List of Figures

1.1 Slice of the 3D map of the distribution of the galaxies in the universe from the Sloan Digital Sky Survey III. The earth is situated at the center and each point represents a galaxy. The radius of the circle is roughly 0.5 Gpc. Super-cluster and voids occur up to a few hundred Mpc, which is the scale on which the cosmological principle can be applied in the universe. The thinning out of the galaxies near the edges which are farthest from us represents the sparser sampling of these faint objects in the survey [START_REF] Keel | The Road to Galaxy Formation[END_REF]. More extensive mapping work will be done in future, for example by the Square Kilometre Array (SKA) [START_REF] Schilizzi | The square kilometer array[END_REF], The shaded, horizontal band corresponds to the 68.3 per cent confidence interval for the result (filled circle) obtained by the project "Weighting the Giants". It is based on weak gravitational lensing measurements constraining the absolute mass scale of X-ray selected clusters detected in the ROSAT All-Sky Survey ( [START_REF] Mantz | Weighing the giants -IV. Cosmology and neutrino mass[END_REF]). .

1.6

The CMB temperature power spectrum as measured by the Planck mission ( [START_REF] Ade | Planck 2015 results. xiii. cosmological parameters[END_REF]). The position of the first peak, at l ∼ 220 (δθ ∼ 1 • ) is related to the cosmological parameters describing the universe at the recombination epoch and the geometry and contents of the universe along the photons path towards us (curvature, dark energy). Superimposed is the best fit to the data, from which the cosmological parameters are inferred. The lower plot is the residual to the upper. The scatter at low l is due to the cosmic variance, that is the lack of possibility to average over sufficiently numerous cells of size δθ on the sky. Image: [START_REF] Ade | Planck 2015 results. xiii. cosmological parameters[END_REF] CMASS and LOWZ SDSS galaxies sample, the 6dFGS survey ( [START_REF] Jones | The 6df galaxy survey: Final redshift release (dr3) and southern large-scale structures[END_REF]) and the WiggleZ survey [START_REF] Blake | The wigglez dark energy survey: mapping the distance-redshift relation with baryon acoustic oscillations[END_REF]). Superimposed is the flat ΛCDM model obtained using the CMB projects Planck and WMAP results ( [START_REF] Ade | Planck 2013 results. xvi. cosmological parameters[END_REF]). There is a remarkable agreement between the different galaxy surveys and the Planck data. Image : [START_REF] Anderson | The clustering of galaxies in the sdss-iii baryon oscillation spectroscopic survey: Baryon acoustic oscillations in the data release 10 and 11 galaxy samples[END_REF] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 The original "discovery data", the Hubble diagram of SNe Ia compiled from the High-z Supernova team and the Supernova Cosmology Project's data. The bottom panel is the residual to the upper (distance modulus) corresponding to an open universe. The points can be seen to lie above the non-accelerating models (dashed and dotted lines). Image : [Fri08] . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10 Constraints on constant w dark energy models from the project Weighing the Giants, which uses weak gravitational lensing measurements to improve the measurements of the galaxy cluster mass function and its evolution, using X-ray selected clusters detected in the ROSAT All-Sky Survey. This measurement is compared with constraints form the CMB (WMAP, ACT and SPT), supernova (from the Union sample including SNLS-1 year data) and BAO (SDSS/BOSS, 6dF galaxy survey) data, and their combination. 3.1 Schematic diagram of gravitational lensing. The reference axis originates from the observer and is orthogonal to the lens mass plane. The position of the lens mass distribution symmetry center L, should it have any, is not specified : it lies somewhere in the lens plane. The light ray originates from a point S which position is set by η in the source plane and travels unperturbed until it hits the lens plane, at the position given by ξ. It then instantaneously changes direction (set by α). In the special case where the lens is spherically symmetric with respect to a symmetry center L, the lens plane can be taken to be orthogonal to the direction (LO). The reference axis can be taken as the optical axis (LO), and ξ is then the impact parameter with respect to L. This is not possible when considering multiple lenses. Image: adapted from Survey (COSMOS), obtained with a tomography technique in 3 bins of redshift at z ∼ 0.3, 0.5, 0.7. The observed shear field is converted into a convergence map (using Kaiser-Squire inversion technique [START_REF] Kaiser | Mapping the dark matter with weak gravitational lensing[END_REF]), which is proportional to the two dimensional, projected mass ( [START_REF] Massey | Dark matter maps reveal cosmic scaffolding[END_REF]). Image credit: NASA . . . . . . . . . . . . vii 3.6 Schematic diagram of the observer's backward ray tracing in the multiple-lensplane approximation. A light ray experiences a deflection α k only when passing through each lens plane (L k ), located at a distance D k from the observer and D kS from the source. Image: [START_REF] Premadi | Light Propagation in Inhomogeneous Universes. I. Methodology and Preliminary Results[END_REF] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Left: growth of matter fluctuations in the linear regime. Non linearity is defined where ∆ 2 (k) = 1. All scales k reaching this threshold collapsed into bound objects, in a bottom-up fashion (smaller scales first). Credits: [START_REF] Norman | Simulating Galaxy Clusters[END_REF]. Right: Schematic diagram of the hierarchical growth. Smaller halos merge into bigger halo as the time increases from top to bottom. The horizontal slice on the tree at any instant gives the halo distribution at the corresponding time. Credits: [START_REF] Lacey | Merger rates in hierarchical models of galaxy formation[END_REF]. . . . . . . 3.8 The difference δ -δ lin as a function of δ lin for two different cosmologies. From down to top, the curves correspond to the Einstein-deSitter case and to the Λ = 0 cosmology (a virial mass of M vir = 3 × 10 12 h -1 M was used in all the cases). The departure from the linear case, leading to the structure collapse and virialization, is clearly seen after δ lin ∼ 1.5. Credits: [START_REF] Sanchez-Conde | The spherical collapse model with shell crossing[END_REF] . . . . . . . . . . . . . . . . . . . . 3.9 The rotational curve measured in the 21-cm line of neutral hydrogen for spiral galaxy NGC6503. The solid line shows the dark halo fitted to the data. The profile is an isothermal sphere with a finite core radius ρ ∝ (r 2 c + r 2 ) -1 . Also shown are the visible (dashed), gas (dotted) and dark halo (dash-dot) components curves. Image : [START_REF] Begeman | Extended rotation curves of spiral galaxies -Dark haloes and modified dynamics[END_REF] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10 Comparison of density profiles of halo models: in green the singular isothermal sphere(SIS) profile, in blue the universal(NFW) profile and in red the truncated isothermal model (TIS) with ρ TIS (r) = (σ 2 v )/(2πGr 2 ) s 2 /(s 2 + r 2 ). Image : Stella Seitz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.11 NFW density profile fit to N-body simulations for eight different cosmologies: the best fitting profile is shown for the low mass (solid) and the high mass (dash) halos. SCDM is the EdS model, Ω 0 is the matter density parameter and n s is the spectral index of the initial density fluctuation power spectrum P (k) ∝ k n S seeding the simulation. The radius is scaled to r 200 and the density to ρ crit (z = 0). Lower mass halos are more concentrated near the center than higher mass halos. As a consequence their concentration parameter is higher. Image : [START_REF] Navarro | A Universal Density Profile from Hierarchical Clustering[END_REF] . . . . 3.12 Faber Jackson relations (top) and Tully Fisher relation (bottom) for 2MASS galaxies in K band. The left panel shows individual galaxies while the right panel shows the mean relations. Note the larger scatter of the Faber Jackson plots than in the Tully Fisher. Image: [START_REF] Kochanek | Gravitational Lensing: Strong, Weak and Micro[END_REF] 4.1 Right: Selection of the stars on the deep stacked frame, using their second order moments, which form a clump in the ∆s x -∆s y plane. The stars are selected in the blue 5-σ ellipse centered at (0,0). Left: in the surface brightness (µ)magnitude plane, the stars selected using the "star clump" method. In red are the stars identified on the deep stack, in blue are the supplementary stars identified using individual CCDs. The bright stars (r ≤ 16) are selected by neither methods. viii 4.2 Stars catalog in the µ -m plane for the four fields. In blue: the stars selected by the star-clump method. In red: the bright stars selected on their magnitude and surface brightness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Upper : CARS mask for star haloes and diffraction spikes. Credit: [START_REF] Erben | Cars: the cfhtls-archiveresearch survey; i. five-band multi-colour data from 37 sq. deg. cfhtls-wide observations[END_REF]. Lower : Final implementation of the masking procedure. In blue the polygons from CARS mask ( [START_REF] Erben | Cars: the cfhtls-archiveresearch survey; i. five-band multi-colour data from 37 sq. deg. cfhtls-wide observations[END_REF]). In magenta K10 bright stars haloes mask. In green and red circles, the bright stars mask, the radius is set according to the star r magnitude. . . . .

5.1

The different SNLS3 SNe samples selected by requiring : that the SN is included in JLA sample ; that the SN was selected in K10 sample ; that the SN is selected using the masking and scoring criteria as described in section 4.3.3. Note that on the first line, for K10 complete sample, we used the magnification computed in this thesis, and the residuals as published in K10 and not the JLA residuals. . . . 5.2 The different SNLS5 SNe samples selected using the masking and scoring criteria as described in section 4.3.3 and by requiring : that the SN is included in SNLS5 sample, with an additional color cut |c| < 0.25 eventually and the exclusion of 03D4gl ; that the SN is included in JLA sample. We used the residuals from SNLS5 analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

Introduction

Towards the end of the last century the discovery of the present time acceleration of the expansion of the Universe [Per97], [START_REF] Riess | Observational evidence from supernovae for an accelerating universe and a cosmological constant[END_REF], [START_REF] Schmidt | The high-z supernova search: Measuring cosmic deceleration and global cur vature of the universe using type ia supernovae[END_REF] changed our perspective of modern cosmology. This discovery was achieved by using distant type Ia supernovae (SNe Ia) explosions as standard candles to estimate their distances and then build a Hubble diagram : the luminosity distance, as a function of the redshift, which is related to the Universe's expansion scale factor at the time of explosion. The observed departure from the Hubble law (valid at small distances) was not compatible with a matter filled Universe, and required an additional constituent of the Universe : the dark energy.

Dark energy can be modeled as a perfect fluid, with an equation of state parameter w = p/ρ, where ρ is the density and p the fluid pressure. It can be equivalent to a cosmological constant Λ -then w = 1 -and is believed to be the responsible factor for the present accelerating expansion. Current observational constraints are in favor of a flat Universe with a cosmological constant Λ, filling up roughly 70% of the Universe.

SNe Ia exhibit homogenous brightness properties. When the intrinsic sources of variations for the peak luminosity of SN Ia are corrected, their corresponding brightness distribution are almost perfectly standard with only 15% peak luminosity variations.

The Supernovae Legacy Survey (SNLS) is a project which main objective was to pursue the nature of dark energy and put a better constraint on the corresponding equation of state parameter w, using high redshift (z) supernova type Ia as a probe. It drew it's data between 2003 -2008 from the CFHTLS1 . Also a joint analysis was carried out using the SNLS 3 years sample (high-z samples) and SDSS II sample (z<0.4 samples) SNe Ia, called the Joint Light Curve Analysis (JLA). With 740 spectroscopically confirmed SN Ia, the JLA result was consistent with other probe's ( e.g. CMB observations by the Planck (and WMAP), the BAO) results of a constant dark energy equation of state parameter in a flat Universe. The best fit value for the equation of state parameter obtained was w = -1.018 ± 0.057 for a flat w-CDM cosmological model [START_REF] Betoule | Improved cosmological constraints from a joint analysis of the sdss-ii and snls supernova samples[END_REF]. The full SNLS 5 year sample analysis, which is underway, will provide ∼ 150 additional spectroscopic SNe Ia.

In this thesis we carried out weak gravitational lensing analysis using SN Ia of the full 5 years data sample of the SNLS. Gravitational lensing is the bending of light rays in the presence of massive gravitational field. It leads to an isotropic alteration in the measured brightness of the source. As a consequence, the supernova light can be either magnified or de-magnified by the presence of dark matter galaxy haloes on their line-of-sight. In relation to the Hubble diagram, this effect will increase it's dispersion.

Our lensing magnification signal detection algorithm was thus based on the computation of the correlation of SN luminosity which is calculated using two separate (and independent) methods i.e. the lensing magnification and the Hubble residual.

The lensing magnification is computed by modelling the dark matter halo (with a prior on the mass-luminosity relation) embedded on foreground galaxies along the line-of-sight. Due to the lensing effect, the SN flux will be slightly increased or decreased with respect to its expected value, leading to a residual in the Hubble diagram, between the distance modulus (or SN flux) obtained from the SN magnitudes and its value predicted from the cosmological model using the SN redshift.

Thus, the detection of the lensing signal relies on the measurement of a positive correlation between the Hubble residual and the lensing magnification. This method was first developed and carried out on SNLS3 data by [START_REF] Kronborg | Gravitational lensing in the supernova legacy survey (SNLS)[END_REF] and in this thesis we present an updated analysis performed with the full spectroscopic sample of the SNLS 5 years data.

In the near future upcoming large scale high-z (deep field) surveys like the LSST, are expected to detect thousands of SNe. With such a statistics at high redshift, SNe gravitational lensing will help probe the dark matter distribution at small scale in a complementary way to the main cosmological probes, such as galaxy-galaxy lensing, carried out by these projects.

The chronology of this monograph is broadly divided into two sections. The first part (chapters 1-3) presents the theoretical framework and the observational background of the subject. In chapter 1, we recap the cosmological background and the recent results relevant to our analysis. The following chapter 2 discusses the supernovae categories and the advantages of the type Ia as a standard candle, along with the description of the SNLS project. Chapter 3 presents the mathematical frameworks of gravitational lensing and the modeling of the galaxy dark matter haloes, along with the previous results obtained in detecting lensing on SNe Ia.

The remaining last two chapters of the thesis are directed towards the analysis and results obtained from the SNLS5 data. Chapter 4 explains the SNe line of sight modeling and selection. We present the galaxy catalogue used to select the line of sight lenses and present the computation process of their photometric redshifts. The identification of the contaminated regions (luminous pollution) by bright stars are described, which determines the selection of the line-of-sight suitable for the analysis. An important step for the SNe selection is the identification of the host galaxy.

In chapter 5 we present the magnification analysis for the SNLS5 sample and the sensitivity to two different halo models (SIS and NFW) and SNe selection. We discuss on the prospects of lensing signal expected with the addition of a complementary ∼ 300 photometric SNe in future. The appendix gives the list of all supernovae selected for the lensing signal detection process and their lensing magnification estimated with two different halo models.

Chapter 1

Cosmology

The twentieth century saw the building of cosmology theoretical frame : Einstein general relativity [Ein52], its solution the Friedman's equation in 1922 [START_REF] Friedmann | On the Curvature of Space[END_REF], the observational discovery of redshift by V. Slipher[Sli13], the proposal of the Big Bang model by Lemaître [Lem33] in 1933 and the establishing by Edwin Hubble of the Hubble's law ([Hub26], [Hub36]). Subsequently is modern cosmology based on the current standard Hot Big Bang Model which states that the universe expanded from an initially hot and dense state, and that it is ever-expanding today.

Most observational evidence agree with the concordance ΛCDM model : a flat universe, where dark energy makes up approximately 70% of the energy density, dark matter about 25%, which leaves 5% for atoms, outnumbered, in 1:6 10 10 by the cold relic black-body photons of the Cosmic Microwave Background radiation at T 2.7255 K 1 .

We present here the cosmological standard model frame, and some of the observational evidences that sustain it. This chapter is based in particular on [Ryd03], [START_REF] Rich | Fundamentals of Cosmology[END_REF], and [START_REF] Hardin | Notes on general cosmology[END_REF] notes.

A homogeneous expanding universe

The Cosmological Principle states that "the universe looks the same whoever and wherever you are" ([Lid03]) or that "there is nothing special about our location in the universe" ([Ryd03]): this implies that is is homogeneous and isotropic. Isotropy refers to the fact that there are no preferred directions, which is sustained by the exceptional isotropy observed by the COBE mission of the cosmic microwave background (CMB) temperature T 0 = 2.7255±0.0006 K ([Fix96], [Fix09]), to roughly one part in 100 000, once the dipole due to the earth motion is subtracted ( [START_REF] Smoot | Structure in the COBE differential microwave radiometer first-year maps[END_REF]). Homogeneity means that there are no preferred location. This holds at large scales 2 , of about 100Mpc ( [START_REF] Hogg | Cosmic Homogeneity Demonstrated with Luminous Red Galaxies[END_REF]), corresponding to a scale larger than super-clusters and voids (see figure 1.1).

The universe geometry is described through its metric, and the homogeneity and isotropy hypothesis lead to the Robertson-Walker (RW) metric :

ds 2 = dt 2 -R 2 (t) dr 2 1 -kr 2 + r 2 (dθ 2 + sin 2 θdφ 2 ) (1.1)
where t is the cosmic time, R(t) is the cosmic scale factor, the coordinate system for a spatial location is given by the spherical coordinates (r, θ, φ), while k = -1, 0, +1 is the sign of the 1 . . . and relic neutrinos 2 And obviously not at scales as small as our Galaxy or our Local Group. 1.1 A homogeneous expanding universe uniform curvature radius of spatial sections and sets the universe geometry :

k =        -1 closed 0 flat +1 open
There are compelling observational evidences that tell us that the universe at present is flat with k = 0 ([Hin13], [START_REF] Ade | Planck 2015 results. xiii. cosmological parameters[END_REF]). The scale factor R(t), or equivalently its normalized expression to present day t 0 value : a(t) ≡ R(t)/R(t 0 ), describes how the distances expand (or contract) with time. The spatial coordinates (r, θ, φ) are said to be comoving as they do not change with the expansion of the universe. The RW metric can also be written using the comoving spatial coordinates (χ, θ, φ), with dχ = dr/ √ 1 -kr 2 so that the relation between the coordinates r and χ depends on k sign : r = S k (χ), with :

S k =        sin(χ) k = +1 χ k = 0 sinh(χ) k = -1

Hubble Law

The proper distance d traveled by a photon between two given points at comoving coordinate χ and χ + dχ is obtained using the RW metric and setting ds 2 = 0 :

d = cdt = R(t)dχ (1.2)
Their distance evolves trough time, due to the expansion. Their relative velocity dv = Ṙ(t)dχ is thus proportional to their distance :

dv = Ṙ(t) R(t) d = H(t)d (1.3)
with H(t) ≡ Ṙ(t)/R(t) being the Hubble parameter, the expansion rate of the universe at time t, and H 0 = H(t 0 ) its present day value. The Hubble law (1.3) was discovered by Edwin Hubble in 1929 ([Hub29]), who interpreted the observed redshift (see below) of nearby galaxies as a (non-relativistic) Doppler shift due to their radial velocity v away from us, and established that it was proportional to their distance d through the linear law : v = H 0 × d. Due to the underestimation of the galaxies distances he was using, the value of H 0 500km s -1 Mpc -1 he obtained was largely overestimated. A recent estimation is ( [START_REF] Ade | Planck 2015 results. xiii. cosmological parameters[END_REF], see section 1.6.1):

H 0 = 67.51 ± 0.64km.s -1 .Mpc -1
H 0 value can also be expressed using the dimensionless number h : H 0 = 100h km s -1 Mpc -1 . The Hubble time H -1 0 ∼ 14 Gyr is the time scale of the expansion, related to the time since the primordial highly dense state of the universe. The Hubble distance c/H 0 ∼ provides a natural distance scale. The Hubble parameter is a function of time, and the next section will establish its dependency, determined by the composition of the universe.

Cosmological redshift

In such a universe, the wavelength λ emt of the light emitted at time t by a given source (located at a radial position r E ), and received at t 0 by an observer (located at r = 0) is redshifted at a wavelength λ obs , through the relation :

R(t 0 ) R(t) = λ obs λ emt
The redshift, defined as : z ≡ λ obs -λ emt λ emt is thus related to the change of the expansion factor between emission and observation of the light :

1 + z = R(t 0 ) R(t)
Vesto Slipher in 1912 was the first to observe this effect on the light emitted by spiral nebulae [Sli15]. It is however not identical to Doppler shift3 .

The Friedman universe

Einstein's equation relates the geometry and the matter and energy content of the universe:

G µν = R µν - 1 2 Rg µν -Λg µν = 8πG c 4 T µν (1.4)
In this famous equation, G µν is the Einstein tensor and g µν , R µν , R, G, Λ, T µν are respectively the metric tensor, the Ricci tensor, the Ricci scalar, the gravitational constant, the cosmological constant, and the energy-momentum tensor, describing the matter-energy contents of the universe.

For a perfect fluid the energy-momentum tensor is given by:

T µ ν = pg µ ν + (p + ρ)U µ U ν (1.5)
where U ν is the fluid four velocity, p the pressure and ρ the energy density.

The cosmological constant Λ was first introduced in 1915 by Einstein to force the existence of solutions describing a static universe. As it came to knowledge that the universe was actually expanding, Einstein would famously call it his 'greatest blunder'. In the late 90's, the cosmological constant came in favor again, as its presence in equation 1.4 could account for the acceleration of the cosmic expansion observed by supernovae cosmology surveys ([Per99], [START_REF] Schmidt | The high-z supernova search: Measuring cosmic deceleration and global cur vature of the universe using type ia supernovae[END_REF]). Equivalently, the cosmological constant can be taken into account by adding a supplementary component entering the energy-momentum tensor, with a constant energy density ρ Λ ≡ c 2 Λ/(8πG) and a negative pressure p Λ = -ρ Λ . Physically it can be interpreted as a "zero point energy" equivalent to the quantum vacuum energy. There is a big mismatch though between its estimated value from the observations of the universe expansion at large scales, and its natural value as suggested by particles physics ([Rug00], [START_REF] Carroll | The cosmological constant[END_REF], [START_REF] Padmanabhan | Cosmological constant -the weight of the vacuum[END_REF]), with a difference in order of 10 120 .

Given the FRW metric (1.4), the 0 -0 component yields the Friedmann equation 1.6a (see e.g. [START_REF] Rich | Fundamentals of Cosmology[END_REF]). The i -i component of the Einstein equation together with eq. 1.6a leads to the acceleration equation 1.6b :

ȧ a 2 = H 2 = 8πG 3 ρ - k R 2 (1.6a) ä a = - 4πG 3 (ρ + 3p) + Λ 3 (1.6b)
The Friedmann's equation 1.6a relates the energy content of the universe to it's geometry. The acceleration equation 1.6b shows that gravitational forces slow down the expansion while a positive cosmological constant can lead to an accelerating expansion, as observed today. Setting 1.2 The Friedman universe k = 0 in 1.6a defines the critical density corresponding to a flat universe :

ρ cr (t) = 3H 2 8πG (1.7)
Numerically, the critical density value is today ρ cr (t 0 ) = 1.88 h 2 × 10 -26 kg.m -3 . The Ω's parameters are defined as the ratio to the critical density of each contribution:

Ω m = ρ m (t 0 ) ρ cr (t 0 ) Ω r = ρ r (t 0 ) ρ cr (t 0 ) Ω Λ = Λ 3H 2 0 Ω k = -k R 2 0 H 2 0
where the subscript m stands for matter and r for radiation. Although they can be defined at anytime t, we refer here and in the following (if not otherwise stated) to their present day value at t = t 0 . Energy conservation is expressed by the vanishing of the covariant divergence of the energymomentum tensor, leading to equation 1.8 :

ρ + 3 ȧ a ρ + p c 2 = 0 (1.8)
which is equivalent to thermodynamics first law d(ρa 3 ) = -pd(a 3 ). The composition of the matter-energy content of the universe is described as an ideal fluid with its density ρ and its pressure p related trough its equation of state parametrised by w :

w = p ρ (1.9)
For non-relativistic particles (p = 0) like dark matter, galaxies, w = 0, and for radiation (relativistic particles) with pressure p = ρ/3, w = 1/3. Using this relation and the fluid equation (1.8) we obtain a general solution of the fluid :

ρ = ρ 0 a(t) -3(1+w)
(1.10) so that ρ r ∝ a(t) -4 for radiation and ρ m ∝ a(t) -3 for matter. As a consequence, the universe started with an era of radiation domination, with radiation density producing almost all the gravitational force. At a -1 eq ≡ (1.68Ω m /Ω γ ) -1 3200 begun the era of matter domination. The value w = -1 corresponds to the cosmological constant, with a negative energy density. More generally, one can postulate the existence of a dark energy (DE) component, with no assumption on w DE value or even its evolution with time or redshift w DE (z), only needing (see equation 1.6b) that w DE < -1/3 so that this dark energy component accelerate the expansion. A range of value of w DE can be obtained in the frame of canonical scalar field models also called quintessence with -1 < w DE < 1 ([Tsu13], [Tak14]). The phantom field energy, with w < -1, can give rise to strange situations including a universe ending in a Big Rip [START_REF] Sami | Phantom field and the fate of universe[END_REF].

Integrating the Friedman equations to obtain the evolution of the expansion rate H as a function of time t, or equivalently the redshift z as 1 + z = 1/a(t), we obtain:

H(z) 2 = H 2 0 Ω r (1 + z) 4 + Ω m (1 + z) 3 + Ω k (1 + z) 2 + Ω Λ (1.11)
leading, when taken at t = t 0 , to the relation :

Ω T = Ω m + Ω r + Ω Λ = 1 -Ω k (1.12) so that we have |1 -Ω T | = c/(R 0 H 0 ). Note that Ω r (t)
Ω m (t) for z z eq and hence Ω r (t) can generally be omitted at later epochs.

Hot Big Bang model and the thermal history of the universe

The Hot Big Bang model is based on the observation of the universe expansion, which might be traced back in time to an originating single point. In this model, the universe started off from a very hot and dense state after the initial 'Big Bang' to its current cool and tenuous state, and is still expanding today.

As the universe expanded adiabatically, its temperature T cooled, scaling as T ∝ a(t) -1 : the mean energy per photons dropped from 10 28 eV at the Planck time, at a time t ∼ 10 -44 s before which no proven theory can correctly describes the universe, to a mean energy of 6. 10 -4 eV today. Salient phases in the universe thermal history are presented in table 1.1.

During the first few microseconds following the Planck epoch, all matter existed as a free sea of quarks and leptons. When the temperature dropped below T 150 MeV, the quarks bounded together into baryons, mostly protons and neutrons. The small excess of quarks over antiquarks led to a small excess of baryons over antibaryons. The radiation was in the form of extremely abundant photons, neutrinos (and anti-neutrinos). The ratio of baryons to photons numbers in the universe η ≡ n B (t)/n γ (t) is constant as n ∝ t -3 for both species, and its value is of order η ∼ 6 10 -10 , very much in favor of the photons. Equivalently, η can be expressed in terms of the baryonic fraction of the critical density Ω B ≡ ρ B (t 0 )/ρ cr (t 0 ), as n γ (t 0 ) is fixed by the present day CMB relic photons temperature T 0 = 2.7255 K : Ω B h 2 3.65 10 -3 × (10 10 η).

