LIGHT ON DARK MATTER WITH WEAK GRAVITATIONAL LENSING 1

Light on Dark Matterwith Weak Gravitational Lensing

S. Pires, J.-L. Starck and A. RefregierLaboratoire AIM, CEA/DSM-CNRS-Universite Paris Diderot, IRFU/SEDI-SAP, CEA Saclay,

Orme des Merisiers, 91191 Gif-sur-Yvette, France

Abstract— This paper reviews statistical methods re-cently developed to reconstruct and analyze dark mattermass maps from weak lensing observations. The field ofweak lensing is motivated by the observations made inthe last decades showing that the visible matter representsonly about 4-5% of the Universe, the rest being dark.The Universe is now thought to be mostly composed byan invisible, pressureless matter -potentially relic fromhigher energy theories- called “dark matter” (20-21%)and by an even more mysterious term, described inEinstein equations as a vacuum energy density, called“dark energy” (70%). This “dark” Universe is not welldescribed or even understood, so this point could be thenext breakthrough in cosmology.

Weak gravitational lensing is believed to be the mostpromising tool to understand the nature of dark matterand to constrain the cosmological model used to describethe Universe. Gravitational lensing is the process in whichlight from distant galaxies is bent by the gravity ofintervening mass in the Universe as it travels towardsus. This bending causes the image of background galaxiesto appear slightly distorted and can be used to extractsignificant results for cosmology.

Future weak lensing surveys are already planned inorder to cover a large fraction of the sky with largeaccuracy. However this increased accuracy also placesgreater demands on the methods used to extract theavailable information. In this paper, we will first describethe important steps of the weak lensing processing toreconstruct the dark matter distribution from shear es-timation. Then we will discuss the problem of statisticalestimation in order to set constraints on the cosmologicalmodel. We review the methods which are currently usedespecially new methods based on sparsity.

Index Terms— Cosmology : Weak Lensing, Methods :Statistics, Data Analysis

I. INTRODUCTION

According to present observations, we believe thatthe majority of the Universe is dark, i.e. does notemit electromagnetic radiations. Its presence is inferredindirectly from its gravitational effects: on the motionsof astronomical objects and on light propagation.

Weak gravitational lensing has been found to be one ofthe most promising tools to probe dark matter and darkenergy because it provides a method to map directly thedistribution of dark matter in the Universe (see [1], [2]).From this dark matter distribution, the nature of darkmatter can be better understood and better constraintscan be set on dark energy because it affects the evolutionof structures. This method is now widely used but, theamplitude of the weak lensing signal is so weak that itsdetection relies on the accuracy of the techniques used toanalyze the data. Each step of the analysis has requiredthe development of advanced techniques dedicated tothese applications.

This paper is organized as follow:Section 2 aims at giving an overview of weak gravi-

tational lensing : the basics of the lensing theory and abrief description of the weak lensing data analysis.

Section 3 will be dedicated to the presentation of theshear estimation problem. It requires the measurementof the shape of millions of galaxies with extremelyhigh accuracy, in the presence of observational problemssuch as anisotropic Point Spread Function, pixelisationand noise. Methods that are currently used to derivethe lensing shear field from the shapes of backgroundgalaxies will be described.

Section 4 will address the following inverse problem :how to derive a dark matter mass map from the measuredshear field. Because of some observational effects suchas noise and complex survey geometry, this problemcan be seen as an ill-posed inverse problem. We willpresent various methods of inversion currently used toreconstruct the dark matter mass map from incompleteshear map especially a recent promising method ofinterpolation, based on the sparse representation of thedata. And we will describe the different filtering methodswhich are used to reduce the noise in these dark mattermass maps such as linear filters, Bayesian techniquesand a recent wavelet method.

Finally, in section 5, we will discuss the problem ofstatistical information extraction in weak lensing datain order to constrain the cosmological model. We will

LIGHT ON DARK MATTER WITH WEAK GRAVITATIONAL LENSING 2

introduce different statistics which are of interest inweak lensing data analysis. A recent approach based onsparse representations will be presented.

II. INTRODUCTION TO WEAK LENSING

A. Gravitational Lensing observations

Fig. 1. Strong Gravitational Lensing effect observed in the Abell2218 cluster (W. Couch et al, 1975 - HST).

