Emanuele Boattini

Frank Smallenburg

Laura Filion

Averaging local structure to predict the dynamic propensity in supercooled liquids

Predicting the local dynamics of supercooled liquids based purely on local structure is a key challenge in our quest for understanding glassy materials. Recent years have seen an explosion of methods for making such a prediction, often via the application of increasingly complex machine learning techniques. The best predictions so far have involved so-called Graph Neural Networks (GNN) whose accuracy comes at a cost of models that involve on the order of 10 5 fit parameters. In this Letter, we propose that the key structural ingredient to the GNN method is its ability to consider not only the local structure around a central particle, but also averaged structural features centered around nearby particles. We demonstrate that this insight can be exploited to design a significantly more efficient model that provides essentially the same predictive power at a fraction of the computational complexity (approximately 1000 fit parameters), and demonstrate its success by fitting the dynamic propensity of Kob-Andersen and binary hard-sphere mixtures. We then use this to make predictions regarding the importance of radial and angular descriptors in the dynamics of both models.

Unraveling the interplay between structure and dynamics in glassy materials is a major challenge in condensed matter science. When a liquid is rapidly cooled down or compressed to the point where it almost turns into a glass, its dynamics slow down by many orders of magnitude, while its structure typically stays largely unchanged. The dynamics of such glassy fluids are heterogeneous, with some regions of particles rearranging much more rapidly than others [1,[START_REF] Berthier | Dynamical heterogeneities in glasses, colloids, and granular media[END_REF]. Key to understanding this phenomenon is identifying structural characteristics that are associated with these heterogeneities [START_REF] Royall | [END_REF]4].

Traditionally, correlating structure and dynamics relied on physical intuition: one can look for local structural motifs [5][6][7][8][9] -such as icosahedra or tetrahedra -or for other (structure-dependent) local physical features of the system [10][11][12][13] -such as local density, or potential energy -that are expected to play a key role in determining the dynamics. A novel approach was pioneered in 2015 by Cubuk et al. [14], who demonstrated that machine learning (ML) techniques could be trained to identify slow and fast regions in a glassy liquid. Since then, a number of works have demonstrated the power of both supervised and unsupervised machine learning for correlating structure and dynamics in a variety of glassy systems [14][15][16][17][START_REF] Schoenholz | Proc. Natl. Acad. Sci. USA[END_REF][START_REF] Landes | [END_REF].

In recent years, many studies attempting to unravel the link between structure and dynamics have tried to reveal structural quantities capable of predicting the dynamic propensity [4-6, 11, 15, 16, 20-24]. The dynamic propensity of a particle is the absolute [33] distance that the particle moves over a given time interval, averaged over many simulated trajectories starting from the same initial configuration. In essence, it provides a measure of the future mobility of a particle, as governed by the structure of its surroundings [25]. To date, the most accurate predictions of the dynamic propensity in glassy systems were achieved by Bapst et al. [23] using a highly advanced machine learning approach: Graph Neural Networks (GNN). While the GNN is quite accurate, its design philosophy draws on physical insights as little as possible, resulting in a high computational complexity, and making drawing physical insights from the fitted model more difficult. Specifically, since the GNN designs its own structural descriptors, training it requires optimizing an enormous number of parameters (∼ 70000 in Ref. [23]). As a result, the training requires both a large data set (to avoid overfitting) and significant computational effort. Nonetheless, the predictive power of GNNs clearly indicates that their model architecture is able to capture the essential physics required to predict dynamics from local structure. This raises two important questions that we address in this letter: can we learn from the success of GNNs what structural features we need to consider to accurately predict local dynamics, and can we exploit these observations to design a significantly more efficient model that performs equally well?

To address the first question, we need to consider the architecture of a GNN and compare it to simpler machine learning approaches. In most ML approaches, such as the support vector machines (SVMs) used in Refs. 14, 17, the environment of each particle is captured via set of handcrafted structure functions centered around the particle under consideration, which provide information about e.g. the radial density profile and bond angle distribution. Subsequently, a model is fitted that relates these structural descriptors to a dynamical descriptor of each particle. In contrast, in the GNN used in Ref. 23, the input is a graph, where each node represents a particle, and particles closer than a certain cutoff distance are connected by an edge which carries as information the vector connecting the two particles. After an encoding step, this information is then passed through a number of recursive iterations. In each iteration, the graph is mapped to a new graph with the same topology, but arXiv:2105.05921v1 [cond-mat.soft] 12 May 2021 with updated information on the nodes and edges, where the mappings are nonlinear functions described by neural networks. Finally, a "decoder" step is used to produce a prediction for the desired dynamical descriptor -in the case of Ref. 23 the dynamic propensity.

The key trick of the GNN lies in the fact that in each successive iteration, information from further away is incorporated into the information for a given node or edgein an average sense. Hence, in contrast to the more standard hand-crafted descriptors which are generally only centered on the particle under consideration, the GNN designs its own descriptors, that can in principle take into account averaged structural features at significant distances away. As shown in the recent publication by Bapst et al [23], this strategy works very well at fitting the dynamics of a supercooled Kob-Andersen mixture, clearly outperforming all previous algorithms. This raises the question: would incorporating the shellaveraging concept of GNNs into handcrafted descriptors lead to similar predictive power? Here we design a set of descriptors that explicitly incorporate information from multiple neighbour shells and fit them to dynamical information using simple linear regression.

We begin with a set of descriptors that encode the structure around each particle i, denoted as the vector X (0) i . Note that many glassy systems consist of particles of two or more different species, and hence these descriptors take into account both the positions of the particles as well as their species. For X (0) i , we design a set of structural descriptors consisting of both radial and angular structure functions. For the radial descriptors, we consider the same type of functions that were used in Refs. [17,23] in combination with SVMs. These functions essentially measure the density of particles at a distance r from a reference particle i in a shell of width 2δ, and are defined as follows:

G (0) i (r, δ, s) = j =i:sj =s e -(r ij -r) 2 2δ 2 (1)
where i is the reference particle, r ij is the distance between particle i and j, s j is the species of particle j, and s is the species of particles whose density we wish to probe. For the angular descriptors, inspired by standard bondorientational-order parameters [26], we use an expansion of the local density in terms of spherical harmonics. First, for any given particle i, we define the complex quantities

q (0) i (l, m, r, δ) = 1 Z j =i e -(r ij -r) 2 2δ 2 Y m l (r ij ), (2)
where Y m l (r ij ) are the spherical harmonics of order l, with m an integer that runs from m = -l to m = +l, and

Z = j =i e -(r ij -r) 2 2δ 2
is a normalization constant. We then construct a rotationally invariant local descriptor

q (0) i (l, r, δ) = 4π 2l + 1 l m=-l |q (0) i (l, m, r, δ)| 2 . (3)
The full vector X (0) i for a given particle i then consists of the values of G (0) i (r, δ, s) and q (0) i (l, r, δ), evaluated for a fixed set of r, δ, and l as specified in the Supplemental Information (SI).

