resulting in twice the contrast achieved by previous facilities. For this reason, MUSE is particularly well suited to study faint sources in great detail.

There are three modes for MUSE running: NFM using AO, WFM with and without using AO. The difference between them is using AO and spatial sampling. NFM coupled with AO allows to correct for atmospheric turbulent above the telescope helping to have sharper images but over a small region for observations (http://www.ens-lyon.

fr/en/article/research/supersharp-images-new-vlt-adaptive-optics). In particular, the system associated with the fore optics is used to select the observing mode, for instance, to change the spatial sampling from 0.2" (in the WFM) to 0.025" (in the NFM) with a reduced field of view of 7.5" × 7.5". Especially, this mode is more significant when working on studying a single object at a very high spatial resolution. For example: studying supermassive black holes, young stellar objects, or the Solar system, etc. Using WFM which has a relatively large field of view allows us to study more extended objects. This mode coupled with AO permits to correct for the effect of atmospheric turbulence up to 1km above the telescope over a wide region of the sky (1 ′ × 1 ′ with a spatial sampling of 0.2" × 0.2"). MUSE has been used extensively in this mode to detect up to 15000 sources by setting up nested surveys of different areas and depths. To date, one of the deepest surveys of MUSE has required a long observing time of around 141 hours reaches a 3 -σ limiting flux of 1 × 10 -19 erg s -1 cm -2 (Bacon 2017; Bacon 2023). These observations can help us solve one of the most challenging studies nowadays in the identification and characterization of the faintest objects formed during the very first billion years in the history of the Universe. For example, studying LAEs redshifted to the wavelength observed by MUSE 2.9 < z < 6.7.

2 The MUSE Lensing Project: from observations of massive clusters to LAE sample selection -2.2 Main scientific goals of MUSE

Main scientific goals of MUSE

Using AO with the possibility to use WFM and NFM makes MUSE become efficient in many different scientific fields since the commission was started, which permits the community to analyze individual sources with high angular spectral resolution such as planetary, small object science to the stellar population in nearby and high redshift galaxies. The main scientific purposes of the MUSE instrument can be listed as follows based on the observing mode.

-Using NFM, which is known to be highly efficient in the analysis of individual sources with very high resolution makes it possible to:

+ Study Super Massive Black Hole (SMBH)s with several issues that can be addressed on the merging of galaxies, SMBHs sinking to the bottom of the potential well, and the formation of binary systems, etc. In addition, it is also possible to study the accretion of the matter to form black holes. However, such observations are rare and difficult to interpret. Therefore, further investigation is necessary to understand the physical properties in these regions.

+ Young stellar objects: Thanks to MUSE we may have the opportunity to study the formation and structure of jets in great detail.

+ Solar system: There are multiple objects within our Solar system that can be observed with the same spatial resolution compared to many space telescopes. -In contrast, we use WFM to detect multiple faint sources, which may be 100 times fainter than those observed with narrow-band imaging to address some scientific goals such as:

+ Study intrinsically faint galaxies at high redshift. This is the main driver for the development of MUSE. It is intended to identify and study the properties of faint high z galaxies observed both in blank and lensing fields. In the latter case, we use massive clusters to study the background population of galaxies by taking advantage of the magnification provided by gravitational lensing (Kneib et al. 2011).

+ Detect Lyman alpha emission out to the EoR, the origin of re-ionization. + The analysis of some physical properties of LBGs such as their wind, and feedback to the IGM. These studies require a combination of MUSE data with deep HST observations. + Find the late-forming population III objects. + Investigate active nuclei at intermediate and high redshifts.

+ Perform a map of the growth of dark matter halos.

Lensing clusters observed with MUSE/VLT

General information of lensing clusters observed with MUSE/VLT

The lensing clusters used in this work were selected to be massive clusters extracted from both the MUSE Lensing Cluster GTO program and the literature, publically available from the ESO archives. All of them were known to be strong lensing clusters, having a high efficiency in terms of magnification power on the population of background sources. They were compiled from different sources. Massive X-ray luminous clusters were mainly selected from ROSAT Brightest Cluster Sample (H. ). Given the wavelength coverage of MUSE, we expect to observe the Lyα emission from background sources at redshift 2.9 < z < 6.7.

The selected clusters have a redshift in the range of 0.2 < z cl < 0.6, some of them were taken from the Hubble Frontier Fields. Therefore, we expect to clearly identify the cluster members by using the absorption (and some emission) lines present in the MUSE spectra. This also helps include the most massive galaxies in the lensing models (see below). An additional criterion is that most of the lensing clusters have a wide Right Ascension (RA) range to adapt to the MUSE GTO observations and a low airmass at the Cerro Paranal Observatory with respect to a declination of range selection -2.3 Lensing clusters observed with MUSE/VLT -60 < Decl i nat i on(DEC ) < +10 degrees. The general information of the clusters observed by MUSE and used for the thesis is presented in Table . 2.1. Taking advantage of HST observations, which are efficient in detecting bright continuum sources, the clusters in the project require at least an available high-resolution broad-band HST image, that is, it requires either F606W or F814W filter to make sure that it overlaps with the MUSE spectral range. Finally, there is a requirement for a mass model based on HST images to help us confirm the existence of at least one multiple-imaged system, that is used to estimate the total mass of the cluster core. From these, one can optimize the field of view in order to cover many of such resolved arcs together with the high magnification region. We briefly present below the sample of lensing clusters used in this work, with the appropriate references. A summary of the relevant information is also provided for convenience in Table . 2.1, 2.2.

-A2390 and A2667. The general information of these clusters can be found in (DLV 2019). An example of spatial image distribution behind A2390 is shown in Fig. 2.6.

-A2744 has been known to have a complex galaxy dynamics with a major merging that happens in the north-south direction [START_REF] Kempner | A Chandra view of the multiple merger in Abell 2744[END_REF]), hosts one of the most luminous known radio haloes within a distance of 1.8 Mpc and a large radio relic at a distance of 2 Mpc from its center [START_REF] Giovannini | Radio halo and relic candidates from the NRAO VLA Sky Survey[END_REF] -A370 is a merging cluster at a redshift of z ∼ 0.375 [START_REF] Mellier | Photometry, spectroscopy and content of the distant cluster of galaxies Abell 370[END_REF]). It is known to exhibit a highly elongated and efficient lens with an Einstein radius of 39" at redshift z ∼ 2, and a mass of M <250kpc = 3.8 × 10 14 M ⊙ (Richard 2010). This cluster is noted for the first confirmed giant gravitational arc at z = 0.725 [START_REF] Hammer | High Redshift and Primeval Galaxies[END_REF] -BULLET is a massive cluster discovered by Tucker et al. 1998 containing two colliding galaxy clusters at z = 0.296 and has been known as one of the hottest galaxy clusters. The name of the cluster referred to a smaller subcluster moving away from the larger one. The majority of the baryons in the cluster is in the form of hot, diffuse X-ray emitting gas and account for a small fraction of the total lensing mass close to the center of the cluster. The mass ratio of the two components -MACS0257 is a special case with 25 background sources confirmed with spectroscopic redshifts, including 81 multiple image systems in total covering a field of view 1 MUSE arcmin 2 . Such systems are too faint to be detected by HST but are readily revealed by MUSE observations. Fig. ) as an X-ray selected system. The MUSE observation was carried out with Observation Blocks (OB) consisting of ∼ 1465s in both two exposures and only one pointing. The total exposure time of this cluster is 2.5 hours conducted with an angle rotation of 90 degrees with respect to the original position angle [START_REF] Caminha | Strong lensing models of eight CLASH clusters from extensive spectroscopy: Accurate total mass reconstructions in the cores[END_REF]. Although the data used in the present work are the same used by Caminha 2019, the number of LAEs behind the lensing cluster is greater than in the previous study. This is due to the improvement of the lensing model, which helped to identify more sources [START_REF] Richard | An atlas of MUSE observations towards twelve massive lensing clusters[END_REF]). The locations of MUSE spectroscopic redshifts over HST F606W-F814W-F160W color composites are shown in Fig. 2.12 (bottom). [START_REF] Maughan | Images, Structural Properties, and Metal Abundances of Galaxy Clusters Observed with Chandra ACIS-I at 0.1 < z < 1.3[END_REF] classified this cluster giving evidence for the presence of substructure in its X-ray surface brightness. Zitrin 2012 studied HST/ACS/WFC3 CLASH data and found that the magnitude of the most magnified image is 24 ± 0.004 AB mag. They also discovered one of the highest redshift (z = 6.18) multiple imaged systems lensed into four images, with magnifications of 11.6 +8.9 -4.1 , 17.6 +6.2 -3.9 , 3.9 +3.0 -1.7 and 3.7 +1. 3 -0.2 . By analyzing the SED using a population synthesis model, they found a demagnified stellar mass of 10 9 M ⊙ , a subsolar metallicity Z /Z ⊙ = 0.5, low dust content, demagnified SFR of 3.2 M ⊙ /year and a specific SFR of 3.4 per Gyr. This suggests that it is a local dwarf galaxy.

-MACS0416 (z = 0.397) was discovered by MAssive Cluster Survey (MACS, (Ebeling et al. 2010)), has been known as a merging system with double peak X-ray surface brightness distribution [START_REF] Mann | X-ray-optical classification of cluster mergers and the evolution of the cluster merger fraction[END_REF]). It was selected as one of five high magnification clusters in the Cluster Lensing And Supernova survey with Hubble (CLASH, (Postman et al. 2012)) because of its large Einstein radius. Aiming to get deep MUSE exposures of the FFs, the GTO program and ESO program are well joined. This allows the data collected from two programs to be merged. As a result, the data in the northern part of MACS0416 were added by the ESO program, while the other were collected by the GTO program. MUSE HFFs observations were conducted using 19 Observing Blocks (OBs). Sixteen of the 19 OBs were successful with an acquirement quality of A or B. Fourteen of the 16 OBs were observed with the assistance of the Ground Layer AO provided by the GALACSI module. An offset was required for each exposure, with a rotation of 90 degrees to improve the sky subtraction. All the observations were conducted under good weather conditions, with a seeing value ranging from 0.4" to 0.8". Under the GTO program, two-hour observation was split into 4 exposures [START_REF] Caminha | A refined mass distribution of the cluster MACS J0416.1-2403 from a new large set of spectroscopic multiply lensed sources[END_REF]). Finally, these data were combined, with a total observing time on the sky of 17.1 hours within an image quality of 0".6. Caminha et al. 2017 improved the strong lensing model of MACS0416 and found 37 multiple imaged sources. Twenty-two of them are newly multiple sources found in the MUSE data set, and the rest of them were detected from previous spectroscopic measurements. They are low luminosity LAEs with redshifts ranging from 3.08 to 6.15. Richard 2021 found 9 new multiple-imaged systems, raising the total number of multiple images in the MUSE cube to 22. Fig. 2.12 (upper panel) displays locations of MUSE spectroscopic redshifts over HST F606W-F814W-F160W color composites. With deeper data, more than 15 multiple systems with 33 multiple images have been found. Eleven of the 15 new systems were discovered with MUSE as faint LAEs. Recently, Vanzella et al. 2021 showed a triple imaged system overlaid between two MUSE fields. They included this system, which is also listed as system 100 in (Richard 2021), for completeness. Balestra et al. 2016 studied morphology and kinematic of the cluster out to 4 Mpc and showed a complex system composed of a main massive cluster with a mass of M ∼ 0.9 10 15 M ⊙ -MACS0520: The general information of this cluster is presented in Table . 2.1 and the lens model is presented in Table . 2.2.

-MACS0940: The lensing model of the cluster can be found in [START_REF] Richard | An atlas of MUSE observations towards twelve massive lensing clusters[END_REF]). An overview of multiple imaged systems behind this cluster is shown in Fig. -RXJ1347 is one of the most massive clusters with M 200cr i t = 3.54 ± 0.51 × 10 15 M ⊙ at redshift z = 0.451 (Caminha 2019), including two very bright central galaxies. This cluster is likely to undergo an initial phase of a significant merging event with a perturbation of intra-cluster medium [START_REF] Ueda | A Cool Core Disturbed: Observational Evidence for the Coexistence of Subsonic Sloshing Gas and Stripped Shock-heated Gas around the Core of RX J1347.5-1145[END_REF]). Based on the previous results, Caminha 2019 improved the mass model and considered two smooth mass components to reproduce the positions of all the multiple images. The results showed that the addition of an external shear improved significantly the lens model with a difference of 0.36". On the other hand, R. E. Johnson et al. 2012 used multiple wavelengths in X-ray and millimeter to show evidence of merging processes in most X-ray cluster galaxies. The result showed that the core gas was undergoing turbulance of gas, which points to the primary and secondary clusters having had at least two prior strong gravitational interactions. This supports the evidence for the second cluster to have been passed by the primary cluster. An overview of multiple imaged systems observed behind this cluster is shown in Fig. -SMACS2131 is a galaxy cluster with a redshift of z=0.44. The general information of this cluster is summarized in Table

Lens models

The signal from background galaxies in the field covered by strong lensing galaxy clusters will be magnified, with a global effect depending on the precise mass distribution within the cluster. The reader can find a detailed description of this phenomenon in [START_REF] Schneider | [END_REF]. As a consequence of this, to measure any parameter about the intrinsic properties of the sources in their respective source planes, we have to model the mass distribution of the lensing clusters. However, the distribution in the galaxy clusters is generally complex due to the presence of dark matter halos that we can not observe directly and which account for 80% of the mass budget. To tackle this problem, one has to introduce several assumptions. One of the most popular ones, used in this work, is to assume that the total mass can be separated into a series of components without any interactions between them. In this parametric approach, the mass components can be described by analytical models, assuming a projected "flat" distribution. This is a reasonable approach given the distances involved between the observer, the lens, and the background sources. For this need, we use the well-known tool LENSTOOL as described below see LENSTOOL proposes several different potential profiles to be used for the fit, but the most commonly used is the PIEMD (for pseudo isothermal elliptical mass distribution) see e.g. [START_REF] Kassiola | Elliptic Mass Distributions versus Elliptic Potentials in Gravitational Lenses[END_REF]. For each potential parameter, the user can freely define a prior parameter, which can be a uniform or Gaussian distribution. Only the bright galaxies have the privilege to be optimized individually as cluster scale mass potentials, whereas the other galaxies have to be linked together by a scale relation parameter that is the only parameter being optimized. Model parameters are constrained through an interactive process, based on as many multiple images as possible, identified by their spectroscopic or photometric redshifts, sometimes with the help of highresolution imaging (HST images). Note that the MUSE spectroscopy helps to improve the pre-existing models by the identification of new multiple images. The more images are provided, the more precise is the result of the model. For the multiple imaged systems, the images will be grouped together as a constraint factor in the same detailed observation. This step is the most important in the modeling process because a poor identification or incorrectly associated redshift can result in a distortion of the mass model. As a consequence, the excellent combination of MUSE and HST, mixing a large number of spectroscopic redshifts with highly spatially resolved images are helped us to improve lens mass modeling. The enhancements presented in the reconstruction mass distribution have been observed by Richard et The procedure for the mass model is performed in the following steps: + Introduce the multiple images and their redshift as a constraint factor by users. + Assume a mass distribution using the visible baryonic matter and an expectation + Project the multiple images of the same system into the source plane using the current mass model. The quality of the model is defined by the coincidence of the source position reconstructed from the given images.

+ Lens the sources, which are found in the previous steps, back to the image plane. The quality of the model is once again estimated as the difference between the expected and observed positions of the images.

+ The procedure is repeated back and forth between source and image plane, source by source, aiming to improve the model.

However, LENSTOOL does not take into account the morphology and flux ratio of the multiple images, only the multiple imaged positions have been used. This can be improved with two following approaches:

+ Direct reconstruction: CLEANLENS mode of LENSTOOL can be used to return values of pixels in the source plane from the image plane by inverting the lens equation. This can be done by isolating the image that we want to reconstruct. By inverting the lens equation pixel by pixel, it is possible to reconstruct the morphology of the background source.

+ Parametric reconstruction: Users can use the CLEANSET 2 function of LENSTOOL to model the background source by using the parametric light distribution. This means that information about source size, source ellipticity, and magnitude are parameters to be optimized to match with observed images. The quality of the model is measured by a chi-squared, which is defined as follows:

χ 2 t ot = i , j (F l ux pr ed i ct [i , j ] -F l ux obser ve [i , j ]) σ i j 2 (2.1)
where F l ux pr ed i ct , F l ux obser ve are the predicted and observed flux for pixel [i, j], σ i j is an estimation of flux error in the pixel [i, j]. The details of the method can be found in the thesis work of G. Mahler. The general information on lens models used for the thesis is presented in Table . 2.2. For each cluster, we provide the position of the cluster-scale DM halos, corresponding to the largest mass distribution, together with the Mass-to-Light scaling for the cluster galaxies, as well as the reference articles where details on the lensing models can be found.

Source detection and catalog building 2.4.1 Source detection with MUSELET

Source visualization and inspection in a 3D datacube, especially when it involves executing on individual channels can be challenging. To address this issue and facilitate the analysis of the MUSE cube looking for emission-line galaxies in an automated way, there is a tool called MUSELET (for MUSE Line Emission Tracker), developed by J. Richard. MUSELET is based on Source EXtractor (Bertin et al. 1996). It is a Python package used to detect emission lines in narrow-band slices of the MUSE cubes. MUSELET is available as part of the MUSE Python Data Analysis Framework (MPDAF) suite at this address https://mpdaf.readthedocs.io/en/latest/muselet.html, where the interested reader can find some instructions and tutorials to run the code. By searching for emission lines, and merging all detections into a single catalog, linking them with continuum sources, the emission lines within a spatial region defined by the "radius" parameter are merged into a single source. Then, one can estimate the redshift of the source based on all the emission lines.

Aiming to obtain emission lines and a high Signal to Noise Ratio (SNR) in each channel of the MUSE cube, a NB cube can be created by subtracting the background continuum. This is done by creating two neighboring cubes of the original cube, each with a width of 5 Å on either side of the wavelength of the Lyman alpha emission line. The continuum image is defined as the average of the two mean images from the blueward and redward subcubes. Finally, the NB image of the source is created by subtracting pixel by pixel between the mean image of the central cube to the continuum image. This is called a background subtracted image or NB image. Running the Source Extractor on an NB image can provide us with several images containing information about the source that is, the RMS image is the background image, the segmentation image shows the source region, the filtered image is the background subtracted and convolved with different seeing values, etc. To judge if a group of pixels could be considered as part of a source, we usually compare pixel by pixel of the filtered image to the RMS image. During this step, we progressively loosen source detection conditions with DETECT_THRESH and MIN_AREA parameters, where MIN_AREA defines the minimum number of connected pixels in the filtered image above the threshold source detection DETECT_THRESH with respect to the RMS image.

Redshift determination using Source Inspector

The following part introduces a new method to identify LAEs behind lensing clusters observed with MUSE/VLT [START_REF] Richard | An atlas of MUSE observations towards twelve massive lensing clusters[END_REF][START_REF] Bacon | The MUSE Hubble Ultra Deep Field surveys: Data release II[END_REF]). This work makes use of a Python-based package named MUSELET, a part of the MPDAF Python package [START_REF] Piqueras | Astronomical Data Analysis Software and Systems XXVI[END_REF]), aiming to detect emission lines in the MUSE datacube. The selection -2.4 Source detection and catalog building datacube is in the form of ∼ 300×300 pixels of 0.2", covering 1 arcmin 2 FoV and ∼3681 frequency channels from 475 to 930nm. The package, in turn, makes use of Source EXtractor package, developed by Bertin et al. 1996, to detect source emission in NB images, created from the MUSE datacube. The main goal is to build a source catalog by detecting emission lines and associating them with the continuum emission obtained either from the MUSE white light or HST images. Using this combination, one can measure the redshift of individual LAE. This process consists of 3 main steps:

-Step 1: Source EXtractor creates the white light and the NB images from the original MUSE datacube. Each NB image is an average of 5 adjacent channels, which improves the signal-to-noise ratio of the NB. The continuum level is estimated from two subcubes, both having the same spatial size and channel width of 25 Å one redward, and one blue-ward with respect to the detection line. The continuum image is then the average of the mean images of these two subcubes. NB images are continuum subtracted.

-Step 2: Source EXtractor runs in each NB image to detect source emission in that wavelength slice.

-Step 3: Source EXtractor tries to merge all sources identified from the previous step and associates the emission lines with continuum sources. The package will automatically match the spatial position of emission lines of the same source using a predefined matching radius, which is typical ∼ 0.8". Based on information on these emission lines, and their features, one can measure the source redshift.

The flowchart describing the process to build the source catalog is shown in Fig. 2.19. Using this package, more than 4000 images having a wide range of redshift from 0 to 6.7 have been identified from 17 lensing clusters observed by MUSE/VLT. However, due to the nature of emission lines, some images are intrinsically bright, while others are fainter. As a result, their redshifts are then estimated at different confidence levels and will need to be discussed further later.

For this work, we use all available HST images covering the same MUSE fields, aiming to create a photometric catalog of continuum sources, i.e., prior sources. This helps to locate source regions for subsequent spectral extraction. The continuum source is then associated/combined with the line emissions detected by MUSELET to form the prior source. HST images obtained using different filters sometimes help to find the dropouts of high redshift galaxies [START_REF] Atek | The extreme faint end of the UV luminosity function at z 6 through gravitational telescopes: a comprehensive assessment of strong lensing uncertainties[END_REF]). During the line detection process, Source EXtractor tends to split large clumpy sources/arcs into multiple images. These are treated as separate sources with slightly different redshifts.

The MUSE consortium has developed a package named Source Inspector, (Bacon 2023) to measure/estimate source redshift. This package makes use of HST, MUSE white light, and MUSE NB images to locate/guide the source position and to extract source's spectrum. Using template spectra, Source EXtractor provides 5 best redshift solutions that match with the source extracted spectrum. Based on the spectral features, source redshift is measured, and a confidence level of redshift is assigned. Confidence 1 means that the detection is tentative, due to low Signal to Noise (SN) or ambiguous lines. These sources, however, are still being kept in the final catalog, aiming to compare with the other observations. Confidence 2 corresponds to a source selection -2.4 Source detection and catalog building that is identified based on a single emission line without any additional information or several low SN absorption lines. Confidence 3 is the case where the identification is obvious, having high SN, clear spectral features, etc. Each source should be investigated manually and independently by at least 3 experts. A reconciliation meeting among them is held to discuss and then resolve the debate cases. An example of how to find the redshift of an LAE and its confidence level is demonstrated in Fig. 2.20. This M54 source does not show up in the HST images, and can only be identified thanks to MUSE. The Lyα line with a typical asymmetry profile is clearly visible together with an OIII emission line with high SN of ∼ 5. For this case, as it is a very reliable source, we assign it the highest confidence for redshift. Only sources having confidence levels of 2 and 3 will be used later for the construction of the galaxy LF.

We often make use of features of bright emission lines to identify source redshift. Lyα usually shows an asymmetric spectral profile. Another good example is the OII doublet, which has a blue peak that is about two-third the size of the red peak. The absorption line features of CaHK, MgII, and other absorption lines are relatively easy to recognize. However, there are also some features that can be easily mistaken for others. One must identify them with much care. selection -2.4 Source detection and catalog building All 17 lensing models have been built using HST images. It may happen that some of the newly identified sources belong to multiple image systems. We then use LENSTOOL to investigate these new sources/images to find possible new counterpart which might be missed during the process described above. As the density of the source at a given redshift is quite low, it is easy to identify and match the multiple images. This helps to increase the number of detected LAEs in the final catalog.

Lensing models sometimes help to improve the source redshift confidence level. For example, if a tentative source detection is located at the position of lensed predicted multiple images, it is likely to be assigned a confidence level 2. Finding more sources/images in turn helps to constrain better the lens model. In the context of this work, we have been studying the sources of cosmic re-ionization as seen by MUSE/VLT, focusing on LAEs. MUSE datacube of 17 lensing clusters provides more than 1400 lensed images of LAEs with redshifts from 2.9 to 6.7, some of them belong to the multiple image systems. To avoid sampling in the source plane several times, only one representative image is chosen for each multiple-image system. The represented image often has high enough SN, moderate magnification error, and less contamination from nearby galaxies. Thanks to lensing clusters, signals from distant galaxies are magnified by a large factor, so one can reach the fainter sources which have not been available in the previous blank field observations. Finally, 600 LAEs have been used for the next step in computing the LF.

LAEs catalog

There are three quality flags for the redshift determinations indicating confidence levels, from 3 (secure redshifts) to 1 (tentative redshifts). After running MUSELET and Source Inspector to identify the redshift of the source detected in the MUSE cube, we have more than 4500 images belonging to the systems, that have a redshift confidence level zcon f > 1 (foreground galaxies, stars, cluster galaxies, etc. at high redshift) behind 17 lensing clusters. More than ∼ 40% objects in this catalog are cluster galaxies. The distribution of the sources is shown in Fig. 2.21. The figure on the left displays the redshift histogram of all the sources detected in the MUSE cube (gray), of all the images which are rated with high confident level zcon f = 2, 3 (pink), and of the representative images for each system having zcon f = 2, 3 (magenta). Fig. 2.21 (right) shows the redshift distribution of all the detected sources (gray), of sources that have been found with line emission only (pink), and of sources detected with only line emission and having a high confident level zcon f = 2, 3 (magenta). All the galaxies from these 17 clusters in the present work are located in a narrow redshift interval, with a prominent peak of an overdensity of galaxies at redshift z ∼ 0.4. The number of sources significantly decreases in the redshift range of 1.5<z<2.9, which is known as the MUSE redshift desert (Fig. 2.21, left). At higher redshifts z > 2.9, LAEs are the main population in detected sources. From choosing image processing, we see that the number of multiple images is found at redshift z > 1.8, the "MUSE redshift desert" can be seen in a range of (1.5 < z < 2.9) with a deficit in redshift, at the higher redshift z > 2.9 the LAEs become main contributors for the redshift distribution. In the present work, we have kept only LAEs at redshift selection -2.4 Source detection and catalog building 2.9 < z < 6.7. For sources with a redshift confidence level of zcon f = 1, most of them belong to the systems having one image with a small magnification (see Fig. 2.22). According to the criteria defined above, these are tentative sources that do not display a clear Lyman alpha profile, have low signal-to-noise, or lack information on association lines, etc. We will ignore them from the list.

To save time during projection back and forth between source and image planes, after having identified the LAEs redshift with source-inspector, we choose a single representative image for each multiple systems based on several criteria: high SN, more isolation compared with other multiple images of a same system, a reasonable magnification uncertainties, and less contamination from other sources. This procedure is done by overplotting source positions on the HST images that were observed in at least two bands and overplotting on the White light image of the MUSE observation (see Fig.

f (λ) = A exp -(λ -λ 0 ) 2 2(a(λ -λ 0 ) + d ) 2 (2.2)
where A is the amplitude of the emission line, λ 0 is the wavelength in which the Lyman alpha emission line reaches the peak and can be converted directly from the redshift of the source, a is the asymmetry parameter of the line, and d is the typical width of the line. These parameters are optimized during fitting by applying the function form above to a subcube with a size of 5"×5"×12. Fig. 2.25 shows the distribution of the asymmetry parameter in the present LAEs sample. 81% of LAEs exhibit an asymmetry towards the red (a > 0), with a medium value of 0.18. This has been confirmed in many recent studies. However, the main reason for the asymmetry of the line profile is still difficult to interpret. The asymmetry generally correlated with FWHM in an asymmetric fit.

-The second method is obtained from SExtractor. In this method, we do not assume a parametric profile for the line, instead, the total flux is directly extracted from the cube. We create a subcube with a size of 10" × 10"× width of the Lyman alpha line and its two neighbor cubes on both sides with a volume of 10" × 10" × 20 Å. We then ask SExtractor to run on NB image to obtain FLUX_AUTO value of the source and its error. This parameter is developed based on Kron's first-moment algorithm, which ensures that 90% of its flux will be contained in the circular aperture of the Kron radius when the source is convolved with the seeing condition. To adapt to different seeing conditions, we relax SExtractor parameters (DETECT_THRESH, MIN_AREA) to ensure that the faint sources also have the opportunity to be detected. The final flux value is multiplied by the width of the emission line.

The first method is more reasonable because it takes into account the shape of line emission, whereas the second method is more efficient in measuring the flux of faint sources. In the present work, the second method is applied for sources found in To compute the LF of LAEs, we have used the non-parametric V max method (Schmidt 1968) to estimate the volume of the survey in which an individual source could be detected. The contribution of the source to the density of galaxies is given by reciprocal this parameter. The present work uses data collected from 17 surveys in lensing clusters meaning that the final value of V max for a given source should be computed from all the fields of the survey, including the fields where the source was present. Working in the image plane to compute the V max value of individual sources may lead to an overestimate of this value. For this reason, the present work has been done in the source plane, based on the creation of a 3D cube in the source plane for each LAE using the pipeline developed in DLV 2019, including some improvements presented in this thesis. The main steps are shown in Fig. 3.1 (DLV 2019).

3D mask cubes in the source plane

Noise level

The noise level of a given layer is computed based on its RMS image combined with the median RMS image of the cube as a normalization factor, so the noise level in the i t h channel is defined as:

Noise level i = Noise level (RMS i ) = < R M S i > x,y < R M S median > x,y (3.1)
where < R M S i > is the RMS image in the i t h layer, < R M S median > is the RMS image of the median layer of the cube obtained by running the Source Extractor on the individual NB images. The signal-to-noise value reflects only the evolution of the noise level with wavelength within a given cube and could not be used to compare the 126 noise level between the cubes. This evolution is shown in Fig. 3.2 for each of the 17 clusters. The noise levels in Fig. 3.2 are similar in all datacubes, except for MACS0257, MACS0416N, MACS0451, MACS0520, MACS0940, MACS2214, and RXJ1347. These cubes show an increase in noise level within the AO filter (around 5890 Å). This can be explained due to the shorter effective exposure time in the WFM-NOAO-N mode of the instrument, which is only 50 minutes instead of 7 or 8 hours elsewhere. However, all of the datacubes show the best sensitivity in the 6800-7800 Å wavelength range, with a detection limit of 3 -σ for an unresolved emission line (Bacon 2023).

3

Signal to noise of a given source

The probability of detecting a given source will be determined by comparing in each pixel the signals from the sources and RMS images. The signal of the source image is defined by a group of the 9 brightest pixels found from the filtered image obtained by running Source Extractor on the NB image of the source. They are found in a circle with a radius of 1" surrounding the source position. When the first pixel was found, we continued looking for the 8 connected pixels to find the second one. The process stops when the 9 brightest pixels of the individual source are found. This probability has been taken into account for all the fields surveyed in the present work, hence the filtered image has to be convolved/de-convolved with different seeing values. Consequently, each source will have one brightest pixel profile found from its parent cube, in which the source is presented and has 17 other brightest pixel profiles due to 3 Computing the Luminosity Function in Lensing Clusters -3.1 3D mask cubes in the source plane the convolving/de-convolving procedure. Fig. 3.3 shows the brightest pixel profiles of M39 detected in A2390 (left) and P8192 detected in A370 (right). M39 was detected under a seeing condition of 0.75" (the thick solid line). When the M39 filtered image has been de-convolved to lower seeing values producing its brightest pixel profiles (the thin solid lines above the thick solid line), better than the value found from the parent cube, while the convolving with higher seeing values will get a lower value (the thin solid lines below the thick solid line). Fig. 3.4 illustrates the 9 Brightest pixel (Bp) that have been found from the filtered image of a source detected in the A2390 MUSE cube. This method allows us to follow the morphology of the source and its detectability without assuming any prior shape. For example, from the profile, we can partly study the spatial profile of the sources because the extended sources tend to display a flattened Bp profile, while compact sources usually show a steep slope profile. The Bp profile of the individual source behind each lensing cluster is shown as the red solid curve of Fig. 3.7.

From the definition of noise level above, the signal-to-noise of each source in a given channel is computed based on its Bp profile and the noise level for that channel, following the equation:

S/N i = max(B p) noi sel evel (R M S i ) (3.2)
Using Eq. 3.2, a given pixel belonging to the source area could be detected and unmasked if its value satisfies the following condition:

B p[M I N _ARE A -1]/DE T EC T _T H RE SH < l oc al _noi se[x, y] (3.3)
where [x, y] is the pixel on the RMS map and B p[MIN_AREA-1] is the faintest value of the individual Bp profile of the source, the DETECT_THRESH is varied to make sure that the source always has the opportunity to be detected. In this case, we are using the faintest value of the Bp profile, meaning that if this value fulfills the criterion, the other bright pixels that have a higher value will also meet the same criteria at the local noise [x, y]. In general, the central pixel of the RMS map is unmasked. This process is applied to all the pixels of the RMS image. When the detectability condition is applied to a given RMS image, a masked image is generated. In principle, this image has to be projected into the source plane. To avoid creating too many images for each source with respect to a given signal-to-noise, we create masked images for a given cube at various signal-to-noise values, ensuring the detection of all sources within the cube. This can be done under two simplifications introduced below:

-First, the RMS map in each channel of the MUSE cube can be modeled and reproduced by scaling from the median map. This assumption has been introduced due to a linear evolution of the pattern with the MUSE slices that have been found during studying a large number of RMS images. The evolution with channels is shown in Fig. 3.5 for A370 (upper panel) and MACS1206 (lower panel) at slice numbers 500, 1000, 2500, 3000, and its median RMS image at channel 1839 (last column image). The different patterns of the RMS maps can be explained by the difference in morphology or the variance of noise with the slices. Fig. 3.6 shows RMS median maps of 18 datacubes used for the present work, the difference in the structure of the instrument, and exposure times are clearly visible in each image, suggesting that the assumption is reasonable for all the channels of RMS cubes. -Second, we use a general bright pixel profile, which may be the representative profile for all sources in a cube named general bright pixel profile (B p g ). The profile is built based on individual brightest pixel profiles of a source that is already convolved/de-convolved to adapt to the seeing conditions. The deconvolution algorithm is computed based on the description of a Weiner filter which is presented in Orieux et al. 2010 combined with the Python Scikit image package in Walt et al. 2014. The final value of B p g in each cube is the median value of the individual bright pixel profiles contained in that cube. At this step, the normalization is presented by max(B p g ) = 1, and the other general bright pixel profiles will be scaled following the value of the maximum general bright pixel profile the source plane before normalization.

Using the two assumptions, the signal-to-noise of a given source in a cube is given by the function:

s/n = max(C × B p g ) Noise level(R M S median ) (3.4)
where:

Noiselevel(R M S median ) = < R M S > median < R M S > median = 1 (3.5)
C and max(B p g ) are independent, so max(C × B p g ) = C × max(B p g ) = C , since max(B p g ) was previously defined as equal to 1. Consequently, the signal-to-noise of a source in a cube is found as the rescaled factor of that cube to the max(B p g ). In another way, we can write as:

s/n = C (3.6)
so now 2D masked images for each cube can be produced based on its RMS median image combined with its signal-to-noise value of s/n × B p g as the bright pixel profile. At this step, we have presented a set of signal-to-noise values (introduced signal-tonoise value) covering a range of 0.6 to 20 with 100 steps in logarithmic scale, which have included all possible values to detect the Lyman alpha sources in the present sample. Therefore, 100 masked images are generated for each cube. In connection with a given value of the signal to noise ratio, we estimate the ratio of masked pixels to the total of pixels in each masked image, named covering fraction. If this ratio is 0, meaning that the possibility of detection happens to any location within the image, a covering fraction of 1 suggests that the input source remains undetected throughout the entire image. The covering factor can vary between 0 and 1, indicating whether the source could be found in certain parts of the image. Therefore, if several images have the same covering fraction, we keep only one of them. Our data sample shows that bright sources tend to have a covering fraction value close to 0, whereas the diffuse sources tend to be 1 in their parent cube. In addition, at a higher redshift, our observations have been affected much by the skylines reducing the number of masked images adapted to the given criteria.

Creating 3D masked images

The previous subsection presented a way to produce 2D masked images for a given cube, that helps us reduce the number of masked images for the source plane projection, save time for computation, and avoid overlapping regions in the computation. However, the main purpose is to compute the total co-moving volume, where individual sources could be detected. This requires 3D masked images, which are associated with individual sources. This subsection presents how to form 3D masked images both in image and source planes.

-In the image plane: From Equation 3.2, the signal to noise of an individual source on all the layers of its the source plane cube can be estimated. They were resampled by 300 values over the MUSE cube, 100 values in the range from 4749 to 7100 Å, and 200 values in the rest. As I mentioned before, at longer wavelengths, the source detection is affected by the skylines, so the resampled signal to noise is denser in this region to achieve higher precision. The final masked images of each source are found by fetching the closest signal-to-noise value from the resampled measurement to the introduced signal-to-noise. -In the source plane:

In this plane, we have applied the same method of constructing the individual 3D mask image of the source. The 2D masked images from the image plane are projected in the source plane at different redshifts. This is done using PYLENSTOOL, which allows us to use LENSTOOL automatically. In practise, we have realized that there is 3 Computing the Luminosity Function in Lensing Clusters -3.2 Computing V max value not a substantial disparity in the pattern of the images at different projected redshift values, therefore there are four redshift samplings chosen for the projection z s = 3.5, 4.5, 5.5, and 6.5 (see Fig. 3.8, 3.9). At this stage, besides matching up the closest signal-to-noise value as the one in the image plane, we attempt to align the projected redshift to the MUSE resampling wavelength to make sure that the z s is the closest value to the (λ sampl i ng /λ Lyα ) -1. The number of source plane projections depends on the number of redshift values used for the procedure, the number of MUSE cubes, and the definition of introduced signal-to-noise values. While the number of sources in the sample and the precision of the sampling signal to noise are independent.

Computing V max value

After the projection of the 2D masked images onto the source plane, we compute the V max value of the sources by associating the proper masks. This is done on the source plane at the pixels in which the value of the magnification exceeds the given limited magnification to make sure that the source can be detected. Therefore, the process will apply to the area with a reasonable magnification. At the parent cube, where the source is present, this limit value is defined as :

µ l i m = µ δF d F d (3.7)
where µ is weighted magnification, F d is the detection flux value, δF d is uncertainty of flux value and µ l i m is the limited magnification of individual source. Following the definition of limited magnification value, the source is only detected at the region in which µ > µ l i m , meaning that the signal we are measuring is high enough to be detected. In addition, the procedure is extended to all the other cubes with different seeing values, Eq 3.7 will be rephrased to be adapted with this condition:

µ l i m,ext = < R M S ext > x,y,λ < R M S p > x,y,λ s 2 ext s 2 p × µ l i m,p (3.8)
where the subscript p and ext correspond to the parent cube, in which the source is present and the external one refers to other cubes. < R M S > x,y,λ is the median value over three axes of the MUSE cube, and s is the seeing value associated with observing condition. The final value of V max is integrated over the individual layers of the 3D source plane masked cube, accounting for the condition of the limited magnification µ > µ l i m . At a given channel the value is estimated following the equation:

dV k = N (µ l i m , k) × ω c H 0 z max (k) z mi n (k) D 2 L (z ′ ) (1 + z ′ ) 2 E (z ′ ) d z ′ (3.9)
where

E (z) = Ω m (1 + z) 3 + (1 -Ω m -Ω Λ )(1 + z) 2 + Ω Λ , D L
is the luminosity distance following the function of redshift, ω is the angular size of the MUSE pixel, z mi n (k) and z max (k) are the redshift limits at k t h channel with respect to its spectral value width, and N (µ l i m , k) is the number of unmasked pixels on the layer k associated with the condition of limited magnification. This function can be applied to compute V max value for the full redshift range 2.9 < z < 6.7, for the effective volume defining the region from the lowest redshift of the survey to the redshift value of the source. From Fig. 3.10, most of the sources in the present work have been detected with a large value of the volume. However, several sources have a large magnification detected in a small volume. The V max value tends to raise with the detection flux, this can be seen in the lower left panel of Fig. 3.10. This trend also can be seen on the lower right panel of that figure, which shows a weak correlation between luminosity and its V max value. The dependence of V max on source magnification is shown in Figs. 3.11 and 3.12. For the highly magnified sources, that may be close to the caustic curve, its magnification deviation value varies largely when the source crosses the line, the uncertainties may reach up to 10% or more compared to its weighted value. When the magnification value is changed within its uncertainties, an exponential function can be used to describe this change. As the source has a modest magnification value, a linear function can be presented. This dependence of V max on lensing magnification has been accounted for in the LF computation process. The total co-moving volume computed following the procedure mentioned above is listed in Tab. 3 is defined as the number of galaxies per a cosmic co-moving volume and a luminosity interval. However, not all the sources in the data sample can be 100% detectable, hence the requirement to correct for the actual probability of finding a source, which is called completeness. In this case, the (in)completeness is an additional correction which is needed to account for the detectability of a source in its parent MUSE cube. It is applied as a multiplicative factor to its previous contribution to the LF (see Sec. 4.2 and Eq. 4.1 below). In the following sub-sections, we present and discuss the determination of the completeness factors for each source, the determination of the LF values, and the fit of the LF points by a Schechter Function. We also discuss the comparison between our results and the previous ones in the literature.

Completeness determination

In a given noisy environment, the probability of successfully detecting the sources depends on their profiles both spatially and spectrally. This means that sources with too large of the noise are not detected. In this section, we introduce a parameter referring to this probability aiming to account for the noise effect. The process to estimate this parameter is conducted in the image plane and applied from source to source to retrieve the detection profile of each LAE and inject it into the masked detection layer. The final completeness value is estimated based on the number of successful outcomes divided by the total number of random injections of the source profile into a mock image.

Reconstruction of the source profile in image plane

The effect of noise on the probability of source detection will be investigated in the image plane where the source has been found first. In the previous steps, the source is always associated with a given channel in a MUSE cube corresponding to a given noise level. As a result, in this section, the probability that I mentioned before should be calculated in the plane where the Lyman alpha emission of the source reaches its peak, called the max-NB image. This image should not be tracked from the redshift of the source, because in some cases there is an offset between the channel which is retrieved from redshift and the one found from the visual source inspection process. For several faint sources, with a complex morphology that varies from channel to channel, the difference in the selection NB layer could be the reason for the underestimation of the probability. The main difficulty is to properly model the light distribution both in the spatial and the spectra dimensions. To assess the quality of the detections, we use the same NB image which is extracted from the source inspection step. To have a more efficient accounting for the noise, the image of the individual source is sized by 80" × 80", to better account for the local noise in the source region. The source recovery is done based on two steps using the same criteria as the original source detection by using the same configuration/parameters of Source Extractor in the MUSELET package.

-From the max NB image of individual sources we run the Source Extractor using the same configuration of the MIN_AREA, DETECT_THRESHOLD, and MATCH-ING_RADIUS aiming to retrieve the segmentation image of the source. These parameters have been progressively loosened to make sure that the source has maximum opportunity to be detected.

-On the other side, we continue running the Source Extractor on the max NB image aiming to obtain the filtered image, that is background subtracted and convolved with different seeing values.

From the segmentation map, we have tried to match the positions in which the source has been detected (in the segmentation image) to the positions in the filtered image. Sometimes the Source Extractor does not work properly when creating the segmentation map image for faint sources. To compensate for this effect, we will search for the connected pixels in the filtered image. Finally, a sub-image containing all the pixels which have been assumed as a part of the source will be kept and matched to the object image to form the source profile in the image plane of individual sources.