Once the temperature drops at T 1 MeV (at t ∼ 100 s), deuterium nuclei could survive disruption by high-energy photons so that the fusion of neutrons and protons into nuclei heavier than hydrogen could proceed. This is the Big Bang Nucleosynthesis (BBN). The primordial nucleosynthesis stopped at t ∼ 10 minutes. It produced essentially helium, no elements heavier than lithium, and left about 75% of hydrogen. The yields of these elements (D, 3 He,4 He, and 7 Li) depends strongly on the baryon-to-photon ration η, as the more the photons, the later the BBN can start. The measurement of the abundance of these elements today gives us a powerful tool for testing the Hot Big Bang model. Indeed (see figure 1.2), the Hot Big Bang model successfully reproduces the observed abundances of the light elements, provided η, or equivalently Ω B h 2 , lies in a very narrow range. By the end of BBN, the universe is filled by a ionized plasma of nuclei (mostly protons), electrons, and photons. The high energy photons prevent the formation of neutral atoms because of photoionization, and interact primarily with the electrons through Compton scattering. So that the universe is opaque as well as ionized. At a time t ∼300,000 years, the temperature has cooled down to T 3000 K, and electrons and protons (or nuclei) can undergo radiative recombination into atoms. Since at the epoch of recombination the number of electrons drops rapidly, photons decouple from the electrons : their mean free path becomes longer than the Hubble distance c/H and the universe becomes transparent. These black-body photons have been streaming (nearly) freely since the recombination-decoupling era, and are at present times observed at a wavelength redshifted into the microwave spectrum, corresponding to a temperature T 0 2.7255 K (see §1.

Distances

Locally in our Galaxy, distances can be measured using the parallax method, by taking advantage of the earth's relative motion about the Sun leading to a shift in the apparent position in the sky. Further away, in the Hubble flow, distances are related to the source redshift through the cosmological parameters. Distance measurement in the expanding universe has many categories and we shall present here the luminosity, angular and comoving volume distances.

For most of the distances we define the common function :

E(z) ≡ H(z)/H 0 = Ω m (1 + z) 3 + Ω k (1 + z) 2 + Ω Λ
where, as z > z eq , we have neglected Ω r . We shall first interest ourselves to the distance of a source, at a spatial position χ (or r E = S k (χ)), emitting light at t E , received, redshifted by a factor (1 + z), at t = t 0 by an observer at a position χ = 0.

Comoving coordinate and proper distance

Following equation 1.2, and integrating along the total line-of-sight the infinitesimal dχ = cdt/R(t) contributions, one obtains the relationship between the comoving coordinate χ and t E , or equivalently, using 1 + z = 1/a(t), with the redshift z :

χ(z) = c R 0 H 0 z 0 dz E(z)
The comoving coordinate is related to the proper distance through = R 0 χ. The time corresponding to the proper distance is the look back time and is simply D P (z)/c. It can be visualized by thinking of a measurement with a tape in any instant of the universe with expansion frozen. Practically it is not feasible since super luminous speed is not possible.

Luminosity distance

The (bolometric) flux F measured by the observer is related to the luminosity L of the source through :

D L = L 4πF
(1.13) 10 1.4 Distances which defines the luminosity distance D L . Taking into account the surface of the sphere of radius χ around the source, and the drop of the photons energy due to their redshift, the luminosity distance reads :

D L = R 0 (1 + z)S k (χ(z)) (1.14)
The dependency of the luminosity distance as a function of redshift, for z < 5, and the cosmological parameters (Ω m , Ω Λ ) is shown on figure 1.3, for different cosmological models : a flat and matter-only (Cold Dark Matter or CDM) universe, also known as the Einstein-de Sitter model, and a flat universe with matter and dark energy (ΛCDM) model. At high redshift, the deviation from the linear Hubble law d ∼ cz/H 0 depends solely on the density parameters Ω m and Ω Λ .

Standard candles are a class of object of supposed constant luminosity L. They can be used to measure relative luminosity distances, their unknown luminosity L canceling out in the ratio : F 1 /F 2 = (D 2 /D 1 ) 2 . Type Ia supernovae explosions have been used to this purpose, and permitted to demonstrate that the universe's expansion at present time is accelerating ([Per99], [START_REF] Schmidt | The high-z supernova search: Measuring cosmic deceleration and global cur vature of the universe using type ia supernovae[END_REF]).

Magnitudes

In practice we don't use the flux F but the apparent magnitude m defined as

m = -2.5 × log F F 0 (1.15)
where F 0 is the flux of a reference source. The absolute magnitude M is the magnitude of a source of same luminosity but situated at a distance of 10pc from the observer. It is thus possible to rewrite the relation 1.15 as:

m = M + 5 × log d L (z) 10pc = M + µ(z) (1.16)
defining the distance modulus µ(z) as :

µ = 5 × log d 1Mpc + 25
(1.17)

Angular Diameter Distance

The object of our interest has now a transverse physical size d , and is observed subtending an angle δθ. The angular diameter distance D A is then:

D A = d δθ (1.18) At emission t = t E , the proper size of the object is d = r E R(t E )δθ, so that D A = S k (χ)R 0 /(1+z)
and we can write :

D A = D L (1 + z) 2 (1.19)
Because of the relation 1.19, there is no point in attempting to separately determine both angular and luminosity distances if the redshift is accurately known, unless there is a need to check the Robertson-Walker metric or the origins of the cosmological redshift. At low redshifts, we retrieve the Hubble law, and the angular diameter distance, the luminosity distance and the proper distance are equal :

D P D A D L cz H 0 for z 0.1
The dependency of the angular distance as a function of redshift, for z < 5, and the cosmological parameters (Ω m , Ω Λ ) is shown on figure 1.3. The relative angular diameter distance between two objects at redshifts z 1 and z 2 is important in gravitational lensing analysis. It is not a straightforward subtraction between two angular distances, and is given by ([Hog99]):

D A12 = 1 1 + z 2 D m2 1 + D 2 m1 R 2 0 -D m1 1 + D 2 m2 R 2 0 , D m (z) = R 0 S(χ(z)) (1.20)
where D m1 and D m2 stand for the corresponding distance D m for the redshifts z 1 and z 2 . This formula is only applicable for Ω k ≥ 0.

A standard ruler is an object which physical scale d is known. Patterns in the distribution of distant galaxies provides such a ruler (see section 1.6.2). In this case, one can observe the pattern in angular coordinates (δθ = d /D A ) as well as in radial -in fact -redshift coordinates: cdz = d H(z). This requires to introduce the comoving volume distance :

D V = (1 + z) 2 D 2 A (z) cz H(z) 1/3
(1.21)

An inhomogeneous universe

Although appearing homogeneous on large scale (>100 Mpc), the universe is clumpy on small scales (<50 Mpc) with voids and clusters of galaxies. We will concern ourselves in this section on the evolution of the matter density fluctuations and the formation on structures larger than galaxies. The formation of bound structures will be described in section 3.2.1.

The matter density ρ( r, t) is now a statistical field depending on the comoving coordinate r as well as time. Averaging the density over a large volume V yields the mean energy density ρ(t). Another point of view would be to average, at a fixed ( r, t), the random field ρ( r, t) over many realizations.

The density contrast is defined as :

δ( r, t) = ρ( r, t) -ρ(t) ρ(t)
Large scale structures will grow through gravitational instability, starting from δ 1 up to δ ∼ 1 where they will leave the linear regime, collapse onto themselves, and in effect detach from the Hubble expansion (see section 3.2.1).

The power spectrum

To study the universe clumpiness as a function of the structures scale, we define the Fourier transform 5 of the density contrast δ, omitting the time dependency :

δ k = 1 V δ( r)e i k. r d 3 r
The comoving wavenumber k is associated to the comoving length through r = 2π/k. If δ( r) is isotropic, we can define without loss of information the power spectrum :

P (k) = |δ( k)| 2
5 following here the conventions from [START_REF] Peacock | Cosmological Physics[END_REF]. The average is again made over many realizations. Besides isotropy, we may also make the assumption that the statistical field δ k is gaussian, so that its characteristics are entirely characterized by its power spectrum. In particular, most inflationary scenarios predicts that inflation seeded at early epochs density fluctuations as a homogeneous, isotropic and gaussian field, via amplified quantum fluctuations from which structures formed at later times. The power spectrum is related to the variance of the total mass M R in a sphere of comoving radius R, measuring the clumpiness at scale R or equivalently k ∼ 1/R. The mean mass is M R = 4π 3 ρ 0 × R 3 , and its variance reads :

M R -M R M R 2 ∼ ∆ 2 k
in which ∆ k is the dimensionless power spectrum per log k :

∆(k) 2 ≡ V (2π) 3 4πk 3 P (k)

The power spectrum law

The power spectrum is often taken as a power law :

P (k) ∝ k n
In this case, the variance of M R becomes δM/M ∝ R -(3+n)/2 . The spectral index n governs the balance between the large and small scale power. The special case n s = 1 is the Harrison-Zel'dovich-Peebles scale invariant spectrum ([Har70], [START_REF] Eisenstein | Baryonic Features in the Matter Transfer Function[END_REF], [START_REF] Zel | Gravitational instability: An approximate theory for large density perturbations[END_REF]), and corresponds to scale invariant fluctuations of the gravitational potential.

In the linear regime, the matter power spectrum is related to its primordial form

P 0 (k) = A S k n S through the transfer function T (k) ∝ δ k (z = 0)/δ k (z = z primordial ) : P (k) lin = P 0 (k)T 2 (k).
During the radiation era, due to the microphysics at work, only density pertubations at scales larger than the Hubble radius could grow as δ ∝ a 2 . When the matter domination era occurred, cold (i.e. non-relativistic particles) dark matter fluctuations could grow as δ ∝ a, whether they were super-Hubble or not. The caracteristical scale involved is the Hubble radius at radiationmatter equality, which translates into a comoving number k 0 ∼ 0.1hMpc -1 × (Ω m h)/(0.3 × 0.7)).

σ 8 at Ω m = 0.3 q q q q q M08 (RASS) M10 (
As a consequence, in the linear regime, starting out e.g. with a scale invariant shape after inflation, the power spectrum for cold dark matter is ∝ k at large scale, and bent over (∝ k -3 ) for wavenumbers k k 0 . The observations corroborate the CDM model (see figure 1.4) and gives a measurement of Ω m h. In the CDM model, the smaller scales structures merge to form the larger ones (bottom-up formation) as is confirmed by the observation of the earliest galaxies at z ∼ 10.

The power spectrum normalization σ 8

The normalization of the power spectrum is customarily expressed in terms of σ 2 8 = (δM R /M R ) 2 for R = 8h -1 Mpc, which corresponds to a cluster size. Observationnally, σ 2 8 ∼1 so that below this scale, linear approximation is not applicable. The normalization of the power spectrum σ 8 can be estimated using several methods:

• measuring the CMB temperature anisotropies as the density perturbations induced the fluctuations in the CMB temperature on the sky. • estimating the two point correlation function ξ(r) = δ( x)δ( x + r) of galaxies. It is related to the probability to find a pair of galaxies at a distance r :

d 2 P (r) = n 2 (1 + ξ(r))dV 1 dV 2
(n is their mean number per unit volume). It expresses the excess of probability over what is expected in the non-clustered case. The correlation function is the Fourier transform of the power spectrum, so that :

ξ(r) = V (2π) 3 P (k) sin kr kr 4πk 2 dk (1.22)
To translates galaxy counts into mass fluctuations, one relies on the assumption that they are related through δ n galaxies = b δ CDM where b is the bias parameter which can be complicated to estimate. • counting clusters. Clusters counts estimated from optical or X-ray data or Sunyaev-Zeldovich(SZ) data ([Sad04], [START_REF] Taburet | The Sunyaev-Zel'dovich contribution in CMB analyses[END_REF]). The number density of clusters forming at a given epoch is related through the gravitational collapse mechanism to the matter density Ω m and σ 8 values at that time. Measurements of σ 8 for different methods are presented on figure 1.5, corresponding to a value of σ 8 0.8.

Cosmological Probes

The parameters of the concordance ΛCDM cosmology model are the density parameters for matter, baryons and dark energy : Ω m , Ω B , Ω DE , the dark energy equation of state parameter w, the Hubble parameter H 0 , the power spectrum spectral index n and its normalization at e.g. 8h -1 Mpc. One can also add : the reionization optical depth τ , which is related to the process that reionized the matter in the universe after the "dark ages", by the first galaxies and quasars at z ∼ 6 -20.

We will present below some of the cosmological probes used to estimate the cosmological parameters.

Cosmic Microwave Background

CMB is one of the major cosmological probe as the properties of our universe at the epoch of the photons emission, and along their path towards us, is imprinted in the CMB features.

Before recombination, the tightly coupled baryons and photons plasma was undergoing acoustic waves caused by two opposing pulls: the gravitational pull into the gravitational potential dark matter wells (dark matter producing almost all the gravitational force as this occurred during the matter domination era), and the outwards pull of radiation pressure. These acoustic oscillations are referred to as the Baryonic Acoustic Oscillations (BAO). After recombination, the universe became neutral and photons free-streamed with a mean free path larger than the Hubble distance, leaving the baryons free to fall in the dark matter gravitational wells. The angular size δθ of a temperature fluctuation in the CMB is related to its physical size on the last scattering surface, at an emission time t ls ∼ t rec corresponding to a redshift z ls ∼ z rec ∼ 1100, through : δθ = /d A (z 1100).

The CMB experiments measure the temperature anisotropies δT /T on the celestial sphere. It is thus natural to write δT /T as :

δT T (θ, φ) = ∞ l=0 m=+l m=-l a lm Y lm (θ, φ) (1.23)
where θ and φ specify the directionality on the sky and Y lm are the spherical harmonic functions.

The dipole l = 1 term corresponds to the Solar system peculiar velocity. The statistical properties of the temperature fluctuations are carried out by the angular spectrum :

δT 2 T 2 = l(l + 1) 2π C l , C l = |a lm | 2 m
The term C l is thus related to the temperature fluctuations at an angular scale δθ ∼ π/l. The angular power spectrum as measured by Planck mission ([Tau04]) is presented on figure 1.6. The highest peak, at l S ∼ 200 or equivalently δθ S ∼ 1 • corresponds to the scale of the sonic horizon at t rec i.e. the distance that a sound wave supported by the photon-baryon-electron plasma could travel between the Big-Bang and the recombination :

S = R(t rec )χ S = R(t rec ) ∞ zrec c s dz H(z) (1.24)
where c s is the sound speed and depends on the baryon-to-photon ratio. The sonic horizon S corresponds to the comoving distance L = R(t 0 )/R(t rec ) S ∼ 150 Mpc. This physical scale is also observed closer to us, in the large scale structures (see section 1.6.2). The measurement of δθ S = S /d A (z) at z ∼ 1100 is essentially sensitive to H 0 and to the curvature of the universe, i.e. Ω T , along, because of the sonic horizon size, with the baryonic content Ω B h 2 , and the matter content Ω m h 2 . CMB photons are polarized at the level of a few microkelvins, as Thomson scattering of radiation on electrons globally falling or escaping from a potential well generates linear polarization. The CMB polarization is described by the two fields E and B, the E modes are generated by the scalar density fluctuations, whereas primordial gravitational waves, foregrounds, and gravitational lensing on the photons line-of-sight generate both E and B modes. The detection of the primordial gravitational waves can only be confirmed from the detection of B modes.

Succeeding to the Wilkinson Microwave Anisotropy Probe (WMAP) ([Pag03], [START_REF] Bennett | The Microwave Anisotropy Probe (MAP) mission[END_REF]) mission, itself a successor to COBE (Cosmic Background Explorer) mission, the ESA (European Space Agency) Planck satellite mission6 ([The06], [Tau04]) was launched in 2009 and had 30 months of observation. The sky map of the temperature anisotropies is presented for these 3 successive missions in figure 1.7. The Planck mission has given indeed an impressive confirmation of the standard cosmological model. Planck TT, TE, and EE cross-and auto-spectra are extremely well fitted by a 6-parameters flat ΛCDM model(Planck 2015 data, [START_REF] Ade | Planck 2015 results. xiii. cosmological parameters[END_REF]). Planck also provides a lensing potential map, which computation is based on the correlation induced at large scale (a few degrees) by the small (a few arcminutes) lensing deflections due to the large structures along the photon line-of-sight. The parameters describing the best fit model are as follows :

1. the density of baryonic dark matter : Ω B h 2 = 0.02225 ± 0.00016 2. the density of cold dark matter: Ω CDM h 2 = 0.1198 ± 0.0015 3. the angle subtended by the sonic horizon at last scattering : 100θ S = 1.04077 ± 0.00032 4. the fraction of the CMB photons scattered by re-ionized matter along their path : τ = 0.079 ± 0.017 5. the amplitude of the initial density fluctuations power spectrum : log(10 10 A S ) = 3.094 ± 0.034 ). There is a remarkable agreement between the different galaxy surveys and the Planck data. Image : [START_REF] Anderson | The clustering of galaxies in the sdss-iii baryon oscillation spectroscopic survey: Baryon acoustic oscillations in the data release 10 and 11 galaxy samples[END_REF] 6. the slope of the initial density fluctuations power spectrum : n S = 0.9645 ± 0.0049

From this analysis are also inferred the value of the Hubble parameter H 0 = 67.51±0.64km.s -1 .Mpc -1 and Ω m = 0.3121 ± 0.0087. Releasing the flat assumption, and adding observational constraints from the large scale structures and supernovae Ia projects, allows to estimate the spatial curvature Ω k = 0.0008 ± 0.004. Together with data from large-scale structure (BAO) and type Ia supernovae, the gravitational lensing potential measurement constrains σ 8 = 0.8159 ± 0.0086 at 68% c.l., in tension with clusters data.

Baryon Acoustic Oscillations

The Baryon Acoustic Oscillation (BAO) is an important cosmological probe for estimating the geometry of the universe and the dark energy content. As the acoustic waves ceased in the primordial baryons-electrons-plasma at recombination, their imprint, of characteristic size S , was left in the matter distribution. As a consequence, galaxies showed a slight preference to form, and thus be later on found at time t, at a separation distance ∼ L = a(t)/a rec S . This length scale L 0 ∼ 150 Mpc is observed in the correlation function of galaxies, and is indeed a standard ruler that can be observed from z ∼ 0 -3 in structures to z ∼ 3200 in the CMB.

Using BAO and comparing the sound horizon size of present with the one at recombination era permits to probe the expansion history of the universe and estimate the cosmological parameters, in particular Ω m . BAO measurements require the observation of millions of galaxies in a huge survey volume. Potential systematic errors arise from bias (i.e. the scale dependent difference between the galaxy and the dark matter clustering) and non linear gravitational evolution of structures.

The Sloan Digital Sky Surveys ( Survey which detected about 500 spectroscopically confirmed type Ia supernovae in the redshift range z = 0.05 -0.4; SDSS-III included the Baryon Oscillation Spectroscopic Survey (BOSS) which mapped the spatial distribution of luminous red galaxies (LRGs) and quasars; SDSS-IV is scheduled to continue up to 2020 and includes the extended Baryon Oscillation Spectroscopic Survey (eBOSS).

[Lov02], [Ken94], [ Yor00
Results from BOSS ( [START_REF] Anderson | The clustering of galaxies in the sdss-iii baryon oscillation spectroscopic survey: Baryon acoustic oscillations in the data release 10 and 11 galaxy samples[END_REF]) which observed 10000 2 degrees of sky at a redshift range of [0.2,0.7] are presented on figure 1.8. Also presented on figure 1.8 are the measurement from the 6dFGS survey ( [START_REF] Jones | The 6df galaxy survey: Final redshift release (dr3) and southern large-scale structures[END_REF]) which observed 17000 2 degrees, and the WiggleZ project ([Bla11]), with a redshift range of [0.4,1] and final data set covering a range of 800 2 degrees of sky. The agreement with Planck data preferred ΛCDM model is remarkable.

Type Ia Supernovae

Type Ia supernovae (SNe Ia) are prime examples of standard candles with a very homogeneous intrinsic brightness. SNe Ia are the end points of stellar evolution when oxygen-carbon accreting white dwarfs reach the Chandrasekhar mass M Ch ∼ 1.4 solar mass, the theoretical limit before electron degeneracy pressure is no longer sufficient to support against the inward gravitational pull. Their remarkable uniformity allows them to be used as distance indicators, and an efficient cosmological probe of the expansion history. Using SNe Ia observations, [Per97], [START_REF] Riess | Observational evidence from supernovae for an accelerating universe and a cosmological constant[END_REF] built the Hubble diagram D L (z) up to z ∼ 0.5 to demonstrate the presence of dark energy and the present time accelerated expansion of the universe (see figure 1.9).

SNe Ia can also be used to measure H 0 value. The challenge lies in the measurement of their absolute luminosity -or equivalently in the measurement of their distance independently of their redshift, using Cepheids distance to nearby galaxies, as did the project SH0ES (Supernova H0 for the Equation of State) with the HST Hubble Space telescope (HST). The analysis of SH0ES results taking into account the slight difference in luminosity for SNe Ia occurring in high or low star forming environment yielded H 0 = 70.6 ± 2.6km.s -1 .Mpc -1 within ∼ 1σ of the measurement based on the CMB power spectrum ( [START_REF] Rigault | Confirmation of a Star Formation Bias in Type Ia Supernova Distances and its Effect on the Measurement of the Hubble Constant[END_REF]).

More on supernova cosmology is being described in the next chapter where the Supernova Legacy Survey (SNLS) project is presented, along with the measurement of the dark energy equation of state parameter w. The joint analysis of the SNLS-3 years and the SDSS-II supernovae data, combined with CMB constraints, yields w = -1.018 ± 0.057 for a flat universe (taking into account the statistic and the systematic errors), compatible with a cosmological constant ([Bet14]).

Gravitational Lensing

Gravitational lensing is an important cosmological probe of the matter distribution in the universe, and its overall geometry. The geometry of space time is dependent on the nature of the mass content of the universe hence the wavefronts of distant objects light traveling through any matter fluctuations during its flight will be distorted.

For most lines of sight in the universe, gravitational lensing occurs in the weak regime, and can be detected in the systematic tangential stretch of background sources around the lensing mass (see section 3.1.5). Cosmic shear is the shape distortion of many galaxies by the foreground large scale structures and is thus a statistical signal. The 2-point correlation function of the image distortions is directly related to the power spectrum of the density perturbations. More precisely, the shear angular power spectrum is related to the expansion history of the universe, the history of the growth of structures, and thus sensitive to the dark energy density. Main sources of systematics in weak lensing analysis originates from the incertitudes on the galaxy shape measurement (notably the point spread function -PSF -of the optical instrument) or on the source or lens populations (e.g. on their photometric redshift estimates).

Some recent weak lensing survey results includes:

• the Hubble Space Telescope Cosmic Evolution Survey (COSMOS), which combined spacebased galaxy shape measurements with ground-based photometric redshifts to constrain cosmological parameters σ 8 and Ω M ([Sch10]).

• the Canada-France-Hawaii Telescope Lensing Survey (CFHTLens) project ( [START_REF] Heymans | CFHTLenS tomographic weak lensing cosmological parameter constraints: Mitigating the impact of intrinsic galaxy alignments[END_REF]) which uses 154 square degrees of deep multi-colour data obtained by the CFHT Legacy Survey. They produced the largest contiguous maps of projected mass density obtained from gravitational lensing shear ( [START_REF] Van Waerbeke | CFHTLenS: mapping the largescale structure with gravitational lensing[END_REF]) and performed detailed cosmology analysis ([Fu14]).

• [START_REF] Demetroullas | Cross-correlation cosmic shear with the SDSS and VLA FIRST surveys[END_REF] did exploratory works on optical/radio weak lensing analysis. They performed cross correlation weak lensing analysis with the VLA first survey and the SDSS, covering approximately 10, 000 square degrees. They probed shear power spectrum using cross power spectrum approach on large scales.

Galaxy Clusters

Galaxies or galaxy clusters number counts as a function of their redshift provides another important cosmological probe that helps in constraining cosmological parameters, such as the density parameters and the equation of state parameter (w) or the density fluctuations power spectrum.

The number counting technique can be based on various effects and methods : the Sunyaev-Zeldovich(SZ) effects which distorts the CMB temperature, due to the Compton scattering of the photons of the intra-cluster hot baryonic gas ; the X-ray observations of this hot gas ; the weak gravitational lensing effect caused by the cluster dark matter halo ; the cluster galaxies velocity functions. The effectiveness of clusters as a probe depends on how robust is the association between the cluster observables with the cluster mass.

The number counts can be formalized as:

dN dzdΩ = dV dzdΩ (z) M min dM dn dM (z, M ) (1.25)
where the left hand side gives the number of objects dN observed in a given redshift slice of dz at a solid angle interval of dΩ. dV is the comoving volume at redshift z, n is the comoving number density at the corresponding redshift, and M min is the minimum mass of objects detected in that redshift slice. The knowledge of the comoving number density n(z) is important and can be expressed with the Press-Schechter semi-analytic formalism:

dn dM (z, M ) = 2 π ρ m M δ c σ 2 (M, z) dσ(M, z) dM exp - δ 2 c 2σ 2 M D 2 (z) (1.26)
where ρ m is the present day matter density, δ c ∼ 1.68 is the linear threshold overdensity collapse while D(z) is the linear perturbation growth factor and σ M is the rms density fluctuations at mass scales of M at present day (z = 0). Various projects are observing clusters detected detected in X or CMB data and performing weak lensing studies, such as the Local Cluster Substructure Survey ([Oka10]) using high quality Subaru Suprime-Cam data or the Cluster Lensing And Supernova survey with Hubble (CLASH) which obtained accurate cluster mass profile measurements ([Ume14]).

Indeed, the main sources of systematics are the cluster mass-observable relations and the selection function, as the clusters selection method are based on redshift dependent observables. Number counts of galaxies and clusters provide a direct cosmological probe complementary to the CMB and the SNe Ia measurements, as shown on figure 1.10.

The Cosmic concordance

It is very remarkable that these various probes all favor the same cosmological model (see figure 1.10). Although very satisfactory, this situation nonetheless present us with challenging questions, such as the true nature of the dark energy, or the identity of the dark matter particle. Ω m w q q q q q Clusters CMB SNIa BAO All q q q q q q q q q q q q q q q 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 Figure 1.10: Constraints on constant w dark energy models from the project Weighing the Giants, which uses weak gravitational lensing measurements to improve the measurements of the galaxy cluster mass function and its evolution, using X-ray selected clusters detected in the ROSAT All-Sky Survey. This measurement is compared with constraints form the CMB (WMAP, ACT and SPT), supernova (from the Union sample including SNLS-1 year data) and BAO (SDSS/BOSS, 6dF galaxy survey) data, and their combination. Dark and light shading respectively indicate the 68.3 and 95.4 per cent confidence regions, accounting for systematic uncertainties. Image : [START_REF] Mantz | Weighing the giants -IV. Cosmology and neutrino mass[END_REF] Chapter 2

Type Ia supernovae and the Supernova Legacy Survey

Type Ia supernovae are one of the most fascinating objects of the Universe. They produce the most violent explosions after the Big Bang, explosion of such magnitude (around 4 × 10 9 solar luminosity) that for a short while a supernova will outshine its entire host galaxy. The occurrences of supernovae are not very frequent at all per galaxy, around 2 to 3 per millennium. They are bright, uniform candles and stays so for a convenient period, about a month, during which they can be detected and observed. They are quasi-ideal calibratable standard candles, which makes them a very attractive and practical cosmological probe.