In order to incorporate the shell-averaging concept from GNNs, we then introduce higher-order descriptors X (n) i , where each consecutive X (n) i is defined as a local average of the previous order X (n-1) i

. Specifically:

X (n) i = 1 C j:rij <rc e -rij /rc X (n-1) j , (4)
where r c is a cutoff radius and C = j:rij <rc e -rij /rc . Here, we choose r c to approximately correspond to the second minimum in the radial distribution function, and we have confirmed that the results are only weakly dependent on the exact value (see SI). The total descriptors for particle i, which we denote X (nmax) , is then the combination of descriptors (angular and radial) from each shell up to a maximum level of n max . Now that we have introduced a new set of descriptors, we explore how well they can be used to fit the dynamics, and whether including multiple generations (i.e. larger n max ) improves the fitting quality. For our fitting approach, we use linear regression including a regularization term, also known as Ridge regression. For this simple fitting algorithm, the number of fit parameters is simply equal to the number of descriptors (plus an offset), and the fit can be performed on a basic workstation in a matter of seconds.

As our dynamic quantity of interest, we use the dynamic propensity [6,11,25], a standard method for quantifying the component of the dynamical heterogeneity that is encoded in the structure. For a given configuration, it is found by performing M simulations of the same configuration, each initialized with a new set of velocities drawn from the Maxwell Boltzmann distribution. The dynamic propensity d(t) of a chosen particle is then its average absolute displacement after time t. Note that in this Letter, all times are measured in units of the structural relaxation time τ α .

As our model system, we first focus on precisely the same system as Ref. [23], i.e. the Kob-Andersen (KA) mixture [27], a well studied glass former that consists of an 80:20 mixture of non-additive Lennard-Jones particles (see SI for further details). We construct a set of descriptors consisting of 200 radial and 192 angular descriptors per generation. We begin by exploring how the number of averages (i.e. n max ) influences our fit quality, as measured by the Pearson correlation coefficient between the predicted and measured propensities (see SI). Clearly, as shown in Fig. 1, for both species in the KA mixture, there is a significant improvement by including the first averaging (LR (1) ), while adding a second averaging step (LR (2) ) only improves the correlations slightly. Adding a third averaging step does not lead to any further improvements.

The question is now: how does this prediction compare to other ML methods? As shown in Fig. 1a, the zero'th order descriptors lead to a prediction that is similar in quality to the SVM fit taken from Ref. [23]. This is not surprising, as both the SVM method and our LR (0) prediction are linear models based on a similar set of descriptors. Much more interesting, however, is that our LR (2) results closely match the GNN results for all time scales. This indicates that we have indeed incorporated in our averaged descriptors the same relevant structural features captured by the GNN.

One clear advantage of our approach compared to GNNs is the interpretability of the fitted model. While GNNs approximate the propensity as a complex nonlinear function of the particles' coordinates and species, our model is a linear combination of structural descriptors with a simple physical interpretation. Both the linearity of the model and the interpretability of the input descriptors can be exploited in order to unveil the structural features that are most relevant for predicting the dynamics. For example, we can ask the question: is radial (or density) information sufficient to accurately predict the dynamics, or do we also need angular information? This question is intriguing in the context of the Kob-Andersen system, as previous studies similar to our LR (0) model have identified radial information to be the most important [16,17]. To answer this question, we separately fit the particles dynamic propensity using only the radial and angular descriptors, and show the results in Fig. 1c. When only non-averaged descriptors (0-th order) are included, we find the radial descriptors to be more informative on the dynamics (especially at long times), in agreement with previous works [16,17]. Interestingly, however, when averaged descriptors (n max = 2) are included, better predictions are obtained with the set of angular functions. One way to interpret this result is that when the region considered is small, the local density is the most important feature, whereas when considering larger length scales, the anisotropy of the structural environment becomes more relevant. This is consistent with the observation that the local environment of particles can show angular ordering over impressively long ranges in KA mixtures [START_REF] Zhang | Proc. Natl. Acad. Sci. USA[END_REF].

In our current approach we have included approximately 10 3 descriptors. However, the linear nature of our model makes it easy to reduce the number significantly at a low cost in terms of accuracy -a highly useful feature for future extensive explorations of the relationship between structure and dynamics of glassy fluids. To this end, we employ the feature selection scheme introduced in Ref. [START_REF] Boattini | [END_REF] in the context of approximating many-body interactions. In this scheme, the most relevant descriptors are iteratively selected from a pool of candidates. At each step, the selected descriptor is the one that maximizes the linear correlation between the currently selected set and a target variable. The selection proceeds until the correlation stops increasing appreciably. Here, we consider the set of all descriptors up to order n max = 2 as the pool of candidates, and use this scheme to select an optimal subset of N s descriptors for predicting the dynamic propensity at time t. In Fig. 2, we report the results of the predictions at t = τ α as a function of the number of selected descriptors. As shown in the figure, the best descriptor in the pool has a correlation of about 0.4 with the dynamic propensity. Moreover, using as few as N s = 6 descriptors, the results already exceed those obtained with SVMs in Ref. [23] (that used 440 descriptors). After that, the results keep improving as N s increases, and do not change appreciably after N s ≈ 100 descriptors have been selected. A table of the first 20 descriptors selected is given in the SI.

The final question we would like to address is how robust this algorithm is -e.g. how well does it perform on a different glass former? To this end, we consider a 30-70 binary mixture of large (A) and small (B) hard spheres with size ratio σ B /σ A = 0.85 and packing fraction η = 0.58. Previous works have shown that the local structure of this system correlates quite strongly with the dynamic propensity at times close to the relaxation time [5,15], with the strongest correlation reported associated with an unsupervised machine-learned (UML) order parameter based on averaged order parameters (somewhat comparable to our LR (1) averaging). For hard spheres, we use the same set of descriptors as for KA, but omit radial descriptors taken at distances where no pairs of particles can exist (resulting in 172 radial descriptors). Using these descriptors with different choices of the maximum order n max , we fit the propensities at different times, and report the results obtained for both species of particles in Fig. 3. Clearly, both the LR (1) and LR (2) outperform the UML over all time frames examined, with the smallest difference appearing near τ α . Interestingly, only approximatley 20 descriptors are necessary to reach approximately the optimum prediction accuracy (see SI, Fig. S3). Similar to the KA system, the radial LR (0) descriptors outperform the angular descriptors at long time scales (see SI, Fig. S4). Overall, as seen in Fig. 3, consistent with previous observations [15], the predictions are much more accurate than those obtained for the KA system, likely due to the simpler dynamics of this system, which lacks both attractions and non-additivity. For the same reason, we also find that the inclusion of averaged descriptors has a weaker impact on the accuracy of the model, and the results essentially stop improving after including descriptors of order n max = 1. An intriguing observation is the fact that the peak in correlation for LR (1) and LR (2) occurs at significantly longer time scales than the peak in correlation associated with the UML approach, and indeed at time scales several times longer than the structural relaxation time τ α .

In conclusion, we have introduced a fast, easy to implement, linear-regression-based model for fitting dynamic propensities in glassy fluids from local structural descriptors. Key to this model was the insight from GNNs that averaged structural features centered around nearby particles carry a significant amount of the necessary information required to predict the heterogeneous dynamics. This observation enabled us to design a significantly more efficient model that provides essentially the same predictive power at a fraction of the computational complexity -from the ∼ 70000 parameters of the GNN to approximately 1000 parameters in the linear regression model at LR (2) . Moreover, we show that by ranking the importance of the descriptors, we can further reduce the number of required descriptors by an order of magnitude. This result not only provides an efficient simple model for fitting the dynamic propensity of glassy fluids, but also suggests that similar local-average-based linear models should be considered in other situation where GNNs are applied to predict structural and dynamical properties of materials [30][31][32].