The quality of the extraction process for individual sources has been quantified using a flag value. As shown in Table 4.1, trustworthy cases are assigned values Flag=1 or Flag=2, whereas a source denoted Flag=3 could be treated as a tentative detection. In the following steps, we keep only the good extractions (Flag=1,2) and ignore the dubious cases. Applying these criteria to the data sample behind 17 lensing clusters, only 2% sources are marked Flag = 3, 5.5% cases are marked Flag = 2, and the rest are Flag = 1. The appearance of Flag = 3 could be explained by: + They are the faint sources, which have only a few pixels having a value above the given threshold to be considered as a part of the source. + There is a difference between the source position found in the catalog and the recovered source position. In practice, the morphology of the source varies from channel to channel, the difference in position between the recovered location and source position in the max-NB channel may be larger than in the other neighbor channels.

+ The size of the image is such that the exploration region is too small to provide a fair sampling of the background. In this work, we have faced several such cases, and for this reason, the size of the window has been enlarged from 30" × 30" used by DLV 2019 to 80" × 80". This results in a significant increase of the Flag=1,2 images with respect to the smaller window. For a source size of 30" × 30", 80% of the sources are marked Flag = 1,2 and the rest are Flag = 3. If we use this condition, we reject 20% of sources that belong to the faintest luminosity bins.

Source recovery

In this section, I use the source profile found in the last subsection to obtain the simulated source profile in the image plane. The advantage of using the source profile is that it takes into account both the spectral profile and spatial profile of the LAE. To recover the source, we inject the source profile into a mocked image, which is built based on its max NB image (see Vieuville 2019 for details). We run Source Extractor on that image and count the number of successful detections. This is different from the previous steps when we ran the Source Extractor on the whole MUSE cube.

We have created a mock image for each LAE, which has been built based on its masked max NB image. This image is tracked following the previous steps using the SN value of the source in its detected layer in the MUSE cube. The injection is done only in pixels that are unmasked to make sure that the completeness value is properly taken into account.

The source profile is injected randomly 500 times on the mock image, each time the source is placed, the pixels are masked to avoid dominating too much at a given position or overlapping with other mock sources, and a new NB image will form. It asks Source Extractor to run using the same values of MIN_AREA and DETECT_THRESH. We match the detected source position in the catalog, which is created by Source Extractor to the input source position. If the distance between the two positions is less than 0.8", then the process is considered successful. The final completeness value is defined by the ratio between the number of recovered sources and the number of injected sources. The uncertainty in the completeness value of an individual source is estimated in the same way, the distribution of the recovered fraction provides the uncertainty value. A large number of sources has completeness value greater than 50%. In most works dealing with the LF in lensing fields, all the images/sources are kept in the sample irrespective of their completeness values (that is, applying Equation 4.1 below without any restriction). For this reason, here we keep as many sources as possible by rejecting only sources having completeness values below 1%. This is discussed in more detail below. Note that DLV 2019 rejected sources with completeness below 10%, making this comparison more difficult. Fig. 4.4 shows the measured fluxes of LAEs versus their redshift. Fig. 4.4 (left) displays the normal case of noise evolution, while Fig. 4.4 (right) shows an abnormal case of noise evolution with a clumpy appearance surrounding z ∼ 3.7, corresponding to a wavelength of ∼ 5500 Å. The low-completeness sources are the faintly detected flux ones, while the high-completeness sources seem to be detected with high signalto-noise. These features are visible in the figure. However, some sources have high detection fluxes and low completeness values, suggesting that source morphology is important, as already mentioned above. From left to right, up to down is the distribution at redshift 2.9 < z < 6.7, 2.9 < z < 4.0, 4.0 < z < 5.0, and 5.0 < z < 6.7.

Discussion

Fig. 4.3 displays the number of sources distributed in each luminosity bin before (blue line) and after (orange line) correction for the completeness values at 4 redshift ranges 2.9 < z < 6.7, 2.9 < z < 4.0, 4.0 < z < 5.0, and 5.0 < z < 6.7. At lower luminosity bins, most of the sources are faint ones with low completeness values. Therefore, when we correct for completeness, the number of sources should be detected in the cube increasing significantly. In contrast, the higher luminosity bins contain a large number of sources having a high completeness value, so the corrections do not have a strong impact.

LF computation

This section presents how to compute LF points in each luminosity bin after obtaining completeness and V max of individual sources. In each luminosity bin, the value is defined as follows: where the index i defines the luminosity bin i t h , index j corresponds to the source j in the data sample, C j and V max,j are the completeness and V max of the j source, ∆log(L i ) refers to the width of luminosity bin i t h in logarithmic scale. To check the possible evolution of LF with redshift, we use four redshift ranges: 2.9 < z < 6.7, 2.9 < z < 4.0, 4.0 < z < 5.0, and 5.0 < z < 6.7. We refer to these ranges as z al l , z 35 , z 45 , and z 60 , respectively. The luminosity bin width has been used as a normalization factor to study the density of the sources in a given co-moving volume and a given luminosity value. However, the number of sources in the faintest luminosity regime decreases quickly, so there may not be enough sources to populate the faintest luminosity bins. For this reason, in the global redshift interval, we divide the explored range into 12 luminosity bins with a width of 0.25, except for the faintest bin, which has a width of 1.25 to have a homogenous source distribution in each luminosity interval. This is applied to other redshift ranges with a larger width of 0.5 covering a luminosity range from 39.0 to 43.0, 39.0 to 43, and 40.0 to 43.0, with respect to the z 35 , z 45 , and z 60 . To estimate the mean and the uncertainty of the LF values, we have employed an MCMC method that takes into account the contributions of completeness, flux measurement, and magnification of individual LAE in the data sample. We compute the LF for each of the 20,000 MCMC iterations. At each iteration, the value of completeness and flux are taken randomly within their error values while the magnification is collected from its magnification distribution in the image plane.

Φ(L i ) = 1 ∆logL i j 1 C j V max,j (4.1)
Faint sources are often highly magnified and have large magnification uncertainties. During each MCMC iteration, the luminosity of the source is recomputed based on the random input values of the source magnification, completeness, and flux, as described above. This means that the MCMC process may spread the contribution of a faint LAE source to several luminosity bins. On the other hand, bright sources with moderate magnifications only contribute to a given luminosity bin during this process. These features are illustrated in Fig. 4.5. Fig. 4.6 shows the LF results for each of the 12 luminosity bins after 20'000 MCMC iterations, together with the LF face value, which 4 Luminosity function -4.2 LF computation is computed directly based on the observed values of LAEs. The mean and statistical uncertainties of LF points are obtained from these histograms.

Apart from the statistical uncertainty obtained by MCMC process, there are two additional uncertainties that contribute to the total uncertainty of the LF points: cosmic variance, which accounts for the limited field of views of the survey and is derived from the calculations in Trenti et al. 2008, and the Poissonian representation of the source distribution within each luminosity bin. The final uncertainties are written under the form:

∆Φ(L i ) 2 = 1 ∆l og L i N i ,c < V max > δ 2 CV + 1 N 2 + δ 2 MC MC (4.2)
where δ CV refers to the cosmic variance uncertainty, δ MC MC is the MCMC uncertainty, N i ,c is the number of LAEs after correcting for their completeness values in i t h bin, < V max > is the average value of V max in i t h bin. The Poissonian uncertainty represents the bias of source distribution in each luminosity bin referred by 1/ N i . Fig. 4.7 shows the total LF uncertainties for different luminosity bins and redshift ranges. It raises the important point that the LF face values obtained directly from observations are subject to large uncertainties, as they may not reflect the "true" values. This shows the importance of properly taking into account different sources of uncertainties when computing the LF. 5.9 5.9

(L) [erg s -1 ] log (φ)(∆(log (L)) = 1) -1 [Mpc -3 ] < N > < N cor r > < V max > [Mpc 3 ] 2.

Binning effect on the LF points

One factor that may affect the shape/evolution of LF at different redshift bins is the width of the luminosity bins, from that it may affect the source density measured in each bin. This can be tackled by testing the stability of the results obtained with different bin widths, as presented in this subsection. Fig. 4.8 displays the effect of luminosity binning on the shape of the LF evolution. At redshift z al l , we have divided into six groups corresponding to six different bin sizes of luminosity and recompute LF values with respect to each bin. This also has been applied to other redshift ranges z 35 , z 45 , and z 60 . The results show that the evolution of the LF does not significantly depend on the binning, as can be seen at low redshift ranges z al l and z 35 . But in the higher redshift range, which contains fewer sources, the shape may vary a bit (Fig. 4.8 bottom right). The figure also shows the changes in uncertainty values allocated in each luminosity bin during the tests. Independently of the binning tests, we find that the contribution of cosmic variance to the total uncertainty budget is about 15 to 20% at redshift z al l and 15 to 30% at other redshift ranges. The MCMC uncertainty significantly dominates at the lower luminosity bins, where the magnification and completeness uncertainties are larger compared to the bright luminosity bins. The Poissonian uncertainty becomes the main contributor at the highest luminosity bins with a smaller source density. There is a drop at log L = 42 erg s -1 , this causes by an appearance of a low completeness source contributed to the previous luminosity bin. -1 .M pc

-3 ])
LF for 2.9 < z < 6.7

Effect of source selection to the evolution of LF points

As I mentioned earlier, in this work we only use LAEs that have been identified with a high secure zcon f = 2, 3, and reject the ones having a lower rate zcon f = 1. In this subsection, I include the sources which are treated as a lower quality to the final data sample and investigate their effect on the shape of the LF. It is worth noting that zcon f = 1 objects have a tentative redshift assigned, with ∼50% chances has been corrected. There are typically two different cases for zcon f = 1 LAEs: faint detections with low SNR, or higher SNR detections with an ambiguous identification of a unique line. LAEs belonging to multiple imaged systems have zcon f = 1 because of lensing considerations, making the redshift determination more secure.

Luminosity function -4.2 LF computation

Before including all of the zcon f = 1 sources, we have made a small test on the sources observed from RXJ1347 to probe their effect and decide what we should do in the next steps. Indeed, there are 46 LAEs listed as zcon f = 1 in this cluster observed with a shallower exposure time, that is two hours in the extended region and three hours in the central one. Most of these zcon f = 1 LAEs are single-imaged, with a relatively small magnification value. Applying the same procedure and same criteria as the zcon f = 2, 3 sources we found that the completeness values of these sources are very small. The faintest sources are the ones having a poor extraction quality which are marked by Flag = 2, 3 in Fig. 4.9. In addition, we find that more than half of the sources have completeness below 10%. Combining these results with the information on the definition of zcon f = 1 leading to a final decision, we don't have 100% confidence to say that they are real LAEs detected from the cube. Based on the preliminary results, I have performed two tests to estimate the effect of source selection on the final results. One includes half of zcon f = 1 and the other is adding all the zcon f = 1 to the final data sample. In both cases, when adding the zcon f = 1 LAEs, the assigned values for the completeness and V max in each luminosity bin were taken from the median values of zcon f = 2, 3 sources in the same bin. This was also applied for the uncertainty values of completeness and V max of the zcon f = 1 sources, meaning that these values of zcon f = 1 sources were set equal to the zcon f = 2, 3 sources corresponding to that bin. ), the number of zcon f = 1 is similar to the number of the zcon f = 2, 3 whereas, the appearance of zcon f = 1 sources at the bright end is almost zero. With the assumptions given above, we find that there is no effect of zcon f = 1 sources on the shape of the LF and its uncertainty values at the bright end. On the contrary, there is a small variation towards the faint end. Fig. 4 Luminosity function -4.3 LF results and comparison to the literature 4.11 shows the results of the LF points when incorporating half (blue) and all (red) zcon f = 1 sources into the data sample. It again confirms that the source selection effect does not significantly affect the obtained LF shape, especially at the faint end. These results are later used to estimate the systematic uncertainties associated with the faint-end slope. -1 .M pc -3 ]) 2.9 < z < 6.7 ( line fit: α = -2.12± 0.15 0.16 ) 2.9 < z < 4.0 ( line fit: α = -1.76± 0.26 0.27 ) 4.0 < z < 5.0 ( line fit: α = -1.99± 0.12 0.17 -1 .M pc

-3 ])
Not used for fit 2.9 < z < 6.7 ( line fit: α = -1.64± 0.12 0.12 ) 2.9 < z < 4.0 ( line fit: α = -1.55± 0.17 0.17 ) 4.0 < z < 5.0 ( line fit: α = -1.63± 0.21 0.21 ) 5.0 < z < 6.7 ( line fit: α = -1.66± 0.22 0.24 )

Figure 4.12: Faint end slope of LF at four redshift ranges 2.9 < z < 4.0 (blue), 4.0 < z < 5.0 (green), 5.0 < z < 6.7 (red), and 2.9 < z < 6.7 (black) when 1% completeness cut (left) and 10% completeness (right) cut have been applied. These fits use only our LF points constructed from the current sample. The open squares are not included in the fit.

LF results and comparison to the literature

To investigate the evolution of LF with redshift, we performed a simple linear fit to the slope on our LF value in four redshift ranges: one covers all the data sample and three others are sub-ranges corresponding to 2.9 < z < 4.0, 4.0 < z < 5.0 and 5.0 < z < 6.7. These results are shown in Fig. 4.12. The open squares at the faint end are not included in the fit. They are often associated with sources that have low completeness, low luminosity, or too high magnification values. In these cases, they are considered less reliable for simple linear fitting. We will discuss the possibility of taking these points into account for Schechter fits later. The best-fit results are plotted in four redshift ranges with a 68% confidence region, shown as a shaded region. It is worth mentioning that, due to lacking statistics in the faint luminosity bins, the width in such cases is larger compared to the others. The final slope values of the LF are -2.12 +0.15 -0.16 , -1.76 +0. 26 -0.27 , -1.99 +0.12 -0.17 and -1.96 +0.5 -0.54 , corresponding to the redshift range z al l , z 35 , z 45 and z 60 . Our results confirmed that there is no evolution of the slope with the redshift, which has been mentioned in previous literature, specifically the work of DLV19, that focused on the LF at redshift 3 < z < 7.

The work has been extended by applying the same completeness threshold (10%) as the one presented in DLV 2019 (Fig. 4.12 right). In both cases, a turnover at log L ∼ 41 [erg s -1 ] is seen. Rejecting sources having completeness below 10% corresponds to

Fitting with of a Schechter function

After having a general view of the linear fitting to the LF points, in this section, I present the procedure to fit our LF points using the Schechter function at four redshift ranges. I compare the results to the literature and discuss the findings. The data used in this work are from lensing clusters, which probe fainter luminosity bins compared to other observations in the blank field. However, the total co-moving volume in the present work is smaller due to being constrained by the magnification. To analyze the LF evolution we use the Schechter function, which is written under the form:

Φ(L)d L = Φ * L L * α exp - L L * d L L * (4.3)
In the log form, this equation translates into:

Φ(L)d L = ln(10)Φ * L L * α+1 exp - L L * d (logL)
where L * is the characteristic luminosity parameter when the Schechter function transfers from exponential law at the bright part to the power law at the faint part, α is the slope value of the Schechter function at the faint end, Φ * is the normalization parameter. The previous studies showed that the characteristic luminosity of the Schechter function is usually around log L * ∼ 42 [erg s -1 ]. Our sample's maximum luminosity of ∼ 43 [erg s -1 ] means that our LF points are less efficient to probe the bright regime. To better constrain the bright part LF, we have used the results from the literature, which cover the same redshift and luminosity ranges as the present work. However, including such numerous data for the fitting could lead to bias that dominates the bright-end data. Therefore, these data have been averaged with the same size of luminosity bin as in the present work. The luminosity log L * found in this work is in line with DLV 2019, a few percent higher than the one obtained in Herenz et al. 2019, 42.20 +0. 22 -0.16 at global redshift range 2.9 < z < 6.7. As mentioned in DLV 2019, the selection of data sets used for the Schechter fitting has an impact on the final results. This has been confirmed in the present work when we applied a different completeness threshold to decide which source would be kept for the LF computing. Even though the data are collected from the literature and used as a constraint at the bright part, a strong degeneracy between the parameters is still being seen. Consequently, we have to be careful to choose reasonable literature data, as performed here. From the present work, the log L * parameter is well measured with different redshift bins. Within the uncertainty, this parameter displays a slight increase with the redshift.

The normalization factor Φ * of the Schechter function is in line with the results obtained from DLV 2019 and Sobral et al. 2018, but it shows a smaller value compared to the one from Herenz et al. 2019. This parameter is a normalization factor. It gives the number density of objects per unit volume and strongly depends on the literature data points used for the fitting.

While previous investigations, which lack data in the faint luminosity regime, usually fixed the slope value α at a fiducial value and tried to find the correlation between the two other parameters, the present work is able to find these parameters without any constraint. The final slope value α varies from -2.0 +0.07 -0.07 at redshift range 2.9 < z < 4.0 to -2.28 +0.12 -0.12 at redshift 5.0 < z < 6.7. At the global interval 2.9 < z < 6.7, the slope value -2.06

log(L) -6 -5 -4 -3 -2 -1 0 1 log(Φ [(∆ log L = 1) -1 .M pc -3 ])
Luminosity function for 2.9 < z < 6.7

log(L) -6 -5 -4 -3 -2 -1 0 1 log(Φ [(∆ log L = 1) -1 .M pc -3 ])
Luminosity function for 2.9 < z < 4.0

-1 0 1 log(Φ [(∆ log L = 1) -1 .M pc -3 ]) 6 -5 -4 -3 -2
Luminosity function for 5.0 < z < 6.7 and z 35 can be explained by the existence of very low completeness sources, which will contribute a large number of sources to the bin after correcting for completeness value. This may be related to the overdensity of background sources at redshift z ∼ 4. Once again, this suggests that a larger cosmic variance than expected may happen in this bin. The present work extends the work performed by DLV 2019, therefore it is worth comparing the results obtained from the two works. The best-fit parameter of the characteristic luminosity displays a consistency within 1 -σ uncertainties in both different redshift bins. However, the slope value obtained in the present work is 20% steeper compared to previous published values. This could be explained by: + The difference in the data sample. The current study includes four times as many sources as those in DLV 2019.

+ The different completeness thresholds. Applying a lower limit in the present work to include as many sources as possible also affects the shape of the LF at the faint end. To illustrate this discussion, we have applied a completeness cut at 10% as performed in the work by DLV 2019. The results are displayed in Fig. 4.16. As expected, the number density of sources decreases significantly compared to those that are cut at 1%. In both cases, the turnover/flattening trend at luminosity logL ∼ 41 ergs -1 is observed.

+ The improved MCMC procedure (with respect to DLV 2019) better captures the relationship between magnification and V max values of the sources.

-Effect of the literature data on the fitting of the linear function and the Schechter function.

As mentioned earlier, the data obtained from the lensing field are not sufficient to constrain the LF in the bright luminosity regime, i.e., above L * . Therefore, literature data from the bright end are needed for the fitting process. However, as the data in the bright end are numerous, including them for the fitting of the Schechter function may affect the final results. In this part, we investigate different combinations between our data points and those from the literature to study a possible impact. Specifically, at the bright end, above log(L) ∼ 43, we rely on literature data points and rely solely on our LF points at the faint end. By doing this, we want to illustrate the importance of the current data sets. For linear fitting, we made the fit in a luminosity interval of 40.5 < log L [erg s -1 ] < 42.5 for 2.9 < z < 6.7, 5.0 < z < 6.7 and 40.0 < log L [erg s -1 ] < 42.5 for two redshift bins 2.9 < z < 4.0, 4.0 < z < 5.0. The results are shown in Fig. 4.17. The faint-end slope values indicate a decrease when compared to the previous case involving a complete data sample (Fig. 4.12, left). The decreases are 16%, 18%, 7% and 23% with respect to the redshift interval z al l , z 35 , z 45 , and z 60 . L * is the parameter describing the "knee" of the Schechter function at the bright end. By removing the "knee" point, which is constrained by bright LF literature points, the faint-end slope decreases, as expected.

We performed the Schechter fitting, using LF data log(L) above 43 [erg s -1 ] from the literature and below 43 [erg s -1 ] from ours. The results are shown in Fig. 4.18. In general, the new results are in line with the original Schechter function fitting within their slope uncertainties. The slope value of the Schechter function when applying a different completeness threshold and including only literature data points that are above 43 [erg s -1 ] to the fit are presented in Table 4.5.

-Effect of source selection In the 17 lensing clusters, 190 LAEs (unique systems) have been classified as zcon f = 1. However, due to the ambiguous nature of their detection line profiles, we have not included them in the computation of the LF presenting in the previous section. As mentioned earlier, at the faint-end,log L< 41 [erg s -1 ], zcon f = 1 sources are just as numerous as zcon f = 2, 3 sources. They may contribute significantly to our faint LF points. To evaluate the impact of zcon f = 1 sources on the final luminosity function (LF) points, we have incorporated them into our LAE sample. By adding two subdatasets (half of zcon f = 1 and all the zcon f = 1) to the present data sample and employing the Schechter function fitting, the faint end slopes obtained from the two categories are -2.14 ± 0.08 and -2.29 ± 0.09, respectively. These values are steeper compared to the original sample (i.e., 5% steeper when half of zcon f = 1 sources have been included, and 10% steeper for the case in which all the zcon f = 1 sources have been added).

-Systematic uncertainties attached to the faint-end slope + Apart from the three types of uncertainties -statistical, cosmic variance, and Poissonian -one should also take into account the systematic uncertainty coming from lensing models, which constitutes another significant factor within the overall error estimation. The adoption of various lensing models might lead to distinct magnification factors for a given source, thereby amplifying the uncertainty associated with the obtained luminosity. This systematic uncertainty has been extensively documented in studies such as R. J. Bouwens + Another source of systematic uncertainties may come from the flux measurement. As mentioned in the previous section, source fluxes are obtained from two different methods. We mostly use fluxes from Claeyssens et al. 2022, which were obtained by fitting source spectra (first method). The remaining 1/3 of the sample was obtained from the second method using SExtractor. To assess the impact caused by the choice of using different methods, we recomputed the LF points by replacing all source fluxes obtained from the first method with those of the second one, i.e., using only detection fluxes. This computation further confirms the robustness of our results, as there is quite good agreement between the two. Using these new LF points, we obtain the Schechter's faint end slope value that is approximately 10% less than the previous slope. These results are useful to estimate the systematic uncertainties attached to the faint end slope.

Taking into account all possible effects that may affect the shape of the LF at the faint end, such as the completeness threshold, source selection, literature selection, and flux measurements, we estimate the uncertainty values of the slope at different redshift bins. The final slope values are -2.00 ± 0.50, -1.97 ± 0.50, -2.28 ± 0.50 and α 2.9 < z < 6.9 2.9 < z < 4 4 < z < 5 5 < z < 6.9 -1 .M pc

-3 ])
Not used for fit 2.9 < z < 6.7 ( line fit: α = -1.79± 0.24 0.48 ) 2.9 < z < 4.0 ( line fit: α = -1.45± 0.11 0.11 ) 4.0 < z < 5.0 ( line fit: α = -1.86± 0.18 0.24 ) 5.0 < z < 6.7 ( line fit: α = -1.52± 0.2 0.32 ) -The probability of a turnover in the faint-end at the higher redshift ranges. Until now, the shape of the LF at the faintest luminosity/magnitude regime and the high redshift ranges is still not well known due to a lack of data. It is possible that there is a turnover, such as the one mentioned in R. J. Bouwens, Illingworth, Oesch, H. Atek, et al. 2017 at M UV > -15 with a faint end slope of α = -1.91 ± 0.04. Atek 2018 presents a probability of such a turnover at a faint end slope of α = -2.01 +0.12 -0.14 , a curvature parameter of β = 0.48 +0.49 -0.25 at a turnover magnitude M T = -14.93 +0.61 -0.52 . However, the recent paper by Bouwens et al. 2022 ruled out this trend at redshift z ∼ 6 when the magnitude reaches M UV = -14. It is also possible that the faint-end slope continues to increase with magnitude, as suggested by Livermore et al. 2017 with a slope value of α = -2.10 ± 0.03 at magnitude M UV < -12.5. Dawoodbhoy et al. 2023 used the CODA simulation to study the UV LF at redshift z ∼ 6 and found strong evidence against a possible turnover at magnitude M UV = -12. In the present work, we do not take the LF points at the faintest luminosity bins into account for the traditional Schechter fitting. We observe a possible turnover at the highest redshift intervals z 45 and z 60 (Fig. 4.13, lower panels). To account for this, we have introduced a modified Schechter function by multiplying the original function by an exponential term, which is now written in the form:

Φ(L)exp(-L T /L) m = Φ * L * L L * α exp(-L/L * )exp(-L T /L) m (4.

Comparison with theoretical predictions

Star formation rate density

One of the greatest interests in the studies of galaxy evolution and reionization is understanding SFRD, which is a fundamental quantity of star formation occurring per unit time per unit volume at a specific redshift. Following the instruction proposed by [START_REF] Donald | Book Review: Astrophysics of Gaseous Nebulae and Active Galactic Nuclei /University Science Books[END_REF], the quantity can be determined within a redshift range by integrating the LF to compute the luminosity density and be transformed using the following formula:

SF RD Ly a [M ⊙ yr -1 Mpc -3 ] = ρ Ly a /1.05 × 10 42 (5.1)
where ρ Ly a is Lyman alpha luminosity density in units of erg s -1 Mpc -3 given by this expression:

ρ Ly a = L sup L i n f Φ(L/L * )d (L/L * ).
The total star formation rate density (SFRD) shows a direct correlation with the density of luminosity and can be estimated by applying the calibration method established by Kennicutt 1998. This estimation involves assuming an intrinsic factor of 8.7 between the intrinsic fluxes of Lyα and Hα. Furthermore, the evaluation adheres to the case B recombination scenario, as outlined by Donald E. Osterbrock 1989, where the assumption is that all newly generated Lyman photons are re-absorbed by the neutral hydrogen atoms within the HII region. On the other hand, one converts the Lyα luminosity into Star Formation Rate (hereafter SFR) by assuming that the escape fraction for Lyman alpha photons is one, meaning that the SFRD above is a lower limit. From the luminosity distribution and the evolution of the LF at log L [erg s -1 ] > 42, the shape of LF fits quite well with the Schechter function. This means that the final value of SFRD will not be affected by the upper limit of luminosity L supp for the integration. On the contrary, the steep slope at the faint end makes the lower limit L inf play an important role in estimating SFRD. This is illustrated in Figure 5.1. The red crosses show the SFRD for a luminosity interval of (39.5 < log L [erg s -1 ] < 43), which covers the full range of present work. The brown-reddish crosses show the SFRD for high-narrower luminosity range of (41.0 < log L [erg s -1 ] < 43.),

5 Star formation rate density and implications for the reionization -5.1 Star formation rate density whic corresponds to the limit ∼ 0.03×log(L * ). This narrower range was chosen to include only the most secure determinations of the LF. It is clear that at a narrower range, the number of sources contributing to the budget is smaller than in the case of a wider luminosity range. The yellow region shows 1 -σ and 2 -σ confidence levels of the SFRD required to fully ionize the Universe, taken from the R. J. Bouwens, Illingworth, [START_REF] Bouwens | UV Luminosity Functions at Redshifts z ∼ 4 to z ∼ 10: 10,000 Galaxies from HST Legacy Fields[END_REF], assuming a clumping factor of 3 (see below) and the conversion to UV luminosity density with an assumption of log(ξ i on f escp ) = 24.5, where f escp is the escape fraction of UV photons, ξ i on is the production efficiency of Lyman-continuum photons per unit UV luminosity. The conversion to SFRD is then calculated as, SF RD Ly a [M ⊙ yr -1 Mpc -3 ] = ρ UV /(8.0 × 10 27 ). . This includes the recombinations that occurr in gas that is a local domination candidate for ionization state. This clumping factor is around 4 at redshift z = 3.5, and decreases at higher redshifts. For example, it is 2.4 at z = 6 and falls between 1.2-1.4 at z > 6. This decrease can be explained by the increase in the neutral fraction of hydrogen in the environment. This clumping factor is even lower than 1 at redshift z > 10, as mentioned in the [START_REF] Finlator | Gas clumping in self-consistent reionization models[END_REF].

Assuming C 30 = 1 and Ω b h 2 50 = 0.08, the critical SFRD is written as:

SF RD cr i t = Ṅ × 10 -53.1 f -1 escp ∼ 0.013 f -1 escp × 1 + z 6 3 (5.3)
where f escp is the escape fraction of ionizing photon, which is defined as the fraction of ionizing photon that escape from the source galaxy to the IGM after correcting for recombination within its halo. If the escape fraction from the host galaxies is low, it is obvious that the star formation will be inefficient to reionize the Universe at high redshift. There are several ways to estimate the escape fraction of ionizing photons using the data collected from observations. One way is to use the HI covering fraction, dust attenuations, and column densities to compute the escape fraction. Another way is to use the Lyman alpha escape fraction. A third way is to compute the flux ratio of OIII/OII. These methods were described in detail by Chisholm 19 -0.12 across an interval of redshift 0.3 < z < 6 (see Figure 5.2). This evolution peaks at a value of unity at redshift z = 11.1. The increasing trend of f Lyα escp with the redshift corresponds to the changing of dust content within the galaxies up to z ∼ 6. Beyond this point, there is an appearance of decline.

5 Star formation rate density and implications for the reionization -5.2 Articles

The quantities of f Lyα escp and f escp are expected to be correlated. This was explored in Dijkstra et al. 2016;[START_REF] Izotov | Diverse properties of Lyα emission in low-redshift compact star-forming galaxies with extremely high [Oiii]/[Oii] ratios[END_REF]. Combining the results obtained from M. Hayes, Daniel Schaerer, et al. 2011, the order of magnitude expected for f escp is ∼ 5% at z ∼ 3, and up to 25% at z ∼ 6. Figure 5.1 shows the critical SFRD value obtained from these two extremes of f escp , displayed as a shaded region using a typical value of the clumping factor of 3, which is mentioned in some studies, such as Pawlik et al. 2009;Robertson, Furlanetto, et al. 2013;Robertson, Ellis, et al. 2015; R. J. Bouwens, Illingworth, [START_REF] Bouwens | UV Luminosity Functions at Redshifts z ∼ 4 to z ∼ 10: 10,000 Galaxies from HST Legacy Fields[END_REF]Gorce, A. et al. 2018. The SFRD points obtained by integrating the luminosity in the range of 41 < log L [erg s -1 ] < 44 are consistent with the critical values obtained from an average value of f escp ∼ 8% and a typical clumping factor of 3 within a redshift range of 3 < z < 6.7. This suggests that the contribution of LAEs to the ionizing flux in the given redshift range is sufficient to keep the hydrogen ionized. At redshift z ∼ 6, this contribution is comparable to the one provided by LBGs.

Using LF face values as presented here implies that the role of the LAEs population in the cosmic reionization might exceed initial expectations. However, there are a few considerations to keep in mind. Firstly, the vast majority of past surveys have focused on the most luminous galaxies in the rest frame UV, which are ideal candidates for spectroscopic follow-up. However, these surveys can not probe the faint luminosity regime, down to 10 39 [erg s -1 ], as the present work does. Moreover, IFU observations can identify LAEs without any preselection. Secondly, the contribution to the SFRD is linked to the steepness of the faint-end slope being obtained, which is directly proportional to the Lyman alpha luminosity density. A slope that is 20% steeper significantly changes the contribution of LAEs to cosmic reionization. Thirdly, the lower boundary of the integration also affects the final contribution. Furthermore, there are significant uncertainties regarding the f Lyα escp and its evolution with redshift, which makes it difficult to compare this study to other studies that use different methodologies.

Articles 1. Introduction

A few hundred thousand years after the Big Bang, the Universe entered the Dark Ages, there were no light sources other than the cosmic microwave background (CMB) radiation. The tempera-ture of the Universe eventually cooled down enough for electrons and protons to combine to form neutral hydrogen atoms. Objects started to collapse under gravity to form the first stars and galaxies. The photons from these first structures, at a redshift range of 6 < z < 12, started to ionize the surrounding environments. The Universe entered the cosmic reionization era and was fully ionized at redshift z ∼ 6, according to observations of the Gunn Peterson trough from quasar spectra (Fan et al. 2006;Goto 2006).

Hydrogen is the most abundant element in the Universe and the Ly-α transition is a major tool in its exploration. Partridge & Peebles (1967) predicted the existence of this transition in high-z galaxies, Lyman-alpha emitters (LAEs) and this was confirmed in the mid-1990s using both large (8-10 m) ground-based telescopes and the Hubble Space Telescope (HST). Since then, the Lyman-α emission has become one of the key tools to explore the Universe at the cosmic reionization stage, and LAEs have been found at a wide range of redshifts, up to z ∼ 9 (see e.g., Oesch et The main sources responsible for cosmic reionization are still a matter of debate. It is evident that quasars are unlikely to play an important role (see, e.g., Willott et al. 2010;McGreer et al. 2013). Star-forming galaxies, such as LAEs and Lyman break galaxies (LBGs), are the more likely sources based on numerous observations (see, e.g., Robertson et al. 2013Robertson et al. , 2015;;Bouwens et al. 2015a). Their contribution to cosmic reionization is quantified via the SFRD. This, in turn, strongly depends on the shape of the galaxy luminosity function, particularly at the faint end, and on the escape fraction of ionizing photons that mainly relates to HI opacity, covering factor, dust content, and geometrical considerations (Giavalisco et al. 1996;Kunth et al. 1998;Hayes et al. 2014).

The galaxy luminosity function (hereafter LF) is the number of galaxies in a given luminosity range per unit co-moving volume. A popular parametric form of the function was proposed by Schechter (1976) and has been used extensively since then. The LF also puts a strong constraint on theoretical models of galaxies formation (Mao et al. 2007;Kobayashi et al. 2007). The Schechter LF has three parameters: Φ * , the normalization factor; L * , the characteristic luminosity defining the transition between the exponential part of the function (at bright luminosity) and the power law part (at faint luminosity); and α, the faint end slope. All three parameters have been studied extensively for LAEs. Fixing the faint end slope at the fiducial values of -1.8 at z = 0.5 and -2.5 at z = 7, the evolution of the other two parameters (Φ * and L * ) as a function of redshift can be roughly summarized as follows: they rise up in the range 0-3 (Deharveng et al. 2008) -0.1 and log(L * ) = 42.91 +0.13 -0.14 , while the slope value of α was fixed at -1.6. At higher redshifts, Konno et al. (2018) constructed the LF from 1266 LAEs for redshifts z = 5.7 and z = 6.6 obtained from Subaru/Hyper Suprime-Cam survey, with luminosities in the range of 10 42.8-43.8 erg s -1 . They measured steep slopes of α = -2.6 +0. 6 -0.4 and -2.5 +0.5 -0.5 at z = 5.7 and z = 6.6, respectively. Itoh et al. (2018) studied 34 LAEs also observed by the Subaru Telescope at an even higher redshift, z ∼ 7.0, covering a sky area of 3.1 deg 2 . However, in most studies the steep faint end slope is considered as a fixed parameter due to the lack of sources observed at faint luminosities, for instance, α = -2.5 in this case. This value was also assumed by Konno et al. (2014) to study LAEs at redshift z ∼ 7.3. More data at faint luminosities are necessary to constrain the faint-end slope.

Recently, 3D/IFU spectroscopy in pencil beam mode, obtained with the Multi Unit Spectroscopic Explorer (MUSE) of the ESO Very Large Telescope (VLT) has allowed for detections of faint LAEs in the redshift range of 2.9 < z < 6. 2022), probing the same luminosity range but with about four times as many sources, namely, about 600 LAEs. With this unprecedented dataset, we expect to set strong constraints on the faint-end shape of the LF, as well as its evolution with redshift, and, hence, on the contribution of LAE population to cosmic reionization.

The structure of the paper is as follows. In Sect. 2, we briefly describe the MUSE cubes used, together with their ancillary HST data. Mass models for lensing clusters and flux measurements of LAEs are presented in Sect. 3. We summarize the method used for the construction of the LF in Sect. 4. Results of LF fitting are presented in Sect. 5. Our results in comparison with previous ones and the implications for cosmic reionization are discussed in Sect. 6. Our conclusions are given in Sect. 7. Throughout the paper, we adopt the following cosmology parameters Ω m = 0.3 and Ω λ = 0.7, with H 0 = 70 km s -1 Mpc -1 .

Data

MUSE data cube

The Multi Unit Spectroscopic Explorer (MUSE) is installed on the Yepun telescope of the ESO Very Large Telescope (VLT; Bacon et al. 2010). It has a spectral resolution of R ∼ 3000 and a field of view (FoV) of ∼1 × 1 in wide field mode (WFM). The MUSE datacubes in the present work have two sky plane coordinates and one wavelength coordinate. The sky plane coordinates have ∼300 × 300 spatial pixels, each measuring 0.2 × 0.2 arcsec 2 . The wavelength coordinate has 3681 channels ranging from 4750 Å to 9350 Å effectively detecting Lyman-α emission at redshifts of 2.9 < z < 6.7.

The data reduction for the first sample of twelve lensing clusters is described in Richard et 2020) with some improvements for crowded fields such as lensing clusters. The same procedure was applied for the five new datacubes (MACS0451, MACS0520, A2390, A2667, and AS1063). These clusters have been observed both with Adaptive Optics and in standard modes (WFM-NOAO-N), except for the BULLET cluster (WFM-NOAO-E), which has a spectral range slightly extended towards the ultraviolet. These observations are well centered on the core regions of the clusters to maximize the chances to detect strong emissions of LAEs. The seeing values, namely, the full width at half maximum (FWHM) of the point spread function (PSF) at 700 nm, vary from 0.52 to 1.02 arcsec among clusters of the sample. Integration times vary from 2 to 14 h. Detailed information of the 17 clusters used in this work is shown in Table 1.

Sample of lensing clusters

The redshifts of the lensing clusters range between 0.2 < z cl < 0.6. These are sufficient to detect LAEs as background sources, with redshifts in the range of 2.9 < z < 6.7. These data are part of the Lensed Lyman-Alpha MUSE Arcs Sample (LLAMAS) project, with LAEs selected from MUSE and HST observations. The HST observations of each lensing cluster have at least one high-resolution broadband filter available to warrant an overlapping with MUSE spectral range. The preliminary mass models of these clusters were built based on HST observations and have been particularly useful in guiding the location of the MUSE fields relatively to the critical lines and the regions hosting multiple images. We refer to R21 and Claeyssens et al. ( 2022), as well as Sect. 3 of this work, for the updated and detailed lensing models adopted for these clusters. The present sample includes 17 cluster fields with 18 datacubes observed by the MUSE GTO program. Three of them, A2390, A2667, and A2744, have been studied before DLV19, but have since been re-reduced to have better quality data in terms of S/N. Three clusters, A2744, A370, and RXJ1347, were observed in mosaic mode with 2 × 2 MUSE FoVs, and MACS1206 was observed with 3×1 MUSE FoVs. The northern part of MACS0416 was observed by an ESO program 0100.A-0764 (PI: Vanzella), which was then combined with the GTO observation to produce an improved datacube in terms of exposure time.

As shown below, with the present sample of lensed LAEs, it is possible to construct a more robust LF than the previous work of DLV19. Our sample probes the same four orders of magnitude in Lyman alpha luminosity, down to 10 39 erg s -1 ; however, we have significantly improved upon the galaxy luminosity distribution in terms of statistics and we provide a better coverage at the faint end. Information on these clusters and the number of LAEs behind them is shown in Table 1.

The complementary HST data, in addition to building lens models of clusters mentioned earlier, are also needed to detect sources. The deep auxiliary HST data are listed in Table 1. Seven clusters belong to the CLASH (Postman & Coe 2012) and Frontier Fields (Lotz et al. 2017) programs, which are high-level science products (HLSP) combining all observations in 12 and 6 filters, respectively. Absolute astrometry calibration was applied for these datasets with an accuracy of <0 .06. The other clusters are part of the MACS survey and follow-up programmes (PIs: Bradac, Egami, and Carton). They were calibrated using source catalogs from Gaia Data Release 2 (Gaia Collaboration 2018). Figure 1 shows MUSE exposure maps of the 17 clusters overlaid on HST/F814W images adapted from R21.

These data have been used for building spectroscopic catalogues providing prior source positions, helping to distinguish overlapping sources detected in the MUSE cubes.

Identification of LAEs

The construction of MUSE spectroscopic catalogues is documented in R21. To detect line emissions, we made use of MUSE Line Emission Tracker (Muselet), a Python package A139, page 4 of 20 written by J. Richard 1 . In turn, this makes use of SExtractor (Bertin & Arnouts 1996) to detect emission line objects on MUSE narrowband continuum subtracted images. To account for low surface brightness sources such as some extended LAEs, a set of given SExtractor detection parameters (DETECT -MINAREA, DETECT -THRESH) have been used to make sure these faint sources are extracted properly. This set of parameters varies among these clusters. Sometimes, a spatial filter is also used by Muselet for smoothing NB images. A spatial filter (5×5 tophat with FWHM = 0 . 8) has been used for A2390, A2667, MACS0257, MACS0451, MACS0520, MACS0940, MACS2214, SMACS2031, and SMACS2131. Another (5 × 5 tophat with FWHM = 0 . 4) has been used for A2744 and A370.

We also made use of the source inspector package, designed by the MUSE collaboration (Bacon et al. 2023), using a combination of both deep HST and MUSE observations to identify sources and their redshifts. Depending on the nature of emission line objects, such as their signal-to-noise ratio (S/N) and the clarity of line absorption and emission features, redshift confidence levels are assigned. A source with zcon f = 1 is considered as a tentative case due to either a low S/N or the presence of ambiguous emission lines that do not display a typical Lyman alpha asymmetry profile or various low-S/N absorption properties. A source with zcon f = 2 is assigned based on emission lines having a higher S/N compared to the previous case. Such sources may display several absorption lines with low S/N. The confidence level will be upgraded if the source belongs to the multiple imaged systems. The highest level zcon f = 3 is for the sources with a high S/N of Lyman-alpha line displaying a typical asymmetry shape and is accompanied by several emission and absorption lines showing the right typical profile. The catalog of sources of the 17 clusters including redshift and spectral line information can be accessed online 2 (see R21, for more details). In the present work, there are such 190 unique LAEs categorized as zcon f = 1. However, for the purposes of our analysis, we only considered LAEs with secure redshifts (zcon f = 2 and 3), excluding all tentative sources with a zcon f of 1. The identification of LAEs is described in detail in DLV19, and is also mentioned in Claeyssens et al. (2022).

It is important to note, with respect to sources detected both by HST, called P (prior) sources and by MUSE M sources, that R21 retained the spatial positions of the P sources. We found an average position offset ∼0.2 between P and M sources. As the LAEs are detected by MUSE, we naturally used the M positions for these sources.