In the beginning of the twentieth century, A. Ein-stein predicted that massive bodies could be seen asgravitational lenses that bend the path of light rays bycreating a local curvature in space-time. One of thefirst confirmations of Einstein’s new theory was theobservation during the 1919 eclipse of the deflectionof light from distant stars by the sun. Since then, awide range of lensing phenomena have been detected.The gravitational deflection of light generated by massconcentrations along light paths produces magnification,multiplication, and distortion of images. These lensingeffects are illustrated by Fig. 1 which shows one of thestrongest lens observed : Abell 2218, a massive clusterof galaxies some 2 billion light years away towards theconstellation Draco. The observed gravitational arcs areactually the magnified and distorted images of galaxiesthat are about 10 times more distant than the cluster.

B. Gravitational lensing theory

The properties of the gravitational lensing effectdepend on all the projected mass density integratedalong the line of sight and on the cosmological angulardistances between the observer, the lens and the source(see Fig. 2).

1) The lens equation: In the thin lens approximation,we consider that the lensing effect comes from a singlematter inhomogeneity located between the source and theobserver. The system is then divided into three planes:the source plane, the lens plane and the observer plane.

Fig. 2. Illustration of the gravitational lensing effect by largescale structures: the light coming from distant galaxies (on the right)traveling toward the observer (on the left) is bent by the structures (inthe middle). This bending causes the image of background galaxiesto appear slightly distorted. The structures causing the deformationsare called gravitational lenses by analogy with classical optics.

The light ray is supposed to travel without deflectionbetween these planes with just a slight deflection α whilecrossing the lens plane (see Fig. 3).

In the limit of a thin lens, all the physics of the gravita-tional lensing effect is contained in the lens equation thatrelates the true position of the source θS to its observedposition(s) on the sky θI :

~θS = ~θI −DLS

DOS~α(~ξ), (1)

with ~ξ = DOL~θI and DOL, DLS and DOS are respec-

tively the distance from the observer to the lens, thelens to the source, and the observer to the source. Thedeflexion angle α is related to the projected gravitationalpotential Ψ obtained by the integration of the 3D New-tonian potential Φ(~r) along the line of sight:

~α(~ξ) =2c2

∫~∇⊥Φ(~r)dz = ~∇⊥

(2c2

∫Φ(~r)dz,

)︸ ︷︷ ︸

Ψ

, (2)

where c is the speed of light and ~∇⊥ is the perpendicularcomponent of the gradient operator.

We can distinguish two regimes of gravitationallensing. In most cases, the bending of light is smalland the background galaxies are just slightly distorted.This corresponds to the weak lensing effect. Sometimes(as seen previously) the bending of light is so extreme,that the light travels along two different paths to theobserver, and multiple images of one single sourceappear on the sky. For this to happen, the lensing effectmust be strong. In this paper, we will only address theweak gravitational lensing regime.

2) The distortion matrix: The weak gravitationallensing effect results in both an isotropic dilation (theconvergence, κ) and an anisotropic distortion (the shear,γ) of the source. To quantify this effect, the lens equation

LIGHT ON DARK MATTER WITH WEAK GRAVITATIONAL LENSING 3

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1,!'

1.,'

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2'

3'

4'

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Fig. 3. The thin lens approximation.

has to be solved. Assuming θI is small, a first orderTaylor series approximation of the distortion operator,given by the lens equation, can be done :

θS,i = AijθI,j , (3)

where i and j correspond respectively to the ith com-ponent in the lens plane and the jth component in thesource plane and :

Ai,j =∂θS,i

∂θI,j= δi,j −

∂αi(θI,i)∂θI,j

= δi,j −∂2Ψ(θI,i)∂θI,i∂θI,j

, (4)

where Ai,j are the elements of the matrix A and δi,j

is the Kronecker delta. All the first order effects (theconvergence κ and the shear γ) can be described by theJacobian matrix A that is called distortion matrix:

A = (1− κ)

(1 00 1

)− γ

(cos 2ϕ sin 2ϕsin 2ϕ − cos 2ϕ

), (5)

where γ1 = γ cos 2ϕ and γ2 = γ sin 2ϕ.