Perhaps the most intriguing observation in this work is that the linear model presented here and the GNNs predict the dynamic propensity to essentially the same accuracy. Given that the dynamic propensity must be completely encoded in the structure by its definition, the new linear model opens the door to asking -what structural information is missing to completely describe the dynamics? The linear model would appear to be missing information related to both anisotropic correlations within the averaged domains, as well as correlations between the averaged domains. This observation should lay the foundation for further extensions and improvements in fitting the dynamics propensity in the future.

FIG. 1 :

 1 FIG.1:(a) Pearson correlation coefficient between predicted and actual propensities for the A particles of the KA system at temperature kBT / AA = 0.44 and pressure P σ 3 AA / AA = 2.93. LR(nmax) refers to linear regression using structural descriptor up to order nmax. For comparisons, the results obtained in Ref.23 using support vector machines (SVMs) and graph neural networks (GNNs) are also shown. (b)Pearson correlation coefficient between predicted and actual propensities for the B particles of the same system. (c) Comparison of the Pearson correlation coefficient between predicted and actual propensities of A particles obtained using only radial descriptors, only angular descriptors, or both.

FIG. 2 :

 2 FIG. 2: Pearson correlation coefficient between predicted and actual propensities of the A particles of the Kob-Andersen mixture (same state point as Fig. 1) at t = τα as a function of the number of selected descriptors. Lines represent the results obtained in Ref. 23 using SVMs and GNNs.

ACKNOWLEDGEMENTS

We would like to thank Giuseppe Foffi for many useful discussions. L.F. and E.B. acknowledge funding from The Netherlands Organisation for Scientific Research (NWO) (Grant No. 16DDS004), and L.F. acknowledges funding from NWO for a Vidi grant (Grant No. VI.VIDI.192.102).


Ilane Schroetter
email: ilane.schroetter@gmail.com

Nicolas F Bouché

Johannes Zabl

Hadi Rahmani

Martin Wendt

Sowgat Muzahid

Thierry Contini

Joop Schaye

Kasper B Schmidt

Lutz Wisotzki

MusE GAs Flow and Wind (MEGAFLOW) VI. A study of C iv and Mg ii absorbing gas surrounding [O ii] emitting galaxies

Keywords: galaxies, evolution -galaxies, formation -galaxies, intergalactic mediumquasars, absorption lines

Using the MEGAFLOW survey, which consists of a combination of MUSE and UVES observations of 22 quasar fields selected to contain strong Mg ii absorbers, we measure the covering fractions of C iv and Mg ii as a function of impact parameter b using a novel Bayesian logistic regression method on unbinned data, appropriate for small samples. We also analyse how the C iv and Mg ii covering fractions evolve with redshift. In the MUSE data, we found 215 z = 1 -1.5 [O ii] emitters with fluxes > 10 -17 erg s -1 cm -2 and within 250 kpc of quasar sightlines. Over this redshift path z = 1 -1.5, we have 19 (32) C iv (Mg ii) absorption systems with rest-frame equivalent width (REW) W r >0.05Å associated with at least one [O ii] emitter. The covering fractions of z ≈ 1.2 C iv (Mg ii) absorbers with mean W r ≈0.7Å (1.0Å), exceeds 50% within 23 +62

-16 (46 +18 -13 ) kpc. Together with published studies, our results suggest that the covering fraction of C iv (Mg ii) becomes larger (smaller) with time, respectively. For absorption systems that have C iv but not Mg ii, we find in 73% of the cases no [O ii] counterpart. This may indicate that C iv comes from the intergalactic medium (IGM), i.e. beyond 250 kpc, or that it is associated with lower-mass or quiescent galaxies.

INTRODUCTION

The surroundings of galaxies, the intergalactic medium (IGM) or the very close environment around galaxies: the circum-galactic medium (CGM), are known to be filled with gas in various phases. This CGM is composed of both ionized and neutral gas. For instance, Mg ii is probing the cool ionized T∼ 10 4 K temperature gas whereas C iv probes warm T∼ 10 5 K gas (see the CGM review of Tumlinson et al. 2017). Because the gas usually has low density compared to the gas in galaxies, it is very difficult to observe di-2013b; [START_REF] Dutta | [END_REF][START_REF] Huang | [END_REF]) and perhaps the largest sample is [START_REF] Lan | [END_REF] who used 15,000 galaxy-quasar pairs. However, concerning C iv (λλ1548, 1550), studies of f c (C iv) are often limited to relatively small samples because of the UV coverage required or the need to find the host galaxies at z > 2 (e.g Fox et al. 2007;Fox & Richter 2016). Among the earlier C iv studies, there is the z < 1 Chen et al. (2001) study. Since the availability of the Cosmic Origin Spectrograph (COS) onboard the Hubble Space Telescope, we note the z < 0.1 work of Bordoloi et al. (2014) and Burchett et al. (2016) on COS quasar sight-lines. In addition, there are other constraints on the C iv CGM with quasar-quasar pairs (Prochaska et al. 2014;Landoni et al. 2016), with lensed quasars (Rauch et al. 2001) and with statistical clustering (e.g. Schaye et al. 2003;Turner et al. 2014) which can be compared to hydrodynamical simulations (e.g. Turner et al. 2017). While these agree that the C iv gas is mainly located outside their host galaxies, there is a lack of consensus on the origin and properties of this highly ionized gas. As a result the origin of this highly ionized gas, C iv, and its distribution around galaxies, is thus still debated.

In this paper, we aim to address the question of how C iv is located around galaxies compared to Mg ii by searching for C iv absorption systems and their possible galaxy counterparts by studying the C iv and Mg ii covering fractions using the MEGAFLOW survey (Schroetter et al. 2016(Schroetter et al. , 2019;;Zabl et al. 2019Zabl et al. , 2020)). The C iv and Mg ii covering fractions are obtained from a novel Bayesian logistic fit to the unbinned data. This method is designed to go beyond the limitations of using binned statistics and the Bayesian fit provides a built-in robustness to outliers.

The paper is organized as follows. We present in § 2 the data and the independent detection of absorption systems and starforming galaxies. We then combine those detections in order to identify possible galaxy counterparts to the absorbers. In § 3 we focus on constraining covering fractions of C iv and Mg ii, we present our results in § 4 and our conclusions in § 5. Throughout, we use a 737 cosmology (H 0 = 70 km s -1 Mpc -1 , Ω m = 0.3, and Ω Λ = 0.7) and a Chabrier (2003) stellar Initial Mass Function (IMF). All wavelengths and redshifts are in vacuum and are corrected to a heliocentric velocity standard. All distances are physical. All stated errors are 2σ, unless otherwise noted.