Around 1350 Lyman alpha images with zcon f >= 1 have been identified behind the 17 lensing clusters. Many of them are multiple image systems. To avoid counting them more than once, for each multiple image system, we chose a representative image. This also helps to avoid de-lensing them which is computationally expensive. Choosing the representative for each multiple system is manually performed by investigating them one by one. The HST images 3 and 4 and MUSE white light images are useful for this selection process. If possible, the chosen representative has high enough S/N, reasonable magnification errors and a position that is well isolated, with little to no contamination from nearby bright sources. Finally, 600 LAEs sources were selected at this stage with, in some cases, several 1 https://mpdaf.readthedocs.io/en/latest/muselet.html 2 https://cral-perso.univ-lyon1.fr/labo/perso/johan. richard/MUSE_data_release/ 3 https://archive.stsci.edu/missions/hlsp/clash/ 4 https://archive.stsci.edu/missions/hlsp/frontier/ different images equally good to represent the parent source. All these sources exhibit zcon f = 2 or 3. Their redshift distribution is shown in Fig. 2, together with those from previous work of DLV19 for comparison. A large number of LAEs in the present work have redshifts in the range of 2.9-4.0. There are two bumps in the distribution at redshifts of ∼4 and ∼5, suggesting some overdensities along the line of sight.

Modeling lensing clusters mass distributions

Mass distributions by LENSTOOL

There are two approaches that are typically adopted in lens modeling: parametric mass modeling and free-form (non-parametric) methods. For the present work, we use the former to build the total mass distribution of lensing clusters, based on Lenstool (see e.g., Kneib et al. 1996;Jullo et al. 2007;Jullo & Kneib 2009). Following Lenstool, as well as the procedure described in DLV19 and R21, we assume two components contribute to the lensing: 1) the large-scale structure of the cluster 2) the contribution from each massive cluster member. The contribution and implementation of each component is well described in Jullo et al. (2007).

Parameters of individual mass profiles are sampled via a Markov chain Monte Carlo (MCMC) method to compare the match between the observed and predicted multiple images. Details of the lens modeling are described in R21. The lens models and their parameters are listed in Table A.1. Twelve out of the seventeen lens models used in the present work come from the work of R21. Thanks to these models, we can work back and forth between the image and source planes for all LAEs. This allows us to correct for the amplification factor, namely, the lensing magnification, for each source, as well as the source plane projected area allowing us to capture the extended morphology of LAEs.

We note that magnification associated with a given source obtained from Lenstool is the magnification of the source centre, not the entire source. If the LAE in question is a compact one and is relatively far from the lens caustic lines, then it is fair to assign this magnification to that source. Such is the case for 90% sources in the sample. However, for extended sources, in particular those close to the lens caustic lines, magnification strongly varies across the source and using the central point magnification is not justified. In such cases, we must take the fluxweighted magnification by averaging the magnification over the entire source. In this work, we use this weighted magnification for all LAEs. We can do so systematically by using the segmentation map which is created from the NB image by Sextractor and is used as a mask for NB fluxes of the LAEs, to switch back and forth between the image and source planes. Figure 3 displays the difference in magnification between the central and the flux-weighted values for the LAEs in the present sample.

Flux measurement

We considered two methods to measure fluxes of LAEs. Fluxes were either extracted by fitting the Lyman-alpha (Lyman α) profile as performed by Claeyssens et al. (2022) or by running SExtractor with FLUX -AUTO on the NB images in which the LAEs are detected as performed by DLV19. The first method was employed for LAEs behind 16 lensing clusters except for the A2744, while the second method is applied for the cases in which the first method fails to fit the Lyman alpha profile. Using the first method, we obtained flux measurements for 425 A139, page 5 of 20 LAEs. However, 47 faint source fluxes could not be measured using this approach. In such cases, we employed the SExtractor package (second method) to extract the fluxes. A large fraction of these failed sources has a low S/N, but they have been upgraded to zcon f = 2 because they are members of multiple imaged systems. Furthermore, we used the second method to extract LAE fluxes for sources located behind the lensing cluster A2744. The reason for this is that the source offset positions of the M sources are not available in the new catalogs provided by

f (x) = A exp -(λ -λ 0 ) 2 2(a asym (λ -λ 0 ) + d) 2 , ( 1
)
where A is the flux amplitude, λ 0 is the wavelength layer in which source's emission reaches maximum, a asym accounts for the asymmetry of the Gaussian function, and d is the width of It is worth mentioning that 31 out of the 600 LAE spectra display similar two-peak profiles. We apply the same procedure as described above, dealing with the blue and red peaks separately, taking into account the amplitude difference between the two. The two peaks are then summed to get the final flux. The flux uncertainty is estimated using python package EMCEE (Foreman-Mackey et al. 2013) with 8 walkers and 10 000 iterations.

For the second method, we made use of the SExtractor software (Bertin & Arnouts 1996) running on NB detection images that host LAEs. This is well described in DLV19. For this purpose, a sub-cube with a size of 10 ×10 × the width of the Lyman-α emission is first extracted from the original cube, then averaged to form the LAE image. Two sub-cubes of the same spatial size, extending 20 Å bluewards and redwards of the Lyman-α line were also created. These were then averaged to form the mean continuum image of each LAE. By subtracting this mean image from the LAE image, pixel by pixel, we can form a new measurement image which is ready for SExtractor. We used SExtractor's FLUX -AUTO parameter to estimate the LAE fluxes together with their uncertainties. This parameter is developed based on Kron's first-moment algorithm: when a given source is convolved with a Gaussian seeing, 90% of its flux will be inside the circular aperture of the Kron radius. This holds even for extended sources. The final flux obtained from the measurement image is then multiplied by the width of the Lyman-α emission.

It may happen that some sources are either faint or have a low extended surface brightness. In such cases, SExtractor does not extract these sources properly. A set of farsighted SExtractor parameters (DETECT -THRESH, DETECT -MINAREA) have been proven to be very useful to overcome this problem. SExtractor progressively releases detection conditions using this set of parameters to make sure that faint sources can be extracted.

Figure 4 shows a comparison between the fluxes of LAEs obtained by using the two methods in the present work. In gen-eral, the two fluxes agree well with each other within a range of a few orders of magnitudes. The deviation spreads out at the faint fluxes, as expected, but no systematic trend is observed. The source fluxes, obtained from the second method, will be used later to estimate the systematic uncertainties attached to the faint end slope.

As mentioned in the previous subsection, to obtain the intrinsic luminosity of a source, its segmentation map is projected into the source plane using Lenstool, then the weighted (averaged) magnification over that source is calculated. The weighted magnifications obtained this way are used throughout this work. The source intrinsic luminosity is computed as:

L Lya = F Lya µ 4πD 2 L , (2)
where F Lya is the flux of the LAE obtained as described above, D L is luminosity distance, and µ is weighted average magnification. The luminosity versus redshift and luminosity histogram of our sample together with the ones from DLV19 are shown in Fig. 2.

Building the LF

The V max method

In order to deal with the complexity of the lensing datacubes obtained by MUSE, we adopt a non-parametric approach, the 1/V max method (Schmidt 1968), allowing us to treat each source individually, to build the LF using detected LAEs behind lensing clusters. The V max of a source is the volume of the survey where that individual source could be detected. Inverting this gives the contribution of that source to a number density of galaxies. Determining V max is the crucial step in building the LF and this is challenging. We note that noise is an important factor. The V max has to take into account noise introduced by the instrument, by sky line variations for different wavelength layers and by local noise variations for different sky directions in the MUSE FOVs. It must take into account the source detectability in all the MUSE lensing datacubes involved in the survey, namely, the whole survey volume. Additionally, it has to be calculated in the source plane to avoid adding the same survey volume for multiple image systems. All of these steps make V max determination complex and computationally expensive.

DLV19 developed a package to compute V max of lensed LAEs behind lensing clusters observed by MUSE. Details of the algorithm are described in DLV19. Here, we briefly summarize the important points and main steps of the procedure.

Detectability masks are an important concept for computing V max . They are built first in the image plane, by comparing a given source brightness distribution pixel by pixel to the RMS maps, channel by channel, in the spectral range of all 18 datacubes. These masks are projected in the source plane using Lenstool, to obtain source plane masks. However, performing this for all sources is extremely computationally expensive. The main important point proposed by DLV19 is to build a set of pre-computed 2D detection masks that cover a wide range of S/Ns for each MUSE datacube. Hence, for a given source, one first computes its S/N in the parent cube, in which the source is detected, then picks up the corresponding 2D mask from the pre-computed list available, which is closest match with the input in terms of S/N. We can do so given that for each lensing datacube root mean square (RMS) maps of different layers display roughly the same type of patterns. One can choose a representative of these RMS maps, namely, their 49 710 median map, then scale it accordingly by a constant factor to get other RMS maps for different layers. Another important simplification is that for each lensing cluster, we can use a representative brightness profile for all the sources, namely, the median of radial source brightness profiles (see Fig. 5). Using both median radial profile and median RMS map, we can then produce masks at wide range of S/N covering the source sample. This is another simplification to limit the number of masks created in both the image and source planes. The Lenstool package is used to perform the reconstruction from image plane to source plane. To account for variations of projected source plane area as a function of redshift, each 2D mask reconstruction is sampled at four different redshifts, z = 3.5, 4.5, 5.5 and 6.5. One of the four source plane masks is then assigned for a given source, namely, the one that is the closest match to its redshift. Source plane magnification maps are also used to eliminate regions with insufficient amplification implying sources could have not been detected there. The 3D masks for a given source are formed in this way. The final volume is integrated from the unmasked pixels of these 3D source plane masks:

V = Nω c H o zmax zmin D 2 L (z) (1 + z) 2 E(z) dz, (3)
where N is number of unmasked pixels counted on all wavelength layers, ω is the angular size of a pixel, D L is the luminosity distance, and

E(z) = Ω m (1 + z) 3 + (1 -Ω m -Ω Λ )(1 + z) 2 + Ω Λ .
The co-volume probed by each cluster of the survey is listed in Table 2.

Completeness value

Completeness is a correction related to the chance that a given source is detected at its own wavelength affected by random variation of noise on the spatial dimension of the channel, namely, the NB layer where the source's emission reaches maximum. Together with V max , it is a crucial correction before deriving LF points. Depending on source morphology, this completeness is computed individually. Details on how to compute completeness are described in DLV19. We condense the process as follows: 1) A source profile of individual LAE is created from a combination of filtered image, object image, and segmentation image obtained by running SExtractor on the peak NB layer. 2) To compute the completeness for a given source, we have to create the simulations by randomly injecting the source on the masked NB image layer where the emission reaches maximum. We use the real source profile to perform this task. The local noise in the region where the source has been injected is likely to decide whether or not a source is detected. SExtractor scans on the masked NB image, with exactly the same parameters as when that source was originally extracted. We iterate 500 times for each source. The completeness is the detection success rate.

3) The quality of extraction by SExtractor is also taken into account. Following DLV19, sources with flags of type 1 and 2 are trustworthy, but sources with flag 3 are doubtful and will not be used for the LF computation afterwards. From the original sample of 600 LAEs, 588 with flags 1 and 2 are kept at this stage (see more below). We found that the size of the masked NB images used to redetect mock sources has an impact on the quality of extractions. When a source has a few neighbors, the local noise is not well represented for that wavelength slice if the size of the image is not large enough. To deal with this, we increased the size from 30 × 30 (as in DLV19) to 80 × 80 . This is more computationally expensive but better accounts for the local noise. The extraction results for the sample are: 555 cases with flag 1; 33 with flag 2; and 12 with flag 3. Additionally, aiming to include as many sources as possible for calculating the contribution to cosmic re-ionization, all sources with completeness values above 1% are retained. Taking this into account, 575 LAEs are used for the next steps.

Figure 6 shows the completeness of each LAE vs. their respective detection flux in the sample. The average completeness value is 0.69 and the median value is 0.90 over the entire sample. The equivalent values from DLV19 are 0.74 and 0.9, respectively. Our average value is a bit smaller than that of the previous work due to numerous faint sources in our sample, which often have low completeness values.

For a given source, one may expect that higher its detection flux leads to higher completeness value. However, it is not always the case as shown in Fig. 7. Some sources still have low completeness values, that is, below 0.2, while their detection fluxes are relatively high. It suggests that source morphology also plays an important role in its completeness, as previously mentioned by DLV19.

Luminosity function determination

We divide the 575 LAEs into four redshift bins in order to study the evolution of the LF with redshift: 2.9 < z < 6.7, 2.9 < z < 4.0, 4.0 < z < 5.0, 5.0 < z < 6.7. The contribution of each individual LAE in each bin is calculated as follows:

φ(L i ) = 1 ∆log L i j 1 C j V max, j , (4)
where i corresponds to the luminosity bin, ∆log L i is the width of the luminosity bin in log scale, j corresponds to the source in the sample, and C j and V max,j are the completeness value and survey volume of the source in bin i. We used MCMC to calculate the mean and the statistical uncertainty for each LF point by randomly generating a set of magnification, flux and completeness values for each individual source. 20 000 catalogs were built from the original catalog. For each LAE, the flux and completeness are randomly sorted following Gaussian distributions having means as their measured values and sigmas as their uncertainties. A random magnification value is taken from the magnification distribution P(µ) of each source, accounting for extended source morphology and uncertainties in the lens model. With respect to DLV19, we have introduced an improved procedure to determine the V max associated to a random value of the magnification. This allows us to better capture the variations in V max taking place for very large values of the magnification. For each iteration, a single value of the LF is obtained for each luminosity bin. The distribution of LF values in each bin, obtained in this way, is used at the end of the process to determine the median of φ(L i ) in each luminosity bin in linear space, and to compute the asymmetric error bars. For the estimation of the cosmic variance, we used the cosmic variance calculator presented in Trenti & Stiavelli (2008). A single compact geometry made of the union of the effective (lensing-corrected) areas of the 18 FoVs is assumed following DLV19.

Schechter function fitting

After obtaining the LF points following the procedure explained above, we fit the results using a Schecter function. The Schechter function, proposed by Schechter (1976), has been extensively used to describe the LF as well as its evolution with redshift. It is often written as:

Φ(L)dL = Φ * L * L L * α exp -L L * dL, (5)
where: Φ * is a normalization parameter, L * is the luminosity at the point where the power law changes to an exponential law at high luminosity, L is the luminosity, α is the faint end slope, and Φ is the number density in a given of logarithmic luminosity interval.

Regarding the fitting method, we adopted the same procedure as described in DLV19, using a Schechter function with three free parameters to vary (Φ * , L * , α). We first minimized these parameters by using the Levenberg-Marquardt algorithm, in particular, using traditional chi-square minimization procedure, provided by the standard package Lmfit.

Faint end of the LF

Faint-end LF

The faint-end slope of the LF function is still a matter of debate and more observations are strongly needed. Strong gravitational lensing by clusters of galaxies is of great help to probe the faint galaxy luminosity regime and hence to constrain the shape of LF in that region. The price to pay is that the cosmic volume of the survey is significantly reduced: the higher the lensing magnification factors, the lower the volume probed. Our sample efficiently probes down to 10 40 erg s -1 , the same faint luminosity regime as studied by DLV19 but with improved statistics and a better coverage at the faint end of the luminosity distribution (see Fig. 2) and 34 000 Mpc 3 larger co-volume explored accounting for the magnification in the range of redshifts 2.9 < z < 6.7. On the contrary, the MUSE Wide and other deep blank fields surveys cover well the bright luminosity part (see Drake et al. 2017;Herenz et al. 2019). These studies are efficient to probe the LF around the log L * regime, while the sample in this work is optimized to probe the faint-end part.

For a first attempt, we try to fit the LF with a linear fit to find the slope at the faint end. We only used our LF points, computed from the 575 LAEs. We considered four different redshift bins, 2.9 < z < 4.0 (z 35 ), 4.0 < z < 5.0 (z 45 ), 5.0 < z < 6.7 (z 60 ), 2.9 < z < 6.7 (z all ) to make these fits. The results are shown in Fig. 9. The respective slopes are -1.76 +0. 26 -0.27 , -1.99 +0.12 -0.17 , -1.96 +0. 5 -0.54 and -2.12 +0.15 -0.16 . In spite of our bigger sample, four times more sources than in DLV19, spanning over four orders of magnitudes in galaxy luminosity, we essentially obtain the same results: no evolution of the faint end slope as a function of redshift is observed. The current sample suggests that (at face value) the faint-end slope in each redshift range is steeper than for DLV19, for the same redshift interval. However, as seen in Fig. 9, a turnover seems to appear at luminosities fainter than

log(L) -3 -2 -1 0 log(Φ [(∆ log L = 1) -1 .Mpc -3 ])
2.9 < z < 6.7 ( line fit: α = -2.12± 0.15 0.16 ) 2.9 < z < 4.0 ( line fit: α = -1.76± 0.26 0.27 ) 4.0 < z < 5.0 ( line fit: α = -1.99± 0.12 0.17 ) 5.0 < z < 6.7 ( line fit: α = -1.96± 0.5 0.54 ) Fig. 9. Faint-end slope of LF: Line (power law) fitting for four different redshift ranges: 2.9 < z < 4.0 (blue), 4.0 < z < 5.0 (green), 5.0 < z < 6.7 (red), and 2.9 < z < 6.7 (black). These fits only use our LF points constructed from the current sample. Open symbols at the faintest luminosity bins are not included in the fitting process.

10 41 erg s -1 , for the two highest redshift bins. This trend is discussed below.

Computing LF parameters

This section presents the fit of our LF points with the Schechter function. As the luminosity range of our sample reaches its maximum 10 43 erg s ), except for the last faintest bin which has a width of 1.25 for redshift bin z all , while in the other bins the width is 0.5. This is meant to avoid an increased weight of this bright sample from the literature on the global fit. The fitting process is performed as described in Sect. 4.4. The results are shown in Fig. 10, with the best fit curves shown as solid lines together with the 68% and 95% confidence area based on the data as indicated in the respective labels. We also check that the shape of the LF for each redshift interval is essentially the same when changing the number of luminosity bins, similarly to DLV19 (Fig. 11).

The best fit parameters are listed in Table 3. The best fit value of log L * is in good agreement with DLV19, and a few percents higher than the value obtained from Herenz et al. (2019), 42.20 +0. 22 -0.16 . Strong degeneracy between these parameters (Φ * , L * , α) is observed, as shown in Fig. 12, and already well documented in Herenz et al. (2019). Also, log L * seems to be well measured from the current work for different redshift intervals. There is a tendency of L * to increase with redshift but this is well within the uncertainties. As both log L * and the steep faint-end slope increase with redshift, it is likely related to the degeneracy mentioned earlier. Then, Φ * is just a normalization factor giving the number density of objects per given volume. Our best fit result gives Φ * [10 -4 Mpc -3 ] = 7.41

log(L) -6 -5 -4 -3 -2
-1 0 1 log(Φ [(∆ log L = 1) -1 .M pc -3 ])
Luminosity function for 2.9 < z < 6.7

Blanc 2011(1.9 < z < 3.8) Drake 2017(2.9 < z < 6.7) Herenz 2019(2.9 < z < 6.7)

log(L) -6 -5 -4 -3 -2
-1 0 1 log(Φ [(∆ log L = 1) -1 .M pc -3 ])
Luminosity function for 2.9 < z < 4.0

-1 0 1 log(Φ [(∆ log L = 1) -1 .M pc -3 ]) 6 -5 -4 -3 -2
Luminosity function for 5.0 < z < 6.7

Drake 2017(5 < z < 6.7) Herenz 2019(5.0 < z < 6.7) Konno 4.

As our sample probes the faint luminosity regime, the slope at the faint-end of LF, α, in principle, is well constrained. We measure steep slopes of α varying from -2.0 ± 0.07 for the redshift interval 2.9 < z < 4.0 to -2.28 ± 0.12 for redshift interval 5.0 < z < 6.7. These results are consistent with the slope measured by Herenz et al. (2019) for the global redshift bin, and Drake et al. (2017) in same redshift bins. The faintest luminosity points in all redshift intervals, having log(L) [erg s -1 ] < 40, are not included in the Schechter's fitting. As sources in this faintest bin are often highly magnified by lensing effects, namely, sources close to the lens caustic lines, this seems to suggest either that lensing systematic uncertainties or that completeness corrections for those sources must be treated with great care. Another possibility is that the faint end of the LF may depart from the traditional Schechter function. More data in this luminosity regime is required to verify this. The enhancement at log(L) around 42 shown in z all is because the number of sources suddenly increases in that luminosity bin. This is also shown in Fig. 8, where a spike appears at the luminosity bin log(L) [erg s -1 ] ∼ 42 after correcting for completeness. This may relate to the over-density of background sources at z ∼ 4 as mention in DLV19. This may also suggest the uncertainty from the cosmic variance is probably larger than expectations. The covolume probed by our survey is ∼50 000 Mpc 3 . It seems that the data points from Drake et al. (2017) and Cassata et al. (2011) within the log(L) range of 41.5-43.0 might influence the fitting results of the faint end slope z 35 and z 60 . However, we checked and found that including or excluding these data points does not substantially alter our results. The faint end slopes at these specific redshift ranges show only minor variations less than a few percent.

Error budget

The uncertainties on LF points are well documented in DLV19. There are three kinds of uncertainties attached to the LF points which are addressed. The statistical uncertainty is derived from the MCMC process. A set of flux, magnification, and completeness associated with each LAE is randomly drawn 20 000 times. The results are then used to estimate the mean and statistical uncertainties attached to each LF point. The second source of uncertainty relates to field-to-field variance for different lensing FoVs observed by MUSE. We use the cosmic variance calculator proposed by Trenti & Stiavelli (2008) to estimate this uncertainty, which is typically about 20%-30% at most. The third one, the Poissonian uncertainty, relates to the number of sources for a given luminosity bin, which is relatively easy to handle. For the bright-end luminosity, log(L) [erg s -1 ] > 42, the Poissonian uncertainty dominates. Its contribution decreases and

0 4 8 1 2 1 6 φ * [10 -4 .Mpc -3 ] - 2 . 4 - 2 . 2 - 2 . 0 - 1 . 8 α 4 2 . 6 4 2 . 8 4 3 . 0 4 3 . 2 4 3 . 4 log(L * ) - 2 . 4 - 2 . 2 - 2 . 0 - 1 . 8 α 2
.9 < z < 6.9 2.9 < z < 4 4 < z < 5 5 < z < 6.9 becomes equivalent to that of cosmic variance in the luminosity range 41 < log(L) [erg s -1 ] < 42. The statistical uncertainty is dominant in the faint-end regime. One would expect systematic uncertainty coming from lensing models to be another important contribution to the total error budget. Different lensing models may give magnification factors that differ by a significant factor for a given source, increasing the uncertainty on the luminosity obtained. This systematic uncertainty is well documented in Bouwens It might play important role in particular to the faint end luminosity regime. As discussed in DLV19 its contribution is ∼15% at log(L) of 40.5 erg s -1 for the case of the lensing field A2744. 2019) by one order of magnitude, allowing us to constrain the faint-end slope of the LF. Although the number of sources in the previous sample by DLV19 is four times smaller, it still covered by the same luminosity range as in the present project. Three lensing clusters, which were previously studied in DLV19, have been re-reduced to achieve better quality in terms of S/N and source positions. In this work, when sources are both detected by HST and MUSE, we used the Muselet positions (M pos.) instead of prior ones from HST observations (P pos.). Choosing these positions has some impact on the LF shape at the faint end. An average spatial offset of ∼0.2 arcsec between the M and P positions has been found in Claeyssens et al. (2022). MUSE observations are sensitive to detect line emissions such as Lyman-alpha while HST ones are sensitive to continuum emission. It may happen A139, page 12 of 20 Thai, T. T., et al.: A&A 678, A139 (2023) Table 4. Luminosity bins and LF points used for Fig. 10.

Discussion

log(L) [erg s

-1 ] log(φ)(∆(log (L)) = 1) -1 [Mpc -3 ] N N corr V max [Mpc 3 ]
2.9 < z < 6.7 39.00 < 39.63 < 40.25 -0.68 5.9 5.9 18 that the LAEs are extended and associated with diffuse gas, hence the offset between the two observations. Moreover, the faint luminosity regime often contains highly magnified sources. These sources are close to the lens caustic lines in the source plane, hence, their positions have to be assessed with great care. The weighted magnifications for these extended sources strongly depend on their positions with respect to the caustic lines. In addition, the magnification uncertainties attached to these sources are also large and this may cause the source contribution to spread to several luminosity bins during the MCMC process to estimate LF point uncertainties. On the contrary, sources in the bright luminosity bins usually have small magnifications and do not suffer from this effect. In principle, the procedure adopted here to compute the LF points captures all these effects.

An important point to note is that DLV19 used a threshold of 10% completeness to reject sources having completeness values below this cut. A small completeness value implies a large correction on the number of detected sources (see Fig. 8). Varying the completeness threshold cut significantly changes the shape of the LF at the faint end as one may loose some fraction of these sources. In the present work, we try to include as many sources A139, page 13 of 20 1% completeness cut -2.00 ± 0.07 -1.97 ± 0.09 -2.28 ± 0.12 -2.06 ± 0.07 10% completeness cut -1.78 ± 0.06 -1.83 ± 0.12 -1.75 ± 0.10 -1.46 ± 0.05 1% completeness cut, fluxes obtained from the second method -2.10 ± 0.06 -1.97 ± 0.08 -2.24 ± 0.11 -1.82 ± 0.03 1% completeness cut zconf1 included -1.83 ± 0.17 -1.92 ± 0.09 -1.94 ± 0.10 -2.29 ± 0.09 Linear fit 1% completeness cut -1.76 ± 0.27 -1.99 ± 0.17 -1.96 ± 0.54 -2.12 ± 0.16 10% completeness cut -1.55 ± 0.17 -1.63 ± 0.21 -1.66 ± 0.24 -1.64 ± 0.12 Faint end slope retain -2.00 ± 0.50 -1.97 ± 0.50 -2.28 ± 0.50 -2.06 ± 0.60 as possible from the sample, only rejecting obvious cases which have a poor extraction quality, namely, flag type 3 as identified from SExtractor. In practice, we used a cut in completeness of 1%, meaning that a final sample of 575 LAEs is used for the LF computation. In the global redshift bin, z all , and z 35 (see Fig. 10), the number density of sources suddenly increases at log(L) [erg s -1 ] ∼ 42; this is also caused by the low completeness values for some sources in that luminosity bin. It is worth noting that usual computations of the LF in lensing fields do not reject any source based on its completeness value (see e.g., Atek et al. 2018). By implementing a 10% completeness, we further removed 62 additional sources from our sample, with 51 of them belonging to the six faintest luminosity bins. As a result, the density of source significantly reduces by a factor of 5 on average. The faint end slope decreases from its original value of -2.06 to a flatter value of -1.46 for the global redshift range Table 5. This reveals insights into the uncertainties associated with the faint end slope, which we discuss in detail in the subsequent subsection.

Comparison with previous results

In the bright luminosity regime (log(L) [erg s -1 ] > 40.5), the data from the literature mainly come from blank field observations. They are numerous and helpful to constrain the LF at the bright end. Indeed, we include some of them from blank fields for the fitting procedure to constraint the bright-end part of the LF. They found the faint-end slope value at redshift 2.9 < z < 6.7 to be α = -1.84 +0.42 -0.41 , log Φ * [Mpc -3 ] = -2.71 and log L * [erg s -1 ] = 42.20 +0.22 +0. 16 . Comparing to our results, the two slopes at the faint end are consistent while their best fit of log L * is a bit smaller. The explanation may relate to the degeneracy between the three best fit parameters. We note that Herenz et al. (2019) constructed LAEs LF in the luminosity range 42.2 < log(L) [erg s -1 ] < 43.5.

It is necessary to compare our results with those obtained by DLV19, which also probe the faint luminosity regime. The characteristic log L * is well measured both in different redshift intervals and in different cluster samples. The best-fit values agree well with each other, within their 1-σ uncertainties. However, the present slope values are steeper by 20% than those from the previous work for the same redshift intervals. The difference between the two may be due to various factors. Firstly, the number of sources in the two samples differ: we have four times more sources in the present sample, giving us better statistics. Our sample has a significant number of sources in the faint luminosity region and more than ten sources in the faintest bin. Secondly, the threshold cuts in completeness are different. DLV19 rejected faint sources with completeness values below 10%, while we have included as many sources as possible, for consistency with other LF determinations in lensing fields, rejecting only those with completeness values below 1%. Thirdly, as explained above, our MCMC procedure to compute the LF points better captures the relationship between magnification and V max with respect to DLV19. Finally, the number of LF literature data points helping to constraint the bright-end part is different between the two works. The results from Herenz et al. (2019) using MUSE-Wide survey to investigate Lyman α LF in the same redshift bin are included and combined with others as a constraint association; the results from Drake et al. (2017) are used for fitting at redshift bin 2.9 < z < 4.0 and for displaying only at others bins. This affects the normalization parameter as well as the faint-end slope, since the two are correlated. We also performed the full analysis with the new improved procedure on the sub-sample of four clusters in DLV19, with the same choices regarding the completeness, and found fully consistent results with DLV19.

One of the important points derived from Fig. 10 is that faint-end slope values of α become steeper at higher redshifts. This seems to suggest the evolution of the faint-end slope as a function of redshift. There is a very good agreement between the slopes obtained from Schechter function fitting and those obtained from line-fitting in Sect. 5.1. However, we do not see the same trend of the slope evolution from line-fitting, which only used our LF points, due to the larger uncertainties associated with them. Moreover, we observe a turnover at luminosities fainter than 10 41 erg s -1 , for the two highest redshift bins. At these luminosities, the LF points are not well described by the Schechter function. A flattening and turnover are observed towards the faint end for the highest redshift bins that still needs further investigation. This turnover is similar to the one observed at M UV = -15 for the UV LF at z ≥ 6 in lensing clusters, with the A139, page 14 of 20 To account for a possible turnover in the two highest redshift ranges 4.0 < z < 5.0 and 5.0 < z < 6.7, two more parameters have been introduced to modify the Schechter function, which is then expressed as:

Φ(L) exp{-(L T /L) m } = Φ * L * L L * α exp(-L/L * ) exp{-(L T /L) m } (6)
where: Φ(L) is the traditional Schechter function together with its three (Φ * , L * , α) parameters, being fixed at their best-fit values, as shown in Table 3. Two new parameters are introduced: L T is the luminosity at the turnover point and m is a power index, which is about unity. Our data suggest that log(L T ) are ∼40 and ∼40.7 erg s -1 for redshift bins 4.0 < z < 5 and 5.0 < z < 6.7, respectively. The values of m are essentially equal to 1 for these two redshift ranges, as expected. Results are shown in Fig. 10 (lower panels).

Regarding the prevalence of a turnover in the LF towards the faint end, it is worth mentioning that Bouwens et al. ( 2022) have recently ruled out this trend in the UV LF down to M UV = -14 mag at z ∼ 6. In addition, Dawoodbhoy et al. (2023), using the CODA simulation, reported no sign of the turnover down to M UV = -12 mag at the same redshifts. The large uncertainty in the faint regime of LAE luminosity prevents us to make much sense of this result. Additional data in this faint region are necessary to improve the statistical significance of the present sample in order to confirm or reject this turnover trend.

Comparison with theoretical predictions

We compared our results on the Lyman-alpha LF with two theoretical models whose predictions at z ∼ 6 can be compared directly with our findings in the highest redshift range of 5.0 < z < 6.7, namely, the predictions of Garel et al. ) uses the Analytic Model of Igm and GAlaxy evolution (AMIGA), a model of galaxy formation and evolution including their feedback on the IGM, to constrain the reionization history of the universe. It provides predictions for two possible scenarios: single and double reionization episodes. The former has hydrogen ionization episode at z ∼ 6, and the latter has two reionization episodes at z ∼ 6 and ∼10, separated by a short recombination period (see Salvador-Solé et al. 2022, for additional details). Their results for the two scenarios are also shown in Fig. 13. In the range of 40 < log(L)[erg s -1 ] < 42, the prediction of the AMIGA double ionization scenario is in good agreement with that of Garel et al. (2021) after IGM correction. Our LF points towards the faint end correctly span the region covered by these models. In general, our LF points are in good agreement with the predictions by both models without any renormalization, as shown in Fig. 13. At the faintest luminosity regime, log(L) < 41, our LF point starts to depart from the theoretical increasing trend by Garel et al. (2021), and is somewhat closer to the single ionization scenario predicted by Salvador-Solé et al. (2022). However, the uncertainties in this regime are very large, preventing us to distinguish between the different theoretical predictions. More data covering this faint regime are badly needed.

Effect of source selection

There are several factors that play important roles in determining the faint end slope. One of them is the source selection process. In the 17 lensing clusters, 190 LAEs (unique system) have been classified as zcon f = 1, and as a consequence, they are not retained for the LF computation process. However, as they are often faint, they may contribute significantly to our faint LF points. To evaluate the impact of zcon f = 1 sources on the final LF points, we have incorporated them into our LAE sample. We assume that zcon f = 1 sources have the same quality of completeness and V max as the zcon f = 2, 3 sources; namely, for a given luminosity bin, zcon f = 1 completeness and V max values are assigned as the mean values for the zcon f = 2, 3. Similarly, the uncertainty values for completeness and V max of the zcon f = 1 sources are set equal to the corresponding uncertainty values of the zcon f = 2, 3 sources in that bin. Including half and all of the zcon f = 1 sources results in a 5% and 10% steeper faint end slope, respectively.

To estimate the systematic uncertainties attached to the faint end slope, we performed various tests to calculate the LF points and measured the slope. We employed different completeness threshold cuts (1% and 10%), utilizing different fitting function forms (Schechter and linear functions) and considering two scenarios: including half or all of the zcon f = 1 sources. We also used source fluxes obtained from the pipeline only (method 2). Our findings are summarized in Table 5. We retained a faintend slope of the 1% completeness cut results and increased the uncertainties. The final slope are -2.00 ± 0.50, -1.97 ± 0.50, -2.28 ± 0.50, -2.06 ± 0.60 for z 35 , z 45 , z 60 , z all , respectively. Figure 14 the uncertainties remain large. Our slopes at various redshift ranges are in good agreement with the findings of other studies, typically within a 1-σ deviation.

Implications for the reionization

One of the most important tools to understand the formation and evolution of galaxies is the cosmic SFRD, providing information on the onset of star formation in the early Universe and its evolution over cosmic time (Madau et al. 1996;Bouwens et al. 2007Bouwens et al. , 2008)). In this section, we present the contribution of LAE population to the cosmic re-ionization by computing the SFRD by integrating the best-fit parameters obtained from the previous section. In addition to the luminosity range of the integral (Eq. ( 7)), log(L) = (39.5, 44.0) as probed by the present sample, we also chose another lower limit luminosity of 0.03L * ∼ 10 41 erg s -1 to facilitate the comparison with previous works. The integrated SFRD is proportional to the luminosity density and can be estimated following the calibration of Kennicutt (1998), assuming an intrinsic factor of 8.7 between the intrinsic Lyα and Hα fluxes, and case B for the recombination (Osterbrock 1989). In that case, all newly formed Lyα photons would be re-absorbed by the neutral hydrogen atoms of the HII region. In the optically thick case, the SFRD is expressed as:

SFRD Lya [M yr -1 Mpc -3 ] = ρ Lya /1.05 × 10 42 , (7)
where ρ Lya is the Lyα luminosity density in units of erg s -1 Mpc -3 . The results are shown in Fig. 15 together with the other literature data from different sources and surveys. The yellow regions show the SFRD needed to fully ionize the entire Universe, taken from the work of Bouwens et al. (2015b) at the level of 1-σ and 2-σ. A clumping factor of 3 is applied to calculate cosmic emissivity, log(ξ ion f escp ) = 24.50, where f escp is escape fraction of ionizing UV photons, ξ ion is the production efficiency of Lyman-continuum photons per unit UV luminosity. The conversion to SFRD is then calculated as, SFRD Lya [M yr -1 Mpc -3 ] = ρ UV /(8.0 × 10 27 ).

Figure 15 shows the evolution of the SFRD with redshift. As the LF decreases steeply toward the bright end, the upper limit of the integration (Eq. ( 7)) does not affect to the final result. However, the steep slope at the faint-end plays an important role. If we take the integral over the full range probed by the present sample, the contribution of LAEs to the cosmic re-ionization at redshift z ∼ 3. ). If we take the lower limit of the integral (Eq. ( 7)) at 0.03L * , results are shown as reddish-brown crosses, which are more consistent with others mentioned above. Then, the contribution of LAEs population at redshift z ∼ 3.5 is 10%, 30% at redshift z ∼ 4.5, and reaches up 100% at higher redshifts. Figure 15 also shows SFRD reported by Sobral et al. (2018). The difference between our results on the evolution of the SFRD and those obtained by Sobral et al.

(2018) can be explained by comparing the detailed shape of the LF and the luminosity range covered by the two studies. Indeed, the shape of the bright end does not affect the SFRD, and our sample better captures the steepening of the LF towards the faint end, which is responsible for the increase of the SFRD with redshift in our case. When using the modified Schechter functions instead of the traditional one to compute the SFRD over the full probed luminosity range, 39.5 < log(L) < 44, in order to better describe the flattening/turnover observed towards the faint end, the contribution is 25%, 50%, and 100%, respectively. Assessing the ability of a population of sources to reionize the Universe is usually done by comparing its ionizing power to the critical value needed to maintain reionization at a given redshift. This value for the critical photon emission rate per unit cosmological comoving volume was introduced by Madau et al. (1999) as follows: Ṅ = (10 51.2 s -1 Mpc -3 )C 30 1

+ z 6 3 Ω b h 2 0.08 2 , (8)
where C 30 is the clumping factor C HII = n 2 HII / n HII 2 , normalized to C HII = 30, and n HII is the mean comoving hydrogen A139, page 16 of 20 density in the Universe. Assuming C 30 = 1 and Ω b h 2 50 = 0.08, the critical SFRD will be written as:

SFRD crit = Ṅ × 10 -53.1 f -1 escp ∼ 0.013 f -1 escp × 1 + z 6 3 . (9)
The clumping factor, C 30 , can be considered as a correction accounting for inhomogeneities in the IGM, and it is supposed to vary as a function of redshift. For example, as reported by Shull et al. (2012), C HII = 2.9 × ((1 + z)/6) -1.1 (i.e., a value ∼3 at z ∼ 5), hence, the critical SFRD obtained from the equation above needs a correction factor of C HII 30 . Here, we adopt the average value of C HII = 3, to facilitate the comparison with previous works (Pawlik et over the redshift range 0.3 < z < 6, reaching a maximum of unity at z = 11.1. The increase in f Lyα escp with redshift follows the evolution of the dust content in galaxies up to z ∼ 6, and then drops above z ∼ 6.5. Using this prescription in the range covered by our study and a reasonable conversion for f escp , following Dijkstra et al. (2016), the order of magnitude expected for f escp is ∼5% at z ∼ 3 and up to some 25% at z ∼ 6. Figure 15 displays the critical SFRDs obtained from these two extremes of f escp as a shaded area, using the same clumping factor of 3 as described above. As seen in the figure, the SFRD points obtained from our Lyα LF with the integration limits 41 < log(L) < 44 are fully consistent with the critical ones for the average value of f escp ∼ 8% and C HII = 3, in the redshift interval of 3 < z < 6.7. In other words, the contribution of LAEs to the ionizing flux in this redshift interval seems to be sufficient to keep the hydrogen ionized. The contribution of LAEs at z ∼ 6 is comparable to the one provided by LBGs.

This result taken at face values suggests that the contribution of the population of LAEs to cosmic re-ionization could be much higher than previously expected. There are, however, a few caveats to mention. Firstly, the vast majority of previous surveys in blank fields have been focused on the most luminous galaxies in their restframe UV that are well suited for a successful spectroscopic follow up. By construction, these samples can not probe to the faint luminosity regime, down to 10 39 erg s -1 , as in the present work. In addition, IFU observations identify LAEs without any pre-selection. Secondly, the contribution to the SFRD is directly related to the steep slope value being obtained, which is directly proportion to Lyα luminosity density. A 20% steeper slope dramatically changes the LAEs contribution to the cosmic re-ionization as illustrated here. Thirdly, the lower limit of the integration also has an impact on the final contribution. Moreover, there are still large uncertainties in the Lyα f escp and its evolution with redshift making difficult the comparison between different works using different prescriptions.

Conclusions

We present a study of the galaxy Lyα LF using a large sample of 17 lensing clusters observed by MUSE. We blindly selected 600 lensed LAEs behind these clusters in the redshift range of 2.9 < z < 6.7. The sample covers four orders of magnitude in galaxy luminosity (39.0 < log(L) < 44.0) and efficiently probes the faint luminosity regime, down to 10 40 erg s -1 . This sample sets a strong constraint on the LF at the faint end as well as the evolution of the slope as a function of redshift.

To deal with the combination of lensing fields and spectroscopic datacubes obtained from MUSE, we adopted the same V max method and the same procedure as described in DLV19 to compute the LF points. Several improvements have been introduced and applied to the original pipeline, allowing us to better account for lensing magnification. The main results and conclusions are listed as follows:

-Regarding the method and the pipeline, several effects have been studied in details, with the corresponding improvement on the final results with respect to DLV19. The new pipeline better accounts for the magnification errors when computing V max . A careful analysis was performed on the effects of source centering and completeness corrections.

-We studied the LF in four redshift bins, 2.9 < z < 4.0, 4.0 < z < 5.0, 5.0 < z < 6.7, and 2.9 < z < 6.7. The total comoving volume of our survey is ∼50 000 Mpc 3 . With respect to the previous work by DLV19, A2744 is still the dominant cluster, with a contribution to the total volume three times above the average.

-The three best-fit parameters of the Schechter function in the redshift range 2.9 < z < 6.7, obtained from our sample, are: α = -2.06 +0.07 -0.05 , Φ * [10 -4 Mpc -3 ] = 7.41 +2.70 -2.20 , log(L * )[erg s -1 ] = 42.85 +0. 10 -0.10 . -The LF values in the bright-end regime (log L > 42) are consistent with previous works, in particular the MUSE-Wide observations, as well as with other surveys in the literature using different techniques.

-In the faint luminosity regime, the contribution of highly magnified sources to the LF points in the faint luminosity bins is significant, as expected. The density of sources is well described by a steep slope, α ∼ -2. The two slopes obtained from linefitting and from the Schechter function are consistent within their uncertainties. The present Schechter slope is consistent with those from Drake et al. (2017) for each redshift interval, following Herenz et al. (2019) for the global redshift range, and is 20% steeper than the slope of DLV19. When taking into account different flux measurements, different completeness threshold cut (1% and 10%), and source selection effects, we obtained faint end slopes of -2.00 ± 0.50, -1.97 ± 0.50, -2.28 ± 0.50, and -2.06 ± 0.60 for z 35 , z 45 , z 60 , and z all , respectively. These results are in line with other studies, typically within a 1-σ deviation.

-The steepening of the faint end slope with redshift, suggested by the earlier work of DLV19 is confirmed here, but the uncertainties remain large. A turnover seems to appear at luminosities fainter than ∼10 41 erg s -1 , for the two highest redshift bins.

-The SFRD depends strongly on the interval of luminosity over which the density is integrated. The steeper slope at the faint end causes the SFRD to dramatically increase between z ∼ 3 and 6, implying that LAEs play a major role in the process of cosmic reionization.

-The contribution of LAEs to the ionizing flux in the redshift interval studied here seems to be sufficient to keep the hydrogen ionized. The contribution of LAEs at z ∼ 6 is comparable to the one provided by LBGs. One of the most powerful tools for studying these intrinsically faint SFGs is Lyman-α emission. Galaxies that exhibit Lyman-α emission are called Lyman-α emitters (LAEs). The strength of the Lyman-α line for equivalent widths (EWs) greater than, for instance, 25 Å allows us to identify intrinsically faint and/or high-redshift galaxies. In recent years, this has been exploited from several different angles in order to learn more about galaxies during this epoch as well as the state of the IGM and hence reionisation itself (see, for example, the reviews by Stark 2016; Dijkstra 2016, and Robertson 2022 and references therein).