This chapter is structured in two sections. The first section presents the properties of type Ia supernovae and their use as a distance indicator. In the second section we shall describe the Supernova Legacy Survey experimental set-up and data analysis technique.

Type Ia supernovae as standard candles

Observations

Supernovae are categorized according to their spectral features near maximum brightness. Type II spectra exhibit hydrogen features, which are lacking in type I spectra. This category is furtherly separated into type Ia and Ib/c according to the presence or absence of silicon features (see figure 2.1).

Besides the clear Si II absorption lines in its spectra at around 4130 Å and 6150 Å, the type Ia spectral signature consists in a 'W' shaped absorption doublet at 5640 Å due to SII, and an absorption doublet at 3934 Å and 3968 Å due to Ca II (figure 2.2). All features exhibit a deep P-Cygni ( [Psk69], [START_REF] Branch | Interpretation of the maximum light spectrum of a Type I supernova[END_REF], [START_REF] Branch | The Type I supernova 1981b in NGC 4536 -The first 100 days[END_REF]) profile characteristic of line formation in an expanding Type Ia supernovae (SNe Ia) form a homogeneous class in term of spectra and luminosity emission as a function of time (light curve). The peak brightness in the B band shows a dispersion of about 40% among "normal" SNe Ia. However, even if SNe light curves are quite similar, they are not identical. There exists correlations between the peak luminosity and some other observables : the luminosity variation time scale, as brighter SNe Ia dim slowlier -this is the so-called Phillips or brighter-slower empirical law ( [START_REF] Phillips | The absolute magnitudes of Type IA supernovae[END_REF]) ; the intrinsic color at peak -brighter SNe Ia are bluer (brighter-bluer law, [START_REF] Tripp | Determination of the hubble constant using a twoparameter luminosity correction for type ia supernovae[END_REF]). The use of SNe Ia as a distance indicator is based on these empirical relations (see section 2.1.3).

Theory

Type Ia supernovae (SN Ia) are believed to be the result of the thermonuclear disruption of a carbon-oxygen white dwarfs which reaches the Chandrasekhar-mass limit of stability (M Ch 1.4M ) by accreting matter from a companion ([Hoy60], [START_REF] Whelan | Binaries and Supernovae of Type I[END_REF])1 .

This model explains the lack of H and He in SNe Ia spectra, their occurring in evolved parent galaxies, and their relative homogeneity, related to the existence of the Chandrasekhar limit. The nuclear energy arising from the explosion is entirely converted into kinetic energy and the luminous emission is powered by the decaying of the nickel 56 Ni produced during the explosion. The reaction 56 Ni -→ 56 Co -→ 56 Fe sets the gradual dimming time-scale ( [START_REF] Colgate | Early Supernova Luminosity[END_REF]). The progenitor system identity (a single or a double WD), the hydrodynamic mechanism of the explosion (detonation or deflagration) are still the object of debate.

The mass of 56 Ni produced is the primary determinant of the peak brightness of the SN Ia. Its variation (M Ni 0.4 -0.9M ) could explain the SNe Ia diversity and the brighter-bluer and brighter-slower relations ([Hoe96]). A higher nickel mass implies a higher temperature and a modification of the opacity : e.g. a decreasing opacity with temperature in the blue B band means a slowlier dimming of the B magnitude ( [START_REF] Kasen | On the Origin of the Type Ia Supernova Width-Luminosity Relation[END_REF]).

The SNe Ia physics is complicated at every step, from the physics of explosion to the processes that convert the energy input into a visible emission through radiative transfer in the envelope, but it results nonetheless in a relatively homogeneous event. Model SN Ia light curves and spectra can fit the observations reasonably well ([Kas09]), but so far, their predictive power is limited: using SNe Ia as cosmological probes rests on empirical relations.

The distance indicator

In order to apply the brighter-slower correction, the light-curve characteristic time scale can be estimated by the stretch factor s : a template light curve in the B (or V) band F (t/s) is stretched to match the observations. Some other parametrization exists such as MLCS/MLCS2k2 ([Jha07]), ∆m15 ( [Phi93]) . . . .

For the brighter-bluer relation, one defines c, the excess color at maximum as c = (B -V ) tmax -(B -V ) max (the average is taken over the SN Ia population).

The parameters s and c can be estimated by fitting a spectro-photometric model of the SN Ia emission φ(λ, t) such as the SALT2 model used in the SNLS data analysis ( [START_REF] Guy | SALT2: using distant supernovae to improve the use of type Ia supernovae as distance indicators[END_REF]).

Taking into account the standardization relations, the distance modulus of a SN Ia is given by :

µ = m B -M B + α × (s -1) -β × c (2.1)
m B is the peak apparent magnitude, estimated in the B band in the the supernova rest-frame (as opposed to the B band in the observer rest-frame). The stretch factor s, the color c and the normalization m B of the spectro-photometric model φ are estimated for each supernova, using its measured magnitudes. The brighter-slower and brighter-bluer relation are parametrized by α and β. M B combines the absolute magnitude of the standard s = 1, c = 0 supernova together with the Hubble parameter value H 0 : M B = M B + 5 log 10 (c/H 0 /10 pc). These three parameters are fitted, along with the cosmological parameters, on the total supernova sample in the Hubble diagram µ(z) (see figure 2.3).

With this parametrization, there remain an intrinsic variation in the SN peak luminosity of ∼ 15% which results in a distance measurement dispersion around 8% ([Ast06]).

Note that the β × c term is equivalent to an extinction correction

A B = R B ×E(V -B) calculated for an excess color E(B -V ) = (B -V ) -(B -V ) 0 , (
the latter term (B -V ) 0 standing for the color the object would have if it was not extincted) and a total to selective extinction ratio R B . However, the estimation of the β parameter with SNe Ia leads to an effective R B < 4.1 ([Ost08]), so that the color corrections of SN Ia are incompatible with known dust properties. This difference could be explained by an intrinsic color variation of the SN Ia combined to dust extinction, and/or a specific extinction law due to the SN environment. Intrinsic and extrinsic color variations are not easily disentangled.

K-correction

In order to compare supernovae, their fluxes have to be expressed in the same way, and in particular using the same rest-frame band, e.g., as proposed above, the B band. A supernova is observed through fixed spectral bands, depending on the telescope set-up. Due to the redshift effect, the effective band-pass the supernova is seen through changes. K-corrections correct for this effect : according to [Ast12], K-correction is "transforming photometric data obtained with some filter into what would have been measured with some other filter". If S(λ) is the rest-frame spectrum of the supernova, observed at a distance d and a redshift z, then the flux measured in the observed pass-band, e.g. the r filter, is related to the flux as it would be observed in a rest-frame pass-band, e.g. the B filter, by :

f B = f r × S(λ)T B (λ)dλ S(λ × (1 + z))T r (λ)dλ
with T being the filter transmission. To compute the rest-frame B band magnitude from the observed magnitudes thus requires a synthetic SED describing the supernova emission at λ and t, a precise knowledge of the intervening effective pass-bands transmission for the instrument, together with its absolute calibration. As a consequence, supernova spectro-photometric modeling and instrument calibration are key-features in their cosmological exploitation.

A third relation

SNe Ia properties correlate with the physical parameters describing its environment, such as it's host galaxy stellar mass, star formation rate, or metallicity. For instance, late type galaxies or galaxies with a higher specific star formation rate (sSFR) preferentially host SN Ia with higher stretch (e.g. [START_REF] Hamuy | A Search for Environmental Effects on Type IA Supernovae[END_REF], [START_REF] Gallagher | Chemistry and Star Formation in the Host Galaxies of Type Ia Supernovae[END_REF], [START_REF] Gallagher | Supernovae in Early-Type Galaxies: Directly Connecting Age and Metallicity with Type Ia Luminosity[END_REF]). This property is illustrated on fig. 2.4 for the Supernova Legacy Survey SNe Ia ([Sul06]). This also holds for the host galaxy stellar mass ([How09]) : massive galaxies host SNe Ia with lower stretch factor (which are then less luminous than the standard SN Ia with s = 1, c = 0). This effect could point out a relation existing between metallicity (higher in massive galaxies) and luminosity via the synthetized 56 Ni mass ([Tim03]), although no conclusive evidence has been obtained ([Gal05], [START_REF] Howell | The Effect of Progenitor Age and Metallicity on Luminosity and 56 Ni Yield in Type Ia Supernovae[END_REF]).

If the correlation between the SNe Ia light curve properties and their host galaxy stellar population are well established, no conclusive trends are seen between SNe Ia color and their host galaxy properties though.

Recently, it was demonstrated ([Kel10], [START_REF] Sullivan | The dependence of Type Ia Supernovae luminosities on their host galaxies[END_REF]) that the standard SN Ia was indeed about 10% more luminous in massive and passive galaxies. As a consequence, the brighter-slower and brighter-bluer relations are not sufficient to describe SN Ia variability. A brighter-heavier relation was thus introduced in recent cosmology analysis ( [START_REF] Conley | Supernova constraints and systematic uncertainties from the first 3 years of the supernova legacy survey[END_REF], [START_REF] Betoule | Improved cosmological constraints from a joint analysis of the sdss-ii and snls supernova samples[END_REF]), to adjust the absolute magnitude parameter M B in eq. 2.1 taking into account the SNe host galaxy properties. The SNe are assigned a host galaxy mass bin, which in turn assigns them an absolute magnitude : M B (resp. M B + ∆M B ) for the SNe occurring in high (resp. low) mass galaxies.

Although SN Ia brightness are clearly linked to their host galaxy properties, the question of which parameter is the most pertinent (sSFR, mass, metallicity), and whether it should be local (e.g. sSFR at the SN location) or global (e.g. mass) ([Rig13]), is still in discussion.

Hubble diagram

Cosmological information are extracted from a Hubble diagram µ(z; Ω M , Ω DE , w . . . ; α, β . . .) where nearby SNe (0.01 < z < 0.1) are compared to more distant ones (z ∼ 0.2 -1). The cosmological parameters Ω M , Ω DE , w . . . are fitted along with the "nuisance" parameters α, β, M B , ∆M B . The joint Supernova Legacy Survey and SDSS Hubble diagram is shown on figure 2.5 and is presented in details in section 2.2.3. With ∼ 500 supernovae, the statistical errors are of the order of the systematics, which must be accounted for and estimated thoroughly.

The main systematics sources are listed below,and their taking into account in the Supernova Legacy Survey will be presented in section 2.2.2.

• Modeling uncertainties : the systematic uncertainties arising from SN spectro-photometric modeling mostly depend on the assumption underlying the model.

• Photometric calibration : this step is the major source of systematics in the cosmology measurement. As explained in section 2.1.4, the comparison of the SNe Ia fluxes measured at different redshift, and in different band-pass, requires the calibration of the different instruments onto a common standard photometric system.

• Contamination from non SNe Ia : for high-z SNe, type Ia can sometimes be contaminated by type Ib and Ic.

• Supernova evolution : supernovae luminosity or standardization relations could be evolving with time, inducing this way a bias in the cosmological parameters estimation. These possibility is checked in cosmology analysis by comparing supernovae characteristics (e.g. spectroscopic features) between low-z and high-z SN, or testing for redshift dependence of the brighter-bluer or brighter-slower laws.

• Malmquist Bias : this selection bias affects any flux limited surveys.

• Peculiar velocities : this systematic uncertainty arises from the local velocity fields and affect the nearby supernovae samples in the Hubble diagram, even so they are required to be in the Hubble flow.

• Gravitational Lensing : gravitational lensing induces an additional scatter to the Hubble diagram (see section 3.3.1).

The Supernova Legacy Survey

The Supernova Legacy Survey has been one of the major projects till date using SNe as a probe to pursue the nature of dark energy and measure precisely its equation of state parameter w.

SNLS experimental set-up is described in section 2.2.1. Recent results, based on the joint data analysis of SNLS first 3 years and SDSS supernovae survey data are presented in section 2.2.3, along with the complete analysis of SNLS 5 years data which is still underway.

Experimental set-up

The Canada France Hawaii Telescope Survey (CFHTLS)

Canada and France joined a large fraction of their dark and grey telescope time from mid-2003 to early 2009 for a large project, the Canada-France-Hawaii Telescope Legacy Survey (CFHTLS). The CFHTLS uses the MegaCam camera ( [START_REF] Boulade | MegaCam: the new Canada-France-Hawaii Telescope wide-field imaging camera[END_REF]) set up on the 3.6-m Canada-France Hawaii telescope (CFHT). It consists of three surveys : • The Very Wide survey for the far ends of the solar system and our Galaxy survey. It covered 1300 square degree in the gri bands only.

• The Wide survey covers 155 square degree, spread over four patches : W1 to W4 in ugriz, reaching a limiting magnitude i AB = 24.5. The scientific goal is the study of the nearby universe and the large scale structures (lensing).

• The Deep survey was designed with four 1 square-degree fields D1 to D4 in ugriz.

The Supernova Legacy Survey (SNLS)

SNLS is mainly a Canada-France collaboration with collaborators from the USA and the UK and some other European partners. SNLS used CFHTLS data for a 5 year period from 2003 to early 2009 with 450 nights of observation. SNLS survey design has been specifically set up so as to control systematics ( [START_REF] Conley | Supernova constraints and systematic uncertainties from the first 3 years of the supernova legacy survey[END_REF]): • SNe are both discovered and photometrically followed with one telescope, so that a special effort could be dedicated to the comprehension of the instrument calibration ([Reg09],

[Bet13]). Within the survey, no intercalibrating between different follow-up telescope was needed.

• The survey four pass-bands griz permit to measure the SNe B rest-frame flux, as well as the B-V and U-B colors in a consistent way over the redshift range z = 0.2 -1.

• The instrument design permit to observe SNe at a deep enough distance so that the Malmquist bias has a limited impact at z 0.6 ([Per10]).

• SNLS is a rolling search, which consists in repeatedly observing the same part of the sky within a span of 2-3 days, thus continuously discovering new supernovae while monitoring the already discovered SNe (fig. 2.8). This method is in contrast to the old technique of observing the same field at different epochs spread over 2 months with the SN detection followed by scheduled photometric follow-up. This strategy ensures an adequate light-curve time sampling (every 3-4 days in the gri filters) as well as very good early-time coverage, which is crucial for the precise determination of each SN lightcurve parameters entering the cosmological fit.

• Each ∼500 SNe entering the cosmology analysis have been spectroscopically identified as a Ia, using data obtained on 8-m telescopes (fig. 2.9). About ∼ 200 were identified at the Very Large Telescope (VLT), ESO, Chile, and ∼ 200 at Gemini-North telescope (Hawaii).

In the D3 field, ∼ 100 were observed at the Keck Telescope (Hawaii). Spectra are published in [START_REF] Howell | Gemini Spectroscopy of Supernovae from the Supernova Legacy Survey: Improving High-Redshift Supernova Selection and Classification[END_REF][START_REF] Bronder | SNLS spectroscopy: testing for evolution in type Ia supernovae[END_REF]Ell08,[START_REF] Balland | The eso/vlt 3rd year type ia supernova data set from the supernova legacy survey[END_REF]. The spectroscopic data also permitted to address the evolution question.

• Because of the survey duration, it is possible to construct very deep SN-free image stacks and get accurate colors of the host galaxies so as to study the relation between SN properties and host-galaxy environment to search for host-dependent systematic effects.

SNLS Data Analysis

We present here the main steps involved in the supernovae data processing for the SNLS 3-years (SNLS3) and five-years (SNLS5) data analysis.

The cosmology parameters are estimated through a a χ 2 minimization between the distance indicator presented in section 2.1.3: µ SN = m B -M B + α × (s -1) -β × c, and the distance modulus µ(z SN ; Ω M , w, . . .) :

χ 2 = i (µ(z SNi ; Ω M , w, . . .) -µ SNi ) 2 σ 2 i + σ 2 int (2.2)
For each supernova, the redshift z and the parameters m B , s, and c must be precisely measured. The cosmological parameters are then obtained from the fit process along with notably α and β, parametrizing the stretch-luminosity and color-luminosity relations. σ i accounts for the photometric uncertainty and σ int is adjusted to account for the remaining intrinsic dispersion.

This diagonal expression for the χ 2 is a simplification, as the covariance matrix V is indeed not diagonal. It will be detailed below.

Detection

After a local pre-processing of the images performed on site using the Elixir pipeline, the r and i images are compared with reference earlier images to detect new transient events using an algorithm based on image subtraction. Two independent detection pipelines (one french and one Canadian), were performing the supernova detection. Although both pipelines uses a different algorithm (the Alard algorithm [START_REF] Alard | A Method for Optimal Image Subtraction[END_REF] or a non-parametric approach), detections from the two different pipelines agree at a 90% level.

All identified transient objects cannot be forwarded for a spectroscopic follow up. Instead a primary photometric checking selects the most promising candidates.

The SNLS images were also submitted to a deferred processing independent of the SNLS real-time detection pipeline. This permits to complete the identification of the transients and perform a photometric analysis and identification of a "photometric" type Ia supernova sample. The photometric sample size is about 50% larger than the spectroscopically selected supernovae ([Baz11]).

Photometry

The supernovae photometry is performed on the griz images. The supernova fluxes are estimated by fitting a point spread function (PSF) source, at the same sky position on the image series, on top of a a time-independent pixelized galaxy model. The local field (tertiaries) stars are measured through the same process (although setting the "galaxy model" to zero). What is measured is thus the ratio of the SN fluxes to those of the tertiary stars.

Two photometry algorithm, the "resampled simultaneous photometry" (RSP) and the "direct simultaneous photometry" (DSP) were implemented ( [START_REF] Astier | Photometry of supernovae in an image series : methods and application to the supernova legacy survey (snls)[END_REF]). SNLS3 analysis is based on RSP : the images are first resampled to a common pixel grid to correct for the difference in telescope pointing, and fitted by a model combining a PSF centered at the SN position and a pixel grid (the underlying galaxy). The correlations introduced between neighboring pixels by resampling results in a sub-optimal estimation of the flux. SNLS5 analysis is based on DSP : to avoid the spatial correlations introduced by the images resampling, the DSP algorithm resamples instead the galaxy model onto the image pixel grid.

Calibration

The purpose of photometric calibration is to relate the instrumental fluxes measured in the image pixels to the physical fluxes of the observed objects. The task is two-fold and thoroughly described in the two reference papers [START_REF] Regnault | Photometric calibration of the Supernova Legacy Survey fields[END_REF] and [Bet13]. First, the effective instrument throughput variation in space and time must be characterized and homogenized to obtain a consistent data set. Secondly, calibration must provide a path to obtain from the homogenized flux measurement of an astrophysical source its physical flux or equivalently its calibrated broadband magnitudes m:

m b ≡ -2.5 log 10 λT b (λ)S(λ) λT b (λ)S ref (λ)
S is the spectral energy density (SED) of the object above the earth's atmosphere in units of erg -1 cm -2 Å -1 , T b is the instrument transmission in the filter b, and S ref is a reference spectrum.

The precise mapping of the spatial non-uniformities of the imager photometric response using dithered observations of dense stellar fields and the modeling of the effective pass-bands of the instrument as a function of the position on the focal plane are explained in [START_REF] Regnault | Photometric calibration of the Supernova Legacy Survey fields[END_REF] and updated in [Bet13]. The effective pass-bands are obtained from laboratory or in situ transmission measurements of the CCD and filter pass-bands, combined with on site measurements of the mean atmospheric absorption.

The calibration process of the supernovae fluxes involved several steps and is described below.

The photometry of supernovae is made relative to the stars, referred to as tertiary standards, surrounding the supernovae in the science fields. This process delivers instrumental fluxes φ for the supernovae and the tertiary standards in consistent but arbitrary units.

To convert the instrumental flux φ SN into a physical flux S SN , one relies on the choice of a fundamental spectro-photometric standard star of known SED S S (λ), and defines the zero point ZP b in the chosen filter b as :

φ b 10 -0.4×ZP b ≡ λT B (λ)S SN (λ λT B (λ)S S (λ) (2.3)
In SNLS3 analysis, the chosen primary standard star was BD+17 4708, a HST spectrophotometric fundamental flux standard of precisely estimated SED 2 . The flux scale now relies on the theoretical modeling of the atmosphere of pure hydrogen WD observed by the HST, transferred to three solar analogs by HST observations with the STIS instrument ([Boh04, Boh10]). These 3 solar analogs were directly observed by MegaCam for the JLA calibration.

The ZP estimation is performed following a path as short as possible. It formerly involved a secondary stars catalog, located in the science fields, of well known magnitudes, to assign magnitudes to the tertiary stars : the Landolt Catalog ([Lan92]) stars, with anchoring to the HST flux scale indirectly provided by [Lan07] observations of HST standards in the Landolt UBVRI photometric system, implying to deal with the transmission of this new set of filters. The BVRI system differs from MegaCam griz (see fig. 2.10), so that piece-wise linear color transformation laws had to be used between the two systems. The color transformation between the Landolt system and the reference MegaCam system are presented in figure 2.11. The use of solar analogs permits now to bypass this step.

The next step is to calibrate tertiary stars using the same photometry than the one performed on the calibration stars observed with MegaCam (aperture photometry), and the transformation laws from MegaCam physical fluxes to the chosen magnitude system. From there, the zero point for a given observation is estimated using: ZP = m aper + 2.5 log 10 φ aper where the mean is taken on the field tertiary stars.

The calibration of the tertiary stars is finally transferred to the SN using the ratio of their fluxes measured with the very same photometry, i.e. the PSF photometry.

For the SDSS survey, the calibration is performed by observing the HST spectro-photometric standard and the SDSS field with a dedicated photometric telescope (PT). The calibration process of both projects is presented on figure 2.12. The short-cut which permits to suppress the use of Landolt secondary stars for SNLS calibration is indicated.

Spectro-photometric modeling: SALT2 and SIFTO

For each SN Ia, the peak flux m B in the rest-frame B band, the color c and the stretch factor s, are estimated by fitting the SN observed magnitudes with a spectro-photometric model (fig.

2.13). The "lightcurve fitters" used in SNLS analysis are SALT2 ( [START_REF] Guy | SALT2: using distant supernovae to improve the use of type Ia supernovae as distance indicators[END_REF]), developed by the french team, and SIFTO ( [START_REF] Conley | SiFTO: An Empirical Method for Fitting SN Ia Light Curves[END_REF]), on the Canadian side.

SALT stands for "Spectral Adaptive Lightcurve Template". It aims at producing an average spectral sequence of SNe Ia with the addition of the fewest degrees of freedom for reciprocating the empirical observations of the SN Ia. The SN rest-frame flux is parametrized by three parameters, X 0 , X 1 , and c :

φ SN Ia (t, λ) = X 0 × [M 0 (t, λ) + X 1 M 1 (t, λ)] ×e c×CL(λ) (2.4)
where t is the phase of the event (t = 0 corresponds to peak luminosity). M 0 (t, λ) is the average spectrum at a given phase t and X 0 is the flux normalization in the rest-frame B-band, equivalent to m B . M 1 (t, λ) is the correction template to the average spectrum, it allows without forcing this behavior for the time-stretching of the light curve. X 1 is equivalent to the stretch parameter s, with roughly X 1 s -1. CL(λ) is the time dependent color law and c is by construction the SN color as it corresponds to the difference in intensity of the rest frame B and the V band at peak luminosity. M 0 (t, λ), M 1 (t, λ), CL(λ) are computed during the model training based on spectro-photometric data from nearby supernovae, and also SNLS supernovae photometry. These latter were quite useful to sample the rest-frame U band. No additional assumptions is made on the wavelength dependency of the color law, and it does indeed differ from the Galactic dust absorption law R λ (e.g. [START_REF] Cardelli | The relationship between infrared, optical, and ultraviolet extinction[END_REF]). The SALT2 model is thus at contrast with the MLCKS2 model ( [START_REF] Jha | Improved Distances to Type Ia Supernovae with Multicolor Light-Curve Shapes: MLCS2k2[END_REF]) which makes strong assumptions on CL wavelength dependency and c, and The SIFTO model uses a time-stretched template time series. The color law is implemented through linear relations relating the colors c = B -V , U -B and the stretch s. The obtained results are similar to SALT2. SALT2 and SIFTO results are mitigated for the SNLS3 analysis, and SALT2 is used for SNLS5 analysis.

β ≡ R B ,

Supernovae host galaxies

The supernova host galaxy identification and photometry is performed on deep stacked images (see chapter 4). The host galaxy stellar mass is computed using PEGASE.2 ([Fio99]), so that each supernova can be assigned a host mass bin, below and above M = 10 10 M . In the process of the cosmology fit, two different absolute magnitudes M B and M B + ∆M B are fitted for two host-mass bins.

Systematic uncertainties

To include the identified systematics in the cosmology fit, eq. 2.2 can be generalized by replacing the diagonal contribution by a more general covariance matrix V combining systematic and statistical errors. The χ 2 equation then reads:

χ 2 = ∆ µ T V ∆ µ (2.5)
Constraints form other cosmological probes can also be taken into account by adding supplementary terms in the χ 2 expression. The intrinsic dispersion σ int is fitted separately on each sample completing the SNLS supernovae in the Hubble diagram : nearby supernovae from nearby searches and intermediate redshift supernovae from the SDSS-II survey. The gravitational lensing supplementary dispersion is accounted for by introducing σ lensing 0.055 × z.

As all SNe share the same photometric model, the statistical part of the covariance matrix V stat dealing with m B , c, s measurements is non-diagonal. The systematic part V syst comprises the calibration uncertainties, which affect the photometric measurement of the supernovae, but also the spectro-photometric model trained with these data. This is to furtherly reduced the contribution of calibration to the systematic uncertainties, and be able to fully exploit the data from SNLS and SDSS survey, that [Bet13] established a common calibration at a level of 0.4% in gri.

Measuring the dark energy equation : JLA and SNLS5

The first set of distance measurements data was released from SNLS 1 st year's observation data [START_REF] Astier | The supernova legacy survey: Measurement of ω m , ω λ and w from the first year data set[END_REF]. With 71 high redshift SNe Ia, together with the constraints from the SDSS baryon acoustic oscillations ( [START_REF] Eisenstein | Detection of the baryon acoustic peak in the large-scale correlation function of sdss luminous red galaxies[END_REF]), the equation of state parameter w was measured at a precision of ∼10%.

SNLS3 (2011)

The SNLS 3 years data analysis ( [START_REF] Conley | Supernova constraints and systematic uncertainties from the first 3 years of the supernova legacy survey[END_REF]) confirmed this measurement. SNLS3 consisted of 242 SNe Ia at 0.3 < z < 1.1. As described in the previous section, two independent analysis were carried out by the Canadian and the french counterpart for the SN photometry, photometric calibration and light curve fitting. Precise photometric calibration and improved SNe Ia light curve modeling were done, models were trained using the SNLS data. Inclusion of a host mass term∆M B was done and the related systematics were added to the cosmological fits.