The convergence term κ enlarges the backgroundobjects by increasing their size, and the shear term γstretches them tangentially around the foreground mass.

3) The gravitational shear (γ): The gravitationalshear γ describes the anisotropic distortions of back-ground galaxy images. It corresponds to a two compo-nents field γ1 and γ2 that can be derived from the shapeof observed galaxies: γ1 describes the shear in the x andy directions and γ2 describes the shear in the x = yand x = −y directions. Using the lens equation, thetwo shear components γ1 and γ2 can be related to the

gravitational potential Ψ by:

γ1 =12

[∂2Ψ(~θI)

∂θ2I,1

− ∂2Ψ(~θI)∂θ2

I,2

]

γ2 =∂2Ψ(~θI)

∂θI,1∂θI,2. (6)

If a galaxy is initially circular with a diameter equal to1, the gravitational shear will change this galaxy in anellipsoid with a major axis a = 1

1−κ−|γ| and a minor axisb = 1

1−κ+|γ| . The eigenvalues of the amplification matrix(corresponding to the inverse of the distortion matrix A)pro- vide the elongation and the orientation producedon the images of lensed sources [2]. The shear γ isfrequently represented by a segment representing theamplitude and the direction of the distortion (see Fig. 4).

4) The convergence (κ): The convergence κ thatcorresponds to the isotropic distortion of backgroundgalaxy images is related to the trace of the distortionmatrix A by:

tr(A) = δ1,1 + δ2,2 −∂2Ψ(~θI)

∂θ2I,1

− ∂2Ψ(~θI)∂θ2

I,2

,

tr(A) = 2−∆Ψ(~θI) = 2(1− κ). (7)

The convergence κ is defined as half the Laplacianof the projected gravitational potential ∆Ψ and is thendirectly proportional to the projected matter densityof the lens (see Fig. 4). For this reason, κ is oftenconsidered as the mass distribution.

.

Fig. 4. Simulated convergence map by [3] covering a 2◦ x 2◦

field with 1024 x 1024 pixels. The shear map is superimposed tothe convergence map. The size and the direction of the segmentsrepresent the amplitude and the direction of the deformation locally.

LIGHT ON DARK MATTER WITH WEAK GRAVITATIONAL LENSING 4

C. Weak lensing data analysis

The dilation and distortion of images of distant galax-ies are directly related to the distribution of the (dark)matter and to the geometry and the dynamics of theUniverse. As a consequence, weak gravitational lensingoffers unique possibilities for probing the statistical prop-erties of dark matter and dark energy in the Universe.In the next sections, we detail the different steps ofthe weak lensing data analysis along with the differenttechniques dedicated to these applications. The followingsub-problems will be addressed:

1) Shear estimation from the measurement of thebackground galaxy ellipticities.

2) Inversion methods to derive a dark matter massmap from a shear map.

3) Filtering of the dark matter mass map to reducethe noise level.

4) Statistical analysis of the weak lensing data toconstrain the cosmological model.

The constraints that can be obtained on cosmology fromthe weak lensing effect relies strongly on the qualityof the techniques used to analyze the data, because theweak lensing signal is very small. In the following, anoverview of the different techniques currently used willbe given along with future prospects.

III. SHEAR ESTIMATION

The weak gravitational lensing effect is so small thatit is not possible to detect it from a single galaxy. Thefundamental problem is that galaxies are not intrinsicallycircular, so the measured ellipticity is a combination oftheir intrinsic ellipticity εint and the gravitational lensingshear γ. By assuming that the orientation of intrinsicellipticities of galaxies are random, any systematic align-ment arises from gravitational lensing. To estimate thegravitational shear locally, the measurements of manybackground galaxies must thus be combined.

From this assumption, to perform an estimation of theshear field, we have to correct each galaxy of the fieldfor the Point Spread Function (PSF) due to instrumentaland atmospheric effects that distort the apparent shapeof background galaxies. A shape determination algorithmhas then to be applied to estimate the gravitational shearfrom background galaxies.