Additionally, the use of lensing allows us to probe fainter galaxies than possible with blank field surveys, down to Lyman-α luminosities of ∼10 39 erg s -1 (Bina et ). This gives us direct access to faint populations, which are currently the favoured candidates for the main contributor to reionisation. This avenue has been explored in recent studies with small to medium sample sizes (dLV20; Fuller et al. 2020). Lensing, however, comes with compromises on sample size and the volume of the Universe probed. Even more recently, samples of more significant sizes (hundreds of objects) have become available, such as the Lensed Lymanalpha MUSE Arcs Sample (LLAMAS; R21,C22).

In order to investigate reionisation and the role of SFGs, the fraction of Lyman break galaxies (LBGs) that exhibit Lyman-α emission (henceforth referred to as the LAE fraction, or X LAE ) is particularly interesting. The Lyman break is caused by the absorption of photons at wavelengths shorter than 912 Å by neutral hydrogen gas around the galaxy and up to 1216 Å by the Lyman forest along the line of sight. This can be used to search for galaxies photometrically using the 'drop-out' technique: galaxies will 'drop out' of filters bluewards of the Lyman break. Lyman-α emission is scattered by neutral hydrogen in the IGM, interstellar medium, and circum-galactic medium, so whether or not Lyman-α emission is detected from LBGs gives us information about the content of these media, in particular how ionised they are, which has the potential to help in reconstructing the timeline and scale of reionisation (see, for example, Mason et However, there are significant uncertainties associated with both the measurement of X LAE and its usage as a probe of the reionisation history. The evolution of X LAE with redshift could also be due to the inherent evolution of one or both of the populations considered, rather than solely a consequence of the changing state of the IGM (Bolton & Haehnelt 2013;Mesinger et al. 2015). Progress has been made in understanding the impact of the interstellar medium, circum-galactic medium, and dust attenuation on Lyman-α emission (Verhamme et Equally important are the possible biases introduced by the different methods of selecting both the UV 'parent sample' and the LAE sample. When collecting the parent sample using the Lyman break, there is no standard way of selecting these galaxies. Some samples are selected via colour-colour cuts (for example, Stark et Kusakabe et al. 2020;dLV20), although this probably does not have a large effect on the sample as the procedure to find the photometric redshift relies on the Lyman break in the same way as the colour-colour cuts (dLV20). The employed colourcolour cuts depend on the instrument and bands used to observe the sample as well as the depth of the observations, leading to different cuts for each study.

The exact selection process using photometric redshifts varies from study to study. For example, authors decide the signal-to-noise required for a detection as well as how to deal with the probability distributions provided by most photometric redshift codes.

The way the search for LAEs is conducted once the parent sample has been selected is not standardised either. Some Using slit spectroscopy to search a UV-selected sample for Lyman-α emission can be problematic since the Lyman-α emission is not always centred on the UV emission; in fact, it has often been found to have an offset. C22 report a median offset of ∆ Lyα-UV = 0.66 ± 0.14 kpc, and Hoag et al. (2019b) find an offset corresponding to 0 . 25 at z = 4.5. Therefore, slit spectroscopy may not see the Lyman-α emission at all, or may miss some flux from extended emission.

The completeness of the different samples is also an important factor to take into account in these types of studies. Completeness is a correction made to the amount of objects observed to account for those present in the field but not observed. This correction depends on many factors and is different for each study. Several different approaches have been employed in the literature. Stark et al. (2010) determined the completeness of their Lyman-α detections by inserting fake emission lines across their spectra and attempting to detect them using the original detection process, a method that has evolved into the complex Lyman-α completeness treatments seen in IFU studies such as Kusakabe et al. (2020) and dLV20. dLV20 and this study use the extra complication engendered by lensing fields (see dLV20 and Thai et al. 2023). dLV20 show that the inclusion of the LAE completeness correction is important for the calculation of X LAE .

The study can be performed on a UV-complete subsample of the LBG population as in Kusakabe et al. ( 2020), but it is common to assume that one's LBG selection is highly complete for the signal-to-noise requirements imposed in the selection process. This is also an assumption sometimes made for the Lyman-α selection. For studies not involving lensing, one can easily calculate the apparent magnitude at which a sample becomes incomplete in the UV selection at the 10% and 50% level, as in Arrabal Haro et al. (2018).

In light of these discrepancies between different studies, it is not surprising that there is significant disagreement on the precise values and evolution of X LAE . There is only a general consensus that X LAE rises from lower redshifts to a redshift of ∼6, after which it decreases (see, among others, Pentericci et In addition to the reasons previously mentioned, the scatter in the results from these various studies can come from issues A174, page 2 of 13 Goovaerts, I., et al.: A&A 678, A174 (2023) related to sample size as well as the exact sample used to calculate the fraction. For example, the inclusion limits on M 1500 are not homogeneous across all studies, nor are the inclusion limits on the Lyman-α EW.

In this paper we investigate faint SFGs towards the epoch of reionisation (2.9 < z < 6.7) observed behind four lensing clusters in the Hubble Frontier Fields (HFF; Lotz 2017), specifically chosen for being efficient enhancers of such highredshift objects. We selected LAEs from Multi-Unit Spectroscopic Explorer (MUSE) IFU spectroscopy and LBGs using deep Hubble photometry and photometric redshifts. We present the largest combined LAE and LBG sample of lensed, intrinsically faint galaxies used for this purpose to date. Our sample reaches as faint as M 1500 ∼ -12 and log(L Lyα /erg s -1 ) ∼ 39.5 after correcting for lensing magnification. This is similar to the latest LBG selection in the HFF from Bouwens et al. (2022). Having blindly selected these populations of galaxies, we compare their UV and Lyman-α-derived properties and explore the fraction of LAEs among the LBG population, its redshift evolution, and its UV magnitude dependence.

In Sect. 2 we cover the data from MUSE and the HFF as well as the specific selection criteria we apply for our samples of LAEs and LBGs. We outline the photometric redshifts used and the blind matching of both populations. Section 3 outlines our results pertaining to derived properties: EWs, UV slopes, and star formation rates (SFRs), as well as the interrelation between the two populations, including several approaches for determining the fraction of LAEs among the LBG population. We discuss the implications for reionisation and the properties of these highredshift galaxies. In Sect. 4 we summarise our findings and offer perspectives for future surveys.

The Hubble constant used throughout this paper is H 0 = 70 km s -1 Mpc -1 , and the cosmology is Ω Λ = 0.7 and Ω m = 0.3. All EWs and UV slopes are converted to their rest-frame values, and all magnitudes are given in the AB system (Oke & Gunn 1983). All values of the UV absolute magnitude (defined at the 1500 Å rest frame, M 1500 ) and Lyman-α luminosities are given corrected for magnification.

Data and population selection

For this study, we combined MUSE (Bacon et al. 2010) IFU observations with the deepest Hubble Space Telescope (HST) photometry available in lensing clusters. The HFF clusters [START_REF] Lotz | The Frontier Fields: Survey Design and Initial Results[END_REF]) are ideal for this work, specifically, we use Abell 2744, Abell 370, Abell S1063, and MACS 0416 (henceforth A2744, A370, AS1063, and M0416). The MUSE observations of these clusters are taken from the data collected in R21, from which the LAEs are selected to form the LLAMA Sample (C22). The complementary HST data are from the HFF-DeepSpace Program (PI: H. Shipley) 1 .

MUSE LAEs: LLAMAS

LAE selection

The MUSE data we use for this work is part of LLAMAS (C22). These observations were part of the MUSE Guaranteed Time Observing programme and are comprehensively described in C22 as well as R21. The catalogues and lens models for the four 1 http://cosmos.phy.tufts.edu/~danilo/HFF/Download. html clusters used in this work are available online2 and at the CDS. We summarise here the details of the reduced MUSE datacubes and the LAE selection process.

The MUSE datacubes contain the flux and variance over a 1 × 1 arcmin 2 field of view, with spatial and spectral pixel scales of 0.2 and 1.25 Å. The spectral range covers 4750 Å-9350 Å, allowing the detection of Lyman-α between redshifts of 2.9 and 6.7. Integration times on the clusters we use for this work vary between 2 and 14 h, with different pointing configurations (see below). The clusters are all in the range 0.3 < z < 0.4, providing magnifications useful to amplify sources in the MUSE Lyman-α redshift range. Details of the four clusters are given in Table 1. Full details covering all 17 LLAMAS clusters can be found in C22 and Thai et al. (2023).

In order to detect line emission sources such as LAEs, R21 follow the prescription laid out in Weilbacher et al. (2020), using the MUSELET software (Piqueras et al. 2019) 3 , which is used on MUSE NB images. Subsequently, the Source Inspector package (Bacon et al. 2023) is used to identify sources and assign their redshifts. This package allows users to cycle through a list of sources with all the relevant information: spectra, NB images, HST counterparts, MUSELET results, and redshift suggestions. With this information, users can decide on a redshift for an object as well as assigning each source a redshift-confidence level from 1 to 3. A confidence level of 1 denotes a tentative redshift and a confidence level of 3 denotes a redshift with a high confidence. For this study we only use LAEs with confidence levels of 2 and 3, indicating secure redshifts. In our sample -A2744, A370, AS1063, M0416 -there are 263 such LAEs.

Magnifications of sources are assigned with the use of the parametric mass distribution models in R21 (for A2744, A370 and M0416, for AS1063, the lens model comes from Beauchesne et al. 2023) and the LENSTOOL software (Kneib et al. 1996(Kneib et al. , 2011;;Jullo et al. 2007;Jullo & Kneib 2009). These models are well constrained by the large number of multiple images with strong spectroscopic redshifts (levels 2 and 3 as described above) in these clusters. R21 estimate a typical statistical error of 1% of the mass profile of these clusters. The models in turn give the magnifications of the sources used in this study, which range from 0.8 (de-magnified) up to 137. Most sources have magnifications between 1.5 and 25. While these lensing models are well understood and benefit from many multiple image systems as constraints, small systematic uncertainties related to the lens model used can still persist due to the particular choice of mass distribution (see, for example, Meneghetti et al. 2017;Acebron et al. 2017Acebron et al. , 2018;;Furtak et al. 2021).

LAE flux determination

The Lyman-α flux for the LAEs in our sample was extracted using one of two different methods.

The main method employed in the LLAMA Sample is the line profile fitting procedure described in C22. For the subsample used in this work (263 LAEs), all those behind A2744 and 20 behind the other three clusters (roughly half the subsample), we use a method involving SExtractor (Bertin & Arnouts 1996) following the procedure outlined in dLV19. We give a brief description of both methods.

The line profile fitting method utilises three steps: spectral fitting, NB image construction and spectral extraction. Notes. We show the number of each sample group in each cluster (see text). The different exposure times for three of the clusters come from the range of exposure times for different MUSE pointings for these clusters. The second method is employed for A2744 and in the three other clusters for the cases where the first method fails to fit the Lyman-α flux. This happens in very low signal-to-noise cases or sources close to the edge of the datacube. This method uses SExtractor on NB images as described in dLV19. The NB images in question are those in which each LAE is detected. Three sub-cubes are extracted from the main datacube, one encompassing the Lyman-α emission and two either side of it (spectrally) each of width 20 Å. These cubes are averaged to form a continuum image, which is subtracted pixel-by-pixel from an image formed by averaging the cube containing the Lymanα emission. SExtractor is then run on this new continuumsubtracted image and the FLUX_AUTO parameter is used to estimate the fluxes of the LAEs. To deal with faint sources or those with an extended, low surface brightness, SExtractor can progressively loosen the detection conditions (using the DETECT_THRESH and DETECT_MINAREA parameters) so that a flux can also be extracted for these sources.

Figure 4 of Thai et al. (2023) shows the comparison between the two methods of extracting the Lyman-α flux. In general, the two methods agree well. In some cases, the line profile fitting method estimates a larger flux due to its enhanced appreciation of the line complexity.

In Fig. 1 we show the properties of our sample of LAEs in red. In order to have a similar derivation of M 1500 for all our LAEs, regardless of detection in HST photometry, values of M 1500 are calculated from the filter closest to the 1500 Å rest frame wavelength (where available, and the 2σ upper limit of the continuum taken from this filter where not), and corrected for magnification. Lyman-α luminosities are derived from the fluxes described above and also magnification-corrected. In terms of Lyman-α luminosity we probe roughly between 39.5 < log 10 (L Lyα /erg s -1 ) < 42.5, with decreased statistics near the faintest and brightest limits. This faint population can at present only be accessed with lensing clusters; the typical limit

HFF data

The HST observations of the clusters used in this work are from the Hubble Frontier Fields programme (ID: 13495, P.I: J. Lotz). In particular, we use the photometric catalogues of the HFF-DeepSpace Program (Shipley et al. 2018). As part of this programme, the authors collected homogeneous photometry across the HFF, including deep K S -band imaging (2.2 µm) from the HAWK-I on the Very Large Telescope (VLT) and Keck-I MOS-FIRE instruments Brammer et al. (2016). Additionally included are all available data from the two Spitzer/IRAC channels at 3.6 µm and 4.5 µm. The two Spitzer/IRAC channels at 5.8 µm and 8 µm were judged to be too noisy and excluded from the spectral energy distribution (SED) fitting process (see Sect. 2.3). The details of the filters used, and their depths, for each cluster can be found in Table 2. The bright cluster galaxies and intracluster light are subtracted by Shipley et al. (2018) for improved photometry of background sources in these very crowded fields. The detection image for each of these clusters is made up of a combination of the F814W, F105W, F125W, F140W and F160W bands (point-spread-function-matched to the F814W band), after the modelling out of the bright cluster galaxies.

The area that MUSE observed for each of these clusters is fully contained within the HST area, so we cut the HST area we consider to that of the MUSE data. The resulting effective (lenscorrected) co-volume, derived from LENSTOOL source-plane projections, is 22 910 Mpc 3 over the redshift range 2.9 < z < 6.7.

LBG selection

We calculate photometric redshifts and probability distributions (hereafter P(z)) in order to perform our LBG selection. We use the package New-Hyperz (Bolzonella et al. 2000) to estimate redshifts and probability distributions. This package uses a standard χ 2 minimisation technique to fit template galaxy SEDs to photometric data points. It has been optimised for the redshift determination of high-redshift galaxies so is ideal for our purpose. We use a suite of template spectra to fit the photometric data: the four template spectra from Coleman et al. (1980), two Starburst99 models with nebular emission (Leitherer et al. 1999): a single burst model and a constant SFR model (where each spans five metallicities and 37 stellar population ages), and seven models adapted from Bruzual & Charlot (2003). Included in these seven are a star formation burst model, five exponentially decaying models with timescales between 1 and 30 Gyr and a constant star formation model. The redshift range we use in New-Hyperz is 0-8 with a step in redshift of 0.03. The Calzetti extinction law (Calzetti et al. 2000) is used to account for internal extinction with values of A v allowed to vary between 0 and 1.5 mag. No priors on galaxy luminosity were introduced during this process, avoiding any bias introduced by lensing magnification.

For a galaxy to be included in the LBG sample for this study we demand that the best solution for the photometric redshift lies in the range 2.9 < z < 6.7. We also accepted any candidates not in this range that have 1σ errors overlapping with it. In practice, this accounts for just 14 objects. To make sure the detection is real, we also demand a 5σ detection in at least one HST filter. New-Hyperz provides a redshift probability distribution, P(z), which can have two peaks, in which case there are two redshift solutions for that particular galaxy. We accept galaxies into the sample if only one of the peaks is in the required range, as long as 60% of the integrated P(z) also lies in this range. We can compare the photometric redshift results for those objects also selected as LAEs to the spectroscopic redshift determined from the peak of the Lyman-α emission. Figure 2 shows this comparison. The dashed lines indicate regions constrained by |z spec -z phot | < 0.15(1+z spec ) and |z spec -z phot | < 0.05(1+z spec ). We encode the apparent magnitude in the F125W filter in the colour bar to demonstrate that the instances in which New-Hyperz performs poorly (i.e., instances outside of the outer region specified above) tend to be fainter objects with magnitudes 28. Finally, we manually inspect our LBG sample using the HST images, photometry, SED fitting from New-Hyperz and the LENSTOOL package with the lens models described in Sect. 2.1.1. This procedure is designed to identify multiple images in the LBG sample caused by the gravitational lensing, as well as to remove any obvious spurious detection; regions of noise or contamination erroneously selected as LBGs. Multiple images are identified using a combination of LENSTOOL predictions, redshift determinations and SEDs from New-Hyperz, and visual inspection of the HST images. LENSTOOL predictions, redshift estimates and colours of objects have to match for those objects to be designated a multiple image system. The colours used are F814W -F606W, F105W -F814W, F140W -F105W, all of which have to match to within 2σ for the objects in question to be designated multiple images. From each system, a representative LBG image is chosen. This image is the least contaminated, or, in the case of a match with an LAE multiple image system, the image chosen to represent the LBG multiple system matches the selected LAE image (see Sect. 2.4).

Matching populations

Having blindly selected our LAE and LBG samples, we compared them, using a matching radius of 0 . 5, to see which objects are selected as both. The results are shown in Table 1.

A174, page 5 of 13 . Almost all of the photometric redshifts agree with their spectroscopic equivalents within the outer of these two bounds. Those that do not tend to be fainter objects, with magnitudes of 28 and above.

We selected a population labelled LAE+continuum, which denotes objects that are selected as LAEs, for which we see a continuum in the HST images, but where this continuum fails to meet the selection criteria for our LBG sample. Mostly, these galaxies fail on the S/N criteria outlined in the previous subsection, indicating a faint continuum or a noisy area in the HST images. Nevertheless, we kept these objects as, thanks to the Lyman-α emission, we know that there is a high-redshift galaxy at these positions. In the previous work of this nature solely on A2744, dLV20 simply included these objects in the LAE+LBG sample, relaxing the signal-to-noise criterion for these objects, seeing as the presence of an object (as well as its redshift) was known thanks to the Lyman-α emission. Here, however, we kept the distinction between these continuum detections with Lyman-α emission and the LBGs in our sample that fulfil all the criteria laid out in the previous subsection. Details on the inclusion of this sample in the analysis of the LAE fraction, X LAE , are given in Sect. 3.2.1.

For the LAEs, a 'best' image is chosen, a process described fully in Thai et al. (2023). This allows us to choose representative images that are minimally impacted by neighbours and have high signal-to-noise and moderate magnification. When selecting LBGs, we kept the image corresponding to the best LAE image selected by Thai et al. (2023) where possible. However, in some cases we chose another image, because that particular LAE image is selected as LBG rather than continuum only. We ensure that the image chosen is always of similar quality. A modification of this nature is rare and we impose it in fewer than ten systems across the whole LAE sample. The original Lyman-α flux and magnification of the source, as given in the LLAMAS catalogues, is used for our analysis on the properties of these objects; however, their designation as LAE+continuum or LAE+LBG can change based on this.

Completeness determination

As covered in Sect. 1, the completeness of the populations considered in such a study is very important. This correction to the number of sources detected aims to reflect the number of sources that are missed in the detection process and thus the number of sources actually present in the field of view.

The completeness methods used for the LAEs in this work are described in Thai et al. (2023) and summarised here. Following the procedure first laid out in dLV19, sources are treated individually in this computation. Each source's brightness distribution profile, both in the spatial and spectral dimensions, is modelled and randomly injected 500 times into the NB layer of the original MUSE datacube where its Lyman-α emission reaches a maximum. This process is performed in the image plane, in order to recreate as closely as possible the actual process of detection with MUSELET. The completeness of the source is the number of times (out of 500) it is successfully detected and extracted.

Since it is the local noise that likely decides whether or not such an injected source is detected, to account for variations in the local noise, Thai et al. (2023) change the size of the NB image used to re-detect the simulated sources from 30 × 30 to 80 × 80 . This was found to improve the extraction of the source when that source has close neighbours. The mean completeness value found for our sample is 0.72 and the standard deviation is 0.34.

Finally, each source's contribution to the LAE fraction is corrected by a value 1/C i , where C i is the completeness value of that source. In effect, a source with a completeness value of 0.5 contributes two LAEs to the calculation of the LAE fraction.

Results

UV properties of the populations

In order to ascertain the similarities and differences between our populations of high-redshift galaxies, it is useful to look at the physical properties we derive from the HST photometry and their relation to Lyman-α emission. This can also help A174, page 6 of 13 disseminate how these properties tie in with the LAE fraction (see Sect. 2.1.2). Additionally, we can appraise any differences from the established trends for these high-redshift galaxies that may appear in our sample of faint, lensed galaxies.

To evaluate this relation to Lyman-α emission, the Lyman-α equivalent width, EW Lyα , is an important property to derive. We can then compare this to UV-derived properties. In order to calculate EW Lyα values for our sample, first we derive UV slopes by fitting the photometric points starting above the location of the Lyman-α emission (irrespective of whether or not we detect it for a given galaxy) and including all the filters up to 2600 Å in the rest-frame, adapting the approach used in Castellano et al. (2012), Schenker et al. (2014), Bouwens et al. (2015a). We fit a power law, F λ ∝ λ β , where β is the UV slope, to the photometry. We chose this method as it does not rely on SED fitting with a set of templates that have specific allowed values of β.

Subsequently, by using this photometrically fitted UV slope for each object, we can ascertain the continuum flux level beneath the Lyman-α emission. Hence, we calculated the Lyman-α equivalent width (EW Lyα ) by dividing the Lyman-α flux by the continuum flux level, corrected to its rest-frame level. In this process we take into account the error on the UV slope resulting from the fitting process, as well as the error associated with the Lyman-α flux (see Sect. 2.1).

For the objects with no associated continuum, the upper limits of the continuum are taken from the filter closest to the 1500 Å rest-frame emission. For the objects with no associated Lyman-α emission, the detection limits of the Lyman-α emission are used.

The EW distribution with redshift for all our sample is shown in Fig. 3. The dashed black line denotes the typical EW inclusion limit for objects to the calculation of X LAE (see Sect. 3.2.1). The percentage of objects above this limit is 44% and 50% for LAE+LBG objects and LAE+continuum objects, respectively. This is to be expected as the objects selected as continuum Orange dots represent UV slopes that, upon inspection, we find to be dubious: photometry that locally does not represent the slope well or has large errors. Red stars represent average values in equally sized bins between β = -3.5 and β = -1.0. The change in EW Lyα across this range is not as significant as the change across the range of probed UV magnitudes (see Fig. 4); however, we see a lightly rising trend in EW Lyα with bluer slopes down to β = -2.5.

but not LBG are in general fainter, giving rise to higher values of EW Lyα .

In Fig. 4 we compare EW Lyα to UV magnitude for our LAE+LBG sample. UV absolute magnitude, M 1500 , is calculated from the filter closest to the 1500 Å rest-frame emission. We see a rise in EW Lyα towards fainter M 1500 , in agreement with results reported in Stark et al. (2010), De Barros et al. (2017), Kusakabe et al. (2020). We note that above M 1500 = -16 this graph is populated by high EW, continuum-undetected LAEs (shown by orange triangles). We did not include these objects in the binned averages as they are not LBGs and the continuum values are estimated upper limits; however, these objects indicate that this trend in EW likely increases to even fainter magnitudes than is populated by our current LBG selection. As the spatial extent of Lyman-α emission correlates with the size of the galaxy and hence M 1500 (Wisotzki et al. 2016;Leclercq et al. 2017; C22), the trend in Fig. 4 is often seen as a natural result.

We also compared EW Lyα to the UV slope (Fig. 5). In general, high-EW LAEs are found to have bluer slopes (Stark et We derive SFRs from Lyman-α (SFR Lyα ) and from the UV continuum (SFR UV ), based on the relations given in Kennicutt (1998) and the factor of 8.7 between L(Hα) and L(Lyα), assuming a Salpeter IMF (Salpeter 1955) and constant star formation. SFR Lyα is a good lower limit on the intrinsic SFR as Lyman-α flux can be lost due to dust attenuation or a more or less opaque IGM (Zheng et al. 2010;Gronke et al. 2021). However, due to the use of an IFU, slit losses have no impact on the Lyman-α flux.

We compare the two measures of SFR for our LAE+LBG sample in Fig. 6, plotting the line of one to one ratio and the actual median ratio (SFR Lyα /SFR UV ) found in our sample. This median ratio (0.35), is well below the one to one ratio, indicating that in most cases the escape fraction of UV photons (the fraction of photons at 1500 Å rest frame that escape the galaxy), f esc,UV , A174, page 7 of 13 exceeds that of Lyman-α photons (the fraction of Lyman-α photons that escape the galaxy), f esc,Lyα . This is less apparent in UV-fainter objects, where there is a greater fraction above the f esc,UV = f esc,Lyα line. This result is in line with previous findings in Ando et al. (2006), Schaerer et al. (2011) and dLV20. The explanation offered previously by Ando et al. (2006) and dLV20 relates to the likely difference between the UV-bright and UV-faint galaxies. If the UV-bright galaxies have had more time to evolve than UV-fainter objects, it is likely that they are chemically more complex and have a larger amount of dust. This would decrease the escape of Lyman-α photons and hence result in a smaller measured SFR Lyα . We return to this point having compared the LAE fraction for the UV-bright and UV-faint halves of our sample in Sect. 3.2.1.

Subsequently, we compare M 1500 and SFR UV to the UV slopes of both samples (LBG and LAE+LBG) between -3.0 < β < -1.0: Fig. 7. In this graph, we calculate binned averages as in Figs. 4 and5. However, for this plot the averages are computed for bins of equal population. For the LBG-only sample, there is a slight increase in SFR UV with steeper UV slopes, this is to be expected as steeper values of β generally indicate populations that are more intensely star forming.

From the binned averages we can see that for the population also selected as LAEs, the galaxies are distributed towards higher values of SFR UV and steeper UV slopes. This result supports previous findings (in various redshift ranges, as well as in simulations) that LAEs are in general dustpoor and are more intensely star forming than objects only selected as LBGs (Verhamme et al. 2008 We extend this genre of analysis to our large sample of intrinsically faint galaxies, finding the same trends; LBGs also selected as LAEs have on average higher SFR UV and steeper UV slopes. High-EW Lyα objects are on average fainter in absolute UV magnitude with steeper UV slopes, and those fainter in UV absolute magnitude tend to have f esc,Lyα approaching f esc,UV or in some cases above, while this is not seen for UV-brighter LAEs. All these trends are seen with significant scatter, which is to be Fig. 7. M 1500 and SFR UV vs. UV slope. The sample selected as LBGonly is shown in grey, and the LAE+LBG sample is colour-coded by redshift. Binned averages in SFR UV for the LBG and LAE+LBG samples are shown with red circles and golden squares, respectively. Each bin contains an equal number of the respective samples, and the horizontal error bars show the width (in terms of β) of each bin. Errors on the LBG-only objects are omitted for readability, and we note that there are LBG-only objects (and four LAE+LBG objects; see Fig. 5) that extend beyond the β limits of this graph; however, they are objects for which the calculation of the UV slope is uncertain. There are LBGonly objects that extend beyond the M 1500 limits of the graph, but these are taken into account when computing the binned averages.

expected given the intrinsically faint nature of the sample, the limits of the photometry (although it is the deepest available), and those objects that have a complex star formation history and physical structure, for which our assumption of constant star formation is less valid. Combined, our results reinforce the picture of high-redshift LAEs as UV-faint, intensely star forming, dustpoor galaxies and show that these trends hold for galaxies down to M 1500 ∼ -12.

The interrelation between the LAE and LBG populations

We define four samples: objects identified as LBG-only, objects identified as LAE-only, objects identified as both LAE and LBG (LAE+LBG), and objects identified as LAE that have a continuum detected in the photometry from HST that does not meet our criteria for the LBG sample (LAE+continuum). These samples have 872, 55, 100, and 108 members, respectively.

The sample is dominated by A2744, with 46% of the LAEs and 35% of the LBGs; this is a large field of view with long MUSE exposures (see R21, C22, and Thai et al. 2023). However, as many fields of view as possible are valuable additions to the sample. The redshift distribution and the M 1500 distribution of the sample are displayed in Fig. 8. We find the same result as dLV20 when considering the relative importance of LAEs that are totally undetected in the continuum. Their increasing presence is clear when looking at the faint region of the M 1500 distribution, even when using the deepest HST photometry. This is crucial to note for any study wishing to catalogue the contribution to reionisation of all star-forming objects at high redshifts. This effect is naturally expected to strengthen with decreasing depth of photometry.

LAE fraction

Before discussing X LAE and its relevance to reionisation, we first outline what is typically meant by X LAE . The typical qualitative A174, page 8 of 13 interpretation is 'the fraction of UV-selected LBGs that display Lyman-α emission'. This stems from the technique of spectroscopic follow-up on a photometric preselection of targets (e.g., Stark et al. 2010Stark et al. , 2011)). In the case of this study, both LAE and LBG selections are performed blindly. With an IFU, we also detect LAEs that have no continuum counterpart, objects that would clearly be missed in the aforementioned type of study.

Additionally, limits are typically placed on the inclusion of objects when calculating X LAE . The most common limits are EW Lyα > 25 Å and M 1500 > -20.25 (Stark et ). These are important distinctions to be aware of when comparing results from the literature. This discrepancy between the exact inclusion limits studies use, as well as the observational differences (as covered in the introduction) is likely a significant factor in the ongoing debate over the exact evolution of X LAE with redshift, EW Lyα and M 1500 (see Fig. 9).

In this work we provide a perspective based on blind selections of both populations with an IFU and deep photometry with which we can assess more fully the landscape of these high-redshift galaxies. We compare the inclusion of objects only selected as LBG as well as the LAE+continuum sample described in Sect. 2.4. Given that we detect LAEs with no continuum counterpart we also discuss the effect this has on X LAE . We investigate different M 1500 and EW Lyα inclusion limits to provide context on the differing results in the literature. This last point is additionally motivated by the use of lensing fields in this study, meaning that we effectively probe a different population than blank field studies. It is therefore appropriate to investigate different regions of the EW Lyα and M 1500 space.

In Fig. 9 we compare our results to results from the literature in the classical manner with inclusion limits of M 1500 > -20.25 and EW Lyα > 25 Å. The redshift bins used are 2.5 < z < 3.5, 3.5 < z < 4.5, 4.5 < z < 5.5 and 5.5 < z < 7.0. For each bin, we include four values of X LAE : the fraction of LAEs among LBGs (yellow dots), the fraction of LAEs among LBGs with the completeness corrections described in Sect. 2.5 (red dots) and the same two measurements including the LAE+continuum objects (yellow and red stars; see Sect. 2.4).

Despite significant uncertainties, we see roughly the same trend as many results in the literature, with X LAE decreasing around z = 6. The decrease of X LAE shown in our data is quicker than many equivalent measurements in the literature. If one A174, page 9 of 13 carries forward the assumption that X LAE efficiently probes the neutral hydrogen content, this result suggests that the IGM quickly becomes neutral after z = 6.

The objects we detect only as LAEs (making up ∼21% of our LAE sample) would push these fractions higher, were they to be included. Considering that these objects are extremely faint in terms of UV magnitude and high in terms of EW Lyα , they would make an appreciable difference in the calculation of the fraction (in a similar way to the LAE+continuum sample shown by red and yellow stars in Fig. 9). There is little reason to suspect that with deeper photometry in the region that sees the rest frame UV emission, the continuum would remain undetected for these objects.

The high values of X LAE that we see in the first (low-redshift) bin in Fig. 9 are likely due to the fact that the Lyman break is harder to detect at these redshifts given the filters at our disposal. In order to check this, we perform a simulation, creating 100 000 fake LBG sources using Starburst99 templates (Leitherer et al. 1999). We add realistic noise to the data and spread them equally throughout the redshift and UV magnitude space probed by our study. We then attempt to select LBGs following the original selection criteria laid out in Sect. 2.3. In the first redshift bin (2.5 < z < 3.5) the amount of LBGs selected versus the number of simulated LBGs is drastically lower than the other bins, at <10%. It is worth noting that this simulation was computed for AS1063: a 'best-case scenario', given that AS1063 has the most filters available in the short wavelength range (<606 nm). For sources at redshifts between 2.5 and 3, the break would appear between ∼320 nm and ∼365 nm. The blue filters have a shallower depth (for AS1063, F225W, F275W, F336W, and F390W have 5σ limiting magnitudes ∼24.0, 25.9, 26.3, and 25.4, respectively) than the redder HST filters and due to this and their scarcity (A2744 only has F275W and F336W), we struggle to recover all the LBGs in this bin (for complete filter depth information, see Table 2). This can also explain why in Fig. 1 we can see a trough in the distribution at the lower end of our redshift range, why X LAE is higher than many literature estimates in the lowest-redshift bin and forms part of the reason why we see little evolution between redshifts of 3 and 5. However, our results are still in agreement with the literature results at 1σ. Effects like this are not unexpected considering the varying nature of LBG (and LAE) selection.

The errors on X LAE as displayed in Figs. 9 and 10 contain several components. As mentioned, there is a big impact on sample inclusion from the EW Lyα limits of 25 Å. Therefore, it is important to quantify an error based on this. We performed 1000 Monte Carlo trials of X LAE , sorting the EW Lyα values in their error bars, assumed to be Gaussian. We evolved an error estimate for X LAE based on the 16th and 84th percentile of the resulting range of results for X LAE .

Secondly, for the fractions to which we applied our LAE completeness correction, this correction comes with an uncertainty (see Sect. 2.5) that is propagated together with the error described above and the Poissonian error count on the number of LBGs and LAEs detected. This Poissonian error in X LAE is given by: σ

P = N LAE /N 2 LBG + N 2 LAE /N 3 LBG 1/2 .
The effect of the sample inclusion limits on EW Lyα is clear in Fig. 9, particularly in the third redshift bin. We reinforce an appreciation of this as a key detail in the calculation of X LAE and the interpretation of plots such as Fig. 9. For this study, this source of error means that it is not possible to describe a statistically significant evolution of X LAE until the highest-redshift bin. Similarly, the inclusion of the LAE+continuum objects in the calculation of X LAE (denoted by red and yellow stars in Fig. 9) has an appreciable effect: an average absolute difference in X LAE of 0.05 for points without completeness correction and 0.07 for those with completeness correction. This is a reflection of the impact of LBG selection in the final calculation of X LAE , particularly the S/N criterion imposed in the selection.

Finally, we can see from Fig. 9 that the completeness correction applied to the number of LAEs plays an important role in the calculation of X LAE (see the red points in Fig. 9). Without this correction, it is also difficult to address the redshift evolution of X LAE .

Evolution of X LAE with UV magnitude

There is interesting and unsettled debate on the relative evolution of X LAE between the UV-bright and UV-faint populations (see Stark consequences at higher redshifts, z 6, on what type of objects can be seen in Lyman-α emission despite an increasingly neutral IGM. Relations between the UV magnitude and Lyman-α emission at these redshifts may hold clues as to how the IGM is reionised and which objects are primarily responsible for this. In order to assess this, we split our sample in UV absolute magnitude at M 1500 = -18 (as this limit splits our sample roughly in half) and recalculate X LAE for the bright and faint portions. Figure 10 shows this comparison. We note that the M 1500 ranges into which our data is split differ from many similar comparisons in the literature due to the lensed nature of our sample. We efficiently probe down to M 1500 ∼ -12, much fainter than studies conducted in blank fields. By contrast, we have few objects as bright as M 1500 = -22.

In the upper panel of Fig. 10, there is no statistical difference between the bright and faint halves of the sample until the final redshift bin (5.5 < z < 7.0); however, we do see a rising trend in the bright half and a decreasing trend in the faint half. In order to investigate this further we remove the EW Lyα limit of 25 Å, resulting in the lower panel of Fig. 10. Based both on the result from Fig. 4 and indications from the literature (Curtis-Lake et al. 2012; Stark et al. 2017), it is likely that UV-bright galaxies exhibit Lyman-α with small EWs: close to or below the 25 Å limit, due to spectral and spatial dispersion of the Lyman-α photons from such systems (Mason et al. 2018b). This is supported by the clear distinction we see in the lower panel of Fig. 10. This result suggests that, while UV-fainter objects generally have higher Lyman-α EWs, the LAE fraction is greater among UV-brighter objects, albeit with those objects exhibiting smaller EW Lyα . Ando et al. 2006). This result is often taken as surprising given that the increasingly neutral IGM should block much of the Lyman-α emission at redshifts greater than 6. However, it is possible that with an increasingly neutral IGM, the more luminous galaxies ionise a surrounding bubble, making it easier for Lyman-α emission to escape. These galaxies are also more likely to be situated in reionised overdensities (Matthee et al. 2015;Mason et al. 2018a;Witten et al. 2023). Witten et al. (2023), cataloguing eight LAEs within the epoch of reionisation, most of which have small EW Lyα (<25 Å), also suggest that mergers and tidal interactions with neighbours may be responsible for enhanced Lyman-α visibility during the epoch of reionisation. reionised bubbles created by these processes and these bright galaxies would allow the Lyman-α photons to redshift out of resonance by the time they encounter significant neutral hydrogen, and hence would not be absorbed or scattered.

More detailed, individual analysis of individual objects would be necessary to firmly support these conclusions for our high-redshift objects but the difference seen in the highestredshift bins of the graphs in Fig. 10 broadly supports the enhanced visibility of UV-bright, low-EW Lyα LAEs during the epoch of reionisation.

We caution the reader that we detect a population of high-EW LAEs undetected in the continuum, and if these objects were to be detected with deeper photometry, they could modify this result as they would be UV-faint with high EW Lyα . However, they are roughly equally spread over the redshift range probed so may leave the trends in Fig. 10 unchanged.

We plot the completeness corrected results in Fig. 10 in grey scale, darker grey representing the UV-bright portion of the sample and lighter grey representing the UV-faint portion. Due to the smaller sample sizes in these magnitude-split bins, these corrections have large errors where one or a few LAEs with a small completeness values, accounting for many completenesscorrected LAEs, can dominate the determination of X LAE for a given bin. The lower panel of Fig. 10 provides an example: X LAE for the bright half of the sample in the bin 4.5 < z < 5.5 is at 1.19; accounting for the completeness correction, there are more UV-bright LAEs than LBGs in this bin. Noting that this is caused by individual highly incomplete objects, we do not interpret this as meaningful in the context of X LAE with respect to the evolution of the IGM and reionisation. On the other hand, we note that the completeness-corrected trends also broadly support the conclusions outlined earlier in this section, particularly in the highest-redshift bins, where the difference between the UV-bright and UV-faint portions of the sample is even more pronounced.

Conclusions

We present an assessment of the interrelation of the faint LAE and LBG population viewed with MUSE IFU spectroscopy and deep HST photometry. We can access faint populations unseen in blank fields through the magnification provided by four lensing clusters, pushing LBG detections down to M 1500 ∼ -12 and LAE detections down to Lyman-α luminosities of log(L Lyα ) ∼ 39 erg s -1 . We find LAEs with no detected continuum counterpart, actors that play an increasingly important role in the regime fainter than M 1500 ∼ -16. We summarise our main results as follows:

-Our results for X LAE agree with findings in the literature, and if we accept that it is unlikely that the evolution of one or both populations considered makes a significant difference to X LAE , our faint sample supports the conclusions of studies conducted on brighter luminosity regimes, namely, that the Universe is reionising at z = 6 and beyond. However, based on the results of this study, we find little to no evolution between redshifts of 3 and 5. This is likely due in part to the greater effect of LBG selection incompleteness in the lower-redshift regions. The scatter in this redshift range in the literature is likely due to issues related to the LBG selection and population completeness, as discussed in Sect. 3.2.1.

-We compared the evolution of X LAE for the bright and faint halves of our sample, split at M 1500 = -18. We find different trends for the two: X LAE for the bright half rises towards z = 6, and X LAE for the faint half falls. The difference in X LAE between the UV-bright and UV-faint halves is statistically significant in the highest-redshift bin, indicating that this effect increases as the Universe becomes more neutral.

-To further investigate this, we removed the 25 Å limit in the calculation of X LAE . Without this limit, a clear distinction is seen between the bright and faint halves of the sample, suggesting that while high-EW LAEs tend to be UV-faint, there is a population of low-EW LAEs among the UV-bright population. This can account for some of the newly detected, low-EW, UV-bright objects around and during the epoch of reionisation.

-When considering the UV properties of our faint samples, we see the typical picture of LAEs as high-SFR, lowdust, UV-faint galaxies, extending previously observed trends to very faint UV magnitudes and Lyman-α luminosities. The A174, page 11 of 13 strongest trend is seen when comparing EW Lyα to M 1500 (Fig. 4). UV-fainter objects on average display larger EW Lyα , a trend that likely continues down to M 1500 ∼ -12 based on our sample.

-For fainter luminosities and with a larger sample, we confirm the trend that UV-brighter galaxies tend to exhibit greater f esc,UV than f esc,Lyα , whereas UV-fainter galaxies are distributed more towards the f esc,UV = f esc,Lyα line, sometimes even displaying f esc,Lyα > f esc,UV . This has been attributed to UV-bright populations being older, dustier, and more chemically evolved, which decreases their Lyman-α emission (Ando et al. 2006;dLV20). This is tentatively reported for our sample, particularly as there are significant uncertainties associated with the SFR UV values of the fainter galaxies.

-Even with the deepest HST photometry available, the presence of continuum-undetected LAEs remains important at UV magnitudes fainter than M 1500 = -16. Surveys with JWST can help shed light on the effect of much deeper photometry, albeit at different wavelengths than the HST.

Using the largest sample of LAEs and LBGs compiled for this purpose to date, we support previous findings on the increasing neutrality of the IGM around a redshift of six. Additionally our sample does not suffer from biases coming from selections with slit spectroscopy.

Moving forward, the outlook for larger samples of a similar nature could be problematic, as they require both deep IFU spectroscopy combined with extremely deep photometry. As the JWST probes a different part of the galaxy spectrum at these redshifts, it remains to be seen what depth, in terms of the photometry will be necessary to detect the same faint populationsor probe even more deeply -with similar sample sizes.

Introduction

The epoch of reionisation (EoR) is a critical period in the history of the Universe, during which the neutral hydrogen in the intergalactic medium was reionised by first generation of galaxies, ending around z = 6, 1 billion years after the Big Bang Among SFGs, the particular importance of Lyman-α emitters (galaxies that display Lyman-α emission at 1215.67 Å rest frame) (henceforth LAEs) has been hypothesised to range from significant (de La Vieuville et al. 2019), to dominant (Drake et al. 2017). LAEs tend to be highly star-forming, young, dust-poor galaxies (Stark et Further constraining the properties of these particular galaxies is crucial, considering their likely importance to the process of reionisation. Now, with JWST/NIRCam data pushing wavelength coverage further into the infrared, to 4.5 µm, we can obtain more robust estimates for the stellar mass of these highredshift star-forming galaxies.