The Joint Light Curve Analysis (2014)

An intermediate update has been presented with the Joint Light Curve Analysis (JLA) which was a joint SNLS3 and SDSS-II data analysis ( [START_REF] Betoule | Improved cosmological constraints from a joint analysis of the sdss-ii and snls supernova samples[END_REF]). The JLA sample consists of 740 spectroscopically confirmed type Ia SNe with high quality light curves. The sample comprised low redshift SNe Ia with z < 0.1 ; mid-range SNe Ia from the SDSS sample at 0.05 < z < 0.4 ; higher redshift SNLS SNe Ia at z < 1 ; a few SNe observed at z ∼ 1 with the HST. The inclusion of the SDSS sample (374 SN Ia) was fully exploited thanks to the inter-calibration of the SNLS and the SDSS data (fig. 2

.5).

The findings of the JLA are consistent with the CMB [START_REF] Ade | Planck 2013 results. XVI. Cosmological parameters[END_REF] observations (e.g. matter density parameter Ω m and dark energy equation of state parameter w). When combining SNe constraints with the the Planck measurement [START_REF] Ade | Planck 2013 results. XVI. Cosmological parameters[END_REF] of the CMB temperature fluctuations and the WMAP measurement of the large-scale fluctuations of the CMB polarization [START_REF] Bennett | Nine-year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Final Maps and Results[END_REF], the precision obtained on w parameter is now less than 6% (see fig. 2

.14).

The previous disagreement on Ω m at a ∼ 2σ level reported by [START_REF] Conley | Supernova constraints and systematic uncertainties from the first 3 years of the supernova legacy survey[END_REF] was greatly reduced with the improvement in the calibration accuracy using the joint recalibration methods (see fig.

2.14).

SNLS5 (2014-)

The SNLS 5 years are based on ∼ 400 spectroscopically confirmed and ∼ 300 photometrically confirmed SNe Ia covering the full period of observation between 2003-2008. The SNLS5 analysis is described in [START_REF] Hage | Measurement of the Dark Energy Equation of State Using the Full SNLS Supernova Sample[END_REF]. With respect to JLA analysis, SNLS5 is now relying on the DSP photometry algorithm, and is using a fully re-trained SALT2 model. We will be using the still cosmologically blinded 3 SNLS5 sample analysis in these thesis. The SNe are selected as follows. To ensure that the supernovae parameters lie within the bulk of the distribution, the condition -3 < X 1 < 3 and -0.3 < c < 0.3 are set. Poorly sampled supernovae are excluded by requiring that it be observed before t max , and that σ(t max ) < 2 and σ(X 1 ) < 1. A cut on the light-curve χ 2 fit is also applied. In addition, all light curves that enter the Hubble diagram have been visually inspected. This resulted in the exclusion of a few SDSS supernovae. The contributions of all sample included in SNLS5 analysis are presented in table 2 The values of the supernovae fluxes have been artificially changed so as to alter the cosmology they describe.

Chapter 3

Gravitational Lensing

Gravitational lensing is a consequence of Einstein's General Relativity1 : the gravitational field of a massive object causes light rays passing close to that object to be bent. As the gravitational deviation of light depends solely on the lens mass distribution, this phenomenon permits to study the dark matter distribution and growth history, from haloes to large scale structures, as well as probing the universe overall geometry and expansion history.

This chapter is essentially based on [Nar96], [Men15], and [Har16b] notes.

Basics of gravitational lensing

We will present here the basics of gravitational lensing. We will be working under three assumptions :

1. the lens mass only weakly perturbs the FRW metric describing the homogeneous universe : the lens mass Newtonian potential Φ is assumed to be small, Φ c2 . As a consequence, it is possible to consider that the individual lenses are embedded in a locally flat (Minkowskian) space-time. For a galaxy cluster where galaxies peculiar velocities are about v ∼ 10 3 km.s -1 , the condition is met, with Φ/c 2 ∼ 10 -5 . Under this assumption, the field equations of General Relativity can be linearised.

2. The lens velocity, and the velocity of its constituents, are slow (v c), which is again the case for galaxies and clusters.

3. Thin lens approximation: the lens physical size L is much smaller than the relative distances between the source, the lens and the observer. This is again the case in this thesis, with distances to galaxies and galaxy clusters of interest being larger that their typical sizes : L c/H 0 . The path of light propagation can then be broadly broken up into 3 parts. From the source to the lens, and the lens to the observer, light travels through an unperturbed space-time described by the Friedman metric. In the lens vicinity, described as locally Minkowskian, the ray, corresponding to an impact parameter ξ, can be approximated as a straight line and its change of direction α can be considered to be instantaneous. Under this condition, the lens mass is assumed to lie in a plane distribution (see figure 3.1) and is described by it's surface density:

Σ( ξ) = ρ( ξ, z) dz (3.1)
where ξ is a two dimensional vector in the lens plane and z is the line-of-sight coordinate.

The deflection angle

The deflection angle is calculated by the integral along the line of sight of the potential gradient along the normal perpendicular to the path of light propagation :

α = 2 c 2 -→ ∇ ⊥ Φdl (3.2)
Because of assumption (1), on expects the deflection angle to be small. So that the gravitational potential along the deflected trajectory can be approximated by the potential along the undeflected trajectory (Born approximation). For a point mass M and an impact parameter ξ, the deflection reads:

α = 4GM c 2 ξ ξ ξ (3.3)
The Schwarzchild radius:

R S = 2GM c 2 (3.4)
sets the scale at which assumption (1) is valid: ξ R S . For M = 10 11 M , R S 10 -2 pc. In the case of the thin lens approximation relation, the deflection angle is obtained by summing all the contributions from all the mass elements:

- → α ( ξ) = 4G c 2 ( ξ -ξ )Σ( ξ ) ξ -ξ 2 d 2 ξ (3.5)
The integral is over ξ in the lens mass distribution plane, and the origin of the coordinates system from which ξ and ξ vectors are taken can be chosen anywhere in this plane. For an axially symmetric lens, the light rays, deflected and undeflected, lay in the same plane than the source-lens-observer and the deflection angle becomes one-dimensional (see figure 3.1):

α(ξ) = 4GM (ξ) c 2 ξ (3.6)
ξ is now the distance from the center of the 2-dimensional mass profile. The deflection is determined by the mass enclosed within the radius ξ, M (ξ) :

M (ξ) = 2π ξ 0 Σ(ξ )ξ dξ (3.7)
where, for the integration in the lens plane, the lens symmetry center is chosen as origin of the coordinates system.

The lens equation

The lensing situation is described on figure 3.1. The reference axis originates from the observer O and is orthogonal to the lens mass plane. The lens equation relates : the angle β between the source direction S and the reference axis; the deflection angle α; the angular separation θ between the source image and the reference axis; the distance D S between the source plane and the observer; the distance D L between the lens plane and the observer; the distance D LS between the source plane and the lens plane. The derivation is obtained explicitly on figure 3.2 and yields:

βD S = θD S -αD LS (3.8)
The distances are angular distances as defined in 1.18 so that η = βD S , ξ = θD L . Note that in general, D L + D LS = D S . However, in a flat universe, The light ray originates from a point S which position is set by η in the source plane and travels unperturbed until it hits the lens plane, at the position given by ξ. It then instantaneously changes direction (set by α). In the special case where the lens is spherically symmetric with respect to a symmetry center L, the lens plane can be taken to be orthogonal to the direction (LO). The reference axis can be taken as the optical axis (LO), and ξ is then the impact parameter with respect to L. This is not possible when considering multiple lenses. Image: adapted from [Bar01]

D S (1 + z s ) = D L (1 + z L ) + D LS (1 + z s ).

Observer

β = θ - D LS D S α = θ -α( θ) (lens equation) (3.9)
This equation in θ may be non-linear and have multiple solutions, so that a given single point source at β has multiple images θ.

The Einstein radius and the critical surface density

In the special case of a point mass M lying exactly behind the lens (β = 0), the image is a ring (see figure 3.3) which angular size is the Einstein radius:

θ E ≡ 4GM c 2 D LS D S D L (3.10)
The Einstein radius sets the characteristic angular scales of lensed images : when multiple images are produced, they are separated by ∼ 2θ E . For a typical galaxy mass and effective distance D = D L D S /D LS , it is of order :

θ E 1" M 10 11 M 1/2 D 1 Gpc -1/2
Σ cr is the critical density2 :

Σ cr ≡ c 2 4πG D S D L D LS (3.11)
Note that the critical density is purely a geometric quantity, and corresponds to the mean surface mass density inside the Einstein radius. Numerically, defining the characteristic distance D = (D L D LS )/D S :

Σ cr 0.35 g.cm -2 D 1 Gpc
The critical density or equivalently the Einstein radius gives roughly the boundary beyond which multiple imaging can occur, and distinguishes between the strong and the weak lensing regime.

Magnification

Because of Liouville theorem3 , gravitational lensing preserves surface brightness but changes the apparent solid angle of the source. The flux of a source of surface brightness I ν and solid angle area δΩ is F ν = I ν δΩ. The lensing alters the solid angle area to δΩ lensed , so that the lensed flux F ν, lensed = I ν δΩ lensed is modified by the magnification factor:

µ ≡ δΩ lensed δΩ (3.12)
The image distortion can be described by the lens mapping Jacobian matrix A:

A ≡ ∂ β ∂ θ = 1 - ∂ α ∂ θ (3.13) so that: µ ≡ 1 detA (3.14)
It is useful to define the effective lensing potential:

ψ( θ) = D LS D L D S 2 c 2 Φ(D L θ, z)dz
Its gradient yields the deflection angle α:

α( θ) = ∇ θ ψ
and its Laplacian is proportional to the surface-mass density at ξ = D L θ (using Poisson equation for Φ):

∇ 2 θ ψ = 2 Σ(D L θ) Σ cr ≡ 2 κ( θ) (3.15)
and introducing the convergence κ. We may then write the Jacobian matrix A in term of the effective lensing potential Hessian :

A = 1 - ∂ 2 ψ( θ) ∂ 2 θ = 1 -κ -γ 1 -γ 2 -γ 2 1 -κ + γ 1 (3.16)
where κ = ψ 11 + ψ 22 according to the definition 3.15, and γ 1 = (ψ 11 -ψ 22 )/2, γ 2 = ψ 12 defines the shear, with γ = γ 1 + iγ 2 = γe 2iφ . In the matrix A, the convergence (1 -κ)1 term produces an isotropic image expansion or contraction, while the shear rotation matrix stretches the intrinsic shape of the source. For an initially circular source of apparent size r, the image is elliptical with a an ellipticity (a -b)/(a + b) = γ/(1 -κ) = g: g is the reduced shear. With these notations, the magnification µ now reads :

µ = [(1 -κ) 2 -γ 2 ] -1 (3.17)
In the weak lensing regime where κ 1 and γ 1, the magnification is simply related to the convergence by:

µ 1 + 2κ 1 + 2 Σ( ξ)
Σ cr (weak regime)

Types of Lensing

One usually distinguishes three regimes : strong lensing, of which microlensing is a special case, and weak lensing. The distinction between these regimes depends on the steepness of the gravitational potential along the light ray path, and the source-lens-observer geometry.

Strong Lensing: strong lensing produces multiple images or highly distorted images (such as arcs) and occurs when the lens system is critical, i.e if κ ≥ 1 (or the source is within an Einstein radius off the optical axis). As a consequence it is very sensitive to the inner lens density profile and its substructure and can be used to compute a detailed model of the mass distribution of the lens.

Microlensing can be regarded as a special case of strong lensing. It was noted by Einstein in 1936 ([Ein36]) that a star could gravitationnaly lens another one that would appear inside its Einstein radius. Given the number of (lensing) stars per unit area of the sky, the probability of observing such an event would be of 1:1 million of observed (source) stars. As the typical image separation distance is ∆θ ≈ 2θ E ∼ 1 mas, far below the observations limiting resolution, only the magnification of the source star can be observed, providing the lens is moving transversely to the line of sight. The background star is progressively magnified as the lens passes in front of it and enters gradually the Einstein radius, exhibiting a characteristic light curve, symmetric in time, and achromatic. Microlensing is suited for looking for planets ( [START_REF] Gould | Microlens OGLE-2005-BLG-169 Implies That Cool Neptune-like Planets Are Common[END_REF]) or the once candidate for Byronic dark matter, the massive halo compact objects (MACHOS), from 10 -3 to 10 3 M Weak Lensing: the most frequent occurrence of gravitational lensing is in the weak regime, when κ 1 and γ 1. The lens is not strong enough to form multiple images or arcs. The source image can still be nonetheless distorted: stretched (shear) and magnified (convergence). Galaxy clusters distort this way the background galaxies shapes, and distant galaxies can also be lensed by galaxies closer to us (galaxy-galaxy lensing).

The galaxies shape distortion is minute: typical image ellipticities induced by weak lensing are of order a few per cent. Moreover, as galaxies are not intrinsically circular, the weak-lensing distortions cannot be inferred from one individual galaxy: several of them need to be averaged, assuming that their intrinsic ellipticities ε are randomly oriented and would thus average to zero in absence of lensing. For these reasons, it is only by studying a large number of sources statistically, that information about the foreground lenses can be inferred (see figure 3.5).

The average (over a finite area of the sky) ellipticity induced by lensing is:

ε = γ 1 -κ
Because in the limit of the weak lensing, the convergence κ( θ) can be inferred from the shear γ( θ), this relation permits to iteratively recover κ( θ) and therefore Σ( θ).

Multiple lens plane method

The dark matter distribution in the Universe is clumpy, so that gravitational lensing by successive local density inhomogeneities along the line of sight have to be considered.

Under the assumption that the light rays originating from the source are undergoing small deflections due to the gravity of the intervening clumps, the inhomogeneities can be taken into account by dividing the space into thin layers, assuming that each lens belong to one lens plane. The plane are perpendicular to the light bundle. The position of the lensing planes can be distributed according to theoretical prescriptions or to N-body simulation, so as to obtain a statistical description. But they can also be set realistically, according to the positions taken in a galaxy and cluster catalog, to study a real lensing situation.

In backward ray-tracing methods, light rays are propagated from the observer to the source with deflections only occurring in successive lens planes (see e.g. [START_REF] Schneider | Gravitational lenses[END_REF]). The lensing effect for the corresponding lens plane is then calculated and the linear summation of all the lens planes gives the net lensing signal.

The multiple-lens situation is described on figure 3.6 for 2 lenses, L 1 and L 2 (L 1 is the closest to the observer). Starting from the source S, one can use the same rationale presented in figure 3.2 (see [START_REF] Schneider | Gravitational lenses[END_REF], chap. 9). So that, generalizing to N lenses:

η = ξ 1 D S D 1 - k α k ( ξ k )D kS
Equivalently, setting ξ 1 = D 1 θ and ξ k = D k θ k , and defining α k as previously:

β = θ - k α k (D k θ k ) D kS D S = θ - k α k ( θ k ) (3.18)
The Jacobian matrix A of the mapping is then: To obtain equations that are as similar as possible to the single lens-plane equations we set:

A = ∂ β ∂ θ = 1 - k ∂ α k ∂ θ k ∂ θ k ∂ θ (3.19)
κ k ( ξ k ) = 4πD i D kS D S Σ k ( ξ k )
In the weak lensing case, where κ k 1 and γ k 1, it can be shown (see again [START_REF] Schneider | Gravitational lenses[END_REF], chap. 9) that at first order:

µ 1 + 2 k κ k (weak approximation)

Lens models

Weak gravitational lensing, whether of type Ia supernovae or galaxies, probes the distribution of matter along the line of sight in the form of dark matter haloes around galaxies or clusters. We shall present in this section two popular halo models, along with the scaling relations used for their parametrization. But first, we introduce some concepts and analytic results from the theory of non-linear structure formation which underlay the use of haloes as cosmological probes.

Non-linear structure formation

The linear growth of density perturbations through gravitational attraction was described in section 1.5 where we introduced the matter density contrast: Let's recall the evolution of a density fluctuation (with δ 1) in the linear regime: it grows as ∝ a 2 (t) (if of super-Hubble size) before radiation-matter equality, and as ∝ a(t) during matter domination (whatever its size). Once Λ starts to dominate, the universe ceases to behave as an Einstein-de Sitter universe, and the linear growth factor is written as δ ∝ ag(a). Most of the evolution of the density perturbation for structure formation has thus occurred during the matter dominated era, in a bottom-up fashion, reaching a non-linearity threshold when δ > 1 (see figure 3.7).

δ = ρ( r, t) -ρ(t) ρ(t)
Using a simple non-linear analytic model, it is possible to describe the spherical collapse of a localized density perturbation, much smaller than the horizon (see [START_REF] Rich | Fundamentals of Cosmology[END_REF] for a detailed study). Considering an overcritical "bubble", embedded in an overall critical Einstein-de Sitter CDM, Ω = 1 universe, it can be shown that, when the initially slightly denser bubble will have reached the state of virialization, where dissipative physics turn the kinetic energy of collapse into random motions and end up satisfying the virial theorem T vir = -V vir /2, its density contrast will be of the order of : at t vir : δ c + 1 178

In a nutshell, over densities at virialization are approximately 100-200 times that of the background cosmic mean density at that epoch. This simple model is consistent with results from N-body simulations. Note using the linear theory would have yield an underestimated value for the density contrast δ lin c 2. The behavior of δ lin -δ is shown on figure 3.8 for an EdS and a FΛCDM cosmology.

These quantities play an important role in defining key parameters in dark matter halo models, such as the "virial" radius r vir and the related "virial" mass M vir through:

M vir = 4π 3 ∆ vir ρ(z) r 3 vir (3.20)
The relation above may be simple, but the variety of choices for ∆ and ρ can create confusion. Following the simple model, the classic choice ∆ = 200 defines M 200 and r 200 :

M 200 ≡ 4π 3 ∆ 200 ρ cr (z) r 3 200 , ρ cr (z) = 3H 2 (z) 8πG , ∆ 200 ≡ 200 (V1)
Figure 3.8: The difference δ -δ lin as a function of δ lin for two different cosmologies. From down to top, the curves correspond to the Einstein-deSitter case and to the Λ = 0 cosmology (a virial mass of M vir = 3 × 10 12 h -1 M was used in all the cases). The departure from the linear case, leading to the structure collapse and virialization, is clearly seen after δ lin ∼ 1.5. Credits: [START_REF] Sanchez-Conde | The spherical collapse model with shell crossing[END_REF] One can also use a fitted formula to the numerical computation corresponding to the spherical collapse model in the frame of e.g. a flat (Ω m , Ω Λ ) universe. [Bry98] propose:

M vir ≡ 4π 3 ∆ c (z) ρ cr (z) r 3 vir (V2) with: ∆ c (z) ≡ 18π 2 + 82x -39x 2 , with x = Ω(z) -1 (3.21)
introducing the ratio Ω(z) of the matter density to the critical density:

Ω(z) ≡ ρ m (z) ρ cr (z) = Ω m (1 + z) 3 E 2 (z) , E 2 (z) ≡ H 2 H 2 0 = Ω m (1 + z) 3 + Ω Λ
Note that 18π 2 178 is the factor we already encountered. For a ΛCDM model, with Ω m = 0.3, ∆ c (z = 0) 100. Once a convention is chosen, there is a one-to-one correspondence between M vir and r vir .

Halo Models

We describe here two popular halo models characteristics, along with their lensing properties.

The Singular Isothermal Sphere (SIS) Model

Spiral galaxies are observed to have flat rotation curves out to a large radii ([Rub78], [START_REF] Begeman | HI rotation curves of spiral galaxies[END_REF], see figure 3.9), suggesting the existence of a spherical halo embedding the galaxy, with a profile density ρ ∝ r -2 (e.g. [START_REF] Ostriker | A Numerical Study of the Stability of Flattened Galaxies: or, can Cold Galaxies Survive?[END_REF]). Stellar dynamics of elliptical galaxies (e.g., [START_REF] Rix | Dynamical Modeling of Velocity Profiles: The Dark Halo around the Elliptical Galaxy NGC 2434[END_REF]) are also consistent with a constant dispersion velocity σ v .

The Singular Isothermal Sphere (SIS) profile is derived by assuming that the halo matter content behaves as an isothermal self-gravitating, equilibrium sphere of collisionless particles. Its density profiles:

ρ(r) = σ 2 v 2πGr 2 (3.22)
Figure 3.9: The rotational curve measured in the 21-cm line of neutral hydrogen for spiral galaxy NGC6503. The solid line shows the dark halo fitted to the data. The profile is an isothermal sphere with a finite core radius ρ ∝ (r 2 c + r 2 ) -1 . Also shown are the visible (dashed), gas (dotted) and dark halo (dash-dot) components curves. Image : [START_REF] Begeman | Extended rotation curves of spiral galaxies -Dark haloes and modified dynamics[END_REF] is parametrized solely by the constant dispersion velocity σ v . The velocity dispersion in a SIS model ranges in 100 -200km.s -1 ([Bri13]). It is singular at r = 04 . The mass M (r) within a given radius r is proportional to r:

M (r) = (2σ 2 v /G) r (3.23)
insuring a flat rotation curve, as:

v 2 rot = GM (r) r = 2σ 2 v M (r
) is divergent at large r, therefore in practice, the model is truncated at some radius, for example at the virial radius r vir defined as M vir = (2σ 2 v /G)r vir . Using the relation between M vir and r vir (depending on the chosen convention, e.g. eqns. V1 and V2), one can compute the virial radius r vir (σ v ) or the virial mass M vir (σ v ) :

r vir = 2 √ ∆ σ H 0 E(z) (3.24)
Some related lensing quantities for the SIS model are ( [START_REF] Gunnarsson | Supernovae under the gravitational lens[END_REF]):

• the (constant) deflection angle:

α = 4π σ 2 v c 2 = 1.15" σ v 200 km.s -1

2

• the Einstein Radius and the lens equation:

θ E = 4π σ v c 2 D LS D S , β = θ -θ E θ |θ| θ = θ E
corresponds to the tangential critical curve of the SIS. For a source at a distance at z = 0.8 and a lens with σ v = 200 km.s -1 at z = 0.4, θ E 0.5".

Σ(ξ) = ∞ -∞ ρ ξ 2 + z 2 dz = σ 2 v 2G 1 ξ
which can be rewritten in term of x = ξ/ξ 0 , choosing the length scale ξ 0 in the lens plane:

ξ 0 = 4π σ 2 v c 2 D LS D L D S (3.25) Σ(x) = Σ cr 2x
For a source at a distance at z = 0.8 and a lens at z = 0.5, ξ 0 2.75 kpc.

• The convergence:

κ = 1 2x • The magnification µ(x) = |x| |x| -1

The Navarro-Frenk-White (NFW) model

In general, structure formation is too complicated to be analytically derived: it is necessary to turn to numerical simulations, that can now include, besides the gravitational interaction of dark matter "macroscopic" particles, hydrodynamic and radiative process. On the basis of Nbody simulation results, Navarro, Frenk and White (NFW, [START_REF] Navarro | The assembly of galaxies in a hierarchically clustering universe[END_REF][START_REF] Navarro | The Structure of Cold Dark Matter Halos[END_REF][START_REF] Navarro | A Universal Density Profile from Hierarchical Clustering[END_REF]) proposed a Figure 3.11: NFW density profile fit to N-body simulations for eight different cosmologies: the best fitting profile is shown for the low mass (solid) and the high mass (dash) halos. SCDM is the EdS model, Ω 0 is the matter density parameter and n s is the spectral index of the initial density fluctuation power spectrum P (k) ∝ k n S seeding the simulation. The radius is scaled to r 200 and the density to ρ crit (z = 0). Lower mass halos are more concentrated near the center than higher mass halos. As a consequence their concentration parameter is higher. Image : [START_REF] Navarro | A Universal Density Profile from Hierarchical Clustering[END_REF] "universal" halo profile, consistent with observational data from the scales of dwarf galaxies to galaxy clusters, over a wide range of masses:

ρ(r) = ρ s r 3 s r(r + r s ) 2 (3.26)
where the parameter are r s the scale radius and and ρ s the amplitude of the density profile.

Another choice is to chose the parameters r s and δ c the relative amplitude of the density profile, defined as:

ρ s = δ c ρ cr (z)
Although singular at r = 0, this profile does not lead to a singularity in mass, being ∝ r -1 , and thus shallower than an isothermal profile. At r r s , the profile is ∝ r -3 , so steeper than an isothermal profile, and the integrated mass rises logarithmically.

The parametrization of the NFW model is worth discussing in details. We shall see that it can be parametrized by M vir and the concentration parameter c: c ≡ r vir r s

A large value for c implies that the halo is highly concentrated in the inner regions. The halo virial mass M vir , enclosed within the virial radius r vir , is computed as:

M vir = r vir 0 ρ(r)4πr 2 dr = 4πρ S r 3 s f r vir r s
with f (x) defined as:

f (x) = x 0 du u(u + 1) 2 = ln(1 + x) - x 1 + x
So that the NFW profile can be written as:

ρ(r) = M vir 4πf (c) 1 r(r + r vir /c) 2 (3.27)
and is now parametrized with M vir and the concentration parameter c. Recall that there is a one-to-one relation between r vir and M vir once a convention is chosen (eqn. V1 or V2). It is worth noting that with a chosen "virial convention" as e.g. V2, the relative amplitude of the density profile δ c defined by ρ s = δ c ρ cr (z) is related simply to the concentration parameter c with:

δ c = ∆ c 3 c 3 f (c)
Simulations show that the concentration index c is related to the halo (virial) mass and redshift ([Nav96], [START_REF] Bullock | Profiles of dark haloes: evolution, scatter and environment[END_REF]): small halos are significantly denser than large halos as a result of the fact that small, low-mass halos formed at higher collapse redshifts when the density of the universe was higher. Observations also show that dark matter halos represent a one-parameter family with self similar density profiles ( [START_REF] Burkert | The structure of dark matter halos. observation versus theory[END_REF]) and that the NFW concentration parameter c decreases with halo mass, from around 10 for galactic halos to 4 for cluster halos ( [START_REF] Mandelbaum | Halo mass -concentration relation from weak lensing[END_REF]). This dependency can be described with a power law:

c(z, M ) = A M M B (1 + z) C (3.28)
where the pivot M is the median halo mass. A value C = -1 can be obtained from analytic considerations ( [START_REF] Bullock | Profiles of dark haloes: evolution, scatter and environment[END_REF]). There is an intrinsic scatter σ(log 10 c) ∼ 0.1 -0.2 about the median c(M ) relation ( [START_REF] Navarro | A Universal Density Profile from Hierarchical Clustering[END_REF][START_REF] Duffy | Dark matter halo concentrations in the wilkinson microwave anisotropy probe year 5 cosmology[END_REF]). Different estimations of law 3.28 parameters are listed in Note that c depends on the chosen virial definition, with c 200 and c vir usually referring respectively to convention V1 and V2. The conversion between different conventions for c are given in [START_REF] Johnston | Cross-correlation Weak Lensing of SDSS galaxy Clusters II: Cluster Density Profiles and the Mass-Richness Relation[END_REF].

Having dealt with the NFW parametrization, we can now present the related lensing quantities ([Gun05]), taking x = r/r s :

• The surface mass density:

Σ(x) = 2ρ s r s F (x) (3.29)
where F (x) reads:

F (x) =    1 x 2 -1 - 2 (x 2 -1) 3/2 arctan x-1 x+1 (x > 1) 1 x 2 -1 + 2 (1-x 2 ) 3/2 artanh 1-x x+1 (x < 1)
• The convergence:

κ(x) = 2κ s F (x), κ s = ρ s r s Σ cr , κ(1) = 2 3 κ s (3.30)

Mass scaling relations

The halo parameters describing the mass distribution of a galaxy are inferred from observational quantities through scaling relations, relating the total mass, or the density profile kinematics properties, to the galaxy luminosity.