MUSE AND UVES OBSERVATIONS

The data we use in this study are part of a MUSE GTO program aiming to study gas flows around galaxies using background quasars. This MusE GAs FLOw and Wind (MEGAFLOW) program combines MUSE [START_REF] Bacon | Society of Photo-Optical Instrumentation Engineers[END_REF] and UVES [START_REF] Dekker | Optical and IR Telescope Instrumentation and Detectors[END_REF] observations of 22 quasar fields and is described in Schroetter et al. (2016); Zabl et al. (2019); Schroetter et al. (2019). The quasar fields were selected based on the presence of at least 3 strong (W λ2796 r > 0.5-0.8Å) Mg ii absorptions in SDSS spectra and observed with MUSE with on average 3 hours of exposure time each. The observations and data reduction are described in Schroetter et al. (2019) and Zabl et al. (2019).

For each field, UVES follow up observations of the quasar have been carried out. In this section, we present, first, the detection of [O ii] emitters throughout all the 22 MUSE fields, and then the detection of C iv and Mg ii absorption systems in the UVES spectra. We will combine those two samples in order to search for absorber counterparts to [O ii] emitters.

2.1 Galaxy detection in all MUSE fields from z ≥ 1.0

The MUSE wavelength range spans from ≈4700 to ≈9300 Å, at a spectral resolution of R≈ 2000 (or 150 km s -1 ). This wavelength range allows for the detection of [O ii] emitters from redshifts 0.3 ≤ z ≤ 1.5.

In order to find star-forming galaxies potentially associated with C iv or Mg ii absorbers, we searched for [O ii] emitters in our MUSE fields, independently of any absorption systems, but to ensure overlapping redshift ranges between C iv and Mg ii in UVES quasar spectra, we restricted the search for [O ii] emitters at redshifts 1.0-1.5, i.e in the 7400-9300 Å MUSE wavelength range. For this, we used narrow band images with the continuum subtracted throughout the cube wavelength range. In the 22 MEGAFLOW fields, we found 215 [O ii] emitters and visually inspected each galaxy, making sure the [O ii] emission doublet was clearly identified down to a flux detection limit of 10 -17 erg s -1 cm -2 . In a forthcoming paper (Bouché et al. in prep.), we will characterize and present the detection completeness for the MEGAFLOW survey using a combination of automatic and visual inspection sources detection. Preliminary results based on inserting fake [O ii] emitters with realistic sizes and kinematic properties give a 50% completeness level at ≈ 8.0 × 10 -18 erg s -1 cm -2 (corresponding to a SFR of ≈ 0.3 M yr -1 at z = 1.0) for a typical depth of 3 hr in the red part of the MUSE wavelength range away from skylines.

C iv and Mg ii absorption systems

Using the MEGAFLOW UVES quasar spectra, we searched for C iv and Mg ii systems at z = 1-1.55 , i.e. in the redshift range where both C iv and Mg ii are covered in the UVES spectra and where [O ii] can be detected in the MUSE wavelength range, down to a rest-frame equivalent width (REW) W r of ∼0.1Å. The detection limit for each field is based on the signal to noise ratio and we find an overall equivalent width limit EW lim ≈0.1 Å for both C iv and Mg ii in the observed frame. Since all the absorption systems have z > 1, the rest EW lim is thus divided by 1+z, leading to REW lim =0.05 Å. Examples of absorption and emission systems are shown in Figure 2. This Figure shows a selected sample of C iv, Mg ii and [O ii] systems detected at high, medium and low SNR. We emphasize here that each panel of this Figure is an independent system (different field, redshift or position).

We visually searched all the 22 QSO's UVES spectra for C iv systems. UVES allows for C iv absorption detection for redshift z > 1.0 with a spectral resolution of R≈ 38000 1 . The UVES spectral coverage is continuous over the redshift range of 1.0 < z < 1.55. Two of the authors performed this search independently and cross-checked the findings. In short, what we call a "confident C iv system" is a system having clear C iv(λλ 1548, 1550) absorption. Both doublet components had to have the same overall shape. Figure 1 shows the redshifts of all the C iv systems for each quasar sight-line.

In the 22 UVES spectra, we find a total of 41 C iv absorption systems with redshifts between 1.0 and 1.55, for which we also have Mg ii coverage. Out of the 22 quasar fields, we only find C iv absorption systems in 13 of them. The 9 remaining fields either do not have UVES wavelength coverage (for 6 of them) or the quasars have lower redshift than 1.55 (for 3 of them). It is worth mentioning that within our selected wavelength coverage from redshifts 1 to 1.55 (from ≈ 3100 Å to ≈ 4000 Å), our UVES data do not have any spectral gap.

For those 41 C iv absorption systems, we also searched for Mg ii absorption lines. We find that out of the 41 C iv systems, 18 have Mg ii absorption (see blue and hatched systems in Figure 3). We emphasize the fact that the MEGAFLOW survey was built based on Mg ii absorption lines detected in quasar spectra from SDSS. Hence, we keep in mind that the numbers we give in this paper may be biased towards more Mg ii absorption systems and that the MEGAFLOW selection may also increase the number of expected C iv systems. However, since this survey has the advantage of having high-resolution quasar spectra for each of the MUSE fields, we begin with this biased sample before building larger ones.

In order to complete our analysis and for the purpose of the covering fraction study later in this paper, we also searched for all Mg ii absorption systems in each of our UVES data, regardless of the presence of C iv absorbers. We first searched for any Mg ii absorption system with redshift between 1.0 and 1.55. We found 52 Mg ii systems in total to begin with. Then, we rejected the ones for which we cannot observe C iv absorption due to the UVES wavelength coverage, resulting in 30 Mg ii systems where the C iv feature is covered. Out of these 30, we detected C iv in 18 systems. The distribution of REW for all the Mg ii systems is represented in Figure 4. Comparing the REW distribution of systems which have only C iv and the ones with the presence of Mg ii (hashed histogram on Figure 4), we find that we are in good agreement with the study of Steidel & Sargent (1992) where they found that C iv-only systems are peaked towards lower REW. This difference could come from the fact that C iv may be more diffuse than Mg ii around galaxies.

Table 1 summarizes all the different absorption systems for which both absorption lines (C iv and Mg ii) can be detected. Table 1 lists the absorption system redshift with the corresponding REW of C iv(λλ1548, 1550) and Mg ii(λλ2796, 2803) (if detected) absorption lines. We note that each C iv or Mg ii doublet is resolved and each corresponding limit was applied to each individual doublet member. Table 2 shows Mg ii absorption system properties for which C iv is not covered in UVES.

Having independently detected the absorption systems in UVES spectra and the [O ii] emitters in MUSE, we next try to match galaxies to absorption systems.

Absorber counterparts

We now look for galaxies which possibly correspond to the C iv absorption systems. For each detected galaxy, we measured the velocity difference2 between C iv absorption and [O ii] emission. For the purpose of this work, a galaxy was considered to be a counterpart if |∆v| < 500 km s -1 .3 With this criterion, we find 39 counterpart galaxies for 19 C iv absorption systems, out to 250 kpc from the quasar. Out of those 19 systems, 13 (≈ 70%) have Mg ii absorption lines.