In order to access the intrinsically faint regime of galaxies at redshifts between 3 and 7, gravitational lensing by clusters of galaxies is a useful, and often necessary, tool. Using the magnification from lensing, LAEs have been detected at these redshifts with Lyman-α luminosities as low as L Lyα ∼ 10 39 ergs 2023), in a sample of 200 galaxies observed with JWST/NIRSpec, included estimations of the MS relation down to a very low mass range (∼ 10 6 M ⊙ ), in redshift ranges 2 < z < 5 and 5 < z < 11, finding the galaxies to be located well above the best estimations of the MS at these redshifts and masses. This study, as well as Atek et al. (2023), clearly demonstrate JWST/NIRCam and JWST/NIRSpec's ability to probe these extremely low masses at these redshifts.

Returning to the most general case, S14 suggest a time evolving form of the MS:

log SFR(M ⋆ , t) = (a + b × t) log M ⋆ -(c + d × t) (2)
where t is the age of the Universe (Gyr) and where they find a = 0.84 ± 0.02, b = -0.026 ± 0.003, c = 6.51 ± 0.24 and d = 0.11 ± 0.03. We chose to adopt this form of the MS to fit to our data. As we do not probe the higher-mass end of the MS, where the linearity of this relation breaks down (see P23 and references therein), this linear form for the MS is safe for us to adopt.

In this work, we present a first derivation of this MS relation for intrinsically-faint, low-mass, lensed LAEs towards the epoch of reionisation (2.9 < z < 6.7). We perform our LAE selection blindly using an Integral Field Unit (IFU): MUSE/VLT, meaning we do not suffer from a biased selection where only the UV-brightest sources are considered. Lensing also helps us to remedy this bias, as well as allowing us to access the intrinsically faint and low-mass population likely responsible for reionisation.

Section 2 presents the spectroscopic and photometric data used. Section 3 covers the details of the SED fitting using secure spectroscopic redshifts from the Lyman-α line. Section 4 presents the results of placing these faint, low-mass LAEs on the galaxy main sequence, compares to previous best estimates of the main sequence relation, and offers conclusions pertinent to these galaxies which likely played a significant role in cosmic reionisation. Throughout this paper, the Hubble constant used is H 0 = 70 km s -1 Mpc -1 , and the cosmology is Ω Λ = 0.7 and Ω m = 0.3. All values of luminosity and UV magnitude are given corrected for magnification and the initial mass function (IMF) adopted is that of Salpeter (1955).

Data: A2744 viewed by MUSE and JWST/NIRCam

We combine MUSE's IFU spectroscopy of the A2744 galaxy cluster with the full range of imaging available, using the catalogs from the UNCOVER Treasury survey (PIs Labbé and Bezanson, JWST-GO-2561; Bezanson et al. (2022)). This photometry is among the deepest available and ideally suited for the detection and characterisation of high-redshift galaxies. Below we give details of both datasets used. The flux for these LAEs is determined using the method detailed in de La Vieuville et al. (2019) and Thai et al. (2023). We briefly summarise this method, which uses SExtractor (Bertin & Arnouts 1996) on a continuum-subtracted image from the Narrow Band (NB) datacube in which each LAE was detected. The continuum level is created by averaging two NB cubes either side (spectrally) of the detection NB cube pixel-by-pixel. SExtractor's FLUX_AUTO parameter is then used to extract the flux from this continuum-subtracted image. The DETECT_THRESH and DETECT_MINAREA parameters are used to loosen detection constraints, should sources be faint or extended.

JWST/NIRCam data

The photometry used for this study comes from the UNCOVER treasury survey (Bezanson et al. 2022). The photometric catalogs themselves are described in Weaver et al. (2023) 3 . This survey combines HST data from the F435W, F606W, F814W, F105W, F115W, F125W, F140W and F160W filters with NIRCam data from the F090W, F150W, F200W, F277W, F356W, F410M and F444W bands. This gives excellent coverage from 15 bands between 400 nm and 4.5 µm.

Weaver et al. (2023) subtract the bright cluster galaxies (BCGs) and intra-cluster light (ICL) in order to better detect faint galaxies. The photometry is then performed on a combination of the F277W, F356W, and F444W bands, PSF (point spread function)-matched to the F444W band. Two photometric catalogs are created with a 0.32 ′′ . -aperture catalog being optimised for faint, compact sources. 5σ depths for all 15 filters for the 0.32 ′′ . catalog are given in Table 1 of Weaver et al. (2023); medians range between 27.16 and 29.56 in AB magnitudes. We use this catalog for our photometry.

The authors also include a USE_PHOT flag, which allows one to filter out objects that are designated as stars, have a signalto-noise in the F444W band of < 3σ. are within 3 ′′ . of a bright cluster galaxy (BCG) centroid, are bad pixels or residuals from the BCG subtraction, have an unphysically large Kron radius, or are flat field artifacts. Applying this flag, there are 23366 objects in the JWST field of view, among which 1987 within the MUSE field of view.

Of the 154 images of MUSE LAEs in the MUSE FoV, 82 are detected in the UNCOVER photometry with the USE_PHOT flag applied, of 70 individual LAEs. Without this flag, 121 images are detected of 99 individual LAEs. We include all available photometry in this analysis in order to extract a maximum of information. Ultimately, we chose a representative for each multiple image system with the least contamination possible.

The range of Lyman-α luminosity probed is 40 < log 10 (L Lyα /erg s -1 ) < 42.5. The UV-magnitude range is -22 ≲ M UV ≲ -13, where this value is derived from the filter closest to the 1500 Å rest frame emission, in the same way as Goovaerts et al. (2023).

SED fitting with CIGALE

CIGALE (Code Investigating GALaxy Emission) (Boquien et al. 2019) is a python code4 designed to extract galaxy properties from photometric data ranging between the far-UV to radio. It takes into account flexible star formation histories (SFHs), nebular emission and different dust attenuation models. We fit the observed SEDs of all the JWST-detected LAE images in A2744 (121 out of 154). We take advantage of the accurately known, secure spectroscopic redshifts available for these LAEs, fixing these values in CIGALE.

We investigate two different star-formation histories (SFHs), a single exponentially decaying burst and a double burst model with a second, delayed burst (Małek et al. 2018;Boquien et al. 2019). A range of ages are taken into account in both models, from 10 -700 Myr. Mass fractions for the delayed burst range from 0.001 to 0.65. The best-fit of these two SFHs is carried forward for analysis.

We use the stellar library of Bruzual & Charlot (2003) with five metallicities ranging between 0.0001 and 0.02. We take into account nebular emission as some high-redshift galaxies can exhibit strong nebular lines (Zackrisson et al. 2008;Schenker et al. 2013;De Barros et al. 2017). The templates used by CIGALE are from Inoue (2011) and model 124 emission lines between 30 nm and 205 µm. Gas metallicities used range between 0.0004 and 0.02. Dust attenuation in our galaxies is modelled based on the modified starburst attenuation curve from Calzetti et al. (2000). Subsequently, dust emission is accounted for using the analytical model of Casey (2012) chosen for its adaptability to highredshift galaxies and computing speed. The choice of this module is of secondary importance as our longest wavelength data is at ∼ 4.5 µm, which at a redshift of 3 corresponds to a rest frame wavelength of ∼ 1.1 µm, meaning we are not sensitive to dust emission. Additionally, there is expected to be very little dust in these low-mass, high-redshift LAEs (Hayes et al. 2013;Santos et al. 2020a).

CIGALE outputs the required physical parameters of each galaxy, in this case stellar mass, star formation rate, age of the stellar population and the χ 2 ν statistic for each fit. In order to check the consistency of these results and the reliability of the CIGALE fitting, we use CIGALE's mock_flag function to generate mock results with simulated noise, to which we compare the actual results in order to assess how reliably we estimate parameters such as the mass and SFR. This comparison is detailed in Appendix A.

Results

Lyman-α Luminosity to Stellar Mass Relation

Following analysis from Santos et al. (2020bSantos et al. ( , 2021) ) we can assess the relationship between Lyman-α luminosity, Lyman-α equivalent width (EW Lyα ), and stellar mass. We find log(M ⋆ /M ⊙ ) = (0.94 ± 0.10) log(L Lyα /ergs -1 ) -(31.3 ± 3.9). This result is in good agreement with that in Santos et al. (2021), especially interesting as the 4000 SC4K LAEs in that study are in a Lyman-α luminosity range of 42.5 ≲ log(L Lyα )(erg s -1 ) ≲ 44, therefore far brighter than our sample. We show that this relationship holds down to intrinsically faint LAEs: when a galaxy displays Lyman-α emission, the connection between the stellar mass of the galaxies and the amount of Lyman-α luminosity they emit remains roughly constant across all currently-accessible luminosities.

When comparing EW Lyα with stellar mass, we find a relationship of the form: log(M ⋆ /M ⊙ ) = (-0.65 ± 0.13) log(EW Lyα ) + (8.5 ± 0.2). This is a tighter relationship than for luminosity, which is unsurprising as the UV magnitude used to calculate the EW Lyα is tightly correlated with the stellar mass (Duncan et al. 2014;Santini et al. 2023). This indicates that Lyman-α photons find it harder to escape higher mass galaxies. A more complete study on these relations necessitates larger samples of lensed zgalaxies which can be binned in terms of stellar mass.

Star-Formation Main Sequence

We plot the stellar masses and SFR derived from CIGALE in Fig. 1. This is done for the first time for LAEs of these masses at these redshifts. Therefore, in order to gain an understanding of how these galaxies compare to galaxies of higher mass and UVselected samples, we incorporate the appropriate best fits from the reviews of S14 and P23, as well as data from S17, a similar study to ours (without JWST data) using Lyman break-selected galaxies (LBGs) in the Hubble Frontier Fields (HFF). Additionally, we compare to the sample of LBGs selected behind A2744 from Goovaerts et al. (2023). These have undergone an identical fitting process to the LAEs, detailed in the previous section (without, of course, the benefit of spectroscopic redshifts).

It is immediately clear that these low-mass LAEs lie above the fits to the main sequence calibrated at higher masses. The fit Article number, page 3 of 7 of P23 when compared to our sample, unsurprising, as it is derived using galaxies down to a lower mass limit of 10 8.5 M ⊙ . The fit of S14, of the form shown in Eq. 2, fits the data slightly better, having been derived using galaxies of mass down to 10 7.3 M ⊙ , however only two studies with galaxies of mass below 10 8 M ⊙ are included, and they do not incorporate the first two Gigayears of galaxy evolution in their fit, limiting their usefulness as a comparative tool to this study.

The fit of the form in Eq. 2 that we find for our LAEs is (0.87 ± 0.07 -0.030 ± 0.031 × t) logM ⋆ -(6.23 ± 0.48 -0.075 ± 0.101). The non time-dependent constants (a and c), are consistent within their uncertainties with the best-fit of S14, and we see a similar but less significant evolution with time (constants b and d). This is to be expected as our sample is much smaller compared to the review of S14. Details of the fitting for the LAEs and LBGs in this work, as well as comparisons to S14 and S17 are given in Table 1.

The discrepancy between our results and the collations of S14 and P23, as mentioned, likely comes from the difference in mass-range probed, but to further verify this, we compare to two UV selected samples in the lower plot of Fig. 1: Santini et al. (2017) (henceforth S17), conducted in the HFF galaxy clusters, and the selection of Lyman break galaxies from Goovaerts et al. (2023), selected behind A2744 itself, in the same volume as the LAEs in this work. The fit to the LBGs (blue dashed line) is (0.89 ± 0.11 -0.024 ± 0.007 × t) logM ⋆ -(6.75 ± 0.78 -0.085 ± 0.592). This relation is consistent with that for the LAEs, further suggesting that it is the mass (and redshift) range probed, rather than the selection of the sample by Lyman-α emission that differentiates these galaxies from MS relations in the literature. The mass-binned data of S17 are in good agreement with our data, particularly in the highest redshift bin (5 < z < 6.6). They also find a similar trend when considering earlier times (higher redshifts): an increasing normalisation but a similar slope compared to later times (lower redshifts). However this evolution is more dramatic in their data than ours. This could be due to their larger sample size, however uncertainties related to the reliance on Spitzer/IRAC photometry for coverage past 1600 Å may have an effect. This photometry can be especially problematic in crowded fields such as lensing clusters (Laporte et al. 2015; Goovaerts et al. 2023). The 2σ-clipping procedure used in S17 and the choice to calculate the MS according to log SFR = α log (M/M 9.7 ) + β (where M 9.7 = 10 9.7 M ⊙ ), may also play a role. Viewed together, these results indicate that, while these lowmass LAEs reside above the typical MS estimates derived from higher-mass samples, they are likely similar to galaxies that are not LAEs, but are of similar mass at similar redshifts. This points to an enhanced specific SFR in these low mass populations, but no special role for LAEs themselves. In the context of cosmic reionisation, this supports conclusions that low-mass galaxies play an important role, however LAEs are not singular in their impact on this process.

Conclusions

With new JWST/NIRCam data, secure spectroscopic redshifts and CIGALE SED fitting, we have obtained reliable estimates for the SFR and mass for intrinsically-faint, lensed, reionisation-era Lyman-α emitters. This allows us to place these galaxies on the star-formation main sequence and compare them to UV-selected samples as well as studies conducted at higher mass. Our conclusions are summarised thus:

-Our results for the relations between stellar mass of the LAEs and the Lyman-α luminosity and equivalent width indicate that previous findings extend from very bright LAEs down to very faint LAEs.

-The MS relation we find for our LAEs between redshifts of 2.9 and 6.7, and masses of 10 6 M ⊙ and 10 10 M ⊙ is (0.87 ± 0.07 -0.030 ± 0.031 × t) logM ⋆ -(6.23 ± 0.48 -0.075 ± 0.101). This relation agrees well with samples of non-LAEs at similar masses and redshifts, but less well with MS relations derived at higher masses and lower redshifts.

-This indicates that while these low mass galaxies lie above typical MS relations, LAEs are not special in this regard, suggesting that while low-mass galaxies likely play a significant or dominant role in reionisation, LAEs themselves do not distinguish themselves in this process. Larger samples of this genre, for more precise estimations of the main sequence relation, both for LAEs and UV-selected samples, will have to wait until further imaging of lensing clusters with JWST/NIRCam. Observing many lensing clusters with the required depth of photometry is expensive, and so far A2744 has been prioritised. However in the coming months and years, we can expect samples of this type to grow significantly, with the addition of further lensing clusters containing hundreds of LAEs in and around the epoch of reionisation. .9 < z < 6.7 0.87 ± 0.07 -0.030 ± 0.031 6.23 ± 0.48 -0.075 ± 0.101 This work: LBGs 2.9 < z < 6.7 0.89 ± 0.11 -0.024 ± 0.007 6.75 ± 0.78 -0.085 ± 0.592

Notes. The constants a, b, c and d refer to the constants from Eq. 2. For S17 a time dependance (constants b and d) is not quoted so have been omitted.

I. INTRODUCTION

After the Big Bang, the temperature of the Universe was so hot that protons and electrons could not combine to form neutral atoms; matter existed in a dense plasma state. As the Universe expanded, its temperature dropped and at ∼ 3000 K electrons and protons can combine to form ©2022 Vietnam Academy of Science and Technology neutral hydrogen. The Universe became transparent and the dark ages began. The second major phase of transition happened when first objects inside hydrogen clouds in the massive dark matter halos were collapsed under gravity to form first stars, galaxies. Their photons having high enough energy started to re-ionize surrounding environments, creating ionized bubbles, getting bigger over time, expanding much further out to interstellar medium (ISM) and to intergalactic medium (IGM). The phase transition happened at redshift beyond 12, and was completed at z ∼6 (Fan et al. 2006).

However, the question about the sources that contributed to cosmic reionization remains unanswered. Star forming galaxies are the favoured candidates to reionize neutral hydrogen, the most abundant element of the Universe which accounts for ∼75% the total baryon mass (see e.g. Robertson et al. 2013, Bouwens et al. 2015a, Ricci et al. 2016). The interstellar medium and circumgalactic medium are the ideal environments for recombination of electrons and ionized hydrogen atoms. The transition between the first excited state and the ground state of neutral hydrogen (n=2-1) is called Lyman alpha, with a wavelength of 121.6 nm in the rest frame. The expansion of Universe makes this spectral line shift to the visible and near infrared ranges, becoming the brightest emission line to be observed with ground or space telescopes (Partridge & Peebles 1967, Charlot & Fall 1993).

Lyman Alpha Emitter (LAE) is a type of distant star forming galaxy that emits Lyman alpha from neutral hydrogen. They can be identified from wide field narrow-band surveys. With the advent of modern spectrographs such as Multi-Unit Spectroscopic Explorer (MUSE), physical properties of LAEs have been extensively studied (see a recent review by Ouchi et al. 2020). They are often found in early stage of their evolution, having moderate masses ranging from 10 8 to 10 9 M ⊙ , more compact in physical size, radii of ∼1kpc and less dust than other types of high-z star forming Lyman Break Galaxies (LBGs) (Taniguchi et al. 2009).

This paper presents a study of the galaxy luminosity function using 530 lensed LAEs in the redshift interval 2.9 < z < 6.7, obtained from a large sample of 17 lensing clusters observed by MUSE. The sample covers four orders of magnitudes (39<log(L)<43 erg s -1 ), efficiently probing luminosities down to 10 40 erg s -1 , aiming to constrain the contribution of LAEs to the density of galaxies at the faintest level hitherto explored.

II. Strong gravitational lensing by clusters of galaxies

A large number of galaxies have been observed thanks to magnification from strong gravitational lensing by clusters of galaxies. This makes it possible to observe high-redshift galaxies belonging to a faint population which could not be observed otherwise. Light traveling from the source towards the observer is bent due to the gravitational attraction of a high-mass object located along the line of sight, producing complex image configurations such as multiple images, arcs or rings (Einstein rings). Multiple image separations can reach up to scales of ∼ 10 arcsec or so. The lensing effect amplifies the brightness of faraway galaxies, an effect which is particularly strong when the source is near the lens caustic lines. These objects near the caustic lines are often very distorted. These distortions must be corrected for to obtain the magnification for each source. We make use of lens models for galaxy clusters, which are built using parametric mass models. An available package, called Lenstool (Kneib et al. 1996, Jullo et al. 2007 The Very Large Telescope (VLT) is located on the Cerro Paranal (2600m above sea level) in the Atacama Desert of Northern Chile. The observatory consists of 4 telescopes, each having a mirror 8.2 m in diameter working in the visible, near infrared and ultraviolet. MUSE was installed on the fourth telescope in 2014. It is a second generation of VLT instrument aiming to achieve high-quality images with a large Field of View (FoV) (1 ′ × 1 ′ in Wide Field Mode and 7" × 7" in Narrow Field Mode). It has excellent spectroscopic capabilities, covering a large wavelength range (475 -935 nm) with spectral resolution R∼3000 and has been facilitated with an Adaptive Optics system to correct for atmospheric turbulence. MUSE is well suited for spectroscopic studies of crowded fields such as lensing clusters, and searching for LAEs with redshifts between 2.9 and 6.7. The datacube is delivered to us in the form of ∼ 300 × 300 pixels of 0.2", covering 1 arcmin 2 FoV and ∼3681 frequency channels from 475 to 930nm.

III.2. Lensing clusters sample

Seventeen lensing clusters, obtained from the MUSE Lensing Survey Guaranteed Time Observations programme, have been used for the present work. About ∼1300 lensed LAE images are identified in redshift range 2.9 < z < 6.7. The data reduction of these lensing clusters were performed by the group of J. Richard, PI of the MUSE Lensing Survey. Twelve such clusters were described in detailed in Richard et al. (2021), following the prescription in Weilbacher et al. (2020). A similar procedure has been applied to the 5 new datacubes (MACS0451, MACS0520, A2390, A2667, A2744). Some technical improvements have been introduced to deal with crowded fields such as the centres of lensing clusters. Seeing values, i.e., full width at half maximum (FWHM) of Point Spread Function at 700 nm, vary from 0.52 to 1.02 and pointing integration times vary from 2 to 14 hours among these clusters. Details of the 17 lensing clusters are presented in Appendix Table A1.

These data belong to the Lensed Lyman Alpha MUSE Arcs Sample (LLAMAS) project. The process to identify LAEs behind lensing clusters is presented in Richard et al. (2021) and Claeyssens et al. (2022). These LAEs are part of a catalog, which is made publicly available and can be accessed at: https://cral-perso.univ-lyon1.fr/labo/perso/johan.richard/ MUSE_data_release/, including all important information for each LAE, such as position (RA, DEC), redshift, measured flux, magnification, etc. It is worth mentioning that magnifications provided by the catalog are for the points at the centre of sources. However, for extended LAEs, in particular sources close to the lens caustic lines or even overlapping them, magnifications across these sources are expected to have a very large gradient. To deal with this effect, the averaged magnification has been used for the present work. In practice, we make use of the Lenstool package to make projections of segmentation maps for each LAE from image plane to source plane and to calculate magnification for each source pixel. We then take the average value for each source.

Figure 1 shows distributions of LAE luminosity and redshift for our lensing cluster sample and those from de la Vieuville et al. (2019), using a smaller subsample of four lensing clusters only. Both samples cover four orders of magnitudes in galaxy luminosity, but the number of LAEs has been increased in our sample by a factor of 4 corresponding to 530 sources compared to 156. Our sample has also provided a better coverage at faint luminosities, which is crucial to constraint the slope of the luminosity function (LF) at faint end.

IV. Luminosity function: results and discussions

IV.1. V max method and completeness correction

In the present work, we use the V max method, introduced by Schmidt 1968, to compute the LF points. This V max is defined as the cosmic volume of the survey where an individual source could be detected. The combination of both MUSE spectroscopic data and strong gravitational lensing by clusters of galaxies makes this V max calculation challenging. Several assumptions have been made to simplify and speed up the process. De la Vieuville et al. 2019 developed a pipeline for this complex procedure. Details of the procedures are well documented in their work. We adopt the same approach, make use of their pipeline, and improve it to handle our sample with four times more LAEs.

The 2D detectability masks play a crucial role in building V max . Such masks are created by comparing the source brightness of a given source pixel by pixel with local noise in a given wavelength layer, then layer by layer through the MUSE spectral range, then field by field. The detectability masks in the source plane are reconstructed from the image plane by using the Lenstool package. This procedure is very computationally expensive, it could take years to complete. The important simplification of de la Vieuville et al. 2019 is to build a set of pre-computed 2D masked images for all ranges of S/N probed by the sample. They used the median source radial profile and median RMS map of each datacube to produce these. One then assigns the corresponding mask for a given source from the list accordingly. To account for dependences of detectability masks on redshift in the source plane, four different redshifts have been sampled: 3.5, 4.5, 5.5 and 6.0. The final V max is calculated in the source plane as:

V = Nω c H o zmax zmin D 2 L (z) (1 + z) 2 E(z) dz (1)
where E z = ω m (1 + z) 3 + (1 -Ω m -Ω λ )(1 + z) 2 + Ω λ , ω is the angular size of a pixel and D L is the luminosity distance, N is number of unmasked pixels counted on all wavelength layers. The total co-moving volume obtained from the present survey is approximately 52,000 Mpc 3 . This value is a few times smaller than that obtained from MUSE blank field observations such as Drake et al. (2017), and Herenz et al. (2019). Lensing magnification has significantly reduced the cosmic volume of the survey but allows us to reach galaxies fainter by two orders of magnitude. In addition to the V max procedure described above, one needs also to correct for the chance that a given source is being detected in its own wavelength layer, where its Lymanalpha line reaches the maximum, i.e. the masked Narrow Band (NB) image. This introduces a correction factor for each source, called the completeness. In order to estimate this, simulations have been used to randomly inject a mock source with the same source brightness radial profile into the masked NB image. Sextractor then tries to detect that mock source. The rate of successful detection, after 500 iterations for each source, measures its completeness. Quality of extraction by Sextractor is classified by different ranks. First and second flags are for trustworthy cases, while

IV.2. Luminosity function construction results

The traditional way to summarize the evolving demographics of high-z star forming galaxies is to use luminosity function, which is the density of galaxies per unit co-moving volume per unit luminosity. Once both V max and completeness for each source are measured, it is straightforward to compute the LF points. A luminosity point in a given luminosity bin i th , is calculated as:

φ (L i ) = 1
∆logL i ∑ j 1 C j V max, j (2)
where j is index of source in that i th luminosity bin, V max, j ,C j are the co-moving volume in the source plane and completeness correction of that source respectively and ∆ log L i is the bin width of the i th luminosity bin. The data sample covers four orders of magnitudes in luminosity in the range: 39<log(L)<43. At the full redshift bin 2.9 < z < 6.7 twelve luminosity bins have been considered, each has a bin width of 0.25 in log(L) except for the faintest one which has a width of 1.25. This is to make sure that both the brightest and faintest bins have a sufficient number of sources.

A set of three parameters: flux, completeness and magnification, for each source has been drawn randomly 20,000 times by employing a MCMC process, in order to estimate statistical uncertainties attached to LF points. Flux and completeness are randomly generated following the standard normal distributions. Source magnification, µ, is randomly picked up from the available set of P(µ) distributions provided by the Lenstool package. During this process, large magnification uncertainties smear source contributions to several luminosity bins. By contrast, moderate magnifications limit source contributions to one bin. We then measure the contribution of each source to each luminosity bin, obtaining in this way the mean and statistical uncertainties of each LF bin.

There are two other uncertainties of a different nature. One accounts for the different fields of the survey, called field to field variance, and one for the number of sources in each luminosity bin, called Poissonian uncertainties. The former contribution is at the level of 20 to 30%. The latter is important for the bright-end, log(L)>42, as bright sources are rare. Its contribution decreases to the level of the former in the moderate luminosity range, 41 < log(L) < 42. Details of these uncertainties are described in de la Vieuvielle et al. (2019). At present, systematic lensing magnification uncertainties have not been taken into account. Luminosity function for 2.9 < z < 6.7

Blanc 2011(1.9 < z < 3.8) Drake 2017(2.9 < z < 6. Sampling sources at different redshifts allows us to track a possible evolution of the density of galaxies with cosmic time. Four redshift ranges have been considered, 2.9 < z < 4.0, 4.0 < z < 5.0, 5.0 < z < 6.7 and 2.9 < z < 6. The minimization process is performed by the standard package, Lmfit, which makes use of the Levenberg Marquardt algorithm, based on the traditional chi-square method. Results are shown in Figure 3 for the four different redshift intervals mentioned above. The three best fit parameters in the full redshift range are α = -1.48 +0.07 -0.07 , φ * (×10 -4 ) = 14.62 +4.28 -3.53 [Mpc -3 ], L * = 42.69 +0.07 -0.07 [erg s -1 ].

IV.3. Discussions

The capabilities of MUSE and strong gravitational lensing help to reach a faint galaxy population, two orders of magnitude below the faintest obtained by other blank field surveys (see Figure 3), making the present work a good complement to these observations. For the bright end, our LF points are consistent with previous results. The L * parameter is well measured, and is in good agreement with Drake et al. (2017) and with those obtained from de la Vieuville et al. (2019). At the faint end, the present sample has provided new measurements which help to set a better constraint on the LF slope in this region. The evolution of the slope as function of redshift, found by de la Vieuville et al. (2019), has been confirmed. However, in each redshift interval, the slope we find is about 10% less. This result seems to suggest a flatter luminosity function in the faint regime, implying that the contribution of the faint galaxy population may be less than previously expected.

Effects which contribute to the slope of the LF must be thoroughly investigated. One of the important points mentioned earlier; faint sources from our sample are often associated with high magnifications. This also implies that their V max volumes have been reduced, by same amount as defined by the magnification factors. This has been treated properly in the process of building V max for each source, as described in Section IV.1. However, this effect has not been taken into account during the MCMC process for estimating the mean LF points and their statistical uncertainties. The set of flux, completeness and magnification values are randomly generated many times but the same V max is kept during each iteration. V max should be re-calculated each time when a new random magnification value is picked up, and this is computationally expensive. In practice, we increase and decrease all source magnifications by their 1-σ and 2-σ uncertainties respectively, calculate their V max , and then LF points (Figure 4). V max values corresponding to other random magnifications are interpolated/extrapolated from these sampling points. For the bright-end part, log(L) > 41 [erg s -1 ], LF points are essentially the same, their deviations are small compared to their uncertainties. On the contrary, results in the faint-end part display large deviations, as expected: the higher the magnifications used, the lower the V max values and LF points obtained. To further demonstrate this effect, once the magnification of a given source is above 25, we saturate it by setting it equal to 25. The faint LF points significantly decrease accordingly, as shown also in Figure 4. Such an effect is not observed for the bright end, as most of sources have moderate magnifications, and their uncertainties are relatively small. This effect must be properly taken into account when discussing about the evolution of LF.

V. Conclusions

We have presented a study of the LAE luminosity function using a large sample of lensed LAEs available: 530 LAEs in the redshift interval 2.9 < z < 6.7, covering four orders of magnitudes in luminosity (39<log(L)<43 [erg s -1 ]). The construction of LF points from the MUSE Lensing Survey is based on the V max method developed by de la Vieuville et al. (2019), by taking into account the contribution of each source to the density of galaxies. The total co-moving volume of this work is ∼ 52,000 Mpc 3 with a significant contribution, one quarter of the total sources, from the cluster A2744. Four redshift intervals have been considered to track a possible evolution of LF as function of redshift. Our LF points are in good agreement with those obtained from other blank fields in the bright part. The three best fit Schechter parameters are α = -1.48 +0.07 -0.07 , φ * (×10 -4 ) = 14.62 +4. 28 -3.53 [Mpc -3 ], L * = 42.69 +0.07 -0.07 [erg s -1 ] roughly consistent with those reported in de la Vieuville et al. (2019). Faint sources, close to the lens caustic lines, that are highly magnified, play an important role in shaping the LF, in particular in the flattening of the faint-end slope. This implies that large uncertainties on the faint LF points must be treated with great care.

Summary and Conclusions

In this thesis, I have presented a study of LAEs LF using a large data sample of sources detected behind 17 lensing clusters observed with MUSE/VLT. Thanks to the lensing effect, the signal from distant galaxies has been magnified by a factor of 4 to 10 allowing us to blindly detect LAEs without any pre-selection and reaching to the faintest luminosity level compared to the blank field observations. Because of the lensing effect, we usually observed multiple images of the same system. In this case, we have to choose one representative image for each system. The image should have a high signal to noise ratio and be more isolated than the other images of the same system. Finally, 600 LAEs have been selected behind 17 lensing clusters, covering four orders of magnitude in luminosity over 39 < logL[erg s -1 ] < 43, within a redshift interval of 2.9 < z < 6.7. In the high redshift regime where they are one order of the magnitude fainter than those of current blank field surveys, in order to investigate a possible evolution of LF with redshift, and to estimate the associated contribution to the total cosmic reionization budget. Consequently, we probed the LF of LAEs over four redshift intervals. To deal with both the data obtained from lensing fields and from MUSE/VLT datacubes, we have adopted the approach described in DLV 2019, which uses the V max method to compute the LF values. The main idea of the method is the creation a 3D detectability mask for each source in the MUSE cubes in the source plane, followed by an evolution of the cosmological volume obtained by integrating over the unmasked pixels in the 3D source plane mask. While maintaining the core framework, we made several changes to the original 6 Conclusions and perspective futures -6.1 Summary and Conclusions pipeline. These changes extended the previous sample to 17 lensing clusters and better accounted for the lensing magnification. The total co-moving volume in the global redshift range 2.9 < z < 6.7 is ∼ 50'000 Mpc 3 . This value is three times larger compared to the previous work of DLV 2019 that probed the LF of LAEs behind 4 lensing clusters (A1689, A2390, A2667, and A2777). However, it is much smaller than the value reported in blank field surveys. This is due to the lensing magnification effect.

To compute a luminosity function point, we must correct for the completeness of each source, in addition to its Vmax. The completeness value is calculated by injecting the real source profile into a mock image and then counting the successful rate of this procedure. Aiming to include as many sources as possible for the LF computation, we used a 1% completeness threshold to reject faint sources. We computed the LF points in each luminosity bin and redshift range and fitted them with the Schechter function.

The LF points at the faintest luminosity bins in the higher redshift ranges 4.0 < z < 5.0 and 5.0 < z < 6.7 were not included in the fit due to low completeness, and high magnification. They seem to suggest a flattening/turnover at the faintest luminosity regime in these two highest redshift ranges. To account for this, we have introduced a modified where L T is turnover luminosity and m the curvature parameter defining the shape, downward or upward.

From the best-fit values of the Schechter function, we integrated the LF to compute the luminosity density and then converted it to the SFRD using an assuming an intrinsic factor between Lyman alpha and Hα fluxes and case B of the recombination [START_REF] Osterbrock | Astrophysics of gaseous nebulae and active galactic nuclei[END_REF]. The results showed that the LAEs contribute significantly to the reionization budget. At the critical value of photon emission rate per unit cosmological comoving volume, we estimate that the escape fraction of Lyman alpha photons is about 8% associated with a typical value of clumpy factor of 3.

In summary, we have presented a study on the LAEs LF using an extensive dataset consisting of 17 lensing clusters from the Lensed Lyman Alpha MUSE Arcs Sample observed with MUSE/VLT. From this dataset, we have selected 600 lensed LAEs behind these clusters within a redshift range of 2.9 < z < 6.7 with a wide range of luminosity spanning four orders of magnitude 39.0 < log L [erg s -1 ] < 44.0, and effectively probing the faint luminosity regime, reaching down to 10 40 [erg s -1 ]. This data sample sets a strong constraint on the LF for faint luminosities and provides insight into the evolution of the slope as a function of redshift. The key outcome and conclusions are presented as follows:

-Regarding the methodology and processing pipeline, we have thoroughly investigated various effects, leading to improvements in the final results compared to the work of DLV 2019. We extended the previous sample to 17 lensing clusters and better accounted for the lensing magnification during the Vmax computation process.

-The evolution of LF with redshift has been investigated in four redshift intervals: 2.9 < z < 6.7, 2.9 < z < 4.0, 4.0 < z < 5.0, and 5.0 < z < 6.7. The total co-moving volume of the survey is ∼ 50000 Mpc 3 , largely dominated by A2744, as presented in DLV 2019.

-The best-fit value of the three parameters of the Schechter function in the redshift range of 2.9 < z < 6.7 are: α = -2.06 +0.07 -0.05 , Φ * [10 -4 Mpc 3 ] = 7.41 +2.70 -2.70 , log L * [erg s -1 ] = 42.85 +0.10 -0.10 . -Toward bright luminosity, log L[erg s -1 ] > 42, the present LF values are consistent with the results of the previous studies which is used the data from the MUSE-Wide observations, as well as with other published results using different observational techniques.

-Toward faint luminosity, the LF values are dominated by highly magnified and low completeness sources as expected. The source density is described by a steep slope of α ∼ -2, obtained from both linear fitting and fitting using the Schechter function. Our results are consistent with the slope values obtained by Drake et al. 2017 for all redshift intervals, by Herenz et al. 2019 for 2.9 < z < 6.7, and 20% steeper than those obtained by DLV 2019. Taking into account various factors, such as different completeness thresholds, different flux measurements, varying fitting models, and source selection, the faint end slopes for four redshift ranges are -2.00 ± 0.50, -1.97 ± 0.50, -2.28 ± 0.50 and -2.06±0.60. These values are consistent with other works within a 1-σ deviation.

-The faint end slope of the LF steepens with redshift, consistent with DLV19, although the uncertainties are large. There may be a turnover at luminosities fainter than log L ∼ 10 41 [erg s -1 ] for the two highest redshift intervals, yet the uncertainties remain large.

-The choice of the luminosity lower limit for the LF integration affects the SFRD value. The increase of SFRD with redshift implies that the LAEs play a major role for the cosmic reionization.

-The ionizing flux from LAEs seems to be sufficient to maintain the hydrogen ionized. At redshift z ∼ 6, this value is comparable to the one provided by LBGs.

Future Perspectives

Luminosity function of line emissions observed with MUSE

The Lensed Lyman Alpha MUSE Arc Sample (LLAMAS) has observed 25 massive clusters with an observing time of ∼150 hours in total. It is expected to find more than 700 LAEs. This will be the largest data sample of lensed LAEs collected by MUSE/VLT at a redshift range 2.9 < z < 6.7. The data from the project yields valuable insights into the luminosity distribution, enhancing the statistical representation of sources in the faintest luminosity regime. In addition, the MUSE cube also includes information on other emission lines, such as Hα, OII, with good SNR. Using the same present method, we can look for the correlation between these emission lines and the total SFR. This will give us a unique opportunity to extend the results obtained in this thesis and improve our understanding of the LF and SFRD at a lower redshift, as well as their evolution with redshift.

Luminosity function using data from JWST and Euclid missions

Nearly 2 years since its launch, JWST has shown its power in collecting data at the EoR. Its observations of several gravitationally lensed clusters with its exquisitely sensitive NIR instruments (NIRSpec, NIRCam, NIRRISS) have provided the strongest constraint at the faintest end of the UV LFs. The galaxy density in such regimes can be estimated from the LF, suggesting that these galaxies may have played a significant role in the cosmic reionization. There are many first galaxies in the redshift range 7 < z < 15 that have been confirmed on either Lyman alpha line emission with NIRSpec or the Gunn-Peterson Trough with deep photometry. I will pursue this topic by using the JWST data to study the evolution of the LF at the higher redshifts compared to the MUSE data. Another opportunity is using Euclid data. Euclid is a space-based visible and nearinfrared observatory with a primary mirror of 1.2 metres in diameter. The Euclid Deep Survey data will cover a region of 40 deg 2 with a limiting flux ∼ 5.10 -17 erg s -1 cm -2 in the wavelength range of 920-1850 nm using the Euclid's blue and red grisms [START_REF] Costille | Modeling effects of common molecular contaminants on the Euclid infrared detectors[END_REF]. This survey would be a great opportunity to study the Lyman alpha population at redshift z ∼ 6.5 -9 by constructing a blind flux-limited sample without any preselection. I will re-use and adapt some of the tools I have developed for this thesis to the Euclid survey to compute the LF of LAEs towards the bright end of the LF

Global escape fraction of Lyman alpha photons as a function of redshift

Using our LF of LAEs combined with the UV LF of all galaxies from R. J. Bouwens, Illingworth, [START_REF] Bouwens | UV Luminosity Functions at Redshifts z ∼ 4 to z ∼ 10: 10,000 Galaxies from HST Legacy Fields[END_REF], we can estimate the global evolution of the escape fraction of Lyman alpha photons from 2.9 to the highest redshift in which we can find the galaxy based on JWST observations.

The ionizing photon production efficiency for LAEs using JWST and MUSE

The excellent combination of JWST and MUSE will open a new window to infer the ionizing photon budget of the Universe using a large data sample of LAEs. I will estimate this parameter's value in two cases: one without accounting for dust attenuation and another with dust attenuation corrections, using the assumption of a Calzetti attenuation law. This estimation will be based on the present MUSE data and the JWST Advanced Deep Extragalactic Survey.

Figure 1 :

 1 Figure 1: Comparison of the present data sample (magenta) to the previous sample used in the DLV 2019 (cyan). The central panel shows luminosity against the redshift, while the top panel displays the redshift distribution, and the right panel illustrates the luminosity distribution.

Figure 2 :

 2 Figure 2: Detection flux and noise level in the sample versus redshift. The left axis displays the detected flux of individual sources, the right axis gives the noise level in wavelength channels, the lower horizontal axis is redshift and the upper one shows the completeness scale. The gray circles represent the evolution of noise level with redshift from the observation through the spectral dimension of the MUSE cube.

16 Flux [erg cm - 2 s - 1 ]Figure 4 :

 16214 Figure 4: Completeness vs. detection flux of LAEs from the present sample. Different colors indicate the quality of extraction from SExtractor. Only sources in the unmasked regions of the detection layer are considered here.

7 )

 7 Schechter fit (α = -2.06) This work This work (Schechter fitting not included)

  2019(4.0 < z < 5.0) Modified Schechter (LT = 40.0) Schechter fit (α = -1.97) This work This work (Schechter fitting not included) z = 5.2, 5.8) de La Vieuville 2019(5.0 < z < 6.7) Modified Schechter (LT = 40.7) Schechter fit (α = -2.28) This work This work (Schechter fitting not included)

Φ

  (L)exp(-L T /L) m = L/L * )exp((-L T /L) m ) (0.3)

Figure 6 :

 6 Figure 6: Comparision between the luminosity function points at the highest redshift interval 5.0 < z < 6.7 and the AMIGA, G21. The pair of red crosses located in the brightest luminosity bins are average values from the literature. The blue and black solid lines are predicted from AMIGA models with respect to the single and double ionization scenarios (Salvador-Solé et al. 2022). The color dashed lines are predicted from SPHINX simulations (Garel et al. 2021) for the intrinsic luminosity value (magenta), correcting for the dust attenuation (blue) and IGM (orange).

Figure 7 :

 7 Figure 7: Evolution of the slope value with redshift. The error bars in the horizontal axis correspond to the redshift range of the survey.

- 1 .Mpc - 3 ]

 13 fescp = 3%, Clumping factor = 3 fescp = 8%, Clumping factor = 3 fescp = 25%, Clumping factor = 3 This work (Linf = 41.0) This work (Linf = 39.5) Cassata(2011) (Linf = 41.3) Drake(2017) (Linf = 41)

FIGURE 1 :

 1 FIGURE 1 : Comparaison entre l'échantillon de données utilisées dans ce travail (magenta) et les données précédentes utilisées dans DLV 2019. Le figure au centre présente la luminosité par rapport au décalage vers le rouge. La figure du haut est la distribution du décalage vers le rouge. La figure de droite est la distribution de luminosité.

FIGURE 2 :

 2 FIGURE 2 : Figure présentant le niveau signal à bruit des objets de l'échantillon en fonction du redshift. L'axe de gauche affiche le flux détecté, l'axe de droite est le niveau de bruit, et l'échelle horizontal supérieure donne la valeur de complétude des sources individuelles. Les cercles gris représentent l'évolution du niveau de bruit avec le décalage vers le rouge à partir de l'observation à travers la dimension spectrale du cube de MUSE.

FIGURE 3 :

 3 FIGURE 3 : Distribution des sources dans l'intervalle de redshift 2.9 < z < 6.7 pour une largeur du l'intervalle de luminosité de 0.25 en échelle logarithmique. Les lignes bleues et orange représentent respectivement la distribution avant et après correction de la complétude.

  2019(4.0 < z < 5.0) Modified Schechter (LT = 40.0) Schechter fit (α = -1.97) This work This work (Schechter fitting not included)

FIGURE 6 :

 6 FIGURE 6 : Comparaison entre les points de fonction de luminosité obtenus dans l'intervalle de décalage vers le rouge le plus élevé 5.0 < z < 6.7, représentés par des croix rouges. Les lignes pleines bleues et noires sont des prédictions à partir des modèles AMIGA en ce qui concerne les scénarios de réionisation simple et double (SALVADOR-SOLÉ et al. 2022). Les lignes en pointillés de couleur sont les prédictions à partir des simulations SPHINX (GAREL et al. 2021) pour la valeur de luminosité intrinsèque (magenta), et en corrigeant l'atténuation de la poussière (bleu) et l'IGM (orange).

FIGURE 7 :

 7 FIGURE 7 : Evolution de la valeur de la pente avec le décalage vers le rouge. La barre d'erreur sur l'axe horizontal correspond à la plage de redshift du relevé.

TABLEAU 3 : 10 - 1 .