The singular isothermal sphere (resp. NFW) halo model presented in section 3.2.2 can be solely parametrized by the dispersion velocity σ (resp. the virial mass M ). The scaling relations thus usually takes the form:

σ σ = L L ησ (3.31) M M = L L η M (3.32)
We shall present here the scaling relations derived from galaxy-galaxy gravitational lensing studies, together with the empirical Tully-Fisher and Faber-Jackson relations for galaxies.

The Tully-Fisher relation

The Tully-Fisher relation ( [START_REF] Tully | A new method of determining distances to galaxies[END_REF]) states that the maximum rotational velocity V max in spiral galaxies are observed to be closely related to their luminosity L (figure 3.12): In the frame of the SIS halo model, the rotational velocity can be converted into an equivalent isotropic velocity dispersion σ using V max = √ 2σ. We shall present here the relation established by [START_REF] Boehm | The tully-fisher relation at intermediate redshift[END_REF], which is used in this thesis and in some previous supernovae gravitational studies [START_REF] Gunnarsson | Corrections for gravitational lensing of supernovae: better than average?[END_REF], [START_REF] Jonsson | Prospects and pitfalls of gravitational lensing in large supernova surveys[END_REF], [START_REF] Kronborg | Gravitational lensing in the supernova legacy survey (SNLS)[END_REF]). Based on the measurement of spatially resolved rotation curves for 77 FORS Deep Field ([Nol04], [START_REF] Heidt | The fors deep field: Field selection, photometric observations and photometric catalog[END_REF]) spiral galaxies with a redshift between z = 0.1 et z = 1 using the Very Large Telescope in Multi Object Spectroscopy, they estimate a T-F relation that takes into account an evolution in redshift. It thus differ from the local relation established by [START_REF] Pierce | Luminosity-line width relations and the extragalactic distance scale. I -Absolute calibration[END_REF]:

L ∝ V α max
M B = -7.48 × log V max -3.52
corresponding to an index η σ 0.33, to which they add a redshift dependent correction:

∆M B = -(1.22 ± 0.56) × z -(0.09 ± 0.24)
so that it is equivalent to:

log V max = -0.134 × (M B + 1.22 × z + 3.61) (3.33)
As a consequence, for the same mass, at higher redshift, a galaxy is more luminous in the Bband, owing to its younger star population. For a magnitude M B = 21, the velocity dispersion is σ 150 km.s -1 . The observed scatter in the relation established by [START_REF] Pierce | Luminosity-line width relations and the extragalactic distance scale. I -Absolute calibration[END_REF] is:

σ M B = 0.41 which results in: σ(log V max ) = 0.055 (3.34)
This scatter arises from the observational errors, a variation from the mass function of stars and an intrinsic contribution ([Eis96]). This result is corroborated by [Fer09] study, based on 612 galaxies from the DEEP-2 spectroscopic survey at a redshift between z = 0.2 and z = 1.2.

The Faber-Jackson relation

The Faber-Jackson ( [START_REF] Faber | Velocity dispersions and mass-to-light ratios for elliptical galaxies[END_REF]) relation expresses the loose correlation between the luminosity L and the velocity dispersion σ observed in the center of early type galaxies (figure 3.12):

L ∝ σ γ (3.35)
The observed aperture-corrected central velocity dispersion is a good estimate for the velocity dispersion of the dark matter halo velocity dispersion when modeled as a spherical isothermal sphere ( [START_REF] Kochanek | The dynamics of luminous galaxies in isothermal halos[END_REF]). The F-J law is a projection of the tighter 3-dimension relation between the effective radius R e , the surface brightness I ∝ L/R 2 e and the velocity dispersion σ, known as the Fundamental Plane: R e ∝ σ 1.4 I -0.85 ([deV82, Djo87]) or equivalently the D n -σ relation ( [START_REF] Dressler | Spectroscopy and photometry of elliptical galaxies. I -A new distance estimator[END_REF]).

We will make use in this thesis of ([Mit05], eq. 33) measurement, following again previous work from [START_REF] Gunnarsson | Corrections for gravitational lensing of supernovae: better than average?[END_REF] and [START_REF] Kronborg | Gravitational lensing in the supernova legacy survey (SNLS)[END_REF]. Using approximately 30,000 SDSS surveyed elliptical galaxies at a redshift between z = 0.01 and z = 0.3 with a spectroscopic measured velocity dispersion, they obtain:

log(σ) = 2.2 -0.091 × (M r + 20.79 + 0.85 × z) (3.36)
corresponding to a Faber-Jackson index γ = 4.4 or η σ = 0.275. For M r -22.34 (corresponding roughly to M B -215 ), the velocity dispersion is σ 220 km.s -1 .

The observed scatter about this relation is taken from [START_REF] Sheth | The Velocity Dispersion Function of Early-Type Galaxies[END_REF]:

σ log σ = 0.079 × [1 + 0.17 × (M r + 21.025 + 0.85 × z)] (3.37)

Galaxy-Galaxy lensing estimation

Galaxy-galaxy lensing is a powerful tool to study the mean radial profile of a ensemble of galaxies, acting as gravitational lenses on a high number density of background galaxies. From the shear of the background sources, one directly measured Σ(R)-Σ(R) for the lensing galaxies as a function of the distance R to the galaxy centers. This stacking or averaging technique yields the mean properties of the sample. It is thus possible, using simple halo profile models, to establish scaling relations describing simultaneously galaxy haloes of various luminosity. Over the last 15 years, high-quality galaxies imaging data together with redshift information (either spectroscopic or photometric redshifts) has become available, leading to progress in the use of galaxy-galaxy lensing, thanks to major surveys such as the COMBO-17 survey (e.g. [START_REF] Kleinheinrich | Weak lensing measurements of dark matter halos of galaxies from COMBO-17[END_REF]), the Red-Sequence Cluster Survey (e.g. [Hoe04], [van11]), the Sloan Digital Sky Survey (e.g. [START_REF] Mandelbaum | Galaxy halo masses and satellite fractions from galaxy-galaxy lensing in the Sloan Digital Sky Survey: stellar mass, luminosity, morphology and environment dependencies[END_REF]), the Canada-France-Hawaii Telescope (CFHT) Survey or the Dark Energy Survey ([Cla16]).

We present here results obtained by [START_REF] Brimioulle | Dark matter halo properties from galaxy-galaxy lensing[END_REF] with data from the CFHT Legacy Wide Survey, based on a high statistics of 12 million objects observed in ugriz, spawning a surface of 124 square degree. Selecting a sample of lenses and sources with 0.05 < z L < 1 (with a mean redshift z L 0.35) and 0.05 < z S < 2, they derive scaling relations for the halo properties, separating the lenses into blue and red lenses according to their color, at B -V = 0.7.

For a SIS model, they obtain, for a fiducial luminosity L r = 1.6 10 10 h -2 L r , corresponding to M r V = 21.7 in the Vega system6 :

σ red = 162 ± 2 km.s -1 L L r 0.24±0.03 σ blue = 115 ± 3 km.s -1 L L r 0.23±0.03
The precision refer to the fitting procedure, and the scatter around the fitted law is indeed larger, of the order of 30 km.s -1 . To take into account that the galaxy luminosity evolves with look-back time and thus redshift, they also considered an evolving luminosity according to L ∝ (1 + z). The fitted amplitude is moderately increased (σ red = 173 ± 2 km.s -1 , σ blue = 123 ± 3 km.s -1 ) in comparison to the values without evolution, but the scaling relation index η σ 0.25 remains almost unaffected. The generic relation with evolution then reads:

σ σ ∼ L L r 0.25 × (1 + z) -0.25
A comparison of [START_REF] Boehm | The tully-fisher relation at intermediate redshift[END_REF], [START_REF] Mitchell | Improved cosmological constraints from gravitational lens statistics[END_REF] and [START_REF] Brimioulle | Dark matter halo properties from galaxy-galaxy lensing[END_REF] results are presented in table 3.2, for a reference redshift z = 0.45 representative of the lenses redshift for the lensing analysis in this thesis. The galaxy-galaxy lensing results are completely compatible with the spectroscopic measurements. does not change, so that the mean magnification stays at the unlensed magnification value µ = 1 (see section XXX-normalization-XXX). The resulting distribution ([Wan99, Wan02, Hol05, Sar08]) is however non-gaussian, with the majority of the supernovae being de-magnified, and rarely substantially magnified (see fig. 3.13). The resulting scatter increases with the redshift, but statistically diminish ∝ 1/ √ N with the number of standard candles involved. This effect has to be considered in cosmology analysis, in the SNe selection (discarding outliers in the Hubble diagram can bias the distribution), and in the statistical treatment of the data. It will not significantly bias the cosmological parameters ([Sar08], [START_REF] Jonsson | Prospects and pitfalls of gravitational lensing in large supernova surveys[END_REF]) but the resulting scatter must be accounted for ( [START_REF] Gunnarsson | Corrections for gravitational lensing of supernovae: better than average?[END_REF]), although within SNLS sample's redshift range, it won't have a significant influence. In SNLS3, a supplementary statistical uncertainty σ 0.05×z was included in the cosmology fit computation.

The magnification distribution p(µ) and especially its skewness is directly related to the large scale structure distribution along the line-of-sight, which will make it a probe for cosmology ([Met99], [START_REF] Dodelson | Learning from the scatter in type Ia supernovae[END_REF]), attainable for future survey such as LSST.

Supernovae lensing signal detection

Supernovae magnification by galaxies along the line of sight is a probe for dark-matter distribution on the haloes scale and its detection has been elusive still then.

The lensing signal is detected by cross-correlating the SN residual r to the cosmological fit in the Hubble diagram:

r = µ SN -µ L (z, cosmology) (3.38)
to the SN expected magnification computed by taking into account the matter distribution on the line of sight. We are of course considering here weak lensing, although interestingly, in 2014, strong lensing of a supernova was detected for the first time ( [START_REF] Kelly | Multiple images of a highly magnified supernova formed by an early-type cluster galaxy lens[END_REF]) in the MACS J1149.6+2223 cluster with four images of the exploding star arrayed as an Einstein cross.

In the weak lensing domain, introductory work were done to detect the lensing signal by [START_REF] Williams | Weak lensing of the high redshift snia sample[END_REF], who used 50 high-z supernovae from the SCP and High-Z projects, and [START_REF] Ménard | Revisiting the magnification of type ia supernovae with sdss[END_REF] and [START_REF] Wang | Observational signatures of the weak lensing magnification of supernovae[END_REF] with [START_REF] Riess | Type ia supernova discoveries at z>1 from the hubble space telescope: Evidence for past deceleration and constraints on dark energy evolution[END_REF] SN sample. It lead to contradictory results : [START_REF] Williams | Weak lensing of the high redshift snia sample[END_REF] found that brighter SNe preferentially lied behind denser foreground galaxies with a statistical significance > 99%, a result not supported by [START_REF] Ménard | Revisiting the magnification of type ia supernovae with sdss[END_REF] and [START_REF] Williams | Weak lensing of the high redshift snia sample[END_REF] owns. At higher redshift (0.4 < z < 0.8), [START_REF] Jonsson | Tentative detection of the gravitational magnification of Type Ia supernovae[END_REF] finally detected a non-zero correlation at 90% confidence with 24 SNe from the GOODS sample.

Regarding SNLS supernovae, [Kro10] estimated a correlation of ρ ∼ 0.2 at ∼ 2.4σ on 171 SNe from the SNLS3 sample. Independently and with the same data, [START_REF] Jonsson | Constraining dark matter halo properties using lensed snls supernovae[END_REF] detected a signal at 92% confidence, constraining the dark matter halo parameters used to assign the galaxies mass.

Recently, [Smi14] with SDSS-II and BOSS survey's combined data set of 608 SNe obtained a 1.7σ significance of a lensing signal detection (fig(3.14).

The aim of this thesis is to extend [START_REF] Kronborg | Gravitational lensing in the supernova legacy survey (SNLS)[END_REF] study to the SNLS5 sample. The data analysis and signal computing will be presented in chapters 4 and 5.

Chapter 4

Line-of-sight modeling and magnification computation

In this chapter we will be presenting the main steps involved in the line-of-sight modeling that lead to the computation of the supernovae magnification.

Following the same road-map as [START_REF] Kronborg | Gravitational lensing in the supernova legacy survey (SNLS)[END_REF] (hereafter K10), the entire analysis chain for the lensing signal detection is outlined below. It consists of :

1. build deep galaxy catalogs (section 4.1).

2. clean catalogers from stars, host galaxies, and polluted areas by e.g. bright stars (section 4.1.5).

3. compute accurate photometric redshifts along with galaxies absolute luminosity and restframe colors (section 4.2).

4. assign each galaxy a halo mass or velocity dispersion using the scaling relations presented in section 3.2.3.

5. select the supernovae in the SNLS5 supernovae sample presented in section 2.2.3. The supernova scoring is detailed in section 4.3.

6. compute the supernovae magnification. The algorithms are presented in section 4.4.

7. compute the magnification normalization using randomly selected line-of-sights (section 4.5).

The last steps below are presented in chapter 5.

8. compute the lensing signal by estimating the correlation between the residual to the cosmology fit in the Hubble diagram and the expected supernova magnification. The lensing signal consists in detecting a positive correlation.

Compared with K10, each involved step (but step 4.) benefits from a re-analysis making use of SNLS 5-years data set, or/and new methods implementation.

Galaxy catalogs building

We present here the key steps involved in the galaxy catalog construction, realized prior to this thesis. This part is based on [START_REF] Hardin | Thèse d'habilitation à diriger des recherches[END_REF] and [Roman, in. prep.]. The observed luminosity of the galaxies (B-band absolute luminosity) situated in close proximity to the line of sight of a supernova are converted to mass using scaling relations, such as the Tully-Fisher and Faber Jackson law, selected according to the galaxy rest-frame color U-V. The galaxy photometric redshift is required to compute the lensing galaxy distance. For these reasons, a deep multi-band galaxy catalog is essential for the lensing analysis.

Making use of the complete SNLS 5-years data set, deep ugriz stacked image frames were built for each field, on which multi-band photometry was then performed.

Image selection

The CFHT-LS ugriz observations of the Deep fields are pre-processed through the Elixir pipeline to correct for the instrumental effects that affect the pixels flux. It includes flat-fielding and fringe-correction in i and z data.

The individual CCD images are then processed through the SNLS pipeline ([Ast13]). A weight map is generated that incorporates bad pixels information. The CCD image catalog is built using SExtractor ( [START_REF] Bertin | Sextractor: Software for source extraction[END_REF]). A smooth sky-background map is subtracted. Each image is calibrated by performing a large diameter (D = 15 × σ seeing ) aperture photometry on the tertiary stars catalog from [START_REF] Regnault | Photometric calibration of the Supernova Legacy Survey fields[END_REF].

The ugriz 36-CCDs frames entering the stacking step are selected using cuts on : the photometric zero point, the mean seeing (FWHM<1.1" for griz bands, 1.3" in u), the sky variance, the mean star shapes, the number of saturated stars. About 60% of the best quality images are kept in griz, which corresponds to 300 to 400 36-CCDs frames.

Image stacking

Each field is observed during a "season" of 6 consecutive months. Depending of the field, there are 5 to 6 observing seasons available. The selected 36-CCDs frames are combined on a per-season basis to construct 1-square-degree griz "per-season" deep stacked images.

The individual CCD images, rescaled to a common photometric zero-point ZP=30, are coadded using SWARP V2.17.1 package1 using the median-filter option, which permits to reject satellites and cosmic rays. A total weight map is also produced.

To avoid supernova light contamination, the supernova host photometry is performed on the "excluded-season" stacked images, obtained by co-adding all seasons but the supernova season stacks.

The deep catalog of the field galaxies, of-interest for the line-of-sight study, is built by using, for the sources detection and the photometry, the "all-seasons" stacked images, combining all the seasons stacks.

Photometry

Getting a very clean galaxy catalog is necessary for obtaining good photometric redshifts. Proper identification of all objects, especially bright stars and SN host galaxies (see following sections) is important.

The detection of the sources and the related photometry are performed using SExtractor ( [START_REF] Bertin | Sextractor: Software for source extraction[END_REF]) in the dual-image mode, with a detection threshold level of 2.5-σ for 3 contiguous pixels. The detection is realized on the i-frame and the photometry on the ugriz frames.

The magnitudes are estimated within SExtractor Kron ([Kro80]) elliptical aperture magnitudes (AUTO-magnitudes) providing a measurement of the galaxies total flux. At the limiting magnitude i 24.8, it corresponds to a magnitude uncertainty of 0.12 mag.

Stars identification

Stars identification is important as they must be excluded from the galaxy catalogs, and bright star light contamination interferes with the detection and the photometry of galaxies in their vicinity.

Star identification is done on the r-band images.

Star cluster method

The star identification method relies on the 2-D profile intensity (PSF) characteristics of the stars and is described in [START_REF] Astier | Photometry of supernovae in an image series : methods and application to the supernova legacy survey (snls)[END_REF]. The Gaussian weighted first and second moments are iteratively calculated for each sources:

M g = 2 × Σ pixels (x i -x c )(x i -x c ) T W g (x i )I i Σ pixels W g (x i )I i (4.1) W g (x i ) = exp - 1 2 (x i -x c ) T M -1 g (x i -x c ) (4.2)
where x i are the pixels coordinates, x c the Gaussian weighted centroid, and I i the sky subtracted image value at the pixel i.

The method is iterative. First a robust stars subset is identified, using the fact that stars tend to cluster in the √ M xx -M yy plane because their characteristic sizes follow the PSF shape. Then a mean star shape model M uu,model (u = x or u = y) is computed on this stars subset. Finally, using the model, the quantity ∆s uu = √ M uu -M uu,model is estimated for every sources, and is used to refine the stars identification, as ∆s uu 0 for star-shaped sources.

This method can be implemented either on the collection of individual CCD images or the deep stacked frames. The method is illustrated on figure 4.1.

Identification on the collection of individual CCD images

All MegaCam single CCD images taken at different epochs undergo a pre-processing in the SNLS pipeline. The star identification process is based on some outputs of this pre-processing, particularly the image catalog of sources, for which the Gaussian-weighted second moments M uu are computed, and the image mean star shape M uu,model model.

We can then, for a given source, collect on every single CCD images it appears on its measurement of ∆s uu . If the source is a star, all its shape measurements will cluster around (0, 0) in the ∆s x -∆s y plane. The source is selected as a star by requiring that its shape measurements cluster within a 2-σ ellipse from the origin.

Identification on the deep stacked frames

It is also possible to estimate the gaussian weighted second moments M uu for each sources throughout the r 1-square-degree deep stacked frame. To take into account the PSF variation across the focal plane, the star shape model is computed on 9 × 9 sub-images, and interpolated by a 4 th order polynomial in x and y to obtain a star shape model valid across the whole frame.

Using this model, ∆s u can be computed for each source detected on the stacked frame. In the ∆s x -∆s y plane, stars lies in a clump centered on the origin (0, 0), and are selected within a 5-σ ellipse (figure 4.1).

Star catalog completion

The star catalog produced by these two methods are identical within 90%. The result of both selection methods are presented on figure 4.1 in the magnitude (r) -surface brightness (µ) plane, where the surface brightness is defined as the logarithm of the source peak value : µ = -3.5 log 10 (peak) + ZP . It shows that bright stars (r ∼ 16) are identified by neither of the two methods.

Indeed, bright stars are often saturated and one can't rely on their shape parameters for their identification. They can be however identified easily in the magnitude -surface brightness plane to complete the star catalog. For this we select: a) all sources with mag= r ≤ 17 for which (µmag) < 4; b) all sources brighter than r ≤ 16.2 for which the peak value is above ≥ 80% of the saturation level value2 on the frame.

The bright stars selection for the four fields is shown on figure 4.2.

Catalog masking

Field edges and areas polluted by bright stars light makes it necessary to mask out some regions in the catalogs because they impaired the photometry accuracy.

The presence of a bright star is causing many defects : a halo (of a roughly constant size) around the star position, which are a consequence of the internal reflections in the optics ; "bleeding" pixels problem which arises when the electrons from the saturated pixel in the CCD sensor overflow to the surrounding pixels ; diffraction spikes due to the support rods of the camera. These spikes sizes depend on the star flux.

All the location and shape of those areas have to be listed, so that catalogs can be cleaned from objects in problematic regions.

Camera Edges

Removal of the 1-square-degree field edges are done straight away to each fields, excluding bout 8-9% of the field surface.

Stars haloes

Haloes positions and sizes had been thoroughly listed for K10 analysis, and masked out by a disk of large radius, positioned according to the star center, shifted according to the star location on the camera field, to take into account the halo spatial shift due to optic features. We have included this list in the SNLS5 mask.

CARS polygons ([Erb09])

To complete the haloes masking and take into account stars spikes, the Deep fields image masks from the CFHTLS-Archive-Research Survey (CARS, [START_REF] Erben | Cars: the cfhtls-archiveresearch survey; i. five-band multi-colour data from 37 sq. deg. cfhtls-wide observations[END_REF]) were used.

They consist of a polygon list built using a semi-automatic procedure. First, SExtractor is run on a deep co-added image r stack, with a fixed background set to zero and a very low detection threshold of 0.6σ. Areas of significant over-densities and strong gradients in the object density distribution are identified and their polygonal boundaries are computed. Secondly, bright stars positions are identified, and cross-checked with published star catalogs (such as the standard stars catalog GSC-1 [START_REF] Lasker | The Guide Star Catalog. I -Astronomical foundations and image processing[END_REF], GSC-2.3.2 [START_REF] Morrison | The Guide Star Catalog, Version 1.2: An Astrometric Recalibration and Other Refinements[END_REF], [Las08] and USNO-A2 [START_REF] Monet | The usno-a2. 0 catalogue[END_REF]) and the extent of their central light halo and their cross-shaped diffraction spikes is modeled as a function of their apparent r magnitude. The very bright stars haloes is modeled in a similar way as in K10, including their radial offset towards the camera center. Camera edges and satellite tracks are also included in the CARS mask as rectangles but we do not take them into account : the edges location is specific to the stacking procedure, and satellites are removed by construction from our deep stacked frames because of our using a median-filter co-adding algorithm.

A example of the CARS mask is presented on figure 4.3.

Bright stars

We make use of the bright stars identification realized in section 4.1.4 to complete the mask by vetoing disk areas around bright stars, which radii are set according to the stars magnitude r : for 15 < r < 17, R = 75 pixels, for 14 < r < 15, R = 100 pixels, and for r < 14,R = 150 pixels.

An example of the masking procedure is presented on figure 4.3. The catalog masking reduce the field effective area by about 20% (see table 4.1).

Photometric redshifts

The galaxies photometric redshifts are estimated by using a template matching algorithm onto their broadband photometry, based on a set of synthetic spectra that are trained on spectroscopic data available on the D3 field. The photometric redshift procedure method is described in K10 and has not been modified for this analysis. The training was re-performed on the SNLS5 stacks photometry.

This work was done prior to this thesis. This part is based on [START_REF] Kronborg | Gravitational lensing in the supernova legacy survey (SNLS)[END_REF], and [START_REF] Hardin | Thèse d'habilitation à diriger des recherches[END_REF].

Spectral template sequence

The photometric redshift algorithm is based on an initial set of galaxies spectra which are used to construct a continuous spectral template sequence. The continuous spectral template sequence F (a , λ) is parametrized using a single parameter which is equivalent to the mean age of the stellar population a , closely related to the galaxies rest-frame color : early type galaxies (elliptical) are redder than the late-type (spiral) galaxies which are bluer.

The spectra F were calculated using the galaxy evolution model PEGASE.2 ([Fio99], specifying an initial zero-metallicity and a galaxy "exponential" law for the star-formation rate : SF R ∝ (t/τ ) × exp(-t/τ ) and a galaxy age also depending on τ , so that the mean age of the stellar population ranges in 50 Myr to 13 Gyr, permitting to reproduce the observed galaxy colors. In magenta K10 bright stars haloes mask. In green and red circles, the bright stars mask, the radius is set according to the star r magnitude.

Spectral template sequence training

The spectral template has to be optimized to described the data better. For this, the template are trained using the magnitudes of ∼ 6000 galaxies in the D3 field catalog, with known spectroscopic redshift from the DEEP-2 spectroscopic survey ( [START_REF] Davis | Science objectives and early results of the deep2 redshift survey[END_REF][START_REF] Davis | The all-wavelength extended groth strip international survey (aegis) data sets[END_REF]).

The training is an iterative process. It consists of fitting for a * on the galaxy magnitudes through a χ 2 -minimization :

χ 2 (z, a ) =

Photometric redshift accuracy

Once the photometric redshift spectral templates have been trained on SNLS5 photometry of Deep-2 galaxies, all four fields galaxies catalogs are processed, yielding a photometric redshift estimation for each galaxy.

To control the accuracy of the obtained photometric redshift, spectroscopic data available on D1, D2, D4 fields are used. D1 is centered in the Deep area of the VVDS field[Le 04], D2 is in the Cosmic Evolution Survey (COSMOS, [START_REF] Scoville | COSMOS: Hubble Space Telescope Observations[END_REF]) area (HST/ACS), and some field galaxies were targeted at the same time as the SN primary target during observations with FORS-2 in multi-slit mode in field D4.

The resolution of the photometric redshift is checked by computing the redshift residuals ∆z = z photo -z spectro and D z = ∆z/(1 + z spectro ), as well as the catastrophic error rate for |D z | > 0.15, at a limiting magnitude i = 24.

The resolution is presented in table 4.3. It is comparable to the one published by [START_REF] Ilbert | Accurate photometric redshifts for the CFHT legacy survey calibrated using the VIMOS VLT deep survey[END_REF] and [START_REF] Ilbert | Cosmos Photometric Redshifts with 30-BANDS for 2-deg 2[END_REF] (dispersion on ∆ z of 0.03, catastrophic error rate of 3.7%).

Note that all spectroscopic redshifts, when available, supersede the photometric estimation in the final galaxy catalog.

Galaxy classification

The scaling relations of Tully-Fisher and Faber Jackson used for the luminosity-mass conversion of the galaxies require to classify galaxies in elliptical or spiral. Since morphological classification was not possible for the galaxies using the SNLS data, a color based classification system was used following K10, as elliptical galaxies are redder while spiral galaxies are bluer. The resolution of the photometric redshift computation in the four fields for galaxy with a Vega magnitude i < 24. The catastrophic error rate at ∆z/(1 + z) > 0.15 is also indicated. For this computation, the rest frame color (U-V) is estimated with the best-fit template obtained in the photometric redshift computation. The distribution of galaxies U -V rest-frame color permits to separate clearly two populations of red and blue galaxies at U -V = 0.54 (see figure 4.6).

The SNLS5 supernovae sample

We will consider here the 439 SNe Ia identified as such in the SNLS5 data base. Note that the cosmology analysis will apply stringent cuts on their light-curve quality and parameters, which will reduce this sample to 389 SNe.

In SNLS3, out of 233 SNe available, only 171 were used for the analysis, and the visual inspection of the sample was important. For the SNLS5 sample was developed a more systematic scoring method.