One of the most interesting results is that out of all the 22 systems with only C iv absorption detected (not Mg ii), we detect counterparts for only 6 systems. This means that we do not detect any counterpart for ≈73% of those "C iv only" systems. This result suggests that the C iv gas is probably located in the IGM rather than in the CGM or that C iv is associated with galaxies with lower SFR than Mg ii, i.e with SFR 0.3 M yr -1 . As already argued by Rahmani et al. (2018), a system having both C iv and Mg ii absorption lines appears to have a higher chance of a counterpart detection, compared to a system with only C iv absorption. Here we can confirm this statement as 70% of our systems for which we find at least 1 counterpart have both C iv and Mg ii absorption lines.

Using the same criterion to identify counterparts of the 52 detected Mg ii systems, we find at least one galaxy counterpart for 324 systems (≈ 63%5 ), within 250 kpc from the quasar. In both Table 1 and 2, the last column shows the number of galaxy counterpart detected for each system.

MEASURING COVERING FRACTIONS

With our sample of [O ii] galaxies, we investigate the covering fractions of C iv and Mg ii at redshifts 1.0 < z < 1.5. The covering fraction f c describes the probability of having an absorber detected (1) or not (0) at a distance r from a galaxy, which is inherently dichotomous. f c can be estimated from the ratio between the number of galaxies at a given impact parameter r ± ∆r which have an absorber detected (above some detection limit) to the total number of galaxies N tot at that impact parameter with or without a detected absorber. If 'D' ('U') represent the detected (undetected) absorbers, respectively, f c (r) ≡ D/D + U. In the notation of Bordoloi et al. (2014), D is N Wr>W r,lim (r), and f c is

f c (r) ≡ N Wr>W r,lim (r) N tot (r) (1)
In general, the probability p for an absorber to be detected (i.e., Y = 1) can be described by any continuous function p = L(t).

Here, we use a logistic function 6 for the probability p to detect an 6 See Wilde et al. ( 2020) for another physical parameterisation. absorber. The logistic function yields a continuous transition of p from 0 to 1:

p(Y = 1) = 1 1 + exp(-t) ≡ L(t) (2)
where t is usually taken to be a linear combination of the independent variables, X n , i.e., t = α + β 1 X 1 + • • • + β n X n .7 Here, the independent variables will be impact parameter r and redshift z.

It is customary to determine f c (r) in radial bins, by binning the data. However, this technique requires samples with large number of galaxies in each bin and thus can lead to large statistical uncertainties when the data do not populate each bin equally. Here, we use a novel unbinned approach to fit the probability function p directly using a logistic function in a Bayesian framework. As discussed in [START_REF] Hosmer | Applied logistic regression[END_REF], this is commonly used to examine the possible relationship between a dichotomous variable (here whether the absorption are detected in the quasar spectra) and independent variables (X n , such as impact parameter, redshift, etc.). This technique has several advantages, namely it uses all the information contained in the data even in bins where there would be few galaxies. The Bayesian framework is natural for this problem and provides also a more robust fitting procedure against outliers which could skew the mean of a particular bin.

A series outcomes O can be generated by the Bernouilli distribution Bern(p), since the observables are dichotomous, with values at 0 or 1, for undetected and detected systems, respectively. In summary, the model is We stress that the purpose of this exercise is to demonstrate the viability of our new technique for determining the covering fractions of metal lines in the regime of small number statistics.

t = f (X i ; θ) (3) p = L(t) (4)

RESULTS

Mg ii covering fraction

The distribution of Mg ii REWs for all detected absorbers is shown in Fig. 4. The Mg ii systems have REWs of [0.1-2.5Å] with a mean of W 2796 r = 0.99 ± 0.8Å. As discussed in Appendix A2, the mean REW W λ2796 r of our sample is biased towards high W λ2796 r compared to random field Mg ii absorbers due to the MEGAFLOW survey selection criteria ( § 2). The mean value for absorbers in the original MEGAFLOW selection is W λ2796 r = 1.41 Å. Fig. 5(bottom) shows the Mg ii covering fraction f c as a function of impact parameter b. The solid line represents a logistic fit with our model (Eq. 7) to the unbinned data (black ticks) with the 95% confidence interval (grey area) derived from the Markov chain. The 50% covering fraction f c (50%) occurs at log b/kpc = 1.66 ± 0.15 (2σ), corresponding to 46 +18 -13 kpc. For comparison, the filled squares with error bars represent the binned data. These error bars are 1σ confidence intervals computed for proportion of a binomial distribution (Wilson 1927;Cameron 2011). Comparing these 1σ errors on the binned data to the 95% confidence interval to the unbinned data, this figure shows the benefit of a parametric fit to the unbinned data. The fitted parameters for the Mg ii covering fractions are listed in Table 3 with (Eq. 7) and without the redshift evolution term (Eq. 6). Fig. 5(top) shows the redshift evolution determined from our model fit (Eq. 7). The solid line shows that f c (50%) evolves as (1 + z) B with B = 1.96 +2.32 -2.68 (2σ). The dotted-dashed line represents the redshift evolution of dark-matter halos R vir ∝ H(z) -1 (e.g. Mo et al. 1998) of a given mass. Comparing the dotted-dashed line to the redshift evolution of f c , one sees that the Mg ii halo is becoming proportionally smaller.

Our results are broadly in agreement with the study of Nielsen et al. (2013a) (grey triangles) which used a sample of 182 galaxies towards 134 quasar sight-lines over a wide redshift range of 0.07≤ z ≤ 1.12. Among other literature results of Mg ii covering fractions, Lan (2020) stands out with a sample of 15,000 Mg ii-galaxy pairs in SDSS with W 2796 r > 0.4 Å (albeit with photometric redshifts). Their f c as a function of radius (redshift) is shown with the dotted line in the bottom (top) panel of Fig. 5, respectively. In spite of our much smaller sample, our results are in good agreement with theirs. The dotted line in Fig. 5(bottom) from Lan (2020) has been calculated for absorbers with W 2796 r ≥ 1Å 8 , at z = 1.2, for starforming galaxies with log(M /M ) = 10.0 assuming a R vir ≈ 200 -250 kpc from their Eqs 7-9. The dotted line in Fig. 5(top) has been calculated by solving numerically for the redshift evolution of f c in [START_REF] Lan | [END_REF]. Overall, our results, from a much smaller sample, are very similar to those of Lan (2020) who used 15,000 pairs.

It is interesting to compare the halo mass for Mg ii absorbers corresponding to R vir ≈ 200 -250 kpc, namely log M h /M ≈ 12.6, to the mass scale obtained by Bouché et al. (2006) from a clustering analysis of 2,500 z = 0.5 absorbers (see also Gauthier et al. 2009;Lundgren et al. 2009Lundgren et al. , 2011)): log M h /M ≈ 12.5 ± 0.4 for their absorbers with 0.3 < W 2796 r < 1.15. The covering fraction could also be weakly dependent on stellar mass (e.g. Chen et al. 2010;Nielsen et al. 2013a;[START_REF] Lan | [END_REF]), but we reserve such an analysis to a forthcoming paper on the full MEGAFLOW sample. Nonetheless, these 8 Lan (2020) does not provide a fit for W 2796 r > 0.4Å absorbers, but the equivalent width dependence seems small compared to the other parameters.

results are a demonstration of the power of our fitting technique on unbinned data.