 3101 Valeur de la pente de meilleur ajustement de la fonction de Schechter pour différentes sélections de données. 46 ± 0.05 1% completeness and using the flux values obtained from source extractor -2.10 ± 0.06 -1.97 ± 0.08 -2.24 ± 0.11 -1.82 ± 0.03 1% completeness and including all the zcon f = 1 source -1.83 ± 0.17

- 1 .Mpc - 3 ]FIGURE 8 :

 138 FIGURE 8 : Evolution de la densité du taux de formation d'étoiles avec le décalage vers le rouge. Les croix rouges et marron présentent les résultats lorsque les limites inférieures d'intégration de log L [erg s -1 ] = 39.5 et log L [erg s -1 ] = 41.0 sont appliquées, respectivement. Les lignes pleines bleues, orange et vertes sont les valeurs critiques obtenues avec une valeur de fraction d'échappement de 3%, 8% et 25% combinée à un facteur d'agglutination (clumping factor) typique de 3.

Hình 1 :

 1 Hình giữa mô tả sự phân bố của độ trưng theo giá trị độ dịch chuyển đỏ, hình bên trên hiển thị sự phân bố giá trị dịch chuyển đỏ, hình bên phải hiển thị phân bố giá trị độ trưng. Phần màu tím hiển thị dữ liệu được sử dụng trong luận án. Phần màu xanh là dữ liệu so sánh từ bài báo DLV 2019 So với dữ liệu nghiên cứu trước đây của DLV 2019 khi nghiên cứu các nguồn phát xạ vạch Lyman alpha phía sau bốn cụm thiên hà thấu kính hấp dẫn, số lượng nguồn trong nghiên cứu hiện tại nhiều gấp bốn lần. So sánh giá trị độ trưng ở những giới hạn mờ nhất logL[ergs -1 ] ∼ 40 cho thấy số lượng nguồn trong nghiên cứu hiện tại nhiều hơn đáng kể so với nghiên cứu trước đó. Vì vậy, dữ liệu được sử dụng trong luận án giúp cải thiện đáng kể về phân bố độ trưng ở những khoảng giới hạn mờ nhất, đồng thời cũng cải thiện thống kê của ở vùng này. Hình 1 so sánh dữ liệu hiện tại và dữ liệu trước đó được sử dụng trong DLV 2019. Hình 2: Thông lượng đo đạc và nhiễu theo dịch chuyển đỏ của các nguồn LAEs trong mẫu khảo sát hiện tại. Trục bên trái hiển thị thông lượng được phát hiện, trục bên phải hiển thị mức nhiễu và thang ngang phía trên cung cấp giá trị hoàn chỉnh cho các nguồn riêng lẻ. Các vòng tròn xám biểu thị sự phụ thuộc của nhiễu theo dịch chuyển đỏ trong khối dữ liệu dạng datacube của MUSE.

  2019(4.0 < z < 5.0) Modified Schechter (LT = 40.0) Schechter fit (α = -1.97) This work This work (Schechter fitting not included)

  vẫn lớn để có thể khẳng định chắc chắn. Giá trị khớp hàm L * và Φ * trong phạm vi sai số phù hợp với các kết quả nghiên cứu trước đó. Ở Hình 5, các dấu chữ thập màu xám không được sử dụng cho quá trình khớp hàm Schechter do đa số các nguồn trong khoảng độ trưng này có độ phóng đại lớn, giá trị hệ số hoàn chỉnh nhỏ. Có điểm đáng chú ý, hàm độ trưng ở những khoảng dịch chuyển đỏ lớn hơn 4.0 < z < 5.0 và 5.0 < z < 6.7 đã cho thấy sự sụt giảm ở những khoảng giới hạn mờ nhất, gợi ý một thay đổi quan trọng, sự quay đầu của hàm độ trưng . Điều này đã được thảo luận trong các nghiên cứu trước đó về hàm độ trưng UV Atek 2018, Bouwens et al. 2022. Để tính tới sự thay đổi này của hàm độ trưng, chúng tôi đã sửa đổi hàm Schechter bằng cách thêm một phần hàm mũ mô tả sự "quay đầu"dạng: Φ(L)exp(-L T /L) m = L/L * )exp(-L T /L) m (0.9) Trong đó L T là giá trị mà tại đó d φ/d L = 0, hay nói một cách khác đây là giá trị mà tại đó hàm Schechter bắt đầu chuyển hướng, m là chỉ số quay đầu hiển thị xu hướng của hàm, m > 0 hàm có xu hướng đi xuống, m < 0 hàm có xu hướng đi lên. Hình dáng của hàm được hiển thị Hình 5 (dưới). Kết quả cho thấy giá trị của m=1 và L T bằng 10 40 và 10 40.7 erg s -1 tương ứng với hai khoảng dịch chuyển đỏ 4.0 < z < 5.0 và với 5.0 < z < 6.7.

Hình 6 :

 6 So sánh giá trị hàm độ trưng trong khoảng dich chuyển đỏ lớn nhất 5.0 < z < 6.7. Hai điểm chữ thập màu đỏ nằm trong vùng có độ trưng lớn nhất là giá trị lấy trung bình từ các kết quả trước đó. Đường nét liền màu xanh và màu đen là kết quả dự đoán từ mô hình AMIGA ứng với quá trình ion hóa đơn và ion hóa kép ( Salvador-Solé et al. 2022). Đường nét đứt hiển thị kết quả dự đoán sự tiến triển giá trị hàm độ trưng từ mô phỏng SPHINX ( Garel et al. 2021) khi tính đến giá trị nội tại của hàm độ trưng (đường màu tím), hiệu chỉnh bởi bụi (đường màu xanh), hiệu chỉnh bởi môi trường giữa các thiên hà (đường màu vàng).

- 1 .Mpc - 3 ]Hình 7 : 1 Introduction 1 . 1

 137111 fescp = 3%, Clumping factor = 3 fescp = 8%, Clumping factor = 3 fescp = 25%, Clumping factor = 3 This work (Linf = 41.0) This work (Linf = 39.5) Cassata(2011) (Linf = 41.3) Drake(2017) (Linf = 41) Tiến triển mật độ hình thành sao theo dịch chuyển đỏ. Dấu thập màu đỏ ứng với giới hạn dưới logL[ergs -1 ] = 39.5, dấu thập đỏ nâu ứng với giới hạn dưới logL[ergs -1 ] = 41.0 trong quá trình lấy tích phân đường của hàm độ trưng. Đường nét liền màu xanh nước biển, vàng, da lá cây ứng với giới hạn mật độ hình thành sao thu được khi giả sử tỉ lệ thoát của các photon ion hóa lần lượt là 3%, 8% và 25% , hệ số cụm (clump) bằng 3 áp dụng cho tất cả các giả thuyết. Chương 6 trình bày thảo luận chung về kết quả cuối cùng và triển vọng tương lai. Tôi hy vọng sẽ tiếp tục công việc nghiên cứu sự tiến triển hàm độ trưng theo dịch chuyển đỏ của các nguồn phát xạ vạch Lyman alpha bằng cách sử dụng dữ liệu được thu thập phía sau 25 cụm thấu kính thuộc chương trình MUSE GTO hoặc đã có trên kho lưu trữ của ESO. Trong quá trình thực hiện luận án, bên cạnh việc xác định giá trị dịch chuyển đỏ của các nguồn phát xạ vach Lyman alpha, danh mục các nguồn phát xạ vạch Balmerr, OII-3727 cũng đã được thiết lập. Do đó tôi sẽ nghiên cứu sự tiến triển hàm độ trưng của các nguồn phát xạ những vạch này nhưng ở khoảng dịch chuyển đỏ thấp hơn. Tôi cũng đã lên kế hoạch làm việc với dữ liệu của JWST và Euclid để tìm kiếm các nguồn phát xạ vạch Lyman alpha ở những khoảng dịch chuyển đỏ lớn hơn. Những kết quả nghiên cứu này cho phép xác định sự tiến triển toàn cục (global evolution) của các nguồn tái ion hóa theo hàm dịch chuyển đỏ. Sự kết hợp tuyệt vời giữa JWST và MUSE cũng mang đến cơ hội phân tích quá trình tạo ra các photon ion hóa với hiệu quả cao. The first structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Circum galactic medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Epoch of re-ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Recombination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Spectral features of star-forming galaxies . . . . . . . . . . . . . . . . . . 1.5.1 The Lyman alpha line . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 The Lyman Break galaxies . . . . . . . . . . . . . . . . . . . . . . . 1.5.3 Lyα forest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.4 The Gunn -Peterson effect . . . . . . . . . . . . . . . . . . . . . . 1.6 Lyman Break Galaxy (LBG) technique and photometric redshifts . . . . 1.7 Selection of Lyman Alpha Emitter (LAE)s: narrow-band technique . . . 1.8 Selection of LAEs: Integral Field Units . . . . . . . . . . . . . . . . . . . . 1.9 The galaxy luminosity function . . . . . . . . . . . . . . . . . . . . . . . . 1.10 Overview of this work . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 Luminosity function 4 . 1

 41 Completeness determination . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Reconstruction of the source profile in image plane . . . . . . . 4.1.2 Source recovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Luminosity Function (LF) computation . . . . . . . . . . . . . . . . . . . 4.2.1 Binning effect on the LF points . . . . . . . . . . . . . . . . . . . . 4.2.2 Effect of source selection to the evolution of LF points . . . . . . 4.3 LF results and comparison to the literature . . . . . . . . . . . . . . . . . 4.4 Fitting with of a Schechter function . . . . . . . . . . . . . . . . . . . . . 4.5 Comparison with theoretical predictions . . . . . . . . . . . . . . . . . . 5 Star formation rate density and implications for the reionization 5.1 Star formation rate density . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Articles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  12 and completed at redshift z ∼ 6 owing to observations of the Gunn-Peterson trough from measured quasars' spectra (Becker et al. 2001; Fan et al. 2006) (see Figure 1.1).

Figure 1 . 1 :

 11 Figure 1.1: A brief history of the Universe from the Big Bang. The horizontal axis traces back time (top). The evolution phases of the Universe are shown from the time when the matter was fully ionized right after the Big Bang; after 380'000 years the recombination happened, at a redshift of 1100; after a few hundred million years the formation of the first structures (first stars, and galaxies) at the redshift of ∼ 10, marked the end of the dark ages; the re-ionization was completed at redshift ∼ 6. Credit: NAOJ.

Figure 1 . 2 :

 12 Figure 1.2: Four categories of morphology galaxies. https://sites.ualberta.ca/ ~pogosyan/teaching/ASTRO_122/lect23/lecture23.html

  methods included smoothed particle hydrodynamics (Ford et al. 2013; Oppenheimer et al. 2016), adaptive mesh refinement (Hummels et al. 2013), moving mesh (Suresh et al. 2016), etc.

1

  Figure 1.3: A schematic of the CGM showing outflow and recycle gas. The center volume and the gas disk are fed by the filamentary accretion from the Intergalactic Medium (IGM) (blue). The outflows have been marked pink and orange colors. The gas that was rejected previously is being recycled. The diffuse gas halo is marked by purple, its probability is contributed by these sources and mixed together over the time. Credit: Tumlinson et al. 2017.

Figure 1 . 4 :

 14 Figure 1.4: Lyman-alpha and other lines are shown together with the Lyman and Balmer series of hydrogen atom (left, credit: https://www.daviddarling. info/encyclopedia/L/Lyman-alpha.html).

Figure 1 . 5 ,

 15 from Ouchi 2020, illustrates the conceptual figure of an LAE.

Figure 1 . 5 :

 15 Figure 1.5: Conceptual figure of an LAE, located at the node of the filament (panel b). The zoom-in of an LAE is shown in panel d). Spectra of light paths 1, 2, and 3 are shown in panels c) and e). Credit: Ouchi 2020.

Figure 1 . 6 :

 16 Figure 1.6: An LAE spectrum obtained from Multi Unit Spectroscopic Explorer/ Very Large Telescope (MUSE/VLT).

Figure 1 . 7 :

 17 Figure 1.7: Left: The evolution of Lyman-alpha EW of instantaneous burst (solid lines) and constant star formation (dashed lines). Credit: D et al. 2008. Right: Fraction of Lyman-alpha emitting galaxies having EW > 25 A • as a function of redshift. Credit: Ouchi 2020 (Figure 18)

Figure 1 . 8 :

 18 Figure 1.8: Mean spectra of LAEs with Lyα EW < 97.2 and > 97.2 (left and right, respectively). The median values of the Lyα EW for the two subsamples are 54.7 and 163.1, respectively. Credit: Anna Feltre et al. 2020.

Figure 1 . 9 :

 19 Figure 1.9: A spectrum at rest frame of a star-forming galaxy at redshift z ∼ 3 ( A. E. Shapley et al. 2006) in the wavelength range of (800 -1500 Å). The break of the UV continuum flux can be seen in the interval of 912 -1215 Å.

Figure 1 . 10 :

 110 Figure 1.10: Observed spectral energy distributions of a quasar at redshift z ∼ 3.6 in the rest-frame. The series of absorption lines (Lyα forest) due to the presence of IGM neutral hydrogen clouds at different redshifts along the line of sight, is clearly seen. The appearance of hundreds of absorption lines in the measured spectra is then referred as Lyα forest. Credit: http: //www.astr.ua.edu/keel/agn/forest.html.

  al. 2006; McGreer et al. 2014). This feature is called a Gunn-Peterson trough from the very first observation made in Gunn et al. 1965. The existence of this effect is shown in Fig. 1.11 as the very latest observation on the quasar spectrum at redshift z = 6.13 showing a clear Gunn-Peterson trough obtained by Becker 2015 and others observations (see Fig. 1.12).

Figure 1 . 11 :

 111 Figure 1.11: Spectrum of a quasar at redshift z = 6.13 (ULAS J1319+0959) taken from Becker 2015 simulation for the existence of Lyα forest and Gunn-Peterson effect.

1

  Figure 1.12: Measured quasar's spectra at different redshifts. The Gunn -Peterson trough is observed for quasar spectra having z > 6. Credit: Fan et al. 2006.

Figure 1 . 14 :

 114 Figure 1.14: Lyman break technique method. Upper panel: SED model of a starforming galaxy at redshift z ∼ 3 is shown together with three broadband filters U, G, R. Lower panel: Galaxy appearances at different filters. The galaxy is detected both in R and G bands but disappears in the U band. Credit: Burgarella 2006.

Figure 1 . 15 :

 115 Figure 1.15: Upper panel: HST filters show a drop-out of a galaxy at short wavelength filters (Lotz et al. 2017). Lower panel: SED fitting technique using HST photometry. Credit: Vieuville 2019.

1 Introduction - 1 . 8

 18 Selection of LAEs: Integral Field Units to study their physical properties in some detail. They are often found to be very compact, with a typical effective radius of 0.6 ± 0.1 kpc at z ∼ 6 (Dow-Hygelund et al. 2007). They have low stellar masses ranging from 10 8 to 10 9 M ⊙ at z ∼ 3, and from 10 6 to 10 10 M ⊙ at z ∼ 5[START_REF] Gawiser | The Physical Nature of Lyα-emitting Galaxies at z=3.1[END_REF]Pirzkal et al. 2007). They host young stellar populations (< 90 Myrs at z ∼ 3) and have moderate star formation rates. Their EW can reach 200 Å(Shimasaku et al. 2006).

Figure 1 . 16 :

 116 Figure 1.16: Illustration of LAE detection at redshift z = 6.96 using the NB technique.In this case, the Lyα emission line is detected using the NB973 filter, but not using the other 5 broadband filters (B, V, R, i', and z') blueward of the line. Credit:[START_REF] Iye | A galaxy at a redshift z = 6.96[END_REF]

  3) as shown by Deharveng et al. 2008, a moderate evolution at intermediate redshifts (z = 3 -6) as shown by Ouchi et al. 2008, and a sudden and strong decrease beyond z ∼ 6 as shown by Kashikawa et al. 2011. At redshifts in the z = 2.2 -3.3 interval, at high luminosities L Lyα > 10 43.3 erg s -1 , Spinoso et al. 2020 measured α = -1.35 ± 0.84 from observations of the Javalambre Photometric Local Universe Survey (J-PLUS) covering a region of ∼ 1000 deg 2 . At redshift z = 2.0 -3.5, and L Lyα = 10 43.

Figure 1 . 17 :

 117 Figure 1.17: Evolution of Lyα LF from z = 0.3 to z = 5.7 (left) and from z = 5.7 to z = 7.3 (right). The curves are the best-fit Schechter functions obtained by L. L. Cowie et al. 2010; Akira Konno et al. 2016; Ouchi et al. 2008; Konno et al. 2018; Itoh et al. 2018

Figure 2 . 1 :

 21 Figure 2.1: The orange glow light of a Laser Guide Star emanates from one of the Very Large Telescope with details of the Milky Way revealed. Credit: ESO/A.Ghizzi Panizza (www.albertoghizzipanizza.com).

Figure 2 . 2 :

 22 Figure 2.2: The MUSE cube structure with 2 spatial dimensions and one wavelength dimension. Credit: Vieuville 2019.

Figure 2 . 3 :

 23 Figure 2.3: One of the recent images of hydrogen filaments (blue) observed by MUSE in the Hubble Ultra Deep Field. (Bacon et al. 2021).

Figure 2 . 4 :

 24 Figure 2.4: Top: The spatial distribution of sources with a secure redshift overlaid on an RGB HST image within the MUSE mosaic observation in the core of lensing cluster A2744. The lower panel shows the redshift distribution of the sources shown in the upper panel. Credit: Mahler et al. 2018.

Figure 2 . 5 :

 25 Figure 2.5: An example of finding extended Lyα haloes around high z SFGs. The first column is the HST image of LAE denoted by its contour on the HST segmentation mask. The second column is MUSE white light image of the LAE. The third column is the Lyα line extracted from the HST segmentation mask convolved with the MUSE Point Spread Function (PSF). The fourth column is NB image of the LAE with surface brightness contour at 10 -17 erg s -1 cm -2 arcsec -2 (central dotted white). The last column is the radial Surface Brightness (SB) profiles of Lyα emission (blue), UV continuum (green), and the PSF (red). Credit: Leclercq et al. 2017.

Figure 2 . 6 :

 26 Figure 2.6: The spatial distribution of secure sources behind A2390 detected by MUSE is limited by the area inside the solid white line.

  ; Govoni, Enßlin, et al. 2001; Govoni, Feretti, et al. 2001). It also has been known for its complex lensing properties through a combination of strong and weak lensing (Merten et al. 2011; Jauzac et al. 2016). This cluster was observed with MUSE under the GTO program (094.A-0115) (PI: Richard) with a 2×2 mosaic of MUSE pointings aiming to cover all the multiple image area, i.e. center at α = 00 h 14 m 20.952 s and δ = -30 o 23 ′ 53.88 " . A simulation produced for the MUSE proposal to observe A2744 overlaid by the HST selection -2.3 Lensing clusters observed with MUSE/VLT observation region is shown in Fig. 2.7. The quadrants observing times are 3.5, 4.0, 4.0, and 5 hours, and an extra observation time of 2 hours at the central cluster. Each pointing is divided into 30-minute individual exposures with a 90-degree rotation applied in between each exposure.

Figure 2 . 7 :

 27 Figure 2.7: A simulation produced for MUSE proposal observation coverage of the first Hubble Frontier Field Abell 2744. The high light red area shows a deep HST coverage with ACS and WFC3. The cyan line delineates the critical line at redshift z ∼ 7.

  ;[START_REF] Soucail | A blue ring-like structure in the center of the A 370 cluster of galaxies[END_REF][START_REF] Patrício | Kinematics, turbulence, and star formation of z 1 strongly lensed galaxies seen with MUSE[END_REF], including a very high boost zone (µ ∼ 5 -10) twice as large as any other Frontier Fields (FF) cluster observation[START_REF] Richard | Mass and magnification maps for the Hubble Space Telescope Frontier Fields clusters: implications for high-redshift studies[END_REF]. A preliminary catalog of background sources behind this cluster observed from the Grism Lens Amplified Survey from Space (GLASS) was reported in Treu et al. 2015. The observation was conducted with a large mosaic covering ∼ 4 arcmin 2 centered on the cluster, covering almost multiple image areas (096A-0710, 094A-0115). Aiming to achieve relatively uniform line sensitivity in the presence of strong intracluster light caused the background to grow for the line searching up to a factor of 2 at the center. For this reason, we use a factor of 3 to 4 larger exposure time in the central portion of the mosaic. This was done with 2 hours on-source (one-fourth of the MUSE image), which is rotated by 28 degrees to cover the bulk of the HST WFC ∼ 5 arcmin 2 footmark, 4 hours on-source spending for the two pointings centered at North-South high boost region, and 2 additional hours on-source focusing in the center. Finally, we have 18-hour observation for the A370 in mosaic mode. An exposure map is shown in Fig.2.8. Both observations were taken in clear and photometric conditions with a DIMM seeing in a range of 0.37" and 1.09", a median airmass of 1.17 within a range of 1.09-1.42(Lagattuta et al. 2019).

Figure 2 .

 2 Figure 2.8: A MUSE exposure map of A370 mosaic observation. The solid black curve delineates the multiple-image zone (Lagattuta et al. 2019).

  is 6:1 (Mastropietro et al. 2008) and displays a prominent bow shock preceding a cool bullet lying between the two. The first lensed model of this cluster was built by Mehlert et al. 2001 using 3 modeled Dark matter (DM) clumps as Single Isothermal Sphere (SIS) and Singular Isothermal Ellipsoid (SIE) based on 6 multiple imaged systems, one of them showing a spectroscopic redshift of z = 3.24. Bradač et al. 2006 improved the model by using 6 multiple imaged systems from Mehlert et al. 2001 combined with 4 new multiple imaged systems detected in the sub-cluster region. The new mass model helped to measure an enclosed mass at r > 250 kpc (∼ 2.8 ± 0.210 14 M ⊙ ) around the main cluster and 2.3 ± 0.2 10 14 M ⊙ around the sub-cluster. Paraficz et al. 2016 reconstructed the mass model by including the intra-cluster gas derived from X-ray observation, which is one with the best Bayess evidence. The new model showed a total rms value of 0.158" of imaged position difference between the predicted and observed positions and confirmed the spatial offset between X-ray mass and dark matter peaks. The fraction mass of the galaxy halo is assumed 2.5 ± 0.110 14 M ⊙ within a 250 kpc radial aperture. With the MUSE observation, this cluster is observed with an Extended mode (WFM-NOAO-E) covering a bluer wavelength. Fig. 2.10 is a Bullet image observed by HST using different filters. Using different mass models helps us find multiple images of the background sources.

  2.11 displays a MUSE Field of View (FoV) overlaid by the HST observation. The dashed cyan line delineates the FoV of MUSE observation, the solid curve highlights the multiple image regions and the solid red line shows the critical curve at redshift z = 4.

Figure 2 . 9 :

 29 Figure 2.9: The red-green-blue HST image of the AS1063 is observed with MUSE. The green square delimits 1 ′ × 1 ′ MUSE FoV. The white circles and accompanying text show a redshift of the source observed by MUSE. The blue, magenta, and red contours refer to the high magnification region µ > 200 at redshift z = 1, 2 and 4, respectively. Credit: Karman et al. 2015.

Figure 2 .

 2 Figure 2.10: The HST observations using different filters of Bullet cluster. The color circles show the multiple images from (Mehlert et al. 2001; Gonzalez et al. 2010), (Bradač 2009) in white squares, (Paraficz et al. 2016) in red circles. The white solid line displays a critical curve at redshift z ∼ 3.24. Credit: Paraficz et al. 2016.

Figure 2 . 11 :

 211 Figure 2.11: An overview of multiple image systems behind MACS0257. The dashed cyan line delineates the FoV of MUSE observation, the solid white line highlights the multiple image regions and the solid red line shows the critical curve at redshift z = 4. Credit: Richard 2021.

2MACS0416Figure 2 . 12 :

 212 Figure 2.12: Source location of MUSE spectroscopic redshifts over HST F606W-F814W-F160W color composites for MACS0416 (top) and MACS0329 (bottom). The cyan dashed lines represent the MUSE observation regions, while the solid white line denotes the multiple imaged area. Credit: Richard 2021.

Figure 2 . 13 :

 213 Figure 2.13: An overview of multiple image systems behind MACS0940. The dashed cyan line delineates the field of view of MUSE observation, the solid curve highlights the multiple image regions, and the solid red line shows the critical curve at redshift z = 4. Credit: Richard 2021.

Figure 2 .

 2 Figure 2.14: An overview of multiple image systems behind MACS1206. The dashed cyan line delineates the field of view of MUSE observations, the solid white curve highlights the multiple image regions, and the solid red line shows the critical curve at redshift z = 4. Credit: Richard 2021.

  2.13. -MACS1206: It was first discovered in the ROSAT All Sky Survey (Böhringer et al. 2001) at redshift z = 0.44. As part of the CLASH project, the mass distribution inside the cluster has been investigated using different techniques, such as exploiting the multiple images created by lensing (Zitrin 2012; Eichner et al. 2013), a combination of strong lensing and weak lensing to obtain information about mass distribution (Umetsu et al. 2012), etc.There is one multiple-image system with a giant arc with respect to its counter-image that was found at redshift z = 1.036[START_REF] Ebeling | A spectacular giant arc in the massive cluster lens MACS J1206.20847*[END_REF]. In this work, we are using data observed from GTO program 095.A-0181(A) and 097.A-0267(A) (PI: J. Richard) in April, May 2015 and April 2016, including 3 pointings, one centered on BCG with a rotation of 20 degrees and the others with an offset of 35 arcsec towards the East and West. The exposure time mode of this cluster is 26×1800 NOAO. An offset of 0.5" combined with a rotational angle of 90 degrees was applied in the different exposures to improve the removal of instrumental signatures and sky subtraction. There is a matching of redshift and multiple image systems between the catalog of[START_REF] Caminha | A refined mass distribution of the cluster MACS J0416.1-2403 from a new large set of spectroscopic multiply lensed sources[END_REF]) and[START_REF] Richard | An atlas of MUSE observations towards twelve massive lensing clusters[END_REF]). The latter found eight new systems more[START_REF] Richard | An atlas of MUSE observations towards twelve massive lensing clusters[END_REF]). An overview of multiple imaged systems behind this cluster is shown in Fig.2.14. Credit: Richard 2021.-MACS2214 was found from Chandra ACIS-I observation as one of the X-ray sources[START_REF] Ebeling | A spectacular giant arc in the massive cluster lens MACS J1206.20847*[END_REF]). It consists of several prominent large blue arcs[START_REF] Zitrin | Full lensing analysis of Abell 1703: comparison of independent lens-modelling techniques[END_REF], four of which are considered to be spiral systems. This cluster is quite close to the ecliptic plane (∼ 2 degrees) allowing us to observe the passing of an asteroid (see Fig. 2.15 (right)). MUSE observed this lensing cluster from 2017 to 2019. During the first exposure, the passing of an object was observed in the field. This was seen again in the last poses in 2019 during the observations of the lensing cluster Claeyssens 2021.

Figure 2 .

 2 Figure 2.15: Left: MACS2214 observed by VLT/MUSE. The cyan dashed line shows the MUSE field of view, the white dashed line denotes the multiple images region, and the red solid line is the critical curve at redshift z = 4. Right: 9 individual exposures of the MACS2214 show the appearance of a satellite crossing the field. Credit: Claeyssens 2021.

  2.16. Credit: Richard 2021. -SMACS2031 (α = 20 : 31 : 47.843, δ = -40 : 36 : 54.76, z = 0.331) was observed with the HST as part of snapshot programs 12884 and 12166 (PI: Ebeling). This cluster has been observed as a part of the MUSE commission (60.A-9100B) with a total observing time of 10.4 hours in the nominal mode (WFM-NOAO-N), focusing on the SE cluster to cover the predicted locus of the multiple images with a small random dithering pattern of a 0.6" box. A rotational angle of 90 degrees has been applied between each exposure of 1200 sec (Richard et al. 2015).

Figure 2 .

 2 Figure 2.16: An overview of multiple image systems behind RXJ1347. The dashed cyan line delineates the field of view of MUSE observation, the solid curve highlights the multiple image regions, and the solid red line shows the critical curve at redshift z = 4. Credit: Richard 2021.

Figure 2 . 17 :

 217 Figure 2.17: An overview of multiple image systems behind SMACS2031. The dashed cyan line delineates the field of view of MUSE observation, the solid curve highlights the multiple image regions, and the solid red line shows the critical curve at redshift z = 4. Credit: Richard 2021.

  .

  2.1. The lens model of this cluster is taken from Richard 2021. An overview of multiple imaged systems observed behind this cluster is shown in Fig. 2.18. Credit: Richard 2021.

Figure 2 . 18 :

 218 Figure 2.18: An overview of multiple image systems behind SMACS2131. The dashed cyan line delineates the field of view of MUSE observation, the solid curve highlights the multiple image regions, and the solid red line shows the critical curve at redshift z = 4. Credit: Richard 2021.

Figure 2 .

 2 Figure 2.19: . Processing to build MUSE spectroscopic catalogs using HST images and MUSE datacubes. Credit: Richard 2021.

Figure 2 .

 2 Figure 2.20: Source Inspector package interface. An example is to find an LAE from the A2667 MUSE lensing datacube and to estimate its redshift. The M54 source has no counterpart in the optical domain and only shows up in the MUSE cube. MUSE white light and its FoV are shown in the upper right. MUSE NB is shown in the upper left. The spectrum of M54 is shown in the lower left, a zoom-in spectrum is shown in the lower right.

Figure 2 . 21 :

 221 Figure 2.21: Redshift distributions of sources in the final sample. Left: the gray histogram shows the redshift distribution of all sources in the sample. The pink histogram is the redshift distribution of all sources having zcon f > 1, and the magenta histogram is for representative images having zcon f > 1. Right: the gray histogram is the redshift distribution of all sources in the sample, the pink histogram is the distribution of line emitters with zcon f > 1 and the magenta histogram is for the representative line emission sources with zcon f > 1. Credit: Claeyssens 2021.

  2.23). Finally, there are 600 Lyman alpha lensed images kept in the final catalog. The number of LAEs behind the lensing clusters is shown in Table.

  2.1.

Figure 2 . 22 :

 222 Figure 2.22: Redshift distribution of Lyman alpha sources with zcon f = 1 behind 17 lensing clusters.

Figure 2 . 23 :

 223 Figure 2.23: An example of choosing a representative image of multiple imaged system behind MACS0451. The image positions (red circles) are overplotted on the White Light MUSE image (left), F140W, and F814W observed by HST.

2. 4 . 4

 44 Lyα flux measurementThousands of LAEs have been observed, showing a variety of Lyα line shapes, mainly in terms of their asymmetry and number of peaks. This can be explained by the different diffusion processes that affect the Lyman alpha photons before they escape from the IGM[START_REF] Verhamme | 3D Lyiation transfer -I. Understanding Lye profile morphologies[END_REF]) as well as the absorption of IGM (M. J.[START_REF] Hayes | Spectral Shapes of the Lyα Emission from Galaxies. I. Blueshifted Emission and Intrinsic Invariance with Redshift[END_REF]. For this reason, various approaches can be employed to measure the total Lyman alpha flux emitted from the galaxies. This section introduces and discusses two methods for measuring this quantity.

Figure 2 . 24 :

 224 Figure 2.24: An example of asymmetry values obtained from the flux fitting method for four different LAEs behind MACS0940. The red dashed curve displays an asymmetry Gaussian fit to the observed profile (black). Right panel shows a double asymmetry Gaussian model (blue). The asymmetry value of a simple model is indicated for each line. Credit: Claeyssens 2021.

5 Å

 5 surrounding the source position. The flux value is integrated along the wavelength channel with a range of [λ 0 -6.25 : λ 0 + 6.25] Å. The mean values of FWHM and asymmetry were found are 7 Å and 0.2, respectively(Claeyssens et al. 2022). Fig.2.24 shows different asymmetry Lyα line profiles, each with a different best-fit a value. The four best fit parameters were obtained by employing a MCMC fitting process using 8 walkers and 10 000 steps, combined with the median values of their posteriors distribution. The uncertainties of individual values are taken at 16th and 84th percentiles. There are 31 sources in the total of 600 that have two peaks in the present work. This pattern can be adapted by 2 The MUSE Lensing Project: from observations of massive clusters to LAE sample selection -2.4 Source detection and catalog building applying the function above on the red peak with a positive asymmetry value and with a negative value for the blue peak. The final flux value of a source will be summed up from the two peaks. The final median flux value is 2 ×10 -18 erg cm -2 s -1 .

Figure 2 . 25 :

 225 Figure 2.25: Left: Asymmetry parameter distribution of Lyman alpha profiles in the present work. A positive value of the asymmetry parameter indicates that the line profile is asymmetric towards the red, while a negative value indicates that the profile is asymmetric towards the blue. A zero value indicates a normal Gaussian profile. Right: Asymmetry parameter versus redshift of individual sources. Credit: Claeyssens 2021.

2

  The MUSE Lensing Project: from observations of massive clusters to LAE sample selection -2.4 Source detection and catalog building A2744 due to a large number of faint sources being included, while the first method is applied for the remaining sources in the catalog. Whenever the first method fails due to a spectral extraction problem, the second method is used. In total, 425 of the 600 sources in the catalog have fluxes measured using the spectral fitting method. The remaining 128 LAEs behind A2744, plus 47 faint sources that failed from the first one, have their fluxes measured using Source Extractor.A comparison of the difference of central flux values for individual sources using two methods is plotted in Fig.2.26 (left). The abscissa axis displays flux values measured from SExtractor and the vertical axis is from spectral fitting, both in the unit of 10 -18 erg s -1 . The solid black line shows a one-to-one ratio between the two methods, and the dashed black lines denote a ratio of 0.5 and 2.It is worth noting that the first method, as described in Claeyssens et al. 2022 tended to use the central magnification value of the individual sources, while the second method used the weighted magnification to correct for the lensing effect. Fig. 2.27 shows distribution of the weighted magnifications of LAEs in the present work. A comparison of magnification values obtained from two methods are shown in Figure 2.26 (right). The flux values obtained from the first method are combined with the central magnification value to compute the luminosity. This luminosity value is then plotted against the value derived using the SExtractor and weighted magnification, as shown in Fig. 2.28. In the remaining part of the work, we will use the weighted magnification values for the computation process.

Figure 2 .

 2 Figure 2.26: Left: Comparison between the flux values obtained from the two methods: line fitting (y-axis) versus SExtractor (x-axis). Right: Comparison of central magnification (ordinate) and weighted one (abscissa) .

Figure 2 . 28 :

 228 Figure 2.28: Comparison of Lyman alpha luminosity between the two methods of flux measurements and magnification computations (see text).

Figure 3 . 1 :

 31 Figure 3.1: Procedure to create the 3D mask images and V max . The main steps are followed by the red boxes and blue arrows. The details are given in DLV 2019.

Figure 3 . 2 :3Figure 3 . 3 :

 3233 Figure 3.2: Evolution of noise with wavelength inside each MUSE cube. From left to right, up to down, the cubes are: A2390, A2667, A2744, A370, AS1063, BULLET, MACS0257, MACS0329, MACS0416N, MACS0416S, MACS0451, MACS0520, MACS0940, MACS1206, MACS2214, RXJ1347, SMACS2031, and SMACS2131.

Figure 3 . 4 :

 34 Figure 3.4: An example of the 9 brightest pixels list of 2 sources behind lensing A2667in the catalog of (DLV 2019). The green circle is the critical region with a radius of 1" which is used to look for the first brightest pixel.

Figure 3 . 5 :

 35 Figure 3.5: The evolution of RMS maps at different channels inside a given cube. From left to right is the evolution of RMS maps at different channels (500, 1000, 2500, 3000) and its median RMS map inside A370 (upper) and MACS1206 (lower).

Figure 3 . 6 :Figure 3 . 7 :

 3637 Figure 3.6: RMS median images of 18 datacubes in the present work. From left to right, up to down corresponding to A2390, A2667, A2744, A370, AS1063, BULLET, MACS0257, MACS0329, MACS0416N, MACS0416S, MACS0451, MACS0520, MACS0940, MACS1206, MACS2214, RXJ1347, SMACS2031, and SMACS2131.

Figure 3 . 8 :

 38 Figure 3.8: A 2D masked image of A2667 in the image plane projected into the source plane at different redshifts. From left to right, up to down, the projected redshift is z s = 3.5, 4.5, 5.5, and 6.5.

Figure 3 . 9 :

 39 Figure 3.9: A 2D masked image of A2744 in the image plane projected into the source plane at different redshifts. From left to right, up to down, the projected redshift is z s = 3.5, 4.5, 5.5, and 6.5.

Figure 3 . 10 :

 310 Figure 3.10: From left to right up to down: Total V max distribution of LAEs in the present sample, the V max versus magnification µ, V max versus detected flux and V max versus luminosity log(L).

Figure 3 . 11 :Figure 3 . 12 :

 311312 Figure 3.11: Dependence of V max on the magnification value of the source, which has a highly magnified value when magnification is varied within its uncertainties ±1σ and ±2σ. The blue points from left to right denote the µ -2σ, µ -1σ, µ, µ + 1σ, µ + 2σ, µ + 3σ, respectively. The orange solid line displays the best-fit exponential function.

Figure 4 . 1 :

 41 Figure 4.1: Detection profiles of four different LAEs. For each LAE case, from left to right, is the max-NB image, filtered image, and detection profile. A2390-96, a bright source (upper left), A2667-30, a bright but diffuse source (upper right), A2744-2956 (lower left), and A2744-12-26, a faint source (lower right). All images are shown in the same spatial scale. Credit: Vieuville 2019.

Fig. 4 . 16 Flux [erg cm - 2 s - 1 ]Figure 4 . 2 :

 4162142 Fig. 4.2 (left) displays the final values of completeness versus the detection flux of individual LAEs in the present data sample. The faintest sources have very low completeness and large uncertainty values, while the bright ones often have very high completeness and small uncertainty values. The appearance of Flag = 2 sources with low detection flux can be attributed, in part, to the population of faint extended sources.Although, the integrated flux of these sources in the max NB image is not high, a sufficient number of pixels meet the detection conditions. Consequently, these

Figure 4 . 3 :

 43 Figure 4.3: Source distribution in each luminosity bin before and after correction for the completeness value.From left to right, up to down is the distribution at redshift 2.9 < z < 6.7, 2.9 < z < 4.0, 4.0 < z < 5.0, and 5.0 < z < 6.7.

Figure 4 . 4 :

 44 Figure 4.4: Detection flux vs redshift for all sources in the present sample. The color bar on the top shows the completeness value of each source in the sample. The grey points show the evolution of noise level as a function of wavelength/redshift or cluster A2667 on the left and AS1063 on the right.

Figure 4 . 5 :

 45 Figure 4.5: Illustration of the contribution of a given source to the luminosity bin during the MCMC process, when completeness, detected flux, and magnification are varied within 1-σ.The ordinate shows the number of sources in the simulation (20'000 in total), and the abscissa shows i t h luminosity bins from the faintest bin (0) to the brightest bin(11). Each panel represents an individual source in one cluster field. Sources contributing to the brightest luminosity bins exhibit little dispersion, whereas sources contributing to the faintest bins are spread into several luminosity bins.

Figure 4 . 6 :

 46 Figure 4.6: From left to right, up to down: LF histograms of each of the 12 luminosity bins (for the global redshift range) obtained from 20'000 MCMC iterations. Their median, median of log and log of the median are shown together with the face value.

Figure 4 . 7 :

 47 Figure 4.7: LF points in each luminosity bin at four redshift ranges obtained after 20'000 MCMC iterations are shown by the cyan lines together with face value LF found from all the Lyman alpha sources in the final data sample. The final uncertainties of LF points have already accounted for the statistical, Poissonian, and cosmic variance uncertainties.

  [(∆ log L = 1)

Figure 4 . 8 :

 48 Figure 4.8: The LF points in each redshift range using different luminosity bin sizes.

17 Flux [erg cm - 2 s - 1 ]

 1721

Figure 4 . 9 :

 49 Figure 4.9: Left: Completeness histogram of LAEs (zcon f = 1) behind RXJ1347. Right: Completeness versus the measured flux of zcon f = 1 sources behind RXJ1347.

Figure 4 . 10 :

 410 Figure 4.10: Source distribution in each luminosity bin when zcon f = 1 sources are taken into account.

Figure 4 . 11 :

 411 Figure 4.11: LF points in each luminosity bin when half and all the zcon f = 1 sources have been included in the data sample.

Fig. 4 .

 4 Fig. 4.10 shows the relative contribution of zcon f = 1 sources to each luminosity bin. At the faint end (log L < 41.25 [erg s -1 ]), the number of zcon f = 1 is similar to the number of the zcon f = 2, 3 whereas, the appearance of zcon f = 1 sources at the bright end is almost zero. With the assumptions given above, we find that there is no effect of zcon f = 1 sources on the shape of the LF and its uncertainty values at the bright end. On the contrary, there is a small variation towards the faint end. Fig.

  39.0 39.5 40.0 40.5 41.0 41.5 42.0 42.5 43.0 log(L)

  ) 5.0 < z < 6.7 ( line fit: α = -1.96± 0.5 0.54 ) 39.0 39.5 40.0 40.5 41.0 41.5 42.0 42.5 43.0 log(L)

  We use the Python package Lm f i t[START_REF] Newville | LMFIT: Non-Linear Least-Square Minimization and Curve-Fitting for Python[END_REF] for the fitting process. Lm f i t makes use of the Levenberg-Marquardt algorithm to minimize the three Schechter parameters. There are several pieces of literature used as a constraint in the bright part such as: +) Blanc et al. 2011 investigated the LF of 89 LAEs within a redshift range of 1.9 < z < 3.8 observed by Hobby Eberly Telescope Dark Energy Experiment Pilot Survey (HETDEX) using the same V max method. +) Drake et al. 2017 constructed the LF of 604 LAEs covering the same redshift range compared to the present work in which the data were collected in the blank field by MUSE/VLT. +) Cassata et al. 2011 used data from Vimos-VLT Deep Survey to study the LF of 217 LAEs at redshift 2 < z < 6.62. +) Sobral et al. 2018 presented the work with a large number of LAEs of ∼ 4000 collected by the Subaru and Isaac Newton Telescope, observed in the ∼ 2 deg 2 COSMOS field that covering a luminosity of 42.2 < logL < 43.0. The best-fit results of the Schechter function fitting in four redshift ranges are shown in Fig. 4.13. The best-fit parameters are shown in Table 4.4. The 68% and 95% confidence regions of the Schechter function fitting are shown in the dark red shaded area. As mentioned before, some open squares at the faintest bins have not been included in the traditional Schechter fit.

  Luminosity function for 4.0 < z < 5.0 Dawson 2007(4.5) Drake 2017(4 < z < 5) Herenz 2019(4.0 < z < 5.0) Sobral 2018(z = 4.6, 4.8) Sobral 2018(4.1) Zheng 2013(4.5) de La Vieuville 2019(4.0 < z < 5.0) Modified Schechter (LT = 40.0) Schechter fit (α = -1.97) This work This work (Schechter fitting not included)

Figure 4 . 13 :

 413 Figure 4.13: Evolution of LF with redshift in different redshift intervals, including previous literature data points. The red squares are the points from the present work. The literature points at the bright end of the LF: Masami Ouchi et al. 2010, Sobral et al. 2018, Zheng et al. 2013, Herenz et al. 2019, and Drake et al. 2017 have been used for fitting. The blue points DLV 2019 are shown for comparison purposes only. The best-fit (Schechter function) are shown as solid lines and the 68% and 95% confidence areas as dark red shaded regions, respectively. The dashed lines shown in lower panels are modified Schechter functions to account for a possible turnover at the faintest luminosity bins.