The D2 field additional HST images were used for a supplementary visual check.

SN host galaxies identification

To compute a supernova magnification, it is necessary to exclude its host galaxy from the line-of-sight galaxy selection. The supernova host identification procedure was primarily set so as to obtain stringent criteria, insuring the accuracy of the derived host galaxy characteristics, such as its stellar mass, that are taken into account in the cosmology fit. As a consequence, a dubious identification of the supernova galaxy implied that it was problematic and tagged as having "no host information". For the lensing analysis, we will pay further attention to these problematic cases, which sometimes correspond to interesting potential lensing situation. These criteria were defined on the SNLS3 supernovae sample prior to this thesis work.

Firstly, the supernova host identification is based on the normalized elliptical distance d, so that d < 1 corresponds to the elliptical aperture, used to estimate the galaxy AUTO-magnitude by the SExtractor software. The equation of the d = 1 ellipse defining the photometry aperture reads (in a coordinate system centered on the galaxy position) :

ax 2 + by 2 + cxy = d 2
where the ellipse is computed using the galaxy second-order moments, scaled with the KRON factor defined in SExtractor to take into account the galaxy light profile. The supernova host galaxy is identified as the galaxy within which the supernova lays at the closest normalized elliptical distance d. When the closest galaxy lays at d > 1.8, the supernova is declared has having "no host". Having the no-host label results in the SN to be assigned in the lowest host stellar mass bin for the cosmology analysis. An example is presented on figure 4.7 where the D2 field HST image is shown along with the deep stacked i image used for the host identification.

Supplementary checks are performed so as to detect problematic cases :

• SNe for which the identified host may be polluted by a nearby bright star are rejected.

• We also require that the closest galaxy photometric redshift be consistent with the supernova spectroscopic redshift : ∆z/(1 + z) < 0.15.

• In the case of many galaxies laying close to the supernovae within d < 1.8, the galaxy with the closest photometric redshift is selected as the host. If more than one of these close galaxies meet the criteria ∆z/(1 + z) < 0.15, then the supernova is declared has having a "dubious host information" and tagged as such.

When the SN host identification is problematic, the SN distance modulus computed for the cosmology analysis is assigned a supplementary error, as the SN cannot be assigned any host stellar mass bin.

These criteria ensuring a consistent cosmology analysis results in ∼ 6% of the SNe being declared as having no host galaxy detected on the deep image stacks, and ∼ 87% as having a

Supernovae scoring for the lensing analysis

SNe classified for the cosmology analysis as not problematic are kept for the lensing analysis, and set to a score = 1. The 27 SNe in a no-host situation are assigned a score = 4 and are also kept for the lensing analysis.

The polluted photometry situation are not kept and assigned a score = 2. The check for the presence of bright stars will moreover be furtherly performed in a thorough automatic way that is presented in section 4.3.3.

The dubious cases classification are re-examined, as some may indicate an interesting lensing situation, such as the presence of a foreground galaxy lying close to the line-of-sight. If the closest galaxy photometric redshift z gal is confirmed by [START_REF] Coupon | Photometric redshifts for the cfhtls t0004 deep and wide fields[END_REF] published estimation, the following cases are distinguished :

• if z gal < z SN : the galaxy is situated in-front-of the SN, so that the SN score is set to score=3 and the SN is kept. Examples of this situation are shown on figure 4.8.

• If z gal > z SN , then the galaxy is behind the SN and will not be included in the line-of-sight modeling anyway. The score is set to score= 1.

• if the closest galaxies lies within z gal z SN , they will not be included in the line-of-sight modeling anyway either. The score is set to score= 1.

As a result of these further checking, all dubious cases were kept with a score= 3 or = 1. This classification, presented in table 4.5, selects 425 SNe, which line-of-sight will be furtherly checked for imagery problems, such as the bright-star light pollution.

Line-of-sight selection and masking

To compute a SN magnification, the involved galaxies are selected on their radial distance R to the SN line-of-sight. In principle all galaxies would have a lensing effect on a given SN, but a reduced sample of galaxies is chosen at a maximum radial distance R LOS . To set R LOS value, the total magnification as a function of the angular radius R has been studied in K10 analysis, concluding that a radius cut off at : R LOS = 60 arcsec. resulted in a corresponding error of less than 1% loss on the total magnification. For a given SN, we will then take into account all galaxies situated at a radial distance less than R LOS , and of course in front of the SN.

The line-of-sight defined this way can include bright stars or polluted area on the image. It is thus necessary to identify and reject them. For this, we have implemented an algorithm aiming at producing in a systematic way a cleaner line of sight. The technique relies on the use of both the mask defined in section 4.1.5, that includes the camera edges, very bright stars haloes, and bright star luminous disks and spikes, and the bright stars list that is entering the mask construction.

For a given SN we define two zones in the LOS disk of radius R LOS (see fig. 4.9) : an inner disk of radius R 1 2 LOS = R LOS /2, completed by an outer annulus between radius R 1 2 LOS and R LOS . Pixels in these two zones that are flagged in the mask are vetoed, and a cut is apply on the number of vetoed pixels. For the D2 and D4 (resp. D1 and D3) fields, the fraction of vetoed pixels must be less than 20% (resp. 30%) in the inner disk and less than 15% (resp. 20%) in the outer annulus. The differences in these two sets of cuts is essentially due the intrinsic size of the [START_REF] Erben | Cars: the cfhtls-archiveresearch survey; i. five-band multi-colour data from 37 sq. deg. cfhtls-wide observations[END_REF] polygons.

Furtherly, a supplementary cut is applied, based on the distance from the bright stars to the center of the line-of-sight. All SNe LOS containing a bright star (magnitude r < 17) within the inner disk, extended to 1/4th of the bright star luminous disk masking radius, is rejected.

The masking cuts were set to obtain a compromise between limiting the rejection rate and obtaining a cleaner sample. In K10 analysis, about 30% of the sample was semi-automatically (automatic masking and visual inspection) rejected, due to bright star luminous contamination or various problems. With the criteria defined above, about ∼ 40% of the SNe are automatically rejected, permitting to obtain a clean sample for this first analysis of the SNLS5 data.

Magnification computation

We describe here the hypothesis and input on which the SNLS5 SNe magnification computation is based.

Weak lensing approximation

Lensing magnification values in K10 analysis were calculated using the publicly available Fortran software called Q-LET (quick lensing estimation tool) by Christopher Gunnarsson [START_REF] Gunnarsson | Supernovae under the gravitational lens[END_REF], [START_REF] Gunnarsson | Q-let -quick lensing estimation tool -an application to sn2003es[END_REF]. Q-LET uses the multiple plane method described in section 3.1.6. It traces the lightrays recursively from the image plane to the source plane, computing the multiple deflections along the light path. K10 used the truncated Singular Isothermal Sphere model and specified the filled beam approximation i.e. that the universe is homogeneously filled with matter as described by the cosmological parameters h 0 = 0.7, Ω m = 0.27, Ω Λ = 0.73, with superimposed dark matter haloes of galaxies. The obtained magnification is always greater than 1., which implies to furtherly apply a normalization factor, as discussed in section 4.5.

Because most line-of-sight are passing away of the intervening haloes, we made use in this analysis of the weak approximation, as suggested in [START_REF] Jonsson | Constraining dark matter halo properties using lensed snls supernovae[END_REF]. As the convergence κ 1, the magnification can be approximated by :

µ = 1 (1 -κ) 2 -|γ| 2 1 + 2κ (4.5)
The contributions of all the haloes met along the line of sight can be cumulatively added to obtain the total magnification :

µ 1 + 2 k κ k (weak approximation)
[Jon10] used the same SNLS SNe Ia sample as K10 to to investigate the properties of dark matter haloes of galaxies. They advocated that the weak lensing approximation has only a 5% margin of deviation from the ray tracing algorithm for magnification computation.

In what follows, we will compare the ray tracing computations from Q-LET and the approximated computation used for this analysis.

Galaxy halo models

For this thesis, we developed a dedicated C++ code for the computation of the convergence of the intervening haloes on the line of sight, which permitted to compute each supernova magnification in a cumulative way. We have set the cosmological parameters to h 0 = 0.7, Ω m = 0.27, Ω Λ = 0.73. All the magnification values are given with respect to a homogeneous Universe and the distances are computed with the filled beam approximation. The impact parameter ξ is computed from the radial distance of the given galaxy (lens) center to the supernova (source).

The halo galaxy model is parametrized within the SIS model frame. The velocity dispersion σ of each galaxy is computed from their absolute magnitude M B , using the Tully-Fisher ([Boe04]) and Faber-Jackson ([Mit05]) redshift dependent scaling laws σ(M B , z) as given in eq. 3.33 and 3.36. The color cut U -V = 0.54 permits to separates galaxies between spiral and ellipticallikes. The value of the velocity dispersion is important as it gives the lensing strength of the corresponding halo.

From there, the virial radius r 200 and mass M 200 are estimated from the velocity dispersion σ using eq. 3.24 :

r vir = 2 √ ∆ σ H 0 E(z)

Truncated SIS halo model

We implemented for the analysis the truncated SIS model, with a truncation radius set at the virial radius value. Following the formalism of section 3.2.2, we define x = ξ/ξ 0 and x t = r 200 /ξ 0 , with the length scale ξ 0 as defined in eq. 3.25.

The truncated SIS convergence reads :

κ(x) =    1 πx arctan √ x 2 t -x 2 x (x ≤ x t ) 0 (x > x t )

Truncated NFW halo model

We also implemented for this analysis the truncated NFW model. We used M 200 (SIS) computed from the velocity dispersion in the frame of the SIS model, as described above. We did not use the NFW profile to convert the velocity dispersion into the virial mass because of the complexity of this relation: the velocity dispersion depends on radius for a NFW profile. However, [START_REF] Gunnarsson | Q-let -quick lensing estimation tool -an application to sn2003es[END_REF] checked that when using an input velocity of σ = 170 km/s to get the parameters for the NFW and then computing the corresponding NFW velocity dispersions, the obtained values vary roughly between 5% to -15% around 170 km/s between radii 0.1r 200 to 2r 200 . Using M 200 (SIS) to parametrize the NFW model provides moreover the possibility to compare easily and consistently the two models.

For the concentration parameter c, we relied on [START_REF] Duffy | Dark matter halo concentrations in the wilkinson microwave anisotropy probe year 5 cosmology[END_REF] law c(M 200 , z) as presented in eq. 3.28, which translates into a c(M B , z) relation when using the TF and FJ σ(M B ) scaling laws. The law is presented on figure 4.10.

We set again the truncation radius at the virial radius value. Defining now x = ξ/r S and x t = r 200 /r S = c, the convergence is given by :

κ(x) = 2κ s F (x), κ s = ρ s r s Σ cr = M 200 c 2 4πr 2 200 f (c) × 1 Σ cr (4.6)
where f (c) = ln(1 + c) -c/(1 + c) as previously defined. The function F (x) is provided by eq. ( 27) in [START_REF] Takada | The three-point correlation function in cosmology[END_REF]:

F (x) =            - √ x 2 t -x 2 (1-x 2 )(1+xt) + 1 (x 2 -1) 3/2 arcosh x 2 +xt x(1+xt) (x < 1 < x t ) √ x 2 t -1 3(1+xt) xt+2 xt+1 (x = 1 < x t ) - √ x 2 t -x 2 (1-x 2 )(1+xt) - 1 (1-x 2 ) 3/2 arcos x 2 +xt x(1+xt) (1 < x ≤ x t )
Note that this expression differs from the relation proposed in [START_REF] Gunnarsson | Supernovae under the gravitational lens[END_REF] for x < 1. Both relations from [START_REF] Gunnarsson | Supernovae under the gravitational lens[END_REF] and [START_REF] Takada | The three-point correlation function in cosmology[END_REF] are strictly equivalent for x > 1 though. Different computation for a typical case are presented on figure 4.11. The impact parameter is set at ξ = 50 kpc, and the source is at z S = 0.8. The lens velocity dispersion is set at σ = 170 km/s, and the lens redshift varies from 0 to z S . The corresponding virial mass, virial radius and concentration parameters, depending on the lens redshift lay within these intervals : M 200 ∼ 3 -4 10 12 M , r 200 ∼ 200 -300 kpc, c ∼ 5 -7.5. The magnification µ is computed with the exact formula for a SIS model, and within the weak approx µ 1 + 2κ for the SIS, the truncated SIS, the NFW and the truncated NFW models. The lensing effect is maximum for a lens at z ∼ 0.3.

We checked for the validity of the weak lensing approximation by comparing, for a truncated SIS model, the values computed with Q-LET algorithm and with our code based on the weak approximation. The result is presented on figure 4.12. Both values are within 5% until the

Chapter 5

The lensing signal detection

The lensing signal detection and prospects

We review here the lensing detection signal method and prospects as described in K10.

The lensing signal computation

The lensing signal detection is based on the computation of the correlation coefficient between the Hubble residuals and the magnification computed as described in chapter 4 : ρ = cov(magnification, residual) var(magnification) var(residual) (5.1)

The magnification term refers to :

µ m = -2.5 log 10 (µ)
The Hubble residual is defined by the difference between the SN distance modulus µ SN from eq. 2.1 and its predicted value µ L by the cosmological model fitted on the Hubble diagram, given the SN redshift :

r = µ SN -µ L (z; Ω m , w)
The weighted covariance cov(x, y) is defined as :

cov(x, y) = w i x i y i w i -x y
where the weighted means x and y are computed as :

x = w i x i w i
It was shown by K10 that weighting with the inverse of the Hubble residual variance, w = 1/σ 2 (r) is maximizing the signal detectability. Indeed, K10 showed that the magnification error is increasing with the magnification value :

σ(µ m ) = 0.008 -0.17 × µ m
The principal contribution to the magnification error is the dispersion on the TF and FJ scaling laws used to parametrized the galaxy haloes. As a consequence, weighting with both the Hubble residual and the magnification variance is sub-optimal, as it lowers the weight of the most magnified events that carry the signal.

Based on the SNLS3 sample of 171 SNe selected for the lensing analysis, K10 obtained the value : ρ = 0.18.

Detected signal significance

The signal significance is assessed by randomly associating Hubble residuals and expected magnifications of the real sample, obtaining this way a "shuffled" sample. The method for evaluating the result is to compute the chance of obtaining, for the N = 10000 "shuffled" samples, a correlation coefficient higher than the actual value obtained on the true sample, ρ = 0.18.

The distribution of the correlation coefficient for these shuffled sample is of course centered on 0, and gives the significance of the true sample correlation coefficient measurement (see fig. 5.1) : 2.3-σ, meaning that 99% of the shuffled samples have a ρ value lower than ρ = 0.18.

Signal detection prospects

We use here the simulation code developed by K10 to assess the lensing signal detection prospects in SNLS5. The simulated SNe redshift interval can be chosen so as to be representative of the experiment. For the SNLS sample, it is a gaussian centered on z = 0.65 and with a RMS of 0.2, and N = 225 SNe where drawn for each sample.

For each individual SN, the un-normalized magnification value µ > 1 is drawn assuming a √ 110 0.276 2.78 √ √ 133 0.252 2.84

Table 5.1: The different SNLS3 SNe samples selected by requiring : that the SN is included in JLA sample ; that the SN was selected in K10 sample ; that the SN is selected using the masking and scoring criteria as described in section 4.3.3. Note that on the first line, for K10 complete sample, we used the magnification computed in this thesis, and the residuals as published in K10 and not the JLA residuals. gaussian distribution for log(µ -1) (see fig. 4.14) which mean and RMS depend on the SN redshift. The magnification value µ > 1 is subsequently normalized as described in section 4.5 to obtain the "true" magnification value µ. The SN residual value r is then drawn from a gaussian centered either on 0 -to obtain an un-correlated sample -or µ m -for a correlated sample -and with a dispersion σ r = 0.16.

The correlation coefficient is computed for each 10 000 sample. For the un-correlated samples, the distribution of the correlation coefficients is of course centered on 0, with a dispersion σ ρ 0.075 (see fig. 5.2). The distribution of the correlation coefficients for the correlated samples peaks on a slightly positive value, ρ corr.

0.21 for a similar dispersion. So that about 42% of the correlated samples obtain a correlation coefficient ρ > 3σ ρ corresponding to a lensing signal detection at 3-σ.

The prospect depends on the number of SNe in the sample, and on the redshift range : with a double statistics of N = 450 SNe, there is a 80% chance of lensing signal detection at 3-σ.

The SNLS5 sample : a preliminary analysis

The SNLS5 cosmology analysis is not finished yet. The first consequence is that the sample of "good for cosmology" SNe is not finalized yet. The second consequence is that the analysis is still blinded, so that we do not use the true residuals. For these reasons, the analysis presented below is still preliminary.

We add a supplementary cut on color to SNLS5 sample so as to further select "good" SNe Ia, demanding that |c| < 0.25. Following [START_REF] Betoule | Improved cosmological constraints from a joint analysis of the sdss-ii and snls supernova samples[END_REF], we also eliminate 03D4gl which has no post-max data. We obtain this way a sample of 384 SNe.

The masking and scoring steps select 225 SNe (60%) out of the 384 SNe in the SNLS5 cosmology sample. This 225 SNe sample can be divided in 128 SNe that were in the JLA sample used in the previous section, and 97 SNe supplementary SNe.

We obtain a correlation coefficient ρ SNLS5;225 = 0.177 which corresponds to a detection level at 2.51-σ. The result is shown on figures 5.5 and 5.6. We can notice that the signal is carried out by the most magnified SNe : by requiring that the magnification µ > 0.99, we select 164 SNe out of 225 (a fraction of 75%), and obtain a correlation coefficient ρ = 0.254.

Note that without a supplementary cut on SNe quality, we obtain, with now 230 SNe, including 3 very red and one very blue SNe plus 03D4gl, a correlation coefficient of ρ SNLS5;230 = 0.138 which corresponds to a detection level at 1.97-σ. This points the necessity to work with a SN sample with a strict selection on color and, in general, quality.

The correlation measured on the 97 SNe added to the previous JLA sample is weak : ρ SNLS5;97 = 0.098, corresponding to a 0.92-σ level. To assess the significance of this performance, we randomly selected N=2000 samples of 97 SNe into the N=225 selected SNe : 25% of the selected samples yield a correlation coefficient lower than ρ SNLS5;97 . Their is thus a reasonable chance when including 97 SNe that they perform as poorly, regarding their ability at producing a lensing detection signal.

For comparison, we can also divide our sample in two halves, according to the fields numbers.

The results are summarized in table 5.2.

Magnification computation method

We have computed the magnification using the truncated SIS and the truncated NFW model. The distribution of the magnification for both models is presented on fig. 5.7. The distribution is indeed skewed, peaking slightly below µ = 1, with a mean value close to 1. The magnification of the SNe as a function of redshift is indicated on figure 5.8. Except for 04D2kr, the results obtained using both models are quite similar. Indeed, the magnification of 04D2kr, computed

• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •
0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 SNLS 5 using Q-LET ray-tracing algorithm, is µ = 1.37, which cannot be considered to be in the weak approximation regime.

µ • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • SIS NFW • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 0
The correlation coefficient computed with the truncated NFW model, ρ NFW = 0.176, does not differ much from the values obtained with the truncated SIS (ρ SIS = 0.177). As previously noted and advocated by [START_REF] Jonsson | Lensing magnification of supernovae in the goods-fields[END_REF] and [START_REF] Jonsson | Constraining dark matter halo properties using lensed snls supernovae[END_REF], the ray tracing algorithm gives similar results within ∼5% to the weak approximation computation (except for 04D2kr) : ρ Q-LET = 0.184.

This permits to conclude that the result is quite robust with regards to the choice of the halo parametrization. The ten most magnified SNe are presented on figure 5.9. Their visual is presented on figure 5.10 to 5.11.

Conclusion and discussion

We conclude that with a new analysis pipeline, we estimate the lensing signal on the preliminary SNLS5 cosmology sample and obtain a consistent result with the one obtained on the JLA SNLS3 sample : the measured correlation coefficient is ρ = 0.177 corresponding to a detection at 2.5-σ. The necessity to work with a SN sample selected with strict quality cuts, and also a "clean" line of sight, was demonstrated.

We would like to point the future steps to complete this analysis.

• Error analysis : the magnification error computation was left aside, as it was already performed in K10. It includes the propagation of the photometric errors on the galaxies luminosity and photo-z computation, but also the observed dispersion on the scaling laws : TF and FJ (see eq. 3.34 and 3.37). In the case of the NFW model, the scaling law for the concentration parameter also exhibits a large dispersion.

• Masking : the masking procedure rejected a large fraction (40%)of the SNe sample. It would be interesting to consider a masking procedure relying on a different algorithm : the computation of the SN magnification with and without the masked areas, and the comparison of the difference to the error on the magnification, to reject or keep the SN line-of-sight.

• Photometric sample : including the SNLS5 photometric SNe sample is a major step to complete the SNLS5 analysis. There will be about 300 photometric SNe Ia. Using a photometric sample prefigures the analysis that will be done within the LSST project which will provide thousands of SNe.

Conclusion

We will conclude the thesis with the current chapter, here we will focus on presenting the advantage of the lensing methodology used in this thesis and suggest a future roadmap. Using the luminosity properties of SNe Ia, cosmological distance measurements have been better constrained in the last 15 years. It also gave impetus to the study of dark energy. In this thesis we worked on the effect of the line of sight dark matter mass distribution on the SNe Ia brightness distributions or typically the effect of gravitational magnification on SN Ia by the foreground galaxies modelled with dark matter halo models. This has been studied in the weak lensing regime. The completion of gravitational lensing study of SNe Ia with 5 years full SNLS data is second in series to the primary lensing analysis started with the three years SNLS lensing data by [START_REF] Kronborg | Gravitational lensing in the supernova legacy survey (SNLS)[END_REF].

Presence of dark matter mass in the line of sight distribution, induces dispersion in the SNe brightness distribution. κ, the convergence parameter in the lensing theory gives a measure of this isotropic brightness variations (responsible for inducing magnification). Thus it was magnification computation that was the primary step in the lensing analysis. Leaving out the shear component (which was justified given the small angular size of the SNe), our lensing magnification computation expression was µ = 1 + 2κ

(5.2) This is evidently the first order approximated expression of the convergence component. The use of the weak lensing approximation formula has been justified by [START_REF] Jonsson | Constraining dark matter halo properties using lensed snls supernovae[END_REF]. We rechecked this approach and our results were found to match the predicted 5% deviation in value within the weak lensing regime as was suggested by [START_REF] Jonsson | Constraining dark matter halo properties using lensed snls supernovae[END_REF].

The presence of lensing signal can be seen as a source of contamination to the original SN brightness distribution, with a reduction in the brightness mode. Consequently this induces a dispersion in the observed Hubble diagram. Making use of this fact, we computed the correlation between the magnification computed and the residuals to the Hubble diagram. Detection of a positive correlation would give us confidence on the prediction of presence/detection of weak lensing signal. We obtained ρ = 0.177 at 2.53σ from 225 SNe Ia at z 1. However it is to be noted that this is not the final computation as the final sample selection of the SNLS 5 SNe is still not set. In addition there is scope to increase the sample size in future, with the use of additional ∼ 300 photometric SNe candidates.

Following the analysis presented in [START_REF] Kronborg | Gravitational lensing in the supernova legacy survey (SNLS)[END_REF], we present in this thesis the comprehensive pipeline for the computation and signal detection of SNe brightness magnification via weak lensing mechanism. This technique could provide a complementary technique to more widely used galaxy-galaxy lensing techniques.

The SNLS 5 lensing analysis was modified from its predecessor with the aim to make it more automatic, as required for future big data analysis. Detection and removal of contaminations in the lensing analysis (masking and scoring) had been upgraded to suit the upcoming bigger surveys. Also the weak lensing analysis code that we developed for the SNLS5 is not based on the rigorous ray-tracing algorithm unlike it's predecessor : QLET and thus saves from program complexity and time. We also made provisions for two halo models in it to increase the acceptability of our code. Together with all these upgrades and modifications we hope that with large scale upcoming surveys such as e.g. the LSST, EUCLID, JWST. There will be more scope for narrowing down on the constraints of weak lensing detection signals from the SNe Ia.

Appendix

  [San15][Maa15]. . . . 1.2 The abundances of light elements as predicted by the BBN as a function of the baryon-to-photon ratio η×10 10 . The bands show the 95% C.L. The narrow vertical band indicates the measurement from CMB for Ω B h 2 = 0.2230±0.00014 ([Ade14]) while the wider band indicates the BBN concordance region. The 4 boxes represent Helium-4, Deuterium, Helium-3 and Lithium-7. The discrepancy between Lithium-7 and the otherwise agreeing abundances could simply reflect difficulty in determining the primordial lithium abundance. Image : [Fie14] . . . . . . . . . . . 1.3 The angular distance H 0 D A /c and the luminosity distance H 0 D L /c as a function of redshift. The solid line shows a FCDM model (Ω m = 1, Ω Λ = 0), the dotted line a solely baryonic matter model (Ω m = 0.05, Ω Λ = 0), and the dashed line a flat ΛCDM model (Ω m = 0.2, Ω Λ = 0.8) close to the concordance (Ω m = 0.3, Ω Λ = 0.7) model. Image: [Hog99] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 SDSS galaxies, CMB (WMAP), cluster, lensing and Ly-α forest constraints on the dimensionless power spectrum ∆(k) as a function of the structures scale 1/k. Credits: [Teg04] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Measurements of σ 8 for different methods (Ω m is set to its concordance value Ω m = 0.3) : results from clusters are shown by circles (X-ray surveys), squares (optical surveys) or triangles (SZsurveys), while crosses show CMB constraints.