C iv covering fraction

In this section, we use the galaxies for which C iv could be detected from our UVES coverage. This lowers the number of [O ii] emitters from 215 to 141. From a total of 141 [O ii] emitters in the MUSE data with redshift >1.0, after looking at the redshift differences between each emitter and C iv absorption systems, we found 39 galaxy counterparts corresponding to 19 C iv absorption systems. Incidentally, 70% (13 out of 19) of those systems have detectable Mg ii absorption lines as well 9 .

Fig. 6(bottom) shows the covering fraction as a function of impact parameter b from our logistic fit (Eq. 7) to the unbinned data (black ticks). The 50% covering fraction occurs at log b/kpc =1.36±0.57(2σ), i.e. 23 +62 -16 kpc. For comparison, the filled squares with error bars represent the binned data. These error bars are 1σ confidence intervals as in § 4.1. Our f c (C iv) appears to be in good agreement with the Bordoloi et al. ( 2014) study (solid triangles), which used a sample of 43 low-mass z ≤ 0.1 galaxies around sight-lines with C iv W 1548 r greater than ∼ 0.1Å. The top panel of Fig. 6 shows the fitted redshift evolution with the 95% predictive interval. The solid line represent the fitted (1 + z) B evolution with B = -0.7 +3.9 -4.0 (2σ), i.e. consistent with no evolution. The tick marks show the location of the unbinned data.The fitted parameters for the C iv covering fractions are listed in Table 3 with (Eq. 7) and without the redshift evolution term (Eq. 6). The solid triangle in this figure represent the C iv covering fraction f c (50%) from the study of Bordoloi et al. (2014) at z = 0.1. This data point is obtained after applying our methodology to their 44 data points which yielded a size of log b/kpc 1.71±0.26 (2σ) for the 50% covering, i.e. 51 +42 -23 kpc.

DISCUSSION AND CONCLUSIONS

This study was divided into two parts: (i) the independent detection of absorption systems in MEGAFLOW UVES and [O ii] emitters in MEGAFLOW MUSE with the identification of galaxy counterparts to the absorbers; (ii) measurement of the covering fractions of C iv and Mg ii. We adopted an innovative Bayesian logistic regression and have shown its ability ( § 3) to determine the covering fractions of absorption lines (such as Mg ii, C iv) with limited samples. This approach allows one to fit a parametric model directly to unbinned binary data (1/0) where 1 represents an absorption detection and 0 its absence. The advantage of our method is that it does not suffer from the limitations inherent when using binned data.

Using the MEGAFLOW UVES quasar spectra, we searched for C iv, Mg ii systems at z = 1-1.5, i.e. in the redshift range where both C iv and Mg ii are covered by UVES and where [O ii] could be detected in the MUSE wavelength coverage, down to a rest-frame equivalent width of 0.05Å. We detected a total of 52 Mg ii absorption systems and 41 C iv absorption systems independently, having large mean REWs of 0.99 and 0.71 Å respectively. 50% and 60% of the 41 C iv and 32 Mg ii systems10 , respectively, have absorption by both ions. Using the 22 MEGAFLOW MUSE fields, we searched for [O ii] emitters at redshift z =1.0-1.5 and found a total of 215 emitters down to a flux limit of ≈ 10 -17 erg s -1 cm -2 (Schroetter et al. 2016(Schroetter et al. , 2019;;Zabl et al. 2019). For each of these 215 [O ii] galaxies, we searched in our catalog of C iv systems with REW > 0.05 Å for matches within 500 km s -1 and b < 250 kpc. We found 39 galaxies associated with 19 C iv systems. For each galaxy, we also searched for corresponding Mg ii systems and found 67 galaxies associated with 32 Mg ii systems.

We find that 70% of the systems for which we detect at least one galaxy counterpart have both C iv and Mg ii absorption lines. Globally, these results point towards a physical picture where Mg ii and C iv (with mean REWs of ≈1Å) are associated with individual L /sub-L galaxies. We also find that 73% of the 'only C iv' systems do not appear to have any galaxy counterpart in our MUSE fields. This suggests that C iv-only gas is more likely to be part of the IGM, i.e. beyond 250 kpc rather than of the CGM, or that C iv is preferentially associated with very low mass galaxies (with SFR 0.3 M yr -1 ).

Then, using the closest associated galaxies in our small sample of 19 (32) C iv (Mg ii) absorber-galaxy pairs, we studied the covering fraction and its redshift evolution using a logistic regression to the unbinned data in a Bayesian framework ( § 3). For Mg ii, we find that f c (r) reaches 50% at r 50 ≈ 50 kpc (46 +18

-13 ; 2σ) (Fig. 5 bottom panel). Whereas for C iv, we find that the size of the warm C iv halo is somewhat smaller with r 50 ≈ 25 kpc (23 +62 -16 ; 2σ) (Fig. 6 bottom panel). However, given the current sample size, this difference is not significant at 2σ.

Regarding the redshift evolution of f c (r, z) for strong Mg ii absorbers with mean W r 1Å (Fig. 5 top panel), we found that f c (50%) evolves as (1 + z) B with B = 1.96 +2.32 -2.68 . Although the uncertainty of B is large due to our small sample size, such that the relation is consistent with no redshift evolution, the best-fit B value agrees with that derived by Lan (2020) ∝ (1 + z) 2 with a sample of 15,000 pairs. Therefore, the two results combined support a physical picture where the size of Mg ii gas/halo decreases over time. We note that whether this redshift index B is ∼ 0 or 2, it appears that the Mg ii gas/halo is becoming smaller with time relative to the dark-matter halo.

Regarding the redshift evolution of f c (r, z) for C iv absorbers (Fig. 6 top panel), our data over z ∼ 1 -1.5 do not show significant redshift evolution. However, extending the redshift baseline to z ∼ 0.1 by including the Bordoloi et al. (2014) data reveals that the C iv covering fraction may in fact increase over time and possibly coevolve with the dark-matter halo evolution (dotted-dashed line in top panel). This would indicate that the warm CGM probed by C iv is growing with the halo size. This co-evolution is expected under a scenario where the warm gas is in hydrostatic equilibrium with the halo.

With the final MEGAFLOW survey and upcoming larger surveys, it will be possible to address this redshift evolution and the potential dependence of covering fraction on azimuthal angle and/or EWs, thus yielding a better understanding of the CGM.

APPENDIX A:

A1 Additional Material Fig. A1 shows two dimensional covering fraction of Mg ii as a function of impact parameter log b and redshift log 1 + z using the logistic regression described in § 3 (Eq. 7). The fitted covering fraction is represented by the background color from low to high covering fractions represented as blue and red respectively. Similarly, Fig. A2 shows two dimensional covering fraction of C iv.

A2 Strong absorbers distribution

In Figure A3, we compare the distribution of W λ2796 r with literature samples. Our sample (in red and grey) has an overabundance of strong absorbers compared to other Mg ii surveys due to the MEGAFLOW selection of multiple strong Mg ii absorbers in quasar spectra. However, the lower REW population (log(W λ2796 r ) ≤ -0.5) of MEGAFLOW absorbers found in this study, which are not pre-selected, are in agreement with other studies.