Figure 4 . 14 :

 414 Figure 4.14: Correlation of three parameters of the Schechter function for four redshift intervals as indicated in the legends. Contours show the 68% confidence level of the fittings.

Figure 4 . 16 :

 416 Figure 4.16: Same as Fig. 4.13 but applying a completeness cut at 10%.

Figure 4 . 17 :

 417 Figure 4.17: Faint end slope of the LF for four redshift intervals. The open squares at the faintest and brightest parts are not included in the fit.

Figure 4 . 18 :

 418 Figure 4.18: Schechter fits for the different redshift intervals, using the constraint that our LF points are at the faint end and literature data points are at logL > 43 erg s -1 . The red squares are the points from this work. The literature points are shown in the insert. The best fits of the Schechter function are shown as a solid line and the 68% and 95% confidence levels as dark red shaded areas, respectively.

Figure 4 . 19 :

 419 Figure 4.19: Faint end slope at different redshift ranges derived from the present work (red) and literature (other colors) as indicated in the legends. The horizontal error bars are the redshift range of the surveys.

4 )

 4 where: Φ(L) is the traditional Schechter function together with its three free parameters (L * , Φ * , α), m is the curvature parameter defining a downward turnover if m > 0 and upward turnover if m < 0, L T denotes the position of turnover, which is defined as the luminosity where the derivative (d Φ/d L ) L=L T = 0. The shape of the modified Schechter function is shown in the lower panels of Fig.4.13, 4.16, 4.17. Our data suggest that m is about unity and l og (L T ) are ∼ 40 and ∼ 40.7 erg s -1 for the redshift bins 4.0 < z < 5.0 and 5.0 < z < 6.7, respectively. The appearance of a turnover at the faintest luminosity bins is explained by star formation inefficiency in small dark matter halos(Jaacks et al. 2013;[START_REF] Nickolay | COSMIC REIONIZATION ON COMPUTERS: THE FAINT END OF THE GALAXY LUMINOSITY FUNCTION[END_REF][START_REF] Yue | On the faint-end of the high-z galaxy luminosity function[END_REF]).

Recently, two models

  have been developed by Garel et al. 2021 and Salvador-Solé et al. 2022 to predict the Lyman alpha LF at redshift z ∼ 6. It is therefore necessary to compare the results of these models to that obtained in the present work for the highest redshift interval 5.0 < z < 6.7. The first model, developed by Garel et al. 2021, applies the SPHINX radiation hydrodynamics cosmological simulation to predict the Lyman alpha LF at the EoR by estimating the radiative transfer of the sources from ISM to the IGM scales. The second model, developed by Salvador-Solé et al. 2022, uses the AMIGA (Analytic Model of Igm and GAlaxy evolution) model to predict the possible scenarios of single and double ionization, taking into account the impact of galaxy formation and its evolution within their feedback on the IGM. The former predicts a single hydrogen ionization stage at redshift z ∼ 6, while the latter predicts two reionization stages at redshift z ∼ 6 and ∼ 10, with two separated phases defined by a short recombination episode. The results are shown in Fig. 4.20. At the luminosity range 40 < logL[ergs -1 ] < 42, the prediction from AMIGA double ionization is in line with the one obtained from the SPHINX simulation after correction for the IGM. In general, the LF points are in good agreement with the two predictions without any normalization requirement. At the faintest luminosity bin, when log L < 41, our LF point tends to depart from the SPHINX predictions but is still close to the single ionization model from Salvador-Solé et al. 2022. However, at these redshift ranges, the uncertainty values are generally large, preventing us from evaluating the difference between the predictions. More observational data at the faint luminosity regime and at high redshift ranges are needed to solve this problem.

Figure 4 . 20 :

 420 Figure 4.20: LF points obtained in the present work (red crosses) at the highest redshift range are compared to the model predictions (Garel et al. 2021) (dashed color lines) and AMIGA models (Salvador-Solé et al. 2022) (solid lines). The two LF points at the brightest part are taken from the literature by averaging their values in the same luminosity and redshift bin.

- 1 .Mpc - 3 ]Figure 5 . 1 :

 1351 Figure 5.1: Cosmic evolution of the SFRD as a function of redshift. The data points are taken from the literature listed in the insert.

Figure 5 .

 5 Figure 5.1 shows the evolution of the SFRD as a function of redshift, by comparing the results obtained in this work with previous results from the literature (see references in the legend). It is worth mentioning that Cassata et al. 2011 computed their SFRD by correcting for the influence of IGM absorption, assuming an escape fraction of ionizing photons that varies from 15% at redshift z=3 to 50% at redshift z=6, combining with a factor of radiative transfer as the one presented in Fan et al.

Figure 5 . 2 :+ z 6 3 Ω b h 2 0.08 2 ( 5 . 2 )

 5262252 Figure 5.2: Evolution of f escp as a function of redshift. The data points are taken from the literature listed in the insert. Credit: (M. Hayes, Daniel Schaerer, et al. 2011).

  , moderate-to-no evolution from z ∼ 3 to z ∼ 6 (Ouchi et al. 2008), and then drop down beyond z ∼ 6 (Kashikawa et al. 2011). Much progress has been made studying the LAE LF in the bright luminosity regime, namely, log(L) [erg s -1 ] > 42 (see a recent review by Ouchi et al. 2020). Spinoso et al. (2020) studied hyper-LAEs (log(L) [erg s -1 ] > 10 43.3 ) in the redshift range of 2.2 < z < 3.3 from Javalambre Photometric Local Universe Survey (J-PLUS), reporting a Schechter slope of α = -1.35 ± 0.84. Using a sample of 166 LAEs at z ∼ 3.1 obtained from Subaru Suprime-Cam, covering luminosity range of log(L)[erg s -1 ] = 42-43.5, Guo et al. (2020) found log (φ * ) = -3.30 +0.09

7 .

 7 With a sample of 604 LAEs probing down to a luminosity of log(L) [erg s -1 ] < 41.5 obtained from the MUSE Hubble Ultra Deep Field Survey, Drake et al. (2017) measured faint end slope values of -2.03 +1.42 -0.07 at z ∼ 3.44 and -2.86 +0.76 -∞ at z ∼ 5.48. The MUSE-Wide survey probed 237 LAEs with luminosities in the range of 42.2 < log(L) [erg s -1 ] < 43.5 (Herenz et al. 2019). These authors measured the three parameters: α = -1.84

A139, page 3 of 20 ThaiFig. 1 .

 201 Fig. 1. MUSE exposure maps overlaid with HST/F814W images adapted from R21.

Fig. 2 .

 2 Fig. 2. Lyman α luminosity vs. redshift of lensed LAEs in the present work (magenta) and those of DLV19 (cyan). Histograms of redshift (top) and of luminosity (right) are also shown, respectively.

Fig. 3 .

 3 Fig. 3. Source central magnification vs. its flux-weighted average one for LAEs in the sample. Extended sources often have averaged magnification values lower than those of their central ones, as expected. The black dashed line denotes a one to one magnification ratio.

  Claeyssens et al. (2022), whereas the results in DLV19 are in line with the other literature. Overall, we obtained 425 source fluxes using the first method and 175 source fluxes using the second method. We expand briefly on both methods below. Regarding the first method, Claeyssens et al. (2022) extract LAE fluxes via three main steps: spectral fitting, NB image construction, and repeating spectral extraction. First, a pseudo NB image of each slice (measuring 5 × 5 ) is constructed from the MUSE datacube, using the formulae introduced by Shibuya et al. (2014) to fit the asymmetric spectral profile of LAEs:

A139, page 6 of 20 ThaiFig. 4 .

 204 Fig. 4. Fluxes of LAEs in the sample, obtained by fitting Lyman-alpha profiles (Claeyssens et al. 2022) vs. those extracted by SExtractor. Solid black line indicates the best linear fit between two the fluxes. the line. The mean value of d is 7 Å, and the mean value of a asym is 0.20. The value of λ 0 is converted directly from source redshift and the flux amplitude A is integrated in the range of [λ 0 -6.25: λ 0 + 6.25] Å.It is worth mentioning that 31 out of the 600 LAE spectra display similar two-peak profiles. We apply the same procedure as described above, dealing with the blue and red peaks separately, taking into account the amplitude difference between the two. The two peaks are then summed to get the final flux. The flux uncertainty is estimated using python package EMCEE (Foreman-Mackey et al. 2013) with 8 walkers and 10 000 iterations.For the second method, we made use of the SExtractor software(Bertin & Arnouts 1996) running on NB detection images that host LAEs. This is well described in DLV19. For this purpose, a sub-cube with a size of 10 ×10 × the width of the Lyman-α emission is first extracted from the original cube, then averaged to form the LAE image. Two sub-cubes of the same spatial size, extending 20 Å bluewards and redwards of the Lyman-α line were also created. These were then averaged to form the mean continuum image of each LAE. By subtracting this mean image from the LAE image, pixel by pixel, we can form a new measurement image which is ready for SExtractor. We used SExtractor's FLUX -AUTO parameter to estimate the LAE fluxes together with their uncertainties. This parameter is developed based on Kron's first-moment algorithm: when a given source is convolved with a Gaussian seeing, 90% of its flux will be inside the circular aperture of the Kron radius. This holds even for extended sources. The final flux obtained from the measurement image is then multiplied by the width of the Lyman-α emission.It may happen that some sources are either faint or have a low extended surface brightness. In such cases, SExtractor does not extract these sources properly. A set of farsighted SExtractor parameters (DETECT -THRESH, DETECT -MINAREA) have been proven to be very useful to overcome this problem. SExtractor progressively releases detection conditions using this set of parameters to make sure that faint sources can be extracted.Figure4shows a comparison between the fluxes of LAEs obtained by using the two methods in the present work. In gen-

A139, page 7 of 20 ThaiFig. 5 .

 205 Fig.5. Examples of normalized bright pixels profiles of sources from clusters AS1063 and BULLET. Individual profiles (red) are shown together with their median (black). Each of these median profiles is then used as a representative for all sources in that MUSE datacube. The different seeings between datacubes are taken into account when producing these profiles. The median profile and the RMS map are used to produce a wide range of S/N values for each MUSE datacube.

A139, page 8 Fig. 6 .

 86 Fig. 6. Completeness vs. detection flux of LAEs from the present sample. Different colors indicate the quality of extraction from SExtractor. Only sources in the unmasked regions of the detection layer are considered here.

Fig. 7 .

 7 Fig. 7. Detection flux vs. redshift for all sources in the present sample. The color bar on the top shows completeness value of each source in the sample. The grey points show the evolution of noise level as function of wavelength/redshift for the cluster MACS0257 as an example.

Fig. 8 .

 8 Fig.8. Source distribution in each luminosity bin (the faintest on the left, the brightest on the right) before and after correcting for the completeness. A larger correction, a factor up to a few tens, has been made at the faint luminosity regime, as expected.

A139, page 9 of 20 Thai

 20 , T. T., et al.: A&A 678, A139 (2023)

  39.0 39.5 40.0 40.5 41.0 41.5 42.0 42.5 43.0

- 1 ,

 1 which is close to values of log L * (Herenz et al. 2019, DLV19), to completely describe the Schechter function, we need to include other data covering the bright end of the LF. The data included are taken from the works of: Dawson et al. (2007), Blanc et al. (2011), Cassata et al. (2011), Zheng et al. (2013), Sobral et al. (2018), Drake et al. (2017) and Herenz et al. (2019), which have been selected to properly cover the redshift and bright part of luminosity ranges. These data from the literature are averaged with the same luminosity bin size of 0.25 (in log 10 (L) [erg s -1 ]

  2019(4.0 < z < 5.0) Modified Schechter (LT = 40.0) Schechter fit (α = -1.97) This work This work (Schechter fitting not included)

Santos 2016( 5 . 7 )Fig. 10 .

 5710 Fig. 10. LF points and their fits for different redshift intervals including previous literature data points. The red squares are the points from the present work. The literature points at the bright end of the LF: Ouchi et al. (2010), Sobral et al. (2018), Zheng et al. (2013), Herenz et al. (2019), and Drake et al. (2017) have been used for the fitting. The blue points DLV19 are shown for comparison purpose only. The best fits (Schechter function) are shown as a solid line and the 68% and 95% confidence areas as dark-red colored regions, respectively. The dashed lines shown in lower panels are modified Schechter functions to account for a possible turnover at faint luminosity bins (see details in the text).

- 2 . 2 5 - 2 - 2 . 7 5 - 2 Fig. 11 .

 25225211 Fig. 11. Correlations between three Schechter's best-fit parameters for different luminosity bins. The tuples denote the lower and upper limit of the luminosity range with respect to the number of the bin. Contours are 68% confidence levels obtained from the Schechter's fitting. Results are shown for 2.9 < z < 6.7 (left) and 4.0 < z < 5.0 (right).

Fig. 12 .

 12 Fig. 12. Correlations between the three parameters, L * , Φ * and α of the Schechter function for four different redshift bins as indicated in the inserted. Contours show 68% confidence levels obtained from fitting procedures.

  et al. (2017), Atek et al. (2018), Priewe et al. (2017), Meneghetti et al. (2017).

6. 1 .

 1 Uncertainties associated to the computation of the LF in strong lensing fields Thanks to the magnification by lensing clusters, we can reach a fainter galaxy population compared to that of Drake et al. (2017) and Herenz et al. (

  Blanc et al. (2011) used 89 LAEs with a redshift range 1.9 < z < 3.8 obtained from Hobby Eberly Telescope Dark Energy Experiment Pilot Survey (HETDEX) to study the LF using the same V max method. Drake et al. (2017) used 604 LAEs in the redshift range 2.91 < z < 6.64 obtained from VLT/MUSE. Cassata et al. (2011) used 217 LAEs in the redshift range 2 < z < 6.62 obtained from Vimos-VLT Deep Survey. Sobral et al. (2018) studied ∼4000 LAEs from z ∼ 2 to 6 covering a luminosity range of 42.4 < log (L) [erg s -1 ] < 43.0 obtained from the Subaru and the Isaac Newton Telescope in the ∼2deg 2 COSMOS field. Our LF points in the bright part (log(L) [erg s -1 ] > 42) are consistent with these results from the literature (Fig. 10). It is worth mentioning the work of Herenz et al. (2019). They used data including 237 LAEs from the MUSE Wide survey to construct the LF in the same redshift ranges as this work.

  Thai, T. T., et al.: A&A 678, A139 (2023) same conclusions regarding the reliability of the current results when we compare, for instance, the work of Atek et al. (2018) to Bouwens et al. (2022; see discussion below).

  (2021) and by Salvador-Solé et al. (2022). The first one (Garel et al. 2021) predicts the Lyman-alpha LF at the epoch of reionization by computing the radiative transfer of Lyman-alpha from ISM to IGM scales, using the SPHINX radiation-hydrodynamics cosmological simulation. Their results, obtained by computing the intrinsic LF function, and the attenuation by dust and then by IGM, are shown in Fig. 13. The second one (Salvador-Solé et al. 2022

Fig. 13 .

 13 Fig. 13. Comparison with model predictions. Red crosses show our reconstructed LF points in the redshift range of 5.0 < z < 6.7. Two additional crosses in the bright end, having log(L)[erg s -1 ] > 43, are taken from the literature, obtained the same way as described in Sect. 5.2. Black and blue solid lines show predictions from the AMIGA models with single-and double-ionization scenarios, respectively. Predicted LFs from the SPHINX simulation (Garel et al. 2021) are shown as dashed lines: intrinsic LF (magenta), attenuation by dust (green), and by IGM (orange).

Fig. 14 .

 14 Fig. 14. Comparison of slope evolution with redshift from this work (red crosses) to the literature which appears in the Fig. 10. The horizontal error bars are the redshift range of the surveys.

- 1 .Mpc - 3 ] 3 Fig. 15 .

 13315 Fig. 15. Evolution of the star formation rate density with redshift using different integration limits for the LF. Brown (red) crosses show results of integrating the best-fit Schechter (modified Schechter) functions log(L) from 41 (39.5) to 44 erg s -1 . Solid blue, orange, and green curves indicate the critical SFRDs obtained by assuming a clumping factor of 3 and f escp = 3%, 8%, and 25%, respectively. The solid green curve, having f escp = 8%, indicates the best likely match with our SFRD points in the luminosity range of 41 < log(L) < 44 (see details in the text).

  5 and z ∼ 4.5 would be ∼80%, compared to that of Bouwens et al. (2015b) using M UV = -17 mag as observational limit. At higher redshifts z ∼ 6 the contribution is much higher, 10% larger than Drake et al. (2017), 4 times larger than Cassata et al. (2011), 12 times larger than Ouchi et al. (2008), and much higher than Bouwens et al. (2015a

  authors select LAEs based on narrow-band (NB) photometry (Arrabal Haro et al. 2018; Yoshioka et al. 2022), some search for Lyman-α emission from UV-selected samples using multiobject slit spectroscopy (Stark et al. 2010, 2011; Pentericci et al. 2011, 2018; De Barros et al. 2017; Fuller et al. 2020), and some use integral field unit (IFU) spectroscopy (Caruana et al. 2018; dLV20; Kusakabe et al. 2020).

Fig. 1 .

 1 Fig. 1. Redshift and M 1500 (and Lyman-α luminosity for the LAEs) distributions of the two samples used in this work. LBGs are shown in blue and LAEs in red.

Fig. 2 .

 2 Fig. 2. Photometric vs. spectroscopic redshift comparison of objects selected as LAEs and LBGs. The red line indicates the one-to-one relation, and the dashed black lines indicate the typical error bounds of |z spec -z phot | < 0.15(1 + z spec ) and |z spec -z phot | < 0.05(1 + z spec ). Almost all of the photometric redshifts agree with their spectroscopic equivalents within the outer of these two bounds. Those that do not tend to be fainter objects, with magnitudes of 28 and above.

Fig. 3 .

 3 Fig. 3. Lyman-α EW (EW Lyα ) distribution for all our samples. Objects selected as LAEs or LBGs are denoted by larger stars, LAE+continuum by smaller stars, LAE-only by upward pointing triangles, and LBGonly by downward pointing triangles. EW Lyα values for the latter two populations are calculated using the upper limits of the continuum and Lyman-α flux, respectively. The objects are colour-coded by M 1500 . The horizontal dashed line demarcates the 25 Å level, above which LAEs are included in the calculation of the LAE fraction. Error bars are omitted for clarity, but shown in Figs. 4 and 5.

Fig. 4 .

 4 Fig. 4. EW Lyα distribution over the range of UV absolute magnitude probed for the LAE+LBG sample. The red stars indicate the average in five equal-size bins between M 1500 = -20 and M 1500 = -15, the region that is well populated. De Barros et al. (2017) provide a similar plot (at z = 6) for a brighter sample with a similar rising average EW Lyα . Our plot extends several magnitudes fainter, and we see that this trend continues down to at least M 1500 = -16. An idea of the region fainter than M 1500 = -15 can be gained by including the LAE-only sample (orange triangles), the vast majority of which are very faint in UV magnitude, and high-EW Lyα objects. For these objects, EW error bars, coming only from the error on the Lyman-α flux, are smaller than the size of the points. The continuum values used are 2σ upper limits estimated from the filter that would see the emission at 1500 Å rest frame.

Fig. 5 .

 5 Fig. 5. UV slope plotted against EW Lyα for the LAE+LBG sample.Orange dots represent UV slopes that, upon inspection, we find to be dubious: photometry that locally does not represent the slope well or has large errors. Red stars represent average values in equally sized bins between β = -3.5 and β = -1.0. The change in EW Lyα across this range is not as significant as the change across the range of probed UV magnitudes (see Fig.4); however, we see a lightly rising trend in EW Lyα with bluer slopes down to β = -2.5.

  al. 2010; Schenker et al. 2014; De Barros et al. 2017). We report a slight increase (<1σ) in average EW in bins between β = -1.0 and β = -3.0. However, similarly to De Barros et al. (2017), we find EW Lyα values as high as ∼100 Å across the sample.

Fig. 6 .

 6 Fig. 6. Comparison of the derived SFRs for our LAE+LBG sample, colour-coded by redshift. The errors on the SFR Lyα are smaller than the size of the points in most cases. The dashed blue line denotes a oneto-one SFR ratio, i.e., SFR UV = SFR Lyα . Galaxies along this line also have equal escape fractions: f esc,UV = f esc,Lyα . Most of our sample lies under this line, and the dashed red line denotes the median SFR ratio of our sample, 0.35. For fainter objects with SFR UV < 1 M yr -1 , a greater fraction lie above the line of equal SFR, a trend previously observed in Ando et al. (2006), Schaerer et al. (2011), and dLV20.

Fig. 8 .

 8 Fig. 8. M 1500 distribution of the populations (left) and redshift distribution of the populations (right). The sample labelled 'LAE+continuum' includes all LAEs matched with continuum objects and LBG-selected galaxies.

Fig. 9 .

 9 Fig. 9. LAE fraction, X LAE , over a redshift range of 2.9-6.7, using the typical literature limits of EW Lyα > 25 Å and M 1500 > -20.25. The coloured circles represent X LAE when just LAE+LBG objects are taken into consideration, and the coloured stars represent the same calculation but including the LAE+continuum sample (see Sect. 2.4). Coloured stars indicate these X LAE values including the completeness correction for the number of LAEs (see Sect. 2.5). Blue and cyan squares represent the results from dLV20, with cyan representing those with the completeness correction. Shorthand is used in the legend for the literature results, but we give the full list here (in the same order as the legend): Kusakabe et al. (2020), Arrabal Haro et al. (2018), De Barros et al. (2017), Stark et al. (2011), Pentericci et al. (2011), Cassata et al. (2015), and Mason et al. (2018b). A small redshift offset is artificially applied to some points for clarity.

  others); however, Caruana et al. (2014) use a limit of 75 Å. Other authors, such as Stark et al. (2010), Cassata et al. (2015), Caruana et al. (2018), Kusakabe et al. (2020), and Fuller et al. (2020), investigate different EW and M 1500 or apparent magnitude ranges to assess the EW and UV magnitude evolution of X LAE , with the exact ranges depending on the study: the depths available and whether or not lensing is involved. In blank fields, most commonly authors investigate the bright and the faint halves of their sample, split at M 1500 = -20.25 (e.g., Cassata et al. 2015; De Barros et al. 2017; Arrabal Haro et al. 2018). Alternatively, where sample sizes allow, studies bin results in M 1500 from -22 to -18 (e.g., Caruana et al. 2018; Kusakabe et al. 2020

Fig. 10 .

 10 Fig. 10. Redshift evolution of X LAE in the bright and faint populations, split at M 1500 = -18. The bright and faint populations are shown in blue and orange, and the completeness corrections in grey scale with the bright population in darker grey. The top panel corresponds to LAEs with EW Lyα > 25 Å, while the lower panel shows the results when this limit is removed. The absence of the EW limit leads to a clearer separation between the two populations, suggesting that UV-bright LBGs exhibit a larger fraction of objects with Lyman-α emission when objects with small values of EW Lyα are taken into account. In the lower panel the UV-bright completeness-corrected point at z ∼ 5, omitted from the plot, is at 1.19 (see the main text).

(

  McGreer et al. 2015; Bouwens et al. 2015; Planck Collaboration et al. 2020). The sources contributing to reionisation are the subject of ongoing debate in the literature with consensus forming around the importance of faint star-forming galaxies (SFGs) (Bouwens et al. 2015; Finkelstein et al. 2015; Livermore et al. 2017; Bouwens et al. 2021).

2. 1 .

 1 MUSE LAEs: LLAMAS The observations of A2744 form part of the MUSE Lensing Cluster GTO data and form part of a data release by Richard et al. (2021). The MUSE Lensing Cluster programme contains clusters specifically chosen for efficient amplification of background sources such as high-redshift LAEs. Among these clusters, A2744 has been particularly well studied (e.g. the well known ASTRODEEP and HFF-DEEPSPACE programmes Lotz et al. 2017; Shipley et al. 2018). A2744 has fourv MUSE pointings forming a 2x2 mosaic with exposures of between 3.5 and 7 hours. Sources are identified in the MUSE datacube using the MUSELET software (Piqueras et al. 2017) 1 , the Marz software (Hinton 2016), and the Source Inspector package (Bacon et al. 2022). These packages allow the user to define redshifts for objects based on identification of emission and absorption features, as well as HST images of sources and redshift suggestions from Marz. The spectroscopic catalog for A2744, originally presented in Mahler et al. (2018), improved by Richard et al. (2021), is publicly available 2 .The LAEs identified behind A2744 form part of the Lensing Lyman Alpha MUSE Arcs Sample (LLAMAS)(Claeyssens et al. 2022). In the creation of this sample, all sources are given a redshift confidence level (zconf =1, 2 or 3). Only highconfidence LAEs (zconf =2 and 3) from the LLAMA sample are used in this work. The lens model used for A2744 comes fromRichard et al. (2021). Magnifications for all sources are calculated using this lens model and the LENSTOOL software(Kneib et al. 1996; Jullo et al. 2007). For A2744, there are 154 images of 121 individual LAEs. It contains the most LAEs of all the LLAMAS sample clusters, making it ideal for this single cluster study.

Fig. 1 :

 1 Fig. 1: Stellar mass vs star formation rate: the galaxy main sequence. Upper: The 99 JWST-detected LAEs plotted on the star-formation main sequence, together with the best fits to the data of P23, S14 and the best fit to the LAEs following Eq. 2. LAEs are colour-coded by "Time", in Gyr, which is the age of the Universe at the observed redshift. The brown vertical line indicates the lowest-mass limit of S14's data and the dotted black line indicates the lowest-mass limit of P23's data. Lower: The same star formation main sequence, compared this time to results in the same mass range but UV-selected: the LBGs from A2744 from Goovaerts et al. (2023) as diamonds and the mass-binned results from S17 as the coloured diamonds with errorbars. Error bars are omitted from the graphs for clarity but are stated here for the LAEs in three mass bins: 10 6 < M/M ⊙ < 2.5 × 10 7 , 2.5 × 10 7 < M/M ⊙ < 5 × 10 8 and 5 × 10 8 < M/M ⊙ < 10 10 . Median mass uncertainties in these bins are 7.2 × 10 6 M ⊙ , 4.1 × 10 7 M ⊙ and 4.3 × 10 8 M ⊙ . Median SFR uncertainties in these bins are 0.5 M ⊙ /yr, 1.8 M ⊙ /yr and 12 M ⊙ /yr.

  , and Jullo & Kneib 2009), has been used for this purpose. Details of methods and most of the lensing models from the present work are described in Richard et al. (2021). Information on these lensing clusters and their lens models can be found in Pello et al. (1991), Richard et al. (2010), Convone et al. (2006), Mahler et al. (2017), Lagattuta et al. (2019), Richard et al. (2021). III. MUSE/VLT and the MUSE Lensing Clusters Survey III.1. Very Large Telescope and MUSE

Fig. 1 .

 1 Fig. 1. Distributions of LAEs luminosity and redshift for the sample of lensed LAEs observed with MUSE. Overplotted in blue is the previous sample from de la Vieuville et al (2019).

16 Flux [erg cm - 2 s - 1 ]Fig. 2 .

 16212 Fig. 2. Completeness vs. detection flux of LAEs from the present sample. Different colors indicate quality of extraction from SExtractor.

7 ) 7 )Fig. 3 .

 773 Fig. 3. Luminosity function points of four different redshift ranges (inserted labels), constructed from the present sample. Their respective Schechter's fittings are shown as solid lines, and their 68% and 95% confidence levels as light brown areas. The gray LF points in the faintest luminosity bins are not included in the fittings. The other data points (see text) are: Ouchi et al. (2010), Sobral et al. (2018), Zheng et al. (2013), Herenz et al. (2019) (coloured diamonds), Drake et al. (2017) (purple squares).

7 .

 7 Measured LF points are often described by the Schechter funtion (Schechter 1976), written as: φ (L)dL = φ * (L/L * ) α exp(-L/L * )dL/L * , where φ * is the normalization density, L * is the characteristic luminosity where the LF turns from an exponential law at the bright end to a power law at the faint end, and α is the power-law slope in the low luminosity regime. As the present sample probes a maximum luminosity around log(L)∼43, other data covering this bright part of the LF are needed to constrain the Schechter luminosity function: Dawson et al. (2007), Blanc et al. (2011), Cassata et al. (2011), Zheng et al. (2013), Sobral et al. (2018), Drake et al. (2017), Herenz et al. (2019). These data are prepared and distributed into the luminosity bins in the same way as our data to balance the two populations of galaxies in the bright and faint regimes.

Fig. 4 .

 4 Fig. 4. Various LF points are computed for different magnifications indicated by different colors. Default averaged magnification values from the pipeline (red), 1-σ magnification increasing (black), 2-σ magnification decreasing (green), magnification saturation at 25 (cyan). The LF points from the previous work of de la Vieuville et al. (2019) are shown in blue. LF points are shown for the faint-end region (left) and bright-end region (right).

  Schechter function: Φ(L)exp(-L T /L) m = Φ * L * L L * α exp(-L/L * )exp(-L T /L) m

Table

	Cluster	Total co-moving volume [Mpc 3 ]
	A2390	735
	A2667	885
	A2744	10500
	A370	5350
	AS1063	1970
	BULLET	895
	MACS0257	730
	MACS0329	1225
	MAC0416N	3420
	MACS0416S	1670
	MACS0451	1210
	MACS0520	765
	MACS0940	5760
	MACS1206	2980
	MACS2214	1100
	RXJ1347	7920
	SMACS2031	1675
	SMACS2131	920
	Total	49710
		source is

Table 2 :

 2 The best-fit values of the Schechter function at different redshift intervals.

Table 3 :

 3 Faint-end slope values when applying different constraints

  bruit, ou ayant une identification ambiguë de la raie Lya. Pour cette raison, nous avons appliqué deux tests, en incluant dans l'échantillon de données actuel la moitié des sources avec zcon f = 1, et outess les sources avec zcon f = 1. Les résultats finaux sont rapportés dans le Tableau 3.Nous avons comparé nos résultats dans le bin de redshift le plus élevé 5.0 < z < 6.7 aux prédictions théoriques/numériques par GAREL et al. 2021, qui corrigent de l'atténuation d'IGM, ainsi qu'à celles de SALVADOR-SOLÉ et al. 2022, qui étudient des scénarios de simple et double réionisation. Cette comparaison montre une grande cohérence entre nos résultats et les prédictions, sans aucune ré-normalisation, sans pour autant pouvoir trancher entre des scénarios différents. Ceci est présenté dans la Figure

  3 , nhiều hơn gấp 3 lần so với DLV 2019. Tuy nhiên, giá trị thể tích này nhỏ hơn đáng kể so với các nghiên

	Cụm thiên hà Tổng thể tích khảo sát [Mpc 3 ]
	A2390	735
	A2667	885
	A2744	10500
	A370	5350
	AS1063	1970
	BULLET	895
	MACS0257	730
	MACS0329	1225
	MAC0416N	3420
	MACS0416S	1670
	MACS0451	1210
	MACS0520	765
	MACS0940	5760
	MACS1206	2980
	MACS2214	1100
	RXJ1347	7920
	SMACS2031	1675
	SMACS2131	920
	Total	49710

Bảng 1: Thể tích khảo sát của 17 cụm thiên hà thấu kính hấp dẫn trong khoảng dịch chuyển đỏ 2.9 < z < 6.7.

  Hình 4: Hệ số hoàn chỉnh so với thông lượng thu được của các nguồn phát xạ vạch Lyman alpha trong công việc hiện tại. Các màu sắc khác nhau hiển thị chất lượng quá trình trích xuất nguồn từ Source Extractor.

	39	40	41	42	43	44
				1.0	Flag ext 1 Flag ext 2
					Flag ext 3
				0.8
			Completeness	0.4 0.6
				0.2
				0.0
				10 -19	10 -18 Flux [erg cm -2 s -1 ] 10 -17	10 -16

Ở đây chỉ xem xét các nguồn trong những vùng không bị loại bỏ, trên kênh bước sóng cực đại mà nguồn được tìm thấy.

  . Kích thước ảnh phải đủ lớn để đặc trưng cho nhiễu trên toàn bộ kênh (channel) mà ở đó phát xạ vạch Lyman alpha của nguồn đạt cực đại. Do đó, để tính toán tốt hơn hệ số hoàn chỉnh đặc trưng cho khả năng tìm thấy nguồn, chúng tôi đã tăng kích thước ảnh gấp 2.5 lần so với công việc của DLV 2019 (từ 30" × 30" lên 80" × 80").

7

. Các dấu thập màu đỏ là kết quả của công việc nghiên cứu hiện tại, các dấu màu khác thu được từ những nghiên cứu trước đây. Đường nét liền màu nâu đỏ hiển thị giá trị khớp hàm Schechter của công việc hiện tại. Phần dấu thập màu xám không được tính đến trong quá trình khớp hàm. Phần màu xanh là kết quả nghiên cứu từ DLV 2019 được sử dụng nhằm mục đích so sánh. Đường nét đứt màu nâu đỏ là hàm đưa vào gợi ý cho hình thái "quay đầu" của hàm độ trưng. chỉnh

  7. Việc sử dụng dữ liệu từ cụm thấu kính hấp dẫn đã cho thấy hiệu quả trong nghiên cứu hàm độ trưng ở phía có độ trưng mờ, tuy nhiên dữ liệu này lại kém hiệu quả khi nghiên cứu ở vùng lân cận giá trị L * và cao hơn. Để xử lý vấn đề này, chúng tôi lấy trung bình các kết quả nghiên cứu trước đó về hàm độ trưng trong cùng khoảng dịch chuyển đỏ, trong cùng khoảng độ trưng. Chúng tôi sử dụng chúng để ràng buộc ở phía độ trưng lớn. Hình 5 hiển thị kết quả khớp hàm Schechter nói trên.Kết quả tốt nhất khi khớp hàm Schechter được trình bày ở Bảng 2, độ dốc của hàm Schechter cho thấy một sự tiến triển nhỏ theo giá trị dịch chuyển đỏ Bảng 2: Giá trị khớp hàm tốt nhất cho hàm Schechter ở bốn khoảng dịch chuyển đỏ.

	Redshift	Φ * [10 -4 Mpc -3 ] log L * [erg s -1 ]	α
	2.9 < z < 6.7

Bảng 3 :

 3 Kết quả độ dốc hàm Schechter với các phép thử khác nhau

		z al l		-2.06 ± 0.07	-1.46 ± 0.05	-1.82 ± 0.03		-2.29 ± 0.09		-2.12 ± 0.16	-1.64 ± 0.12	-2.06 ± 0.60
		z 60		-2.28 ± 0.12	-1.75 ± 0.10	-2.24 ± 0.11		-1.94 ± 0.10		-1.96 ± 0.54	-1.66 ± 0.24	-2.28 ± 0.50
	0	z 45 2		-1.97 ± 0.09 4	-1.83 ± 0.12	-1.97 ± 0.08	6	-1.92 ± 0.09		-1.99 ± 0.17 8	-1.63 ± 0.21	-1.97 ± 0.50 10
		z 35	Schechter fitting	Hiệu chỉnh 1% -2.00 ± 0.07	Hiệu chỉnh 10% -1.78 ± 0.06	Hiệu chỉnh 1% sử dụng giá trị độ sáng đo được từ Source extractor -2.10 ± 0.06	Hiệu chỉnh 1% và thêm vào tất cả các nguồn zcon f = 1 -1.83 ± 0.17	Linear fit	Hiệu chỉnh 1% -1.76 ± 0.27	Hiệu chỉnh 10% -1.55 ± 0.17	Giá trị độ dốc cuối cùng -2.00 ± 0.50

2 The Multi Unit Spectroscopic Explorer (MUSE) Lensing Project: from observations of massive clusters to LAE sample selection 91 2

  .1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 2.2 Main scientific goals of MUSE . . . . . . . . . . . . . . . . . . . . . . . . 94 2.3 Lensing clusters observed with MUSE/VLT . . . . . . . . . . . . . . . . . 97 2.3.1 General information of lensing clusters observed with MUSE/VLT 97 2.3.2 Lens models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Source detection and catalog building . . . . . . . . . . . . . . . . . . . . 2.4.1 Source detection with MUSE Line Emission Tracker (MUSELET) 2.4.2 Redshift determination using Source Inspector . . . . . . . . . . 2.4.3 LAEs catalog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.4 Lyα flux measurement . . . . . . . . . . . . . . . . . . . . . . . . .

3 Computing the Luminosity Function in Lensing Clusters

3.1 3D mask cubes in the source plane . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Noise level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Signal to noise of a given source . . . . . . . . . . . . . . . . . . . 3.1.3 Creating 3D masked images . . . . . . . . . . . . . . . . . . . . . . 3.2 Computing V max value . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

232 Bibliography 233 List of Figures

  6.1 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Future Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Luminosity function of line emissions observed with MUSE . . . 6.2.2 Luminosity function using data from James Web Space Telescope (JWST) and Euclid missions . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Global escape fraction of Lyman alpha photons as a function of redshift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.4 The ionizing photon production efficiency for LAEs using JWST and MUSE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conceptual figure of an LAE . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 An LAE spectrum obtained from MUSE/VLT . . . . . . . . . . . . . . . . 1.7 The evolution of Lyman-alpha Equivalent Width (EW) of instantaneous burst . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Mean spectra of LAEs with Lyα EW < 97.2 Å and > 97.2 Å . . . . . . . . 1.9 A rest frame spectrum of a star-forming galaxy at redshift z ∼ 3 ( A. E. Shapley et al. 2006) in the wavelength range of (800 -1500 Å). The break of the Ultra Violet (UV) continuum flux can be seen in the interval of 912 -1215 Å. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10 Observed spectral energy distributions of a quasar at redshift z ∼3.6 in the rest-frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.11 Spectrum of a quasar at redshift z = 6.13 . . . . . . . . . . . . . . . . . . 1.12 Measured quasar's spectra at different redshifts . . . . . . . . . . . . . . 1.13 Typical spectrum of a high redshift galaxy . . . . . . . . . . . . . . . . . Processing to build MUSE spectroscopic catalog using HST images and MUSE cube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.20 Source Inspector package interface . . . . . . . . . . . . . . . . . . . . . 2.21 Redshift distributions of sources in the sample . . . . . . . . . . . . . . . 2.22 Redshift distribution of zcon f = 1 sources behind 17 lensing clusters . 2.23 An example of choosing a representative image of multiple imaged system behind MACS0451 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.24 An example of asymetry value obtained from flux fitting method . . . . 2.25 The asymmetry distribution of Lyman alpha profiles . . . . . . . . . . . 2.26 A comparison between flux values obtained from fitting spectra and from Source Extractor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.27 A weighted magnification distribution of the present data sample . . . The LF points in each luminosity bin at four redshift ranges obtained after MCMC iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 The LF points in each redshift range using different bin sizes. . . . . . . 4.9 The completeness histogram of Lyman alpha sources (zcon f = 1) behind RXJ1347 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.10 Source distribution in each luminosity bin when zcon f = 1 sources are taken into account. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.11 LF points in each luminosity bin when half and all the zcon f = 1 sources have been included in the data sample. . . . . . . . . . . . . . . . . . . . Evolution of f escp as a function of redshift. The data points are taken from the literature listed in the insert. Credit: (M. Hayes, Daniel Schaerer, et al. 2011). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