  Dark and light shading respectively indicate the 68.3 and 95.4 per cent confidence regions, accounting for systematic uncertainties. Image : [Man15] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Supernovae categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Example of a supernovae Ia spectrum near it's peak brightness. The clear dip in the spectra near the 6000Å indicates the presence of Si II which is the clear signature of a type Ia SN. Almost 3/4 th of the emitted light is in the optical range (3800-7500 Å). Image : Daniel Kasen, LBL . . . . . . . . . . . . . . . . . . . . . . 2.3 brighter-slower relation and the brighter-bluer relation : residual plots of µ B -M B -m B (see eq. 2.1) showing the the relationships between the SN Ia luminosity and their light curve stretch factor s or color c.The y-axis goes from brighter to fainter. Image [Ast06] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 For for the supernovae of SNLS survey : light curve stretch parameter s as a function of the host galaxy specific star formation rate, from active (top) to passive (bottom) galaxies. Vertical lines represent median stretch in the respective histograms. Image : [Sul06] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Hubble diagram from the joint sample of SNLS3, SDSS II, HST and low-z SNe (see §2.2.3). The bottom panel is the residual to ΛCDM cosmology fit. Image : Betoule [Bet14] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Positions of the Deep and the Wide fields on a full sky map. Image : CFHTLS website [Meg05] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Left : The Canada-France-Hawaii Telescope. Right : The MegaCam CCD mosaic. Image : CFHTLS website [Meg05] . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Rolling search : gri supernovae light curves as observed in spring 2004 to winter 2005. The newly detected SNe light-curves overlap with those of the already discovered SNe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 The spectroscopic follow up telescopes : UT1 Very Large Telescope, Chile (upper middle); Gemini-North (bottom middle) and Keck I and II telescopes (right and left sides), Mauna Kea, Hawaii. Image : CFHTLS . . . . . . . . . . . . . . . . . . vi 2.10 The transmission distribution as a function of the wavelength of the commonly used filter system: UBVRI and ugriz. Image : [Ast12] . . . . . . . . . . . . . . . . 2.11 Color transformation between the Landolt system and the reference MegaCam system. Open black circles show the MegaCam and Landolt measurements of the Landolt secondary standards. The black line represents the average color transformation determined from the secondary stars measurements. The solid red square indicates the primary spectro-photometric standard star BD+17 4708 colors. Other primary standards from [Lan07] are displayed as red crosses. Image : [Bet13] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.12 Flow chart of the calibration data process. The solid black line represents the SNLS3 calibration transfer scheme, while the red dotted line represents the new calibration path taken into account in JLA and SNLS5, and detailed in [Bet13]. Each box represents a set of standard stars established in that photometric system. The Instrument names indicate that both sets of stars on either side were measured using the same instrument hence making transfer of flux calibration data possible. Image : [Bet13] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.13 Typical SN Ia light curve in different bands. The points are observations done with the griz pass-band filters using the MegaCam (see §2.2) for the supernova 04D3fk, while the curves are the model fits from the SALT2 lightcurve fitting software. . . 2.14 Left: Phase plot of dark energy equation of state parameter w vs matter density Ω m . Right: Comparison of various measurements of Ω m from different collaborations. The JLA and the Planck measurements are in good agreement with Planck (2013) measurements. They are also in good agreement with the final Planck (2015) data release [Ade15]. Image : [Bet14] . . . . . . . . . . . . . . . . . . . . .

  Bar01] . . . . . . . . . . . . . . . . . . . . . 3.2 Derivation of the lensing equation. The rationale is done between the source plane and the lens plane using Thales theorem: (D L /D S )ξ = η + D LS α. Image: adapted from [Ama15] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Einstein Ring Image. The point-like source and the point mass lens are nearly perfectly aligned with the observer. Credit : NASA/HST . . . . . . . . . . . . . . 3.4 Microlensing light curves for a point-like source and a point-mass lens (right). The curves are color coded to the corresponding minimum impact parameters (left) (y 0 = 0.1 (top, red), 0.3,...1.1 (bottom, black)) given by their trajectories across the Einstein ring (θ E ). Image : Professor Penny D Sackett & [Koc06] XX reference correspond pas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Dark matter distribution mapped with Hubble Space Telescope's Cosmic Evolution

  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.13 Lensing magnification distributions for a perfect standard candles sample at z = 1.5 in a ΛCDM cosmology. Because of flux conservation the mean magnification value is µ = 1. As more sources are observed, the distribution approaches a Gaussian and eventually converges on a δ-function. Image: [Hol05]. . . . . . . . . 3.14 The residual to the Hubble diagram for 608 Sne from the SDSS II-BOSS sample vs the corresponding convergence κ. The red line is the line of best fit with a correlation coefficient ρ = -0.068 ± 0.41. The anticipated range of correlation is in blue shade. Image : [Smi14] . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4. 4 x

 4 The trained spectral templates SED for different values of the mean stellar age a * . The flux is in erg.Å -1 .s -1 .M -1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Redshift residual as a function of the spectroscopic redshift obtained from the VVDS data set [Le 04]. No redshift dependent systematic bias is seen. . . . . . . 4.6 Distribution of the galaxies rest-frame U-V, which is used to separate the galaxy types into red and blue : below (resp. above) U -V = 0.54 lay the bluer spirals (resp. redder elliptical) galaxies. Image: K10 . . . . . . . . . . . . . . . . . . . . . 4.7 SN 04D2iu at z = 0.7 with no identified host. Left : deep stacked i image used for the host identification. Right : HST image of the same location. . . . . . . . . . . 4.8 Examples of SNe for which the closest galaxy is a foreground galaxy on the lineof-sight (score=3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9 Line of sight inner disk and outer annulus at R LOS = 60 arcsec and R 1 2 LOS = R LOS /2, defined for the fractional contamination estimation. . . . . . . . . . . . . . 4.10 Parametrization of the halo models : the SIS velocity dispersion is modeled with the TF and FJ relations from [Boe04] and [Mit05]. The velocity dispersion is then converted into a virial mass M 200 and radius r 200 values. For the NFW concentration parameter c, we used the published [Duf08] c(M 200 , z) law, converted into a c(M B , z) law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.11 Magnification computation for a typical lensing situation and different models. . . 4.12 Comparison between ray tracing (Q-LET) and the weak lensing approximation adopted for this work for a truncated SIS model. . . . . . . . . . . . . . . . . . . . 4.13 Normalization factor computed for the four fields, for the truncated SIS model (Q-LET and weak approximation), and within the weak approximation for the truncated NFW model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.14 Upper left : for the D1 field , distribution of the magnification computed for 1000 simulated SNe at z = 1. and the truncated SIS (weak approximation). The mean magnification for this redshift bin is 1.03957. Upper right : distribution of the logarithm of the un-normalized magnification : log 10 (µ -1), which follows approximately a gaussian distribution, with a RMS of 0.43. Middle: same with the NFW truncated SIS model (weak approximation). The RMS of the gaussian distribution is 0.47. Lower: Normalized magnification distribution for the two models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Distributions of the weighted correlations coefficient. ρ = 0.18 at 99% C.L. detection. Image : [Kro10] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Left: Correlation coefficient distribution for 10 000 sample mimicking SNLS5 data, in the un-correlated case, where the SNe residuals are centered on 0, and in the correlated case, where the SN residual is supposed to equate the SN magnification. Each simulated sample contains N = 225 SNe. Right: same with a double statistics N = 450. Note the accordingly lower RMS of the distribution.. . . . . . . . . . . . 5.3 Measurement of the correlation on 133 SNe selected in the JLA sample. The histogram of the correlation coefficient computed on the "shuffled" samples. The measured value ρ = 0.252 is indicated. . . . . . . . . . . . . . . . . . . . . . . . . ix 5.4 Measurement of the correlation on 133 SNe selected in the JLA sample. The dots indicate the 133 SNe. In blue, the 23 SNe that were not included in K10 sample. Stars in grey: the 45 SNe from K10 sample that were rejected by the masking process described in this thesis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Measurement of the correlation on 225 SNe selected in the SNLS5 sample. The histogram of the correlation coefficient computed on the "shuffled" samples. The measured value ρ = 0.177 is indicated. . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Measurement of the correlation on 225 SNe selected in the SNLS5 sample. . . . . 5.7 Normalized magnification of SNLS5 SNe for the truncated SIS and the truncated NFW halo model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Normalized magnification of SNLS5 SNe for the truncated SIS and the truncated NFW halo model as a function of redshift. Except for 04D2kr which is not in the weak regime, the obtained values are very similar. . . . . . . . . . . . . . . . . . . 5.9 The 10 most magnified SNe from the SNLS5 sample . . . . . . . . . . . . . . . . . 5.10 Ten most magnified SNe (grouped according to their Field). The SN position is indicated by a red diamond. A blue ellipse indicates a foreground galaxies, a green one a background galaxy, and the red one the host galaxy. A yellow circle indicates a star. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.11 The ten most magnified SNe (grouped according to their Field), cont. from fig. 5.10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . List of Tables 1.1 Brief history of the thermal regimes of the universe. Credits: [Bau13] . . . . . . . . 1.2 Major upcoming weak lensing surveys. n gal is the effective galaxy count, f sky is the fraction of the sky covered by the survey. Source : [Liu15] . . . . . . . . . . . 2.1 Summary of the four Deep fields. The last 5 columns show the amount of observing time in hours ensuring a deep coverage. . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Summary of MegaCam specifications . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Break up of the JLA SNe Ia sample . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Break up of the SNLS5 SNe Ia sample . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Parametrization of the concentration parameter law c(M, z) = A(M/M ) B (1+z) C . The values are referring to c vir , except for [Net07] which gives c 200 . . . . . . . . . 3.2 A comparison of velocity dispersion values σ for the fiducial luminosity L r = 10 10 h -2 L r , roughly equivalent to L B = 1.6 10 10 h -2 L B , using the Vega solar magnitude M B = 5.47, and, following [Gun06], a mean galactic AB color M r -M B = -1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Summary of SNLS four galaxy field catalogs . . . . . . . . . . . . . . . . . . . . . 4.2 Magnitude offsets which were computed during the training process. . . . . . . . . 4.3 The resolution of the photometric redshift computation in the four fields for galaxy with a Vega magnitude i < 24. The catastrophic error rate at ∆z/(1 + z) > 0.15 is also indicated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Summary of SNLS5 SNe Ia host galaxy situation classification. Note that 6 SNe classified as having a dubious host identification are also classified as having a redshift inconsistent with the closest galaxy photometric redshift. . . . . . . . . . 4.5 Summary of SNLS5 SNe Ia lensing situation classification. The selected SNe lineof-sights will then be furtherly checked for imagery problems, such as the brightstar light pollution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 1 . 1 :

 11 Figure 1.1: Slice of the 3D map of the distribution of the galaxies in the universe from the Sloan Digital Sky Survey III. The earth is situated at the center and each point represents a galaxy. The radius of the circle is roughly 0.5 Gpc. Super-cluster and voids occur up to a few hundred Mpc, which is the scale on which the cosmological principle can be applied in the universe. The thinning out of the galaxies near the edges which are farthest from us represents the sparser sampling of these faint objects in the survey[START_REF] Keel | The Road to Galaxy Formation[END_REF]. More extensive mapping work will be done in future, for example by the Square Kilometre Array (SKA)[START_REF] Schilizzi | The square kilometer array[END_REF], [San15][START_REF] Maartens | Cosmology with the ska -overview[END_REF].

Figure 1 . 2 :

 12 Figure 1.2: The abundances of light elements as predicted by the BBN as a function of the baryon-to-photon ratio η×10 10 . The bands show the 95% C.L. The narrow vertical band indicates the measurement from CMB for Ω B h 2 = 0.2230±0.00014 ([Ade14]) while the wider band indicates the BBN concordance region. The 4 boxes represent Helium-4, Deuterium, Helium-3 and Lithium-7.The discrepancy between Lithium-7 and the otherwise agreeing abundances could simply reflect difficulty in determining the primordial lithium abundance. Image :[START_REF] Fields | Big-Bang Nucleosynthesis[END_REF]

  6.1). The discovery of the Cosmic Microwave Background (CMB) by Arno Penzias and Robert Wilson in 1965 ([Pen65]) established another observational pillar of the Hot Big-Bang model.

Figure 1 . 3 :

 13 Figure 1.3: The angular distance H 0 D A /c and the luminosity distance H 0 D L /c as a function of redshift. The solid line shows a FCDM model (Ω m = 1, Ω Λ = 0), the dotted line a solely baryonic matter model (Ω m = 0.05, Ω Λ = 0), and the dashed line a flat ΛCDM model (Ω m = 0.2, Ω Λ = 0.8) close to the concordance (Ω m = 0.3, Ω Λ = 0.7) model. Image: [Hog99]

Figure 1 . 4 :

 14 Figure1.4: SDSS galaxies, CMB (WMAP), cluster, lensing and Ly-α forest constraints on the dimensionless power spectrum ∆(k) as a function of the structures scale 1/k. Credits:[START_REF] Tegmark | The Three-Dimensional Power Spectrum of Galaxies from the Sloan Digital Sky Survey[END_REF]

Figure 1 . 5 :

 15 Figure 1.5: Measurements of σ 8 for different methods (Ω m is set to its concordance value Ω m = 0.3) : results from clusters are shown by circles (X-ray surveys), squares (optical surveys) or triangles (SZsurveys), while crosses show CMB constraints. The shaded, horizontal band corresponds to the 68.3 per cent confidence interval for the result (filled circle) obtained by the project "Weighting the Giants". It is based on weak gravitational lensing measurements constraining the absolute mass scale of X-ray selected clusters detected in the ROSAT All-Sky Survey ([Man15]).

Figure 1 . 6 :Figure 1 . 7 :

 1617 Figure 1.6:The CMB temperature power spectrum as measured by the Planck mission ([START_REF] Ade | Planck 2015 results. xiii. cosmological parameters[END_REF]). The position of the first peak, at l ∼ 220 (δθ ∼ 1 • ) is related to the cosmological parameters describing the universe at the recombination epoch and the geometry and contents of the universe along the photons path towards us (curvature, dark energy). Superimposed is the best fit to the data, from which the cosmological parameters are inferred. The lower plot is the residual to the upper. The scatter at low l is due to the cosmic variance, that is the lack of possibility to average over sufficiently numerous cells of size δθ on the sky. Image:[START_REF] Ade | Planck 2015 results. xiii. cosmological parameters[END_REF]

Figure 1 . 8 :

 18 Figure 1.8: The distance D V as a function of redshift from BAO experiments : BOSS, on the CMASS and LOWZ SDSS galaxies sample, the 6dFGS survey ( [Jon09]) and the WiggleZ survey ([Bla11]). Superimposed is the flat ΛCDM model obtained using the CMB projects Planck and WMAP results ([Ade14]). There is a remarkable agreement between the different galaxy surveys and the Planck data. Image :[START_REF] Anderson | The clustering of galaxies in the sdss-iii baryon oscillation spectroscopic survey: Baryon acoustic oscillations in the data release 10 and 11 galaxy samples[END_REF]

Figure 1 . 9 :

 19 Figure1.9: The original "discovery data", the Hubble diagram of SNe Ia compiled from the High-z Supernova team and the Supernova Cosmology Project's data. The bottom panel is the residual to the upper (distance modulus) corresponding to an open universe. The points can be seen to lie above the non-accelerating models (dashed and dotted lines). Image :[Fri08]

Figure 2

 2 Figure 2.1: Supernovae categories

Figure 2 . 3 :

 23 Figure 2.3: brighter-slower relation and the brighter-bluer relation : residual plots of µ B -M B -m B (see eq. 2.1) showing the the relationships between the SN Ia luminosity and their light curve stretch factor s or color c. The y-axis goes from brighter to fainter. Image [Ast06]

Figure 2 . 4 :

 24 Figure 2.4: For for the supernovae of SNLS survey : light curve stretch parameter s as a function of the host galaxy specific star formation rate, from active (top) to passive (bottom) galaxies. Vertical lines represent median stretch in the respective histograms. Image : [Sul06]

Figure 2 . 5 :

 25 Figure 2.5: Hubble diagram from the joint sample of SNLS3, SDSS II, HST and low-z SNe (see §2.2.3). The bottom panel is the residual to ΛCDM cosmology fit. Image : Betoule [Bet14]

Figure 2 . 6 :

 26 Figure 2.6: Positions of the Deep and the Wide fields on a full sky map. Image : CFHTLS website [Meg05]

Figure 2 . 7 :

 27 Figure 2.7: Left : The Canada-France-Hawaii Telescope. Right : The MegaCam CCD mosaic. Image : CFHTLS website [Meg05]

Figure 2 . 8 :

 28 Figure 2.8: Rolling search : gri supernovae light curves as observed in spring 2004 to winter 2005. The newly detected SNe light-curves overlap with those of the already discovered SNe.

Figure 2 . 9 :

 29 Figure 2.9: The spectroscopic follow up telescopes : UT1 Very Large Telescope, Chile (upper middle); Gemini-North (bottom middle) and Keck I and II telescopes (right and left sides), Mauna Kea, Hawaii. Image : CFHTLS

Figure 2 . 10 :

 210 Figure 2.10: The transmission distribution as a function of the wavelength of the commonly used filter system: UBVRI and ugriz. Image : [Ast12]

Figure 2 . 11 :

 211 Figure 2.11: Color transformation between the Landolt system and the reference MegaCam system. Open black circles show the MegaCam and Landolt measurements of the Landolt secondary standards. The black line represents the average color transformation determined from the secondary stars measurements. The solid red square indicates the primary spectro-photometric standard star BD+17 4708 colors. Other primary standards from [Lan07] are displayed as red crosses. Image : [Bet13]

Figure 2 . 12 :

 212 Figure 2.12: Flow chart of the calibration data process. The solid black line represents the SNLS3 calibration transfer scheme, while the red dotted line represents the new calibration path taken into account in JLA and SNLS5, and detailed in [Bet13]. Each box represents a set of standard stars established in that photometric system. The Instrument names indicate that both sets of stars on either side were measured using the same instrument hence making transfer of flux calibration data possible. Image : [Bet13]

Figure 2 .

 2 Figure 2.13: Typical SN Ia light curve in different bands. The points are observations done with the griz pass-band filters using the MegaCam (see §2.2) for the supernova 04D3fk, while the curves are the model fits from the SALT2 lightcurve fitting software.

Figure 3 . 1 :

 31 Figure 3.1: Schematic diagram of gravitational lensing. The reference axis originates from the observer and is orthogonal to the lens mass plane. The position of the lens mass distribution symmetry center L, should it have any, is not specified : it lies somewhere in the lens plane. The light ray originates from a point S which position is set by η in the source plane and travels unperturbed until it hits the lens plane, at the position given by ξ. It then instantaneously changes direction (set by α). In the special case where the lens is spherically symmetric with respect to a symmetry center L, the lens plane can be taken to be orthogonal to the direction (LO). The reference axis can be taken as the optical axis (LO), and ξ is then the impact parameter with respect to L. This is not possible when considering multiple lenses. Image: adapted from[START_REF] Bartelmann | Weak gravitational lensing[END_REF]

Figure 3 . 2 :

 32 Figure 3.2: Derivation of the lensing equation. The rationale is done between the source plane and the lens plane using Thales theorem: (D L /D S )ξ = η + D LS α. Image: adapted from [Ama15]

Figure 3 . 4 :

 34 Figure 3.4: Microlensing light curves for a point-like source and a point-mass lens (right). The curves are color coded to the corresponding minimum impact parameters (left) (y 0 = 0.1 (top, red), 0.3,...1.1 (bottom, black)) given by their trajectories across the Einstein ring (θ E ). Image : Professor Penny D Sackett & [Koc06] XX reference correspond pas.

Figure 3 . 5 :

 35 Figure 3.5: Dark matter distribution mapped with Hubble Space Telescope's Cosmic Evolution Survey (COSMOS), obtained with a tomography technique in 3 bins of redshift at z ∼ 0.3, 0.5, 0.7. The observed shear field is converted into a convergence map (using Kaiser-Squire inversion technique [Kai93]), which is proportional to the two dimensional, projected mass ([Mas07]). Image credit: NASA

Figure 3 . 6 :

 36 Figure 3.6: Schematic diagram of the observer's backward ray tracing in the multiple-lens-plane approximation.A light ray experiences a deflection α k only when passing through each lens plane (L k ), located at a distance D k from the observer and D kS from the source. Image:[START_REF] Premadi | Light Propagation in Inhomogeneous Universes. I. Methodology and Preliminary Results[END_REF]

Figure 3 . 7 :

 37 Figure 3.7: Left: growth of matter fluctuations in the linear regime. Non linearity is defined where ∆ 2 (k) = 1. All scales k reaching this threshold collapsed into bound objects, in a bottomup fashion (smaller scales first). Credits: [Nor10]. Right: Schematic diagram of the hierarchical growth. Smaller halos merge into bigger halo as the time increases from top to bottom. The horizontal slice on the tree at any instant gives the halo distribution at the corresponding time. Credits: [Lac93].

Figure 3 . 10 :

 310 Figure 3.10: Comparison of density profiles of halo models: in green the singular isothermal sphere(SIS) profile, in blue the universal(NFW) profile and in red the truncated isothermal model (TIS) with ρ TIS (r) = (σ 2 v )/(2πGr 2 ) s 2 /(s 2 + r 2 ). Image : Stella Seitz

Figure 3 .

 3 Figure 3.12: Faber Jackson relations (top) and Tully Fisher relation (bottom) for 2MASS galaxies in K band. The left panel shows individual galaxies while the right panel shows the mean relations. Note the larger scatter of the Faber Jackson plots than in the Tully Fisher. Image: [Koc06].

Figure 3 . 13 :

 313 Figure 3.13: Lensing magnification distributions for a perfect standard candles sample at z = 1.5 in a ΛCDM cosmology. Because of flux conservation the mean magnification value is µ = 1. As more sources are observed, the distribution approaches a Gaussian and eventually converges on a δ-function. Image: [Hol05].

Figure 3 . 14 :

 314 Figure 3.14: The residual to the Hubble diagram for 608 Sne from the SDSS II-BOSS sample vs the corresponding convergence κ. The red line is the line of best fit with a correlation coefficient ρ = -0.068 ± 0.41. The anticipated range of correlation is in blue shade. Image : [Smi14]

Figure 4 . 1 :

 41 Figure 4.1: Right: Selection of the stars on the deep stacked frame, using their second order moments, which form a clump in the ∆s x -∆s y plane. The stars are selected in the blue 5-σ ellipse centered at (0,0). Left: in the surface brightness (µ) -magnitude plane, the stars selected using the "star clump" method. In red are the stars identified on the deep stack, in blue are the supplementary stars identified using individual CCDs. The bright stars (r ≤ 16) are selected by neither methods.

Figure 4 . 2 :

 42 Figure 4.2: Stars catalog in the µ -m plane for the four fields. In blue: the stars selected by the star-clump method. In red: the bright stars selected on their magnitude and surface brightness.

Figure 4 . 3 :

 43 Figure 4.3: Upper : CARS mask for star haloes and diffraction spikes. Credit: [Erb09]. Lower : Final implementation of the masking procedure. In blue the polygons from CARS mask ([Erb09]).In magenta K10 bright stars haloes mask. In green and red circles, the bright stars mask, the radius is set according to the star r magnitude.

  bands b m b (obs.) -m b (z, a ) residuals, i.e. the difference between the galaxy observed magnitudes m b (obs. and the synthetic model magnitudes m b (z, a ) computed by redshifting and integrating F (a , λ). A third-order spline correction f (a , λ) to F (a , λ) is then computed, again by χ 2minimization, to minimize the magnitude residuals to the fitted model. A set of magnitude offsets is also computed in this iterative process (4.2). u g r i z ∆m -0.0033 0.0006 0 -0.0141 -0.0040 Table 4.2: Magnitude offsets which were computed during the training process. The trained spectral F: F(a , λ) = F (a , λ)×f (a , λ) (4.4) is presented on figure 4.4. The correction f can be as high as 30% of the initial template value, alleviating the importance of the initial template set choice.

Figure 4 . 4 :Figure 4 . 5 :

 4445 Figure 4.4: The trained spectral templates SED for different values of the mean stellar age a * . The flux is in erg.Å -1 .s -1 .M -1 .

Field

Figure 4 . 6 :

 46 Figure 4.6: Distribution of the galaxies rest-frame U-V, which is used to separate the galaxy types into red and blue : below (resp. above) U -V = 0.54 lay the bluer spirals (resp. redder elliptical) galaxies. Image: K10

Figure 4 . 7 :

 47 Figure 4.7: SN 04D2iu at z = 0.7 with no identified host. Left : deep stacked i image used for the host identification. Right : HST image of the same location.

  (a) SN 04D3cp at z = 0.83, the closest galaxy is at z = 0.29. (b) SN 07D3bt at z = 0.91, the closest galaxies are at z = 0.32 and z = 0.35. (c) SN 05D3km at z = 0.96, the closest galaxy is at z = 0.43. (d) SN 04D4bq at z = 0.55, the closest galaxies are at z ∼ 0.35.

Figure 4 . 8 :

 48 Figure 4.8: Examples of SNe for which the closest galaxy is a foreground galaxy on the line-ofsight (score=3).

Figure 4 . 9 :

 49 Figure 4.9: Line of sight inner disk and outer annulus at R LOS = 60 arcsec and R 1 2 LOS = R LOS /2, defined for the fractional contamination estimation.

Figure 4 . 10 :

 410 Figure 4.10: Parametrization of the halo models : the SIS velocity dispersion is modeled with the TF and FJ relations from [Boe04] and [Mit05]. The velocity dispersion is then converted into a virial mass M 200 and radius r 200 values. For the NFW concentration parameter c, we used the published [Duf08] c(M 200 , z) law, converted into a c(M B , z) law.

Figure 4 . 11 :Figure 4 . 12 :

 411412 Figure 4.11: Magnification computation for a typical lensing situation and different models.

Figure 4 . 13 :Figure 4 . 14 :

 413414 Figure 4.13: Normalization factor computed for the four fields, for the truncated SIS model (Q-LET and weak approximation), and within the weak approximation for the truncated NFW model.

Figure 5 . 1 :Figure 5 . 2 :

 5152 Figure 5.1: Distributions of the weighted correlations coefficient. ρ = 0.18 at 99% C.L. detection. Image : [Kro10]

Figure 5 . 3 :

 53 Figure 5.3: Measurement of the correlation on 133 SNe selected in the JLA sample. The histogram of the correlation coefficient computed on the "shuffled" samples. The measured value ρ = 0.252 is indicated.

Figure 5 . 5 :Figure 5 . 6 :

 5556 Figure 5.5: Measurement of the correlation on 225 SNe selected in the SNLS5 sample. The histogram of the correlation coefficient computed on the "shuffled" samples. The measured value ρ = 0.177 is indicated.

Figure 5 . 7 :

 57 Figure 5.7: Normalized magnification of SNLS5 SNe for the truncated SIS and the truncated NFW halo model.

Figure 5 . 8 :Figure 5 . 9 :

 5859 Figure 5.8: Normalized magnification of SNLS5 SNe for the truncated SIS and the truncated NFW halo model as a function of redshift. Except for 04D2kr which is not in the weak regime, the obtained values are very similar.

Figure 5 . 11 :

 511 Figure 5.11: The ten most magnified SNe (grouped according to their Field), cont. from fig. 5.10.

  Sky map temperature anisotropies as measured by the successive COBE (Cosmic Background Explorer, launched in 1989), WMAP (Wilkinson Microwave Anisotropy Probe, launched in 2001) and Planck mission (launched in 2009). . . . . . . . . . . 1.8 The distance D V as a function of redshift from BAO experiments : BOSS, on the

. . . . . . . . . . . v 1.7

Table 1 .1: Brief

 1 history of the thermal regimes of the universe. Credits:[START_REF] Bauman | Thermal history[END_REF]

	Temperature	Redshift	Time	Era
	10 19 GeV		0 -10 -43 s	Planck epoch
	10 15 -10 16 GeV		10 -43 -10 -38 s	GUT scale
			10 -34 s ?	Inflation?
	GeV		t µs	Quark-hadrons transition
	?	?	?	Dark matter freeze-out
	MeV	z ∼ 6 × 10 9	1 s	Neutrinos decouple
	500 KeV	z ∼ 2 × 10 9	6 s	Electron-positrons annihilate
	100 KeV	z BBN ∼ 4 × 10 8	t 3 minutes	Big Bang Nucleosynthesis
	0.75 eV	z eq ∼ 3400	t 55kyr	Matter radiation equality
	(0.25 -0.33) eV z rec ∼ (1100 -1400) t (260 -380)kyr Recombination epoch
	(0.23 -0.28) eV z rec ∼ (1000 -1200) t 380kyr	Photon decoupling 4
	(2.6 -7.0) meV z ∼ (11 -30)	t 100 -400Myr Reionization
	0.33 meV	z ∼ 0.4	t 9.5 Gyr	Matter-Λ equality
	0.24 meV	z = 0	t 13.8 Gyr	Now

Table 1 . 2 :

 12 Major upcoming weak lensing surveys. n gal is the effective galaxy count, f sky is the fraction of the sky covered by the survey. Source :[START_REF] Liu | Cross-correlation of Planck CMB lensing and CFHTLenS galaxy weak lensing maps[END_REF]

	Surveys	when	where	caracteristics	n gal f sky
						arcmin 2
	Dark Energy Survey now	CTIO, Chile	4-m Blanco	10	0.1
	Hyper Suprime Cam now	Mauna Kea, Hawaii 8.2-m Subaru	20	0.048
	LSST	2019	Cerro Pachón, Chile 8.4-m	40	0.25
	EUCLID (ESA)	>2020 space	1.2-m, visible & NIR 35	0.20
	WFIRST (NASA)	>2020 space	2.4-m, NIR	45	0.05

Some of the principal on-going and future lensing surveys are listed in table 1.2.