Figure 1 .

 1 Figure 1. Observed wavelengths of C iv systems in each MEGAFLOW QSO UVES spectra. The top x-axis represents the corresponding redshift. The blue crosses represent the detected C iv absorption system. The red circles represent the expected λ obs (C iv) for systems with detected Mg ii.The red vertical lines represents the UVES lowest wavelength coverage and the black vertical line the z = 1.55 redshift limit. The 3 horizontal black dashed lines are the fields for which the lowest UVES wavelength is above 4000 Å where C iv is not covered at z ≈1-1.5. The left black triangles show the quasar redshift positions with z QSO < 1.5.

Figure 2 .

 2 Figure 2. Examples of C iv (left) and Mg ii (middle) absorption systems as well as [O ii] (right) emission for high, medium and low Signal to Noise Ratio (SNR) in the top, middle and bottom rows, respectively. Each panel is independent and they were chosen arbitrary for examples purpose. Each SNR is shown for each panel as well as each field name. The x-axis for [O ii] emission (right column) corresponds to observed wavelength.

Figure 3 .

 3 Figure 3. Histogram showing number of C iv absorption systems as a function of their REW for all the C iv absorbers (in blue), the ones with galaxy counterparts detected (in orange) and the number of absorbers for which we also detect Mg ii absorption lines (hatched). The REW detection limit is shown by the black dashed vertical line

  t) is given by Eq. 2, f (X i ; θ) linearly combines the data and map it to L(t), X i are the independent variables, θ are the model parameters, and O are the simulated observables which can be compared to the observed data Y i .For the model function t, we choose a simple linear functionf of impact parameter b, f (b; θ) = A (log b -C) (6)where A describes the slope of the covering fraction and C the zeropoint at 50% covering fraction f c = 0.5, since L(0) = 0.5. One could add a redshift dependent zero-pointf (b, z; θ) = A (log b -ZP(z))(7) where ZP(z) = B log(1 + z) + C. This second model is discussed/shown in § 4. We used a Markov Chain Monte Carlo (MCMC) algorithm to estimate the best fitting parameters θ for our model. Because traditional MCMC algorithms are somewhat sensitive to the step size and the desired number of steps, in what follows we use the No-U Turn Sampler (NUTS) of Hoffman & Gelman (2014) implemented in Pymc3 (Salvatier et al. 2016), a self-tuning variant of Hamiltonian Monte Carlo (HMC). We typically used 2 MCMC chains per run and 4,500 iterations per chain.

  Fitted parameters (A, B, C) for f c for Mg ii and C iv in relation to Eqs. 6-7 along with their 95% confidence interval. The log f c column shows the distance at which f c reaches 50% at z = 1.2.

Figure 5 .

 5 Figure 5. Bottom: Mg ii covering fraction f c as a function of impact parameter b. The fit to the unbinned data (black ticks) is shown (for z = 1.2) with the solid blue line and its 2σ confidence region is shown by the shaded region. The 50% covering fraction is found to be at log b/kpc 1.7 or b 50 kpc. The dotted red line represents the fit found by Lan (2020) for z = 1.2, log(M /M ) = 10 and R vir = 250 kpc. The solid black (green) triangles represent the Nielsen et al. (2013a) (Huang et al. 2021) results, respectively, both at z ∼ 0.5. The solid blue squares with error bars show the binned data with 1σ confidence intervals for binomial proportions (see text). The error bars are shown only with at least 2 galaxies contributing. Top: The redshift evolution of the 50% covering fraction with the 2σ predictive interval shown in grey. The redshift evolution found by Lan (2020) is shown with the dotted red line. The dotted-dashed line shows the expected redshift evolution of halo sizes (e.g. Mo et al. 1998).

Figure 6 .

 6 Figure 6. Bottom: C iv covering fraction f c as a function of impact parameter b. The fit to the unbinned data (black ticks) is shown (for z = 1.2) with the solid orange line and its 95% (2σ) confidence region is shown by the shaded region. The black triangles represent the Bordoloi et al. (2014) z = 0 results. The solid red squares with error bars show the binned data with 1σ confidence intervals for binomial proportions (see text). The 50% covering fraction is found to be at log b/kpc 1.4 or b 25 kpc. Top: The redshift evolution of the 50% covering fraction with the 2σ predictive interval shown in grey. The dotted-dashed line shows the expected redshift evolution of halo sizes (e.g. Mo et al. 1998).

Figure A1 .

 A1 Figure A1. Two dimensional covering fraction of Mg ii as a function of impact parameter log b and redshift log 1 + z. The unbinned data is shown as blue (red) squares representing the undetected (detected) systems, respectively. The Mg ii covering fraction from the logistic fit is represented by the background color with low (high) covering fractions represented as blue (red) respectively.

Figure A2 .

 A2 Figure A2. Two dimensional covering fraction of C iv as in Fig. A1.

Figure A3 .

 A3 Figure A3. Rest-frame W λ2796 r distribution of the 32 Mg ii systems with C iv (red histogram) and the 52 Mg ii systems (grey histogram) using in Fig. 5. The cyan (blue) curve represent the W r distribution from Nestor et al. (2005) (Zhu & Ménard 2013) at z = 1.2, respectively. For comparison, the MAGIICAT data is shown with the green histogram (Nielsen et al. 2013b).

Table 1 .