	6 2.12 Source location of MUSE spectroscopic redshifts for the MACS0416 and So sánh các điểm hàm độ trưng ở khoảng dịch chuyển đỏ cao 3.12 Correlation between surveyed volume and magnification value of low
	nhất 5, 0 < z < 6, 7 với các dự đoán lý thuyết/số . . . . . . . . . . . . MACS0329 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . magnified sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
	1 2 3 4 5 6 7 8 1 2 3 7 1.1 A brief history of the Universe from the Big Bang . . . . . . . . . . . . . Comparison of the present data sample (magenta) to the previous sam-ple used in the DLV 2019 . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure showing the signal-to-noise level of objects detected in the sam-ple as a function of redshift. . . . . . . . . . . . . . . . . . . . . . . . . . . The source distribution with a redshift interval of 2.9 < z < 6.7 for a luminosity bin width of 0.25 in log scale . . . . . . . . . . . . . . . . . . . Completeness vs. detection flux of LAEs from the present sample. . . . The evolution of luminosity function with different redshift intervals . Comparison the luminosity function points at the highest redshift inter-val 5.0 < z < 6.7 to the theoretical/numerical predictions . . . . . . . . . The evolution of the slope value with redshift . . . . . . . . . . . . . . . The evolution of star formation rate density with redshift . . . . . . . . Comparaison entre l'échantillon de données utilisées dans ce travail (magenta) et les données précédentes utilisées dans DLV 2019 . . . . . Figure présentant le niveau signal à bruit des objets de l'échantillon en fonction du redshift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Distribution des sources dans l'intervalle de redshift 2.9 < z < 6.7 pour Tiến triển mật độ hình thành sao theo giá trị dịch chuyển đỏ . . . 1.2 Four categories of morphology of galaxies . . . . . . . . . . . . . . . . . 1.3 A schematic of the Circum Galactic Medium (CGM) . . . . . . . . . . . 1.4 Lyman-alpha and other lines are shown together with the Lyman and Balmer series of hydrogen atom . . . . . . . . . . . . . . . . . . . . . . . 1.5 1.14 Lyman break technique method . . . . . . . . . . . . . . . . . . . . . . . 1.15 Spectral Energy Distribution (SED) fitting technique using Hubble Space 2.13 An overview of multiple image systems behind MACS0940 . . . . . . . 4.1 Detection profiles of four different LAEs . . . . . . . . . . . . . . . . . . 2.14 An overview of multiple image systems behind MACS1206 . . . . . . . 4.2 Completeness vs. detection flux of LAEs from the present sample . . . 2.15 MUSE observation on MACS2214 . . . . . . . . . . . . . . . . . . . . . . 4.3 Source distribution in each luminosity bin before and after correction 2.16 An overview of multiple image systems behind RXJ1347 . . . . . . . . . for the completeness value . . . . . . . . . . . . . . . . . . . . . . . . . . 2.17 An overview of multiple image systems behind SMACS2031 . . . . . . . 4.4 Detection flux vs redshift for all sources in the present sample . . . . . 2.18 An overview of multiple image systems behind SMACS2131 . . . . . . . 4.5 Illustration of the contribution of a given source to the luminosity bin using Markov Chain Monte Carlo (MCMC) . . . . . . . . . . . . . . . . . 4.6 LF histograms of each luminosity bin obtained from 20'000 MCMC integrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.19 2.28 A comparison between luminosities obtained by applying central mag-4.7 4.12 The line fitting for four redshift ranges when 1% completeness cut (left) nification, flux fitting and weighted magnification combined with mea-and 10% completeness cut have been applied . . . . . . . . . . . . . . . sured flux from Source Extractor . . . . . . . . . . . . . . . . . . . . . . . 4.13 The evolution of LF with redshift . . . . . . . . . . . . . . . . . . . . . . . 2.29 A comparison of the present data sample with the one in the DLV 2019 4.14 Correlation of three parameters of the Schechter function for four red-une largeur du l'intervalle de luminosité de 0.25 en échelle logarithmique 4 Telescope (HST) photometry . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The procedure to create the 3D mask images and V max . . . . . . . . . . shift intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Corrélation entre la valeur de complétude et le flux détecté de sources 1.16 Illustration of LAE detection at redshift z = 6.96 using the Narrow Band 3.2 Evolution of noise level with wavelength inside each MUSE cube . . . . 4.15 Correlation between three free parameters of the Schechter function individuelles dans l'échantillon de données. . . . . . . . . . . . . . . . . 5 (NB) technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 An example of an individual bright pixel profile after convolved/de-within 68% confident level using different luminosity bin width values De gauche à droite, et de haut en bas, l'évolution de la fonction de 1.17 Evolution of Lyα LF with redshift z from 0.3 to 5.7 and 5.7 to 7.3 . . . . convolved to account for the different seeing conditions . . . . . . . . . 4.16 The evolution of LF with redshift using a completeness cut at 10% . . . luminosité avec différents intervalles de redshift . . . . . . . . . . . . . 6 Comparaison entre les points de fonction de luminosité obtenus dans l'intervalle de décalage vers le rouge le plus élevé 5.0 < z < 6.7 . . . . . 7 Evolution de la valeur de la pente avec le décalage vers le rouge . . . . . 8 Evolution de la densité du taux de formation d'étoiles avec le décalage vers le rouge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 So sánh dữ liệu trong luận án với dữ liệu của DLV 2019 . . . . . . 2 Thông lượng đo đạc và nhiễu theo dịch chuyển đỏ của các nguồn LAEs trong mẫu khảo sát hiện tại . . . . . . . . . . . . . . . . . . . . 3 Sự phân bố của nguồn có dịch chuyển đỏ trong khoảng 2.9 < z < 6.7 trong mỗi khoảng độ trưng có độ rộng 0.25 trong qui mô hàm logarit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Hệ số hoàn chỉnh với thông lượng thu được của các nguồn phát xạ vạch Lyman alpha trong công việc hiện tại. . . . . . . . . . . . . 5 Sự tiến triển hàm độ trưng theo dịch chuyển đỏ . . . . . . . . . . . 3.4 An example of the 9 brightest pixels found from a filtered image of a 4.17 Faint end slope of the LF for four redshift intervals excluding the LF 2.1 One of the telescope of the Very Large Telescope 8.2-metre . . . . . . . source detected behind cluster A2667 . . . . . . . . . . . . . . . . . . . . points at the faintest and brightest parts . . . . . . . . . . . . . . . . . . 2.2 The MUSE cube structure with 2 spatial dimensions and one wavelength 3.5 The evolution of Root Mean Square (RMS) maps at different channels 4.18 The LF points and their fits for the different redshift intervals including dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . inside a given cube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . previous literature data points that are at l og L > 43erg s -1 . . . . . . . . 2.3 The hydrogen filaments observed by MUSE in the Hubble Ultra Deep Field 3.6 RMS median images of 18 datacubes in the present work . . . . . . . . 4.19 Faint end slope at different redshift ranges derived from the present 2.4 The spatial distribution sources behind lensing cluster A2744 . . . . . . 3.7 The individual and general brightest pixel profiles are represented for work and literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 An example of finding extended Lyα haloes around high z SFGs . . . . each MUSE cube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.20 The LF points obtained in the present work at the highest redshift range 2.6 The spatial source distribution behind A2390 detected by MUSE . . . . 3.8 A 2D masked image of A2667 in the image plane is projected into the are compared to the model predictions . . . . . . . . . . . . . . . . . . . 2.7 Simulation of a MUSE observation covering the first Hubble Frontier Filed A2744 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 A MUSE exposure map of A370 mosaic observation . . . . . . . . . . . . 100 2.9 The RGB HST image of the AS1063 observed with MUSE . . . . . . . . . 101 2.10 HST observations using different filters of BULLET cluster . . . . . . . . 102 magnified sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (MACS)0257 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 3.11 Correlation between surveyed volume and magnification value of highly 2.11 An overview of multiple image systems behind MAssive Cluster Survey source plane at different redshifts . . . . . . . . . . . . . . . . . . . . . . 5.1 Cosmic evolution of the Star Formation Rate Density (SFRD) as a func-3.9 A 2D masked image of A2744 in the image plane is projected into the tion of redshift. The data points are taken from the literature listed in source plane at different redshifts . . . . . . . . . . . . . . . . . . . . . . the insert. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10 The V max distribution and its correlation with magnification, detected flux and luminosity value of individual source . . . . . . . . . . . . . . . 5.2
	53

  The first structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 1.2 Circum galactic medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 1.3 Epoch of re-ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 1.4 Recombination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 1.5 Spectral features of star-forming galaxies . . . . . . . . . . . . . . . . . . 72 1.5.1 The Lyman alpha line . . . . . . . . . . . . . . . . . . . . . . . . . 72 1.5.2 The Lyman Break galaxies . . . . . . . . . . . . . . . . . . . . . . . 77 1.5.3 Lyα forest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 1.5.4 The Gunn -Peterson effect . . . . . . . . . . . . . . . . . . . . . . 79 1.6 LBG technique and photometric redshifts . . . . . . . . . . . . . . . . . 81 1.7 Selection of LAEs: narrow-band technique . . . . . . . . . . . . . . . . . 84 1.8 Selection of LAEs: Integral Field Units . . . . . . . . . . . . . . . . . . . . 85 1.9 The galaxy luminosity function . . . . . . . . . . . . . . . . . . . . . . . . 86 1.10 Overview of this work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

	EW OB
		Equivalent Width. 54, 67, 73-76, 84, 85, 97 Observation Blocks. 102, 103
	FF PSF 1 Introduction Frontier Fields. 99, 100, 103, 110 Point Spread Function. 96
	1 FIR QSO Contents Total co-moving volume of 17 gravitational lensing clusters in the present work within a redshift range of 2.9 < z < 6.7. . . . . . . . . . . . . . . . . 17 Far Infrared. 105 Quasi-Stellar Object. 68, 72, 80
	2 3 1 2 3 SB GALACSI The best-fit values of the Schechter function at different redshift intervals. 20 Slope values of the Schechter function when applying different data samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 pour les 17 amas de galaxies utilisés comme lentilles gravitationnelles dans le présent travail. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 différents intervalles de redshift. . . . . . . . . . . . . . . . . . . . . . . . 33 Valeur de la pente de meilleur ajustement de la fonction de Schechter pour différentes sélections de données. . . . . . . . . . . . . . . . . . . . 36 103 Surface Brightness. 96, 108 Ground Atmospheric Layer Adaptive Corrector for Spectroscopic Imaging. 93, Les meilleures valeurs d'ajustement de la fonction de Schechter dans Root Mean Square. 55, 115, 126-128, 131, 133, 136, 137 Full Width at Half Maximum. 108, 121, 122 RMS FWHM Volume comobile effectif total, dans la plage de redshift de 2.9 < z < 6.7 FoV Right Ascension. 97, 110 Field of View. 101, 102, 116, 118 RA 1.1
	1 GLASS SED	Thể tích khảo sát của 17 cụm thiên hà thấu kính hấp dẫn trong khoảng dịch chuyển đỏ 2.9 < z < 6.7. . . . . . . . . . . . . . . . . . . 41
	2	Giá trị khớp hàm tốt nhất cho hàm Schechter ở bốn khoảng dịch Grism Lens Amplified Survey from Space. 99 Spectral Energy Distribution. 54, 72, 82-84, 103, 105
	3 GTO SFR		chuyển đỏ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Kết quả độ dốc hàm Schechter với các phép thử khác nhau . . . . 48
	Star formation Rate. 68, 103 Guarantee Time of Observation. 88, 91, 97, 98, 103, 106 2.1 Information of 17 lensing clusters (18 fields) observed by MUSE/VLT. . 2.2 Best fit parameters of mass distribution in each cluster . . . . . . . . . . 3.1 Total co-volume of 17 clusters at redshift 2.9 < z < 6.7 . . . . . . . . . . SFRD HDFS Hubble Deep Field Survey. 75 Star Formation Rate Density. 56, 71, 87, 90, 173-177, 230, 231
	4.1 Flag values based on a set of SExtractor parameters used for source recovery simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SIE HST Singular Isothermal Ellipsoid. 100 Hubble Space Telescope. 54, 55, 83, 89, 95, 96, 98, 99, 101-104, 107, 111, 116-118, 4.2 Luminosity bins and LF points with respect to different redshift intervals and different luminosity ranges . . . . . . . . . . . . . . . . . . . . . . . . 120 SIS
	4.3 Comparison slope values obtained from linear fitting using different completeness thresholds. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Best-fit parameter values for the Schechter function . . . . . . . . . . . Single Isothermal Sphere. 100 IFU Integral Field Unit. 85, 92 SMBH
	4.5 Summary of the best fit values of the faint end slope using different constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Results of the faint end slope α from different tests . . . . . . . . . . . . Super Massive Black Hole. 94 IGM Intergalactic Medium. 69-73, 77, 78, 80, 81, 87, 95, 121, 171, 172 SN
	ISM SNR	Signal to Noise. 116-118, 120, 147 Interstellar Medium. 72, 73, 77, 81, 171
		Signal to Noise Ratio. 115, 156

  ; J. .-. Paardekooper et al. 2013). Other sources, such as X-ray binaries and Active Galactic Nuclei (AGN), were then considered, but their abundance prevented them from making a significant contribution (Massimo Ricotti et al. 2004; McGreer 2013). There is a minimum contribution to the ionizing budget coming from AGN (Cowie 2009; Madau and Haardt 2015; Trebitsch et al. 2021). This is because the number density of AGN decreases dramatically at redshift z > 3 (Masters et al. 2012), and the escape fraction from ionizing photons seems not to be as high as needed (Micheva et al. 2017). The contribution of AGN to the Epoch of Reionization (EoR) becomes significant for a large number of AGN having a low luminosity. This was proposed by Giallongo et al. 2015; Grazian et al. 2018, and has been confirmed by recent observations conducted with JWST (Kocevski et al. 2023; Übler et al. 2023; Harikane et al. 2023). In addition, the contribution of AGN to the EoR can be discussed via its contribution to the LyC emission at the redshift range z = 2 -3 (Becker 2015; Madau and Haardt 2015;

1

  Introduction -1.5 Spectral features of star-forming galaxies M ⊙ with z ∼ 1 -4 (Shapley 2001; Elsner et al. 2008), dustier (Gawiser 2006; Gawiser 2007; Kornei et al. 2010), more extended in size, R ∼1.8 kpc at z ∼ 3 and decrease to R ∼0.8 at z ∼ 6 (Akiyama et al. 2008; Bouwens 2006).

2 The MUSE Lensing Project: from observations of massive clusters to LAE sample selection Contents

2.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 2.2 Main scientific goals of MUSE . . . . . . . . . . . . . . . . . . . . . . . . 94 2.3 Lensing clusters observed with MUSE/VLT . . . . . . . . . . . . . . . . . 97 2.3.1 General information of lensing clusters observed with MUSE/VLT 97 2.3.2 Lens models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 2.4 Source detection and catalog building . . . . . . . . . . . . . . . . . . . . 115 2.4.1 Source detection with MUSELET . . . . . . . . . . . . . . . . . . . 115 2.4.2 Redshift determination using Source Inspector . . . . . . . . . . 115 2.4.3 LAEs catalog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 2.4.4 Lyα flux measurement . . . . . . . . . . . . . . . . . . . . . . . . . 121

2

  The MUSE Lensing Project: from observations of massive clusters to LAE sample selection -2.3 Lensing clusters observed with MUSE/VLT

Table 2 .

 2

	N.	LAEs	7	14	128	41	20	11	24	16	45	32	21	19	48	49	17	72	20	16	600
	Seeing MUSE	value depth (hrs)	0.75 2	0.62 2	0.61 3.5-7	0.66 1.5-8.5	1.02 3.9	0.56 2	0.52 8	0.69 2.5	0.53 17	0.65 11-15	0.58 8	0.57 8	0.571 8	0.521 4-9	0.55 7	0.551 2-3	0.79 10	0.59 7	107-128
	Programme ID Notes		094.A-0115	094.A-0115	094.A-0115, 095.A-0181 MACS, FF	094.A-0115, 096.A-0710 FF	60.A-9345, 095.A-0653 FF	094.A-0115	099.A-0292, 0100.A-0249 MACS	096.A-0105 MACS, CLASH	094.A-0115, 0100.A-0763 MACS, CLASH	094.A-0525 MACS, CLASH	098.A-0502, 0104.A-0489 MACS	098.A-0502, 0104.A-0489 MACS	098.A-0502, 0101.A-0506 MACS	095.A-0181, 097.A-0269 MACS, CLASH	099.A-0292, 0101.A-0506 MACS	095.A-0525, 097.A-0909 MACS, CLASH	60.A-9100 MACS	0101.A-0506, 0102.A-0135 MACS
	Cluster RA DEC z		A2390 21:53:36.823 +17:41:43.59 0.228	A2667 23:52:28.400 -26:05:08.00 0.233	A2744 00:14:20.702 -30:24:00.63 0.308	A370 02:29:53.122 -01:34:56.14 0.375	AS1063 22:48:43.975 -44:31:51.16 0.348	BULLET 06:58:38.126 -55:57:25.87 0.296	MACS0257 02:57:41.070 -22:09:17.70 0.322	MACS0329 03:29:41.568 -02:11:46.41 0.450	MACS0416N 04:16:09.144 -24:04:02.95 0.397	MACS0416S 04:16:09.144 -24:04:02.95 0.397	MACS0451 04:51:54.647 +00:06:18.21 0.430	MACS0520 05:20:42.046 -13:28:47.58 0.336	MACS0940 09:40:53.698 +07:44:25.31 0.335	MACS1206 12:06:12.149 -08:48:03.37 0.438	MACS2214 22:14:57.292 -14:00:12.91 0.502	RXJ1347 13:47:30.617 -11:45:09.51 0.451	SMACS2031 20:31:53.256 -40:37:30.79 0.331	SMACS2131 21:31:04.831 -40:19:20.92 0.442	Total:

1: Information of 17 lensing clusters (18 fields) observed by MUSE/VLT.

  Kneib et al. 2011, for more details. In the following section, I will introduce a general picture of the mass model technique. The details of the method referrto the recent work presented inMahler et al. 2018; Lagattuta et al. 2019, etc., with an original review performed in Schneider et al. 1992; Kneib et al. 2011.

  al. 2015; Bina et al. 2016; Mahler et al. 2018; Lagattuta et al. 2019, among others.

2

  The MUSE Lensing Project: from observations of massive clusters to LAE sample selection -2.3 Lensing clusters observed with MUSE/VLT about the DM distribution.

3 Computing the Luminosity Function in Lensing Clusters Contents 3

  .1 3D mask cubes in the source plane . . . . . . . . . . . . . . . . . . . . . . 126 3.1.1 Noise level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 3.1.2 Signal to noise of a given source . . . . . . . . . . . . . . . . . . . 127 3.1.3 Creating 3D masked images . . . . . . . . . . . . . . . . . . . . . . 137 3.2 Computing V max value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

Table 3 .

 3 .1. 3 Computing the Luminosity Function in Lensing Clusters -3.2 Computing V max value 1: Total co-volume of 17 clusters at redshift 2.9 < z < 6.7. Completeness determination . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Reconstruction of the source profile in image plane . . . . . . . 4.1.2 Source recovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 LF computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Binning effect on the LF points . . . . . . . . . . . . . . . . . . . . 4.2.2 Effect of source selection to the evolution of LF points . . . . . . 4.3 LF results and comparison to the literature . . . . . . . . . . . . . . . . . 4.4 Fitting with of a Schechter function . . . . . . . . . . . . . . . . . . . . . 4.5 Comparison with theoretical predictions . . . . . . . . . . . . . . . . . . By definition, the LF

	Cluster	Total co-volume [Mpc 3 ]
	A2390	735
	A2667	885
	A2744	10500
	A370	5350
	AS1063	1970
	BULLET	895
	MACS0257	730
	MACS0329	1225
	MACS0416N	3420
	MACS0416S	1670
	MACS0451	1210
	MACS0520	765
	MACS0940	5760
	MACS1206	2980
	MACS2214	1100
	RXJ1347	7920
	SMACS2031	1675
	SMACS2131	920
	Total	49710

Table 4 .

 4 1: Flag values based on a set of SExtractor parameters used for source recovery simulation.

		4 Luminosity function -4.1 Completeness determination
	Flag value MIN_AREA DETECT_THRESH MATCHING_RADIUS
	1	6	1.3	4
	1	3	2.5	4
	1	6	1.1	4
	2	6	1.0	4
	2	6	1.0	5
	2	6	0.9	5
	2	5	0.9	5
	2	6	0.8	5
	2	5	0.8	5
	2	6	0.8	6
	2	5	0.8	6
	2	4	0.8	5
	3	5	0.8	7
	3	5	0.7	7
	3	2	0.3	4

Table 4 .

 4 2: Luminosity bins and LF points used for Fig. 4.13 log

Table 4 .

 4 3: Comparison slope values obtained from linear fitting using different completeness thresholds. repudiating 62 additional sources from the present sample, with 51 of them belonging to the six faintest luminosity bins. This significantly reduces the number of sources in these bins. Comparing the slope value in both cases, there is a noticeable decrease at a higher completeness threshold, i.e., 23%, 22%, 19%, and 16% with respect to the redshift z al l , z 35 , z 45 and z 60 . The results are presented in Table 4.3.

	4 Luminosity function -4.4 Fitting with of a Schechter function
	Redshift range 1% completeness cut 10% completeness cut
	2.9 < z < 6.7 2.9 < z < 4.0 4.0 < z < 5.0 5.0 < z < 6.7	-2.12 +0.15 -0.16 -1.76 +0.26 -0.27 -1.99 +0.12 -0.17 -1.96 +0.50 -0.54	-1.64 +0.12 -0.12 -1.55 +0.17 -0.17 -1.63 +0.21 -0.21 -1.66 +0.22 -0.24

  +0.07 -0.05 is consistent with the values measured by Herenz et al. 2019 and Drake et al. 2017. The sudden enhancement at the luminosity bin log L ∼ 42 at redshift ranges z al l

	39	40	41	42	43	44

Table 4 .

 4

	4: Best fit parameter values for the Schechter function
	Redshift	Φ * [10 -4 Mpc -3 ] log L * [erg s -1 ]	α
	2.9 < z < 6.7 2.9 < z < 4.0 4.0 < z < 5.0 5.0 < z < 6.7	7.41 +2.70 -2.20 6.56 +3.20 -2.40 4.06 +2.70 -1.70 3.49 +2.11 -1.50	42.85 +0.10 -0.10 42.87 +0.11 -0.1 42.97 +0.13 -0.11 43.09 +0.10 -0.08	-2.06 +0.07 -0.05 -2.00 +0.07 -0.07 -1.97 +0.09 -0.08 -2.28 +0.12 -0.12

  , Illingworth, Oesch, H. Atek, et al. 2017; Atek 2018; Priewe et al. 2017; Meneghetti et al. 2017. Its impact could be particularly notable within the faint end luminosity regime. As discussed in DLV 2019, its contribution reaches approximately 15% at log(L) of 40.5 erg s -1 , particularly in the case of the lensing field A2744.

4

  Luminosity function -4.4 Fitting with of a Schechter function -2.06±0.60 corresponding to the redshift ranges z 35 , z 45 , z 60 and z al l . The final results are shown in Fig. 4.19 together with results from the literature. The horizontal error bars represent the redshift range of the survey. The results display a slight increase in the slope with redshift, but the uncertainties remain large. The results are presented in Table 4.6.

		4 3 . 4
	4 3 . 2 * )
	log(L	4 3 . 0
		4 2 . 8
		4 2 . 6
		1 . 8
	-
		2 . 0
	-α
		2 . 2
	-
		2 . 4
	0 φ -4	8	1 2	1 6 4 2 . 6	4 2 . 8	log(L * ) 4 3 . 0 4 3 . 2	-4 3 . 4 2 . 4 -2 . 2 -2 . 0 -1 . 8

* [10 -4 .Mpc

-3 ]

5 Star formation rate density and implications for the reionization Contents 5

  .1 Star formation rate density . . . . . . . . . . . . . . . . . . . . . . . . . . 173 5.2 Articles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

  In addition, hydrodynamical simulations suggest that f escp is probably anisotropic and time-dependent (Renyue[START_REF] Cen | QUANTIFYING DISTRIBUTIONS OF THE LYMAN CONTINUUM ESCAPE FRACTION[END_REF][START_REF] Ma | The difficulty of getting high escape fractions of ionizing photons from high-redshift galaxies: a view from the FIRE cosmological simulations[END_REF]. It depends on the characteristics of galaxies, such as a potentially higher f escp for galaxies with lower mass, as demonstrated by J.-P.[START_REF] Paardekooper | The First Billion Years project: the escape fraction of ionizing photons in the epoch of reionization[END_REF]. This could impact the process of reionization[START_REF] Sharma | The brighter galaxies reionized the Universe[END_REF] The relationship between the Lyman alpha escape fraction and redshift is important for constraining the reionization history of the Universe. M. Hayes,Daniel Schaerer, et al. 2011 observed an evolution trend of f

	where ξ = 2.57 +0.	Lyα escp with redshift, described by f	Lyα escp ∝ (1+z) ξ ,

, J. et al. 2018. Recent results show that the escape fraction of ionizing photon f escp , typically falls within the range of 10 to 20%. This value remains consistent regardless of the mass or luminosity of the galaxy (Choudhury et al. 2015; Robertson, Ellis, et al. 2015).

It is difficult to directly measure f escp at high redshift (z > 6) due to the substantial opacity of the IGM towards ionizing photons

[START_REF] Akio | An updated analytic model for attenuation by the intergalactic medium[END_REF]

). However, computing the escape fraction of the ionizing photons is still a subject of debate, and more data is needed to constrain its value. In theory, this can be done by focusing on the absorption of the photons as they travel through the interstellar medium. Some recent results include: Wood et al. 2000 found a rough value of f escp < 1% at redshift z ∼ 10, suggesting that this value decreases with increasing redshift due to the increasing density of galactic discs.

[START_REF] Razoumov | Escape of Ionizing Radiation from Star-forming Regions in Young Galaxies[END_REF]

found that f escp ∼ 1 -2% at z = 2.39 and f escp = 6 -10% at z = 3.6 using the galaxy formation simulations combined with 3D radiative transfer.

  al. 2015; Claeyssens et al. 2022; de la Vieuville et al. 2019; Richard et al. 2021).

  +0.42 -0.41 , logL * [erg s -1 ] = 42.2 +0.22 -0.16 and logΦ * [Mpc -3 ] = -2.71 and found no evolution of the LF in the explored redshift range of 2.9 < z < 6.7. The Lensed Lyman-Alpha MUSE Arcs Sample (LLAMAS) project, which is part of MUSE Lensing Cluster Survey (PI: J. Richard, Richard et al. 2021, Claeyssens et al. 2022), helps to reach the fainter luminosity regime, thanks to magnification from strong gravitational lensing. DLV19 studied a sample of four lensing clusters, including 156 LAEs with luminosities in the range 39 < log(L)[erg s -1 ] < 43. The present work extends the previous study to a sample of 17 clusters presented in Claeyssens et al. (

Table 1 .

 1 al. (2021; hereafter R21). Information of 17 lensing clusters (18 fields) observed by MUSE/VLT.

	A139, page 2 of 20

  Thai, T. T., et al.: A&A 678, A139 (2023)

Table 2 .

 2 Total co-volume of 17 clusters (2.9 < z < 6.7).

	Cluster	Total co-volume [Mpc 3 ]
	A2390 A2667 A2744 A370 AS1063 BULLET MACS0257 MACS0329 MACS0416N MACS0416S MACS0451 MACS0520 MACS0940 MACS1206 MACS2214 RXJ1347 SMACS2031 SMACS2131 Total:	735 885 10500 5350 1970 895 730 1225 3420 1670 1210 765 5760 2980 1100 7920 1675 920

Table 3 .

 3 Best-fit parameter values for the Schechter function. Redshift Φ * [10 -4 Mpc -3 ] log L * [erg s -1 ] α Herenz et al. (2019). The Φ * value strongly depends on the literature data points used for the fitting procedure. Luminosity bins and LF points for each redshift interval are given in Table

	2.9 < z < 6.7 2.9 < z < 4.0 4.0 < z < 5.0 5.0 < z < 6.7	7.41 +2.70 -2.20 6.56 +3.20 -2.40 4.06 +2.70 -1.70 3.49 +2.11 -1.50	42.85 +0.10 -0.10 42.87 +0.11 -0.1 42.97 +0.13 -0.11 43.09 +0.10 -0.08	-2.06 +0.07 -0.05 -2.00 +0.07 -0.07 -1.97 +0.09 -0.08 -2.28 +0.12 -0.12
	consistent with DLV19 and Sobral et al. (2018), but smaller than that of

Table 5 .

 5 Thai, T. T., et al.: A&A 678, A139 (2023) Results of the faint end slope α from different tests.

	z 35	z 45	z 60	z all

  Thai, T. T., et al.: A&A 678, A139 (2023)

  al. 2009; Robertson et al. 2013, 2015; Bouwens et al. 2015b; Gorce et al. 2018). The dependence of Lyα escape fraction on redshift is an important quantity as it helps to constrain the reionization history of the Universe. The Lyα escape fraction f Lyα escp and f escp are expected to be correlated (see e.g., Dijkstra et al. 2016; Izotov et al. 2020), Hayes et al. (2011) reported an evolution as a function of redshift as follows f

	Lyα escp ∝ (1 + z) ξ , with ξ = 2.57 +0.19 -0.12

  1 and 2 are available at the CDS via anonymous ftp to cdsarc.cds.unistra.fr (130.79.128.5) or via https:// cdsarc.cds.unistra.fr/viz-bin/cat/J/A+A/678/A174 Currently, the favoured candidate is SFGs, particularly faint SFGs (Robertson et al. 2013; Bouwens et al. 2015b; Stark 2016), although the possibly significant contribution of bright SFGs is still debated (Naidu et al. 2020; Matthee et al. 2022).

	1. Introduction
	After the Dark Ages, the neutral gas in the intergalactic medium (IGM) was reionised: the last phase transition undergone by the Universe. This process ended with the hydrogen in the IGM ionised, around z = 6 (Fan et al. 2006; McGreer et al. 2015; Planck Collaboration VI 2020; Lu et al. 2022). There are two main candidates thought to be responsible for this process, star-forming galaxies (SFGs; Bouwens et al. 2015a; Finkelstein et al. 2015; Livermore et al. 2017) and active galac-tic nuclei (Madau & Haardt 2015; Grazian et al. 2018). The influence of active galactic nuclei is likely small (Onoue et al. 2017; Parsa et al. 2018; McGreer et al. 2018; Jiang et al. 2022).

Tables

  al. 2016; Smit et al. 2017; de La Vieuville et al. 2019, 2020; Richard et al. 2021; Claeyssens et al. 2022, the last four of which are henceforth dLV19, dLV20, R21, and C22

  al. 2018b; Arrabal Haro et al. 2018; Kusakabe et al. 2020; Leonova et al. 2022; Bolan et al. 2022). Studies show a drop in the prevalence of LAEs among LBGs (X LAE ) above z = 6, suggesting an increasing neutrality of the IGM before this time and supporting the established reionisation timeline (Stark et al. 2010, 2011; Pentericci et al. 2011, 2018; Caruana et al. 2014, 2018; De Barros et al. 2017; Hoag et al. 2019a; dLV20).

  al. 2008; Zheng et al. 2010; Dijkstra et al. 2011; Kakiichi et al. 2016); however, there is still significant debate about the physics of Lyman-α photon escape, in simulations as well as observations, at different redshifts and how this impacts LAE visibility and hence X LAE (Dayal et al. 2011; Matthee et al. 2016; Hutter et al. 2014; Sobral & Matthee 2019; Smith et al. 2022). Additionally, X LAE has shown some dependence on the absolute rest-frame UV magnitude (Stark et al. 2010, 2011; Schaerer et al. 2011; Schenker et al. 2014; Kusakabe et al. 2020) and a significant dependence on the Lyman-α EW cut above which the X LAE is calculated (Stark et al. 2010; Caruana et al. 2018; Kusakabe et al. 2020). It is crucial to be aware of these factors when comparing results in the literature.

  al. 2010; Pentericci et al. 2011; Bouwens et al. 2015a, 2021; Pentericci et al. 2018; Yoshioka et al. 2022), and some by photometric redshifts (Caruana et al. 2018; Fuller et al. 2020;

  al. 2011; Stark et al. 2011; De Barros et al. 2017; Arrabal Haro et al. 2018; Caruana et al. 2018; Fuller et al. 2020; dLV20).

Table 1 .

 1 Details of each lensing cluster used in this study.

	Cluster	RA	Dec	Redshift MUSE exposure time (h) Co-volume (Mpc 3 ) LAE-only LBG-only LAE+LBG LAE+continuum
	A2744 A370 AS1063 22:48:43.975 -26:05:08.00 0.348 00:14:20.702 -30:24:00.63 0.308 02:29:53.122 -01:34:56.14 0.375 M0416S 04:16:09.144 -24:04:02.95 0.397 M0416N 04:16:09.144 -24:04:02.95 0.397 Total	3.5-7 1.5-8.5 3.9 11-15 17	10 500 5350 1970 1670 3420 22 910	24 8 6 4 13 55	294 267 159 81 71 872	47 20 7 14 12 100	50 14 8 16 20 108

  A174, page 4 of 13 Goovaerts, I., et al.: A&A 678, A174 (2023) in blank fields lies around log 10 (L Lyα ) ∼ 42 (see, for example, Herenz et al. 2019), down to log 10 (L Lyα ) ∼ 41.5 in the MUSE Ultra Deep Field (Drake et al. 2017).

Table 2 .

 2 5σ limiting magnitudes of each filter for the four clusters.Notes.Values have been calculated from the properties of the sample and adopted for the SED fitting described in Sect. 2.3. Dashes indicate missing filters for a particular cluster.

	Filter	A2744 A370 AS1063 M0416
	F225W F275W F336W F390W F435W F475W F606W F625W F775W F814W F850LP F105W F110W F125W F140W F160W K S IRAC1 IRAC2	-26.0 26.7 -27.5 -27.7 --27.8 -27.9 -27.4 27.4 26.8 25.1 24.1 24.3	-25.7 26.2 -27.2 26.6 27.3 25.6 -27.5 -27.2 26.6 27.0 27.1 26.9 24.9 23.6 23.7	24.0 25.9 26.3 25.4 27.3 25.8 27.5 25.5 25.5 27.7 25.2 27.6 26.4 27.4 27.3 26.9 25.1 23.6 24.2	24.4 25.7 26.2 25.7 27.5 26.2 27.7 25.9 25.6 27.9 25.2 27.7 26.3 27.6 27.7 27.4 25.1 24.7 24.2

  ; Hayes et al. 2011, 2013; Sobral & Matthee 2019; Santos et al. 2020).

  al. 2011; Schenker et al. 2014; De Barros et al. 2017; Mason et al. 2018b; Pentericci et al. 2018; Arrabal Haro et al. 2018; dLV20 among

  et al. 2010, 2017; Oesch et al. 2015; De Barros et al. 2017; Kusakabe et al. 2020). In particular, this debate has A174, page 10 of 13 Goovaerts, I., et al.: A&A 678, A174 (2023)

  Our findings are especially interesting in the context of the number of low EW Lyα , UV-bright objects reported towards and into the epoch of reionisation (Curtis-Lake et al. 2012; Oesch et al. 2015; Stark et al. 2017; Mason et al. 2018a,b; Larson et al. 2022; Bunker et al. 2023; Witten et al. 2023, see also

  al. 2010, 2011; De Barros et al. 2017; Goovaerts et al. 2023). They have been used extensively as probes of the IGM neutral fraction and therefore the progression of reionisation (see, in addition to the studies above: Stark et al. 2010; Pentericci et al. 2011; Arrabal Haro et al. 2018; Pen-tericci et al. 2018; Caruana et al. 2018; de La Vieuville et al. 2020; Kusakabe et al. 2020; Bolan et al. 2022).

  -1 (Richard et al. 2021; Claeyssens et al. 2022) and UV-selected sources have been detected down to M UV ∼ -14 (de La Vieuville et al. 2020; Bouwens et al. 2022; Goovaerts et al. 2023). Using gravitational lensing and the wavelength coverage of NIRCam now allows us to place intrinsically faint, very low-mass LAEs on the star formation main sequence in a robust manner.The star formation (or galaxy) main sequence (henceforth MS), is a typically tight relation between the stellar mass and star formation rate of galaxies with the form: the slope of the relation, and β, the normalisation, being fit to the data: SFR, the star formation rate, and M ⋆ , the stellar mass. Between 10 8 < M ⋆ /M ⊙ < 10 12 and 0 < z < 6 the MS is well understood, even taking into account calibrations between different studies related to chosen IMF, chosen cosmology, selection biases and SFR indicator (see primarily the reviews of Previous studies have made inroads into the low-mass end of the galaxy main sequence, using lensing, such asSantini et al. (2009Santini et al. ( , 2017) ) and at low redshift, such as Boogaard et al. (2018). The high-redshift studies however, were performed without NIRCam data, relying on Spitzer/IRAC for their coverage past 1600 Å. More recently, Looser et al. (

		A&A proofs: manuscript no. output
	log SFR = α log M ⋆ + β	(1)
	with α, Speagle et al. 2014 (S14) and Popesso et al. 2023 (P23), as well as Brinchmann et al. (2004); Daddi et al. (2007); Elbaz et al. (2007); Santini et al. (2009); Elbaz et al. (2011); Sobral et al. (2014); Steinhardt et al. (2014)). Plotting galaxies on the MS gives an indication of what kind of star-formation a given galaxy is undergoing, be it a starbursting galaxy or a quenched galaxy with little to no ongoing star formation.
		Article number, page 1 of 7

Table 1 :

 1 Results for the galaxy main sequence from the literature and from this study

	Study S14 S17 S17 S17 This work: LAEs 25 < z < 6.6 Redshift 0 < z < 6 3 < z < 4 4 < z < 5	a 0.84 ± 0.02 -0.026 ± 0.003 6.51 ± 0.24 b c 1.02 ± 0.04 -8.52 ± 0.42 0.94 ± 0.06 -7.75 ± 0.63 0.92 ± 0.15 -6.93 ± 1.59	d -0.11 ± 0.03 ---

6 Conclusions and perspective futures Contents 6

  .1 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 229 6.2 Future Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 6.2.1 Luminosity function of line emissions observed with MUSE . . . 231 6.2.2 Luminosity function using data from JWST and Euclid missions 232 6.2.3 Global escape fraction of Lyman alpha photons as a function of redshift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 6.2.4 The ionizing photon production efficiency for LAEs using JWST and MUSE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

2023 Asia-Pacific Regional IAU Meeting (APRIM). Japan, August 2023, oral presentation.

2. Shedding the new light on the first billion years of the Universe. France, July 2023, oral

presentation.3. The 16th International Symposium on Origin of Matter and Evolution of Galaxies.Vietnam, October 2022, oral presentation.

The MUSE Lensing Project: from observations of massive clusters to LAE sample selection -2.1 General

The MUSE Lensing Project: from observations of massive clusters to LAE sample

The MUSE Lensing Project: from observations of massive clusters to LAE sample selection -2.2 Main scientific goals of MUSE

The MUSE Lensing Project: from observations of massive clusters to LAE sample selection -2.3 Lensing clusters observed with MUSE/VLT

The MUSE Lensing Project: from observations of massive clusters to LAE sample selection -2.3 Lensing clusters observed with MUSE/VLTTable 2.2: Best fit parameters of mass distribution in each cluster Continued on next page

The MUSE Lensing Project: from observations of massive clusters to LAE sample selection -2.3 Lensing clusters observed with MUSE/VLTTable 2.2 continued from previous page

The MUSE Lensing Project: from observations of massive clusters to LAE sample selection -2.4 Source detection and catalog building

Star formation rate density and implications for the reionization -5.1 Star

https://cral-perso.univ-lyon1.fr/labo/perso/johan. richard/MUSE_data_release/

https://mpdaf.readthedocs.io/en/latest/muselet.html A174, page 3 of 13

https://mpdaf.readthedocs.io/en/latest/muselet.html

https://cral-perso.univ-lyon1.fr/labo/perso/johan. richard/MUSE_data_release/

https://jwst-uncover.github.io Article number, page 2 of 7 I. Goovaerts et al.: Properties of faint SFGs

https://cigale.lam.fr

Acknowledgements. We are grateful to the referee for a very careful reading of the manuscript and pertinent comments that helped greatly with improving the substance of the present article. This work is done based on observations made with ESO Telescopes at the La Silla Paranal Observatory under programme IDs 060.A-9345, 094.A-0115, 095.A-0181, 096.A-0710, 097.A0269, 100.A-0249, A139, page 17 of 20 Thai, T. T., et al.: A&A 678, A139 (2023) and 294.A-5032. Also based on observations obtained with the NASA/ESA Hubble Space Telescope, retrieved from the Mikulski Archive for Space Telescopes (MAST) at the Space Telescope Science Institute (STScI). STScI is operated by the Association of Universities for Research in Astronomy, Inc. under NASA contract NAS 5-26555. This research made use of Astropy, a communitydeveloped core Python package for Astronomy (Astropy Collaboration 2013). All plots in this paper were created using Matplotlib (Hunter 2007). Financial support from the World Laboratory, the Odon Vallet Foundation and VNSC is gratefully acknowledged. Tran Thi Thai was funded by Vingroup JSC and supported by the Master, PhD Scholarship Programme of Vingroup Innovation Foundation (VINIF), Institute of Big Data, code VINIF.2022.TS.107. This work received support from the French government under the France 2030 investment plan, as part of the Excellence Initiative of Aix-Marseille University -A*MIDEX (AMX-19-IET-008 -IPhU). A139, page 18 of 20 Thai, T. T., et al.: A&A 678, A139 (2023) Acknowledgements. This work is done based on observations made with ESO Telescopes at the La Silla Paranal Observatory under programme IDs 060.A-9345, 092.A-0472, 094.A-0115, 095.A-0181, 096.A-0710, 097.A0269, 100.A-0249, and 294.A-5032. Also based on observations obtained with the NASA/ESA Hubble Space Telescope, retrieved from the Mikulski Archive for Space Telescopes (MAST) at the Space Telescope Science Institute (STScI). STScI is operated by the Association of Universities for Research in Astronomy, Inc. under NASA contract NAS 5-26555. All plots in this paper were created using Matplotlib (Hunter 2007). Part of this work was supported by the French CNRS, the Aix-Marseille University, the French Programme National de Cosmologie et Galaxies (PNCG) of CNRS/INSU with INP and IN2P3, co-funded by CEA and CNES. This work also received support from the French government under the France 2030 investment plan, as part of the Excellence Initiative of Aix-Marseille University -A*MIDEX (AMX-19-IET-008 -IPhU). Financial support from the World Laboratory, the Odon Vallet Foundation and VNSC is gratefully acknowledged. Tran Thi Thai was funded by Vingroup JSC and supported by the Master, PhD Scholarship Programme of Vingroup Innovation Foundation (VINIF), Institute of Big Data, code VINIF.2022.TS.107. Acknowledgements. This work is done based on observations made with ESO Telescopes at the La Silla Paranal Observatory under programme IDs 060.A-9345, 092.A-0472, 094.A-0115, 095.A-0181, 096.A-0710, 097.A0269, 100.A-0249, and 294.A-5032. Also based on observations obtained with the NASA/ESA Hubble Space Telescope, retrieved from the Mikulski Archive for Space Telescopes (MAST) at the Space Telescope Science Institute (STScI). STScI is operated by the Association of Universities for Research in Astronomy, Inc. under NASA contract NAS 5-26555. All plots in this paper were created using Matplotlib (Hunter 2007). Part of this work was supported by the French CNRS, the Aix-Marseille University, the French Programme National de Cosmologie et Galaxies (PNCG) of CNRS/INSU with INP and IN2P3, cofunded by CEA and CNES. This work also received support from the French government under the France 2030 investment plan, as part of the Excellence Initiative of Aix-Marseille University -A*MIDEX (AMX-19-IET-008 -IPhU). Financial support from the World Laboratory, the Odon Vallet Foundation and VNSC is gratefully acknowledged. Tran Thi Thai was funded by Vingroup JSC and supported by the Master, PhD Scholarship Programme of Vingroup Innovation Foundation (VINIF), Institute of Big Data, code VINIF.2022.TS.107. ACKNOWLEDGMENT This work has been performed using observations by the ESO Telescopes at the La Silla Paranal Observatory under programme IDs 060.A-9345, 094.A-0115, 095.A-0181, 096.A-0710, 097.A0269, 100.A-0249, and 294.A-5032 and by the NASA/ESA Hubble Space Telescope, retrieved from the Mikulski Archive for Space Telescopes (MAST) at the Space Telescope Science Institute (STScI). STScI is operated by the Association of Universities for Research in Astronomy, Inc. under NASA contract NAS 5-26555. This research made use of Matplotlib (Hunter 2007) for plotting figures. Financial support from the World Laboratory, the Odon Vallet Foundation and VNSC is gratefully acknowledged. This research is funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 103.99-2019.368. Tran Thi Thai was funded by Vingroup JSC and supported by the Master, PhD Scholarship Programme of Vingroup Innovation Foundation (VINIF), Institute of Big Data, code VINIF.2022.TS.107.

work presented in this manuscript is my own work, carried out under the scientific supervision of Roser PELLO and Tuan-Anh PHAM, in accordance with the principles of honesty, integrity and responsibility inherent to the research mission. The research work and the writing of this manuscript have been carried out in compliance with both the french national charter for Research Integrity and the Aix-Marseille University

This work (Linf = 41.0) This work (Linf = 39.5) Cassata(2011) (Linf = 41.3) Drake(2017) (Linf = 41) Ouchi(2008) (full integration) Wyder(2005) VanderBurg(2010) McLure(2009) Reddy(2009) Bouwens(2015,Mlim = -17) Bouwens(2015,Mlim = -13) Infante(2015) Schiminovich(2005)

this data, we expected to have better coverage towards the faint part of the LF, as well as an improved the statistic in this domain. The correlation of three free parameters of the Schechter function within a 68% confidence level. From left to right, up to down corresponds to the redshift intervals 2.9 < z < 6.7, 2.9 < z < 4.0, 4.0 < z < 5.0, and 5.0 < z < 6.7, respectively. The tuples denote the lower and upper limits of the luminosity range with respect to the number of the bin. In descending order, the graphs depict the comparison of the age of the main stellar population, the stellar mass, the SFR and the χ 2 ν statistic. Grey dots indicate galaxies for which the single burst model is a better fit and coloured dots indicate galaxies for which the double burst model is better. Red dots indicate galaxies for which the double burst model is significantly better than the single burst model. Error bars on these values are described in the caption of Fig. 1.

Here we give further details of the SED fitting we performed with CIGALE, in particular, the checks we performed to ensure that this fitting process results in reliable stellar mass and SFR estimates. First, we depict in