  Example of a supernovae Ia spectrum near it's peak brightness. The clear dip in the spectra near the 6000Å indicates the presence of Si II which is the clear signature of a type Ia SN. Almost 3/4 th of the emitted light is in the optical range (3800-7500 Å).

			Type 1a Line Identifications
			spectrum of SN1981b, a normal type1a near max
					Daniel Kasen, LBL
	1.5e-14				http://panisse.lbl.gov/~dnkasen
		SiII
	1e-14	4128,4131 CoII 4145,4161
	flux
		MgII triplet
	5e-15	4481	FeII blend 4923,5018,5169,etc..	SiII 5958,5979
		CaII H&K 3934,3968	SII 5958,5978	SII 6347,6371	atmospheric absorption
				SiII '6150' feature
				6347,6371	OI triplet
	0				7771,774,7775	8498,8542,8662 CaII IR triplet
		4000	5000	6000	7000	8000
			Wavelength (Angstroms)
	Figure 2.2: Image : Daniel
	Kasen, LBL
	photosphere.

Table 2 . 1 :

 21 Summary of the four Deep fields. The last 5 columns show the amount of observing time in hours ensuring a deep coverage.

	Other Observations	u	g	r	i	z

Table 2.2: Summary of MegaCam specifications

  The Wide and Deep fields are indicated on fig. 2.6 and the filter transmissions on fig.2.10. Post observational data processing took place at the Terapix (http://terapix.iap.fr/cplt/T0007/ doc/T0007-doc.html) center. The Deep survey fields coverage are presented in table 2.1. MegaCam was the wide-field (1 square degree) optical imaging facility used at CFHT. The MegaCam camera specifications are summarized in table 2.2. The CFHT and Mega-Cam are presented on fig. 2.7.

	MegaPrime
	MegaPrime/

Table 2 . 3 :

 23 possible values. Break up of the JLA SNe Ia sample

	Redshift Number	Survey
	[0, 0.1]	239	SNLS
	[0.03, 0.4]	374	SDSS
	[0.1, 1.1]	118	Several Low-z samples
	[0.8, 1.3]	9	HST

  Left: Phase plot of dark energy equation of state parameter w vs matter density Ω m . Right: Comparison of various measurements of Ω m from different collaborations. The JLA and the Planck measurements are in good agreement with Planck (2013) measurements. They are also in good agreement with the final Planck (2015) data release[START_REF] Ade | Planck 2015 results. xiii. cosmological parameters[END_REF]. Image :[START_REF] Betoule | Improved cosmological constraints from a joint analysis of the sdss-ii and snls supernova samples[END_REF]

		-1.0 -0.8 -0.6 -0.4										C11 (combined) C11 (SALT2) JLA
	w	-1.2
		-1.4											Union 2.1 (Suzuki et al. 2012)
		-1.6				JLA		WMAP9					WMAP9 (Hinshaw et al. 2013)
		-1.8				C11 Planck+WP		Planck+WP+JLA Planck+WP+BAO				Planck (2013)
		-2.0	0.15	0.20	0.25	0.30	0.35	0.40	0.45	0.15	0.20	0.25	0.30	0.35	0.40	0.45	0.50
						Ω m							Ω m
	Figure 2.14:

Table 3 . 1 :

 31 Parametrization of the concentration parameter law c(M, z) = A(M/M ) B (1 + z) C . The values are referring to c vir , except for [Net07] which gives c 200 .

	Author	hM /M	A	B	C
	Seljak, 2000 ([Sel00])	3.42 10 12	10	-0.2	-1
	Neto et al., 2007 ([Net07])	10 14 4.67 -0.11 (z = 0)
	A. Duffy et al., 2008 ([Duf08])	2. 10 12 7.85 -0.081	-0.71
	A. Maccio et al., 2008 ([Mac08])	10 12 6.76 -0.098	-1

table 3.1. concentration parameter c(z, M ) = A (M/M ) B (1 + z) C

  The summary of SNLS four fields catalogs is presented in table 4.1.

	Field effective surface Number of galaxies	Number of galaxies	Limiting magnitude
		square degrees		per arc minutes squared	i-band (S/N>9)
	D1	0.78	133476	48	24.8
	D2	0.8	129112	45	24.7
	D3	0.87	153520	49	24.9
	D4	0.77	117591	42	24.7
		Table 4.1: Summary of SNLS four galaxy field catalogs

Table 4 .3:

 4 N gal. i lim. % cata.

				∆z	RMS(∆z) ∆z/(1 + z) RMS(∆z/(1+z))
	D1	3691	24	3.62% 0.0035	0.0551	0.0033	0.0323
	D2	5537	24	2.85% 0.0076	0.0474	0.0062	0.0295
	D3	6006	24	3.17% 0.0091	0.0544	0.0064	0.0311
	D4	200	24	3.06% 0.0169	0.0339	0.0119	0.0253

  Summary of SNLS5 SNe Ia host galaxy situation classification. Note that 6 SNe classified as having a dubious host identification are also classified as having a redshift inconsistent with the closest galaxy photometric redshift. detected host galaxy. The remaining SNe are either dubious or problematic cases. This classification, indicated in table 4.4, is furtherly refined for the lensing analysis.

	Host galaxy identification N SN %
	All	439	100
	problematic (all)	30	6.8
	polluted photom.	10	2.3
	inconsistent photo.-z	17	3.9
	dubious host id.	9	2.1
	no id. host	27	6.2
	Table 4.4:

  ). Summary of SNLS5 SNe Ia lensing situation classification. The selected SNe line-ofsights will then be furtherly checked for imagery problems, such as the bright-star light pollution.

	Selection for lensing analysis	N SN	%
	All	439	100
	selected (all)	425	96.8
	closest gal. is a foreground gal.	6	1.4
	no host gal. is identified	27	6.2
	problematic	14	3.2
	Table 4.5:

Table 5 . 2 :

 52 The different SNLS5 SNe samples selected using the masking and scoring criteria as described in section 4.3.3 and by requiring : that the SN is included in SNLS5 sample, with an additional color cut |c| < 0.25 eventually and the exclusion of 03D4gl ; that the SN is included in JLA sample. We used the residuals from SNLS5 analysis.

	SNLS5 sample √ color cut	in JLA sample Mask+Score √ √ √	N 128 0.228 2.47 ρ n-σ 225 0.177 2.51
	color cut color cut, D1 & D2 color cut, D3 & D4 color cut, µ > 0.99	excluded	√ √ √	97 0.098 0.92 109 0.122 1.19 116 0.218 2.23 164 0.254 3.04

A SNLS5 SNe Ia magnifications

  Table A.1 -Continued from previous page No. redshift µ SIS µ N F W Table A.1 -Continued from previous page No. redshift µ SIS µ N F W Table A.1 -Continued from previous page No. redshift µ SIS µ N F W Table A.1 -Continued from previous page No. redshift µ SIS µ N F W

	Table A.1 -Continued from previous page
	No. redshift µ SIS µ N F W	name name name name name
	35 81 127 219	0.248 1.000 0.813 0.669 0.769	0.998 0.998 1.000 0.999 0.985 0.983 04D3mk 05D1iy 06D2cb 1.010 1.010 07D3gt 1.060 1.080 06D4dr
	36 82 128 220	0.760 0.532 0.810 0.391 0.677	1.010 1.010 1.030 1.020 0.997 0.994 0.995 0.995 07D3gw 06D1bg 06D2cc 04D3ny 0.981 0.980 06D4fc
	37 83 129 221	0.620 0.552 0.643 0.670 0.566	1.040 1.050 0.989 0.990 1.030 1.030 1.010 1.010 0.986 0.983	06D1bo 06D2ck 05D3ax 07D3hl 06D4jh
	38 84 130 222	0.820 0.345 0.419 0.572 0.760	0.985 0.981 1.000 1.000 0.996 0.996 0.992 0.991 0.984 0.982	06D1bt 06D2ff 05D3cf 07D3hu 06D4jt
	39 85 131 223	0.619 0.560 0.890 0.351 0.456	0.990 0.988 06D1cm 1.030 1.030 06D2hm 1.020 1.020 05D3cq 1.010 1.010 07D3hv 0.995 0.993 07D4cy
	40 86 132 224	0.860 0.850 0.805 0.748 0.743	1.000 0.998 1.010 1.010 1.120 1.140 1.010 1.000 07D3hw 06D1cx 06D2iz 05D3cx 1.010 1.000 07D4dp
	41 87 133 225	0.767 0.726 0.805 0.505 0.554	0.993 0.991 1.020 1.020 0.978 0.977 0.992 0.992 0.983 0.981	06D1dc 06D2ja 05D3ha 07D3hz 07D4dq
	42 88 134 226	0.240 0.900 0.766 0.680 0.772	1.000 1.000 0.983 0.986 1.090 1.120 1.000 0.999 0.981 0.977	06D1du 06D2ji 05D3hh 07D3ib 07D4dr
	43 89 135 227	0.350 0.600 0.338 0.928 0.653	1.010 1.010 0.993 0.992 1.000 1.000 1.030 1.030 0.990 0.985	06D1fd 06D2js 05D3hq 08D3dx 07D4ec
	44 90 136 228	0.524 0.927 0.664 0.170 0.520	0.989 0.988 1.060 1.070 0.999 0.995 1.000 1.000 1.010 1.010	06D1fx 06D2ju 05D3hs 08D3gb 07D4ed
	45 91 137 229	0.980 0.900 0.901 0.352 0.370	1.030 1.040 1.040 1.050 1.030 1.030 1.010 1.010 0.989 0.988	06D1gl 06D2jw 05D3ht 08D3gf 07D4ei
	46 92 138 230	0.345 0.780 0.745 0.767 0.503	0.994 0.994 1.020 1.030 0.996 0.995 0.990 0.987 1.000 1.000	06D1hf 07D2ah 05D3jb 08D3gu 07D4fl
	No. redshift µ SIS µ N F W 1 0.408 0.996 0.995 47 0.327 0.994 0.994 93 0.793 1.060 1.070 139 0.736 1.020 1.030 0.468 1.020 1.020 2 0.504 0.996 0.994 48 0.641 0.998 1.000 94 0.551 0.989 0.986 140 0.579 0.988 0.989 0.270 0.994 0.994 3 0.582 49 0.561 1.020 1.010 95 0.535 1.010 1.010 141 0.850 0.990 0.992 0.949 1.170 1.190 1.010 1.010 03D1aw name 06D1hj 07D2be 05D3jk 03D4au 03D1ar 06D1jf 07D2bi 05D3jq 03D4cj 03D1au 06D1kf 07D2bq 05D3kp 03D4cx 4 0.496 0.987 0.987 50 0.320 1.000 1.000 06D1kg 96 0.694 1.010 1.010 07D2cb 142 0.936 0.993 0.993 05D3la 0.610 0.982 0.979 03D4dy 03D1ax 5 0.865 1.070 1.060 51 0.328 0.998 0.998 07D1ab 97 0.749 1.010 1.020 07D2cl 143 0.647 0.989 0.988 05D3lb 0.592 0.995 0.992 03D4gg 03D1bk 6 0.679 0.988 0.985 52 0.617 0.997 0.996 07D1bs 98 0.738 1.000 0.998 07D2co 144 0.461 1.000 1.000 05D3lc 0.571 0.979 0.978 03D4gl 03D1co 7 0.612 1.070 1.080 53 0.626 0.987 0.985 07D1bu 99 0.746 1.010 1.010 07D2cq 145 0.670 1.030 1.030 05D3mh 0.613 0.986 0.984 04D4an 03D1dt 8 0.868 54 0.853 0.991 0.989 07D1cc 0.538 0.983 0.982 07D2du 146 0.760 0.988 0.985 05D3mn 0.880 0.978 0.977 04D4bk 0.993 0.987 03D1ew 9 0.332 1.000 1.000 55 0.500 1.010 1.010 07D1cf 0.270 0.998 0.998 07D2ec 147 0.246 0.999 0.999 05D3mq 0.550 1.170 1.200 04D4bq 03D1fc 10 0.800 0.995 0.995 56 0.705 0.977 0.976 07D1cl 0.871 1.010 1.010 07D2fv 148 0.470 1.020 1.020 05D3mx 0.811 0.973 0.971 04D4dm 03D1fq 11 0.211 0.999 0.999 57 0.730 0.999 0.996 07D1cx 0.720 0.980 0.978 07D2fy 149 0.169 1.000 1.000 05D3ne 1.031 1.030 1.050 04D4dw 04D1dc 12 0.768 1.000 0.998 58 0.540 1.000 1.000 08D1aa 0.354 0.996 0.995 07D2kc 150 0.232 1.000 1.000 06D3cn 0.629 1.020 1.020 04D4fx 04D1de 13 0.369 0.996 0.995 59 0.836 1.000 0.995 04D2al 0.731 0.984 0.981 07D2kh 151 0.442 0.993 0.993 06D3df 0.936 1.020 1.010 04D4hf 04D1hd 14 0.560 0.990 0.990 60 0.220 0.997 0.997 04D2bt 0.554 0.989 0.987 08D2ad 152 0.726 1.020 1.020 06D3do 0.699 0.994 0.993 04D4ib 04D1hx 15 0.850 1.000 1.000 61 0.568 1.010 1.010 04D2cw 0.689 0.987 0.988 08D2ag 153 0.282 0.999 0.999 06D3dt 0.687 0.987 0.984 04D4ic 04D1hy 16 0.998 1.120 1.150 62 0.732 0.987 0.988 04D2gp 0.474 1.030 1.040 08D2ch 154 0.519 0.995 0.994 06D3el 0.866 0.997 1.000 04D4ii 04D1iv 17 0.585 0.988 0.987 63 0.740 0.984 0.981 04D2ja 0.831 0.981 0.979 08D2cl 155 0.576 0.987 0.987 06D3et 0.751 1.030 1.040 04D4im 04D1kj 18 0.590 0.994 0.993 64 0.744 1.260 1.220 04D2kr 0.355 0.997 0.996 08D2dr 156 0.268 0.999 0.998 06D3fp 0.472 1.000 1.000 04D4ju 04D1oh 19 0.915 65 0.513 0.990 0.988 04D2mj 0.650 0.978 0.977 08D2dz 157 0.720 1.020 1.020 06D3gh 0.961 0.988 0.985 04D4jw 0.982 0.979 04D1ow 20 0.515 0.999 0.998 66 0.323 0.996 0.995 05D2ab 0.833 0.980 0.977 08D2id 158 0.250 0.999 0.999 06D3gn 0.930 0.966 0.964 04D4jy 04D1pg 21 0.639 0.980 0.979 67 0.479 0.992 0.992 05D2ac 0.291 0.998 0.998 03D3ba 159 0.760 1.010 1.000 06D3gx 0.640 0.982 0.979 05D4ag 04D1pu 22 0.767 0.973 0.973 68 0.184 0.998 0.998 05D2ah 0.355 1.020 1.020 03D3bl 160 0.237 0.999 0.999 07D3ae 0.509 0.982 0.981 05D4av 04D1qd 23 0.436 0.990 0.990 69 0.920 1.000 1.010 05D2ay 0.715 0.993 0.993 03D3bn 161 0.355 0.997 0.996 07D3af 0.701 1.060 1.070 05D4bj 04D1rh 24 0.985 1.030 1.030 70 0.680 1.110 1.140 05D2bt 0.156 1.000 1.000 04D3bf 162 0.451 0.994 0.994 07D3ap 0.372 0.993 0.993 05D4bm 04D1rx 25 0.702 0.985 0.983 71 0.891 1.070 1.080 05D2by 0.620 1.050 1.060 04D3co 163 0.920 0.996 0.996 07D3bo 0.790 1.030 1.030 05D4cs 04D1si 26 0.663 1.030 1.030 72 0.427 0.998 0.995 05D2cb 0.643 1.010 1.010 04D3cy 164 0.910 1.130 1.150 07D3bt 0.855 0.990 0.991 05D4dw 04D1sk 27 0.842 0.974 0.972 73 0.574 0.986 0.985 05D2dt 1.002 0.997 0.996 04D3dd 165 0.708 0.999 0.996 07D3cc 0.810 0.979 0.975 05D4dy 05D1az 28 0.632 0.995 0.992 74 0.510 1.000 1.000 05D2dy 0.470 0.991 0.991 04D3df 166 0.807 0.997 0.995 07D3cp 0.605 1.030 1.030 05D4ef 05D1cb 29 0.830 0.996 0.993 75 0.733 1.010 0.999 05D2fq 0.610 0.994 0.992 04D3do 167 0.808 0.976 0.976 07D3cs 0.536 0.982 0.980 05D4ek 05D1cl 30 0.580 1.010 1.010 76 0.348 1.000 1.000 05D2ie 0.263 0.999 0.999 04D3ez 168 0.512 0.993 0.993 07D3cu 0.402 0.988 0.987 05D4ff 05D1dx 31 0.312 0.995 0.995 77 0.354 0.993 0.993 05D2mp 0.358 0.997 0.997 04D3fk 169 0.837 0.973 0.973 07D3da 0.808 0.975 0.972 05D4gw 05D1ej 32 0.860 1.030 1.030 78 0.924 0.968 0.966 05D2ob 0.910 1.060 1.060 04D3gx 170 1.020 1.030 1.030 07D3do 0.552 0.989 0.989 06D4bo 05D1er 33 0.763 1.010 1.010 79 0.310 0.999 0.999 06D2ag 0.552 1.070 1.080 04D3hn 171 0.740 1.020 1.020 07D3ey 1.000 1.070 1.080 06D4cl 05D1if 34 0.490 1.010 1.000 80 0.499 0.990 0.988 06D2bk 0.983 0.988 0.984 04D3lp 172 0.830 1.010 1.000 07D3gm 0.303 1.010 1.020 06D4dh 05D1ix Continued on next page Continued on next page Continued on next page Continued on next page
			Continued on next page

Table A . 1 :

 A1 List of magnifications for all selected SNe from SNLS5 sample computed with two halo models

http://www.cfht.hawaii.edu/Science/CFHTLS/

Doppler shift depends on the speed of the emitter while in cosmological redshift the stretching of wavelength depends on the expansion of the intermediate space. In a locally flat space, at a small distance (z 1) it will be a consequence of the Doppler shift and z = v c . With space time curvature, gravitational shift comes into play and the redshift caused is not due to Doppler shift[Bun09]

Recombination and photon decoupling are two different events, coinciding only if recombination is instantaneous.

http://www.cosmos.esa.int/web/planck

http://www.sdss.org/

The other supernovae type originate from the gravitational collapse of the iron core of massive stars M > 8M ([Ibe83],[START_REF] Woosley | The evolution and explosion of massive stars[END_REF]) that cannot be supported by further exothermal thermonuclear reactions (core-collapse SNe).

https://www.eso.org/sci/facilities/paranal/instruments/xshooter/tools/specphot_list.html

Newton pointed out about the possibility of deflection of light rays. Later on Eddington calculated it to be roughly twice the deflection of Newtonian formalism, the missing factor of

from Newtonian calculations was due to the time dilation factor which is present in G.R. See for example:[START_REF] Rich | Fundamentals of Cosmology[END_REF] or[START_REF] Schneider | Gravitational lenses[END_REF].

For a lens with constant surface density Σ = Σcr, β = 0 for any θ, so that the lens focuses perfectly.

See e.g. [Ser10] for a demonstration.

This can be overcome by introducing a core radius rc so that ρ ∝ (r 2 c + r 2 ) -1 , see figure 3.9.

Assuming, following[START_REF] Gunnarsson | Corrections for gravitational lensing of supernovae: better than average?[END_REF], a mean galactic AB color Mr -MB = -1.2 and a conversion from AB to Vega system B AB = B Vega -0.12.

Taking h = 0.72, a solar magnitude Mr AB = 4.68 and a conversion from AB to Vega system r AB = r Vega + 0.125.

http://terapix.iap.fr/soft/swarp/

The saturation level is the minimum of the saturation levels of all the CCDs entering the stacking process. Its value ranges in the r frame from 7000 to 8000 on the stack normalized to a ZP =

Author

L redshift scaling index η σ σ (km.s -1 )

[Boe04] (TF) 3.2: A comparison of velocity dispersion values σ for the fiducial luminosity L r = 10 10 h -2 L r , roughly equivalent to L B = 1.6 10 10 h -2 L B , using the Vega solar magnitude M B = 5.47, and, following [START_REF] Gunnarsson | Corrections for gravitational lensing of supernovae: better than average?[END_REF], a mean galactic AB color M r -M B = -1.2 .

Gravitational Lensing and Supernovae Ia

The impact of gravitational lensing by matter inhomogeneity on standard-candle use as distance indicators for cosmology has been considered since the 90's. It was noted along that standard candles could also serve as a probe of the dark matter distribution along their line of sight. [Smi14] can be consulted for a recent thorough review on this subject.

Lensing impact on the Hubble diagram

The gravitational impact of lensing on the Hubble diagram is to increase the scatter of the distance measurement around the value predicted in a homogeneous universe model ([Fri96], [Hol05], [START_REF] Gunnarsson | Corrections for gravitational lensing of supernovae: better than average?[END_REF], [START_REF] Sarkar | Lensing and Supernovae: Quantifying the Bias on the Dark Energy Equation of State[END_REF]). Flux conservation guarantees that the mean flux of the SN population magnification starts to get bigger than 1 + 10 -1.5 , and start to deviate when µ > 1 + 10 -1 , which marks the gradual limit towards a stronger lensing regime.

Magnification normalization

The magnification computed as explained in previous section needs to be normalized, to insure that µ = 1 at all redshifts.

Following [START_REF] Schneider | Gravitational lenses[END_REF] (but see also [START_REF] Babul | Gravitational lensing by smooth inhomogeneities in the universe[END_REF]), the magnification is given by :

where F is the flux in the inhomogeneous clumpy universe and F 0 is the flux in the homogeneous universe, related to the source luminosity L through :

The distance D L (z) is the luminosity distance as defined in eq. 1.14 for a smooth Friedmann-Lemaitre model, fitted to the inhomogeneous universe [START_REF] Schneider | Gravitational lenses[END_REF]). Because of flux conservation, the average of the flux F z , obtained by "moving" the source of luminosity L over the sphere corresponding to a fixed redshift z, is equal to F 0 . This implies that the average magnification is

When computing the magnification, we superimposed haloes on a universe already homogeneously filled with matter, obtaining systematically a magnification µ > 1. As a consequence, it is necessary to compute µ z > 1 to normalize the computed magnification per redshift bin to 1.

Following K10, we thus carry out for each field a simulation, randomly positioning line-ofsight pointing towards fictitious sources, distributed over 12 redshift bins from z = 0.1 to z = 1.2 in steps of ∆z = 0.1. We simulate 1000 sources per redshift bin, a choice justified by a required precision of 10 -3 on the computed magnification mean. The magnification distribution for the redshift bin z = 1 is shown on 4.14. Note that the distribution of log 10 (µ -1) is roughly gaussian, with a compatible RMS for the two considered model, truncated SIS and NFW in the weak approximation.

For each field, we compute the mean magnification per redshift bin, and fit the 12 points by a 3rd-order polynomial to obtain a normalization N (z) valid across the redshift interval 0 < z < 1.2. The normalization function N (z) is presented on figure 4.13 for the four fields and the three models : truncated SIS with ray tracing (Q-LET) and weak approximation, truncated NFW with the weak approximation.

The normalized magnification for the z = 1 bin is presented on 4.14. The distributions peaks slightly below 1 (i.e. for SNe slightly de-magnified), keeping its skewed tail towards magnification values greater than 1.

In the next chapters we will analyze the results obtained from the magnification computation pipeline, including their significance, their dependence on the halo model choice. the 45 SNe from K10 sample that were rejected by the masking process described in this thesis.

SNLS5 lensing signal detection

The SNLS3 sample

We first checked our newly computed SNe magnifications against K10 previous results, restricting thus ourselves to SNLS3 data. We will be using the cosmology residuals from the JLA analysis. Note that K10 SNe are not all included in the JLA sample and vice-versa, as the criteria for the selection of SNe Ia "good for cosmology" were slightly different for these two analysis : only 155 of the 170 K10 SNe are included in the JLA cosmology sample.

We use here the magnification computed using the weak approximation and the truncated SIS model.

The masking and scoring steps select 133 SNe out of the 239 SNe in the JLA cosmology sample, which correspond to a rejection of 45% of the SNe, within 1-σ of the announced rejection rate of 40%.

Of the 133 JLA SNe selected by our masking and scoring procedure, 110 are included in K10 sample. Compared to K10 selection, we reject 45 SNe, with one third being on the edge of the camera, which were not masked out in K10 ; two-third because of bright star light contamination. On the reverse, we kept 23 supplementary SNe : 8 SNe were masked out in K10 selection, plus 15 SNe were not included in the preliminary SNLS3 cosmology sample used for K10.

The samples summary and their associated correlation coefficients are presented on table 5.1. Note that rejecting 45 SNe from K10 sample helps to strengthen the signal, which does not change when the supplementary 23 SNe are added. As a conclusion, the result from K10 is confirmed by this new analysis. The results for the 133 SNe selected in JLA sample is shown on figures 5.3 and 5.4. A correlation coefficient ρ JLA;133 = 0.252 is measured, corresponding to a detection level at 2.8-σ.

Abstract

The presence of mass inhomogeneities along the line of sight of propagation of light from distant objects can induce deflection in the flight path of the photon. This phenomenon is called as gravitational lensing. Lensing can have both distortion (shear) and isotropic magnification effects on the source. We studied the effect of lensing magnification on supernova (SN) Ia in this thesis.

Presence of lensing would introduce a source of contamination to the brightness distribution of the source (SN Ia in our case). Thus it also enables one to compute the lensing effect indirectly from the Hubble diagram (i.e. from the residual to the Hubble diagram). In this thesis we computed the correlation between these two effects : the Hubble residual and the computed lensing magnification for the SN by the line of sight foreground dark matter haloes. A detection of positive correlation between these two would signify the positivity of lensing signal detection.

The data sample is the spectroscopic SNe Ia sample from the five years full SNLS data and the Hubble resiudals are those of the preliminary cosmology analysis performed on SNLS5 data. We obtain a signal of ρ = 0.177 at 2.51σ. This result is consistent with the previous SNLS three years data lensing analysis results.