 1 Absorption system properties (both C iv and Mg ii covered).

	Field name	z abs	W λ1548 r	W λ1550 r	W λ2796 r	W λ2802 r	N c
	(1)	(2)	(3)	(4)	(5)	(6)	(7)
	J0014m0028 1.16444	≤ 0.05	≤ 0.05	0.3251	0.2721	0
	• • •	1.25638	0.6108	0.3913	≤ 0.05	≤ 0.05	0
	• • •	1.35566	0.0655	0.0762	≤ 0.05	≤ 0.05	0
	J0014p0912	1.17550	≤ 0.05	≤ 0.05	0.9277	0.8453	0
	• • •	1.22115	≤ 0.05	≤ 0.05	1.2521	0.9717	5
	• • •	1.26575	1.5459	0.9266	0.3772	0.1909	0
	• • •	1.45844	≤ 0.05	≤ 0.05	0.2831	0.1541	0
	• • •	1.48212	0.1778	0.3613	≤ 0.05	≤ 0.05	0
	J0103p1332	1.24646	0.5493	0.5053	0.2369	0.1577	0
	• • •	1.25029	0.5005	0.1065	≤ 0.05	≤ 0.05	0
	• • •	1.33451	0.4467	0.3516	0.3347	0.2231	0
	• • •	1.34053	≤ 0.05	≤ 0.05	0.1687	0.1107	1
	• • •	1.36112	≤ 0.05	≤ 0.05	0.3727	0.2989	2
	• • •	1.40648	0.3491	0.2141	≤ 0.05	≤ 0.05	0
	• • •	1.41847	0.9428	0.6973	≤ 0.05	≤ 0.05	0
	J0131p1303	1.22845	0.2819	0.2384	≤ 0.05	≤ 0.05	0
	• • •	1.33945	0.7591	0.5660	0.1567	0.1024	1
	• • •	1.36075	0.3734	0.3213	0.3582	0.2862	2
	• • •	1.50130	0.0947	0.0695	≤ 0.05	≤ 0.05	0
	J0134p0051	1.27246	≤ 0.05	≤ 0.05	0.3312	0.2847	0
	• • •	1.27411	≤ 0.05	≤ 0.05	0.3500	0.2266	0
	• • •	1.44917	≤ 0.05	≤ 0.05	0.4287	0.2362	1
	J0838p0257	1.33042	1.4164	1.1392	1.5306	1.2020	1
	• • •	1.39601	0.5442	0.5272	0.0598	0.0228	1
	• • •	1.41490	0.7320	0.3133	0.8686	0.6654	2
	• • •	1.44629	1.4876	1.2233	0.7595	0.5980	0
	J0937p0656	1.28260	1.0395	0.7001	≤ 0.05	≤ 0.05	0
	• • •	1.49647	1.0355	1.0112	≤ 0.05	≤ 0.05	0
	J1039p0714	1.35888	1.1127	1.0100	2.4970	2.2989	3
	J1107p1021	1.14907	≤ 0.05	≤ 0.05	0.7981	0.515	0
	• • •	1.18750	0.2123	0.1948	≤ 0.05	≤ 0.05	2
	• • •	1.19557	0.4752	0.1473	≤ 0.05	≤ 0.05	0
	• • •	1.28363	0.2810	0.1408	≤ 0.05	≤ 0.05	2
	• • •	1.29562	2.1644	1.9257	0.5323	0.2984	6
	• • •	1.32523	1.6618	1.4281	2.7651	2.6038	3
	• • •	1.34105	1.2897	1.0869	0.1009	0.0524	3
	J1236p0725	1.44338	0.4736	0.3960	1.7089	1.5374	0
	J1314p0657	1.29867	0.2399	0.1287	≤ 0.05	≤ 0.05	0
	• • •	1.51345	0.1353	0.1080	≤ 0.05	≤ 0.05	0
	J1358p1145	1.03320	1.1543	0.9007	≤ 0.05	≤ 0.05	1
	• • •	1.21902	0.1625	0.1520	≤ 0.05	≤ 0.05	2
	• • •	1.24627	0.0615	0.0350	≤ 0.05	≤ 0.05	0
	• • •	1.41709	0.9830	0.7056	2.5651	2.3938	1
	J2137p0012	1.15191	0.2559	0.1254	≤ 0.05	≤ 0.05	0
	• • •	1.18860	0.6404	0.4889	0.2686	0.1671	2
	• • •	1.21472	≤ 0.05	≤ 0.05	1.1219	1.0573	4
	• • •	1.30023	0.4736	0.1542	0.0731	0.0590	1
	J2152p0625	1.17651	≤ 0.05	≤ 0.05	0.496	0.3924	0
	• • •	1.31870	1.3851	0.9527	1.3841	1.1038	2
	• • •	1.39149	0.8075	0.7151	≤ 0.05	≤ 0.05	1
	• • •	1.43101	1.0550	0.9595	0.9884 †	≤ 0.05	3
	• • •	1.46349	0.4458	0.1634	≤ 0.05	≤ 0.05	0
	• • •	1.47132	0.5880	0.4002	≤ 0.05	≤ 0.05	0

1: Field name; 2: Absorption system redshift; 3: C iv (λ1548) REW (Å); 4: C iv (λ1550) REW (Å); 5: Mg ii (λ2796) REW (Å); 6: Mg ii (λ2802) REW (Å); 7: Number of detected counterparts. Typical EW 1-σ errors are 0.01 Å. †: this system is not considered as a detection since we do not detect the λ2802 component which falls in the UVES coverage gap.

Table 2 .

 2 Mg ii absorption system properties (C iv not covered).

	Field name	z abs	W λ2796 r	W λ2802 r	N c
	(1)	(2)	(3)	(4)	(5)
	J0014m0028 1.05265	2.1232	1.7896	1
	J0058p0111	1.06528	1.5822	1.3268	0
	J0103p1332	1.04836	2.9956	2.7317	2
	J0134p0051	1.07010	0.1906	0.0534	2
	J0131p1303	1.01045	1.3197	0.9514	2
	• • •	1.10436	1.0906	0.8421	2
	J0800p1849	1.12153	0.5252	0.5156	0
	J0838p0257	1.09958	0.1474	0.0538	1
	J1107p1021	1.01464	0.6477	0.5716	0
	• • •	1.01643	1.1665	1.0339	4
	• • •	1.04815	0.4274	0.3120	1
	J1107p1757	1.04801	2.1392	1.6854	0
	• • •	1.06296	3.2741	2.6220	1
	• • •	1.16320	1.5809	1.4023	1
	• • •	1.32678	0.7279	0.5251	0
	• • •	1.33079	0.9160	0.7724	2
	J1236p0725	1.33128	0.3908	0.2291	3
	J1352p0614	1.13736	1.4189	1.0794	0
	• • •	1.20049	1.0326	0.9159	0
	J1509p1506	1.04659	1.5000	1.3575	0
	J2137p0012	1.04423	0.8518	0.8741	2
	J2152p0625	1.05366	0.5688	0.4714	2
	1: Field name; 2: Absorption system redshift; 3: Mg ii (λ2796) REW (Å);
	4: Mg ii (λ2802) REW (Å); 5: Number of detected counterparts. Typical
		EW 1-σ errors are 0.01 Å.

With a slit width of 1.2 and a CCD readout with

2x2 binning. UVES observation settings and data quality are detailed in Zabl et al. (2019) and Schroetter et al. (2019).MNRAS 000, 1-8 (0000)

The velocity difference between the absorber redshift and the galaxy systemic redshift.

 3 We also searched for systems with ∆v < 1000 km s -1 but found no more associations.MNRAS 000, 1-8 (0000)

Note that 21 of those 32 systems were part of the original MEGAFLOW selection.

Note that this is the detection rate for z > 1 galaxies, compared to the ≈ 70% detection of the whole MEGAFLOW survey for redshifts 0.4 < z < 1.5 MNRAS 000, 1-8 (0000)

This assumes that all variables are independent of one another, but covariance terms can be included.MNRAS 000, 1-8 (0000)

MEGAFLOW was not designed around C iv absorption system, this 70% might be indirectly biased by our Mg ii selection.

The 32 Mg ii systems are the ones with C iv coverage.MNRAS 000, 1-8 (0000)

MNRAS 000, 1-8 (0000)

ACKNOWLEDGMENTS

We thank the referee for his/her in depth comments which led to an improved paper. This work received funding from the grants 3DGasFlows (ANR-17-CE31-0017), and the OCEVU Labex (ANR-11-LABX-0060) from the French National Research Agency (ANR).

Software: This work made use of the following open-source software: Numpy (Van Der Walt et al. 2011), Scipy (Jones et al. 2001), Matplotlib (Hunter 2007), and Pymc3 (Salvatier et al. 2016). SM is supported by the Alexander von Humboldt-Stiftung via the Experienced Researchers fellowship.

DATA AVAILABILITY

The data underlying this article are available in the ESO archive (http://archive.eso.org). The reduced data will be shared on reasonable request to the corresponding author.