A. Correction of the PSF and shape measurements

A major challenge for weak lensing is the correctionfor the PSF. Each background galaxy (S) is convolvedby the PSF of the imagery system (H) to produce theimage that is seen by the instrument (Iobs):

Iobs(θ) = S(θ) ∗H(θ). (8)

This effect can contaminate a true lensing signaland even for the most modern telescopes, this effectis usually at least of the same order of magnitudeas the weak gravitational lensing shear, and is oftenmuch larger. Then, we must calibrate the PSF usingthe images of stars. Indeed, stars present in the fieldand which correspond to point sources, provide a directmeasurement of the PSF, and these can be used to modelthe variation of the PSF across the field by doing aninterpolation between the points where stars appear onthe image.

In this section, we briefly review the different methodsto correct for instrumental and atmospheric distortions.These methods are broadly distinguished in two classes.The first class of methods subtract the ellipticity of thePSF from that of each galaxy while the second classof methods attempt to deconvolve each galaxy from thePSF.

The most used method is the KSB+ method thatbelongs to the first category. This method is the result ofa series of successive improvements of the original KSBmethod proposed by Kaiser, Squires & Broadhurst [9].The core of the method is based on the measurement ofthe ellipticity of the background galaxies. The weightedellipticity of an object is defined as:(

ε1ε2

)=

1Q1,1 + Q2,2

(Q1,1 −Q2,2

2Q1,2

), (9)

where:

Qi,j =∫

d2θW (θ)I(θ)θiθj∫d2θW (θ)I(θ)

(10)

are the quadrupole moments weighted by a Gaussianfunction W of scale length r estimated from the objectsize, I is the surface brightness of the object and θ isthe angular distance from the object center.

The PSF correction is obtained by subtracting thestar weighted ellipticity ε∗i from the observed galaxyweighted ellipticity εobs

i . The corrected galaxy ellipticityεi is given by:

εi = εobsi − P sm(P sm∗)−1ε∗i , (11)

where i=1,2 and P sm and P sm∗) is the smear suscepti-bility tensors for the galaxy and star, that can be derivedfrom higher-order moments of the images.

This method has been used by many authorsalthough different interpretations of the method haveintroduced differences between each implementations.One drawback of the KSB+ method is that for non-Gaussian PSFs, the PSF correction is defined poorlymathematically.

LIGHT ON DARK MATTER WITH WEAK GRAVITATIONAL LENSING 5

In [11], the authors propose a method to account forrealistic PSF better by convolving with an additionalkernel to eliminate the anisotropic component of the PSF.

The other class of methods attempt a deconvolutionof each galaxy from the PSF. The direct deconvolution(of equation 8) requires an inversion matrix and thatbecomes an ill-posed inverse problem if the matrix His singular (i.e. can not be inverted). Some methodshave been developed to correct for the PSF without adirect deconvolution. These methods try to reproducethe observed galaxies by modeling each unconvolvedbackground galaxy (the background galaxy as it wouldbe seen without a PSF):

Imod(θ) = Smod(θ) ∗H∗(θ). (12)

The modeled galaxy (Smod) is then convolved by thePSF estimated from the stars present in the field (H∗)and the galaxy model is tuned so that the convolvedmodel (Imod) reproduces the observed galaxy (Iobs).One major problem of these methods is that a parametricGaussian shape is assumed for the PSF and dependingon the survey the Gaussian functions are sometimesbadly suited to represent PSF shapes.

All the methods work by estimating for each galaxyan ellipticity ε (after PSF correction) whose definitioncan vary between the different methods. The accuracy ofthe shear measurement method depends on the techniqueused to estimate the ellipticity of the galaxies in presenceof observational effects such as noise, poor pixelisation,etc. In the KSB+ method, the ellipticity is derived fromquadrupole moments weighted by a Gaussian function.This method has been used by many authors but it is notsufficiently accurate for future surveys. The extensionof KSB+ to higher order moments has been done toallow more complex galaxy shapes. Other methods [6],[7], based on the “shapelet” formalism (see [8]) aremore accurate but many shapelet coefficients are neededto represent a galaxy. Indeed, the basis functions ofshapelets representation constructed from Hermite poly-nomials weighted by a Gaussian function are not optimalto represent galaxy shapes that are closer to exponentialfunctions. By consequence, in presence of noise, theaccuracy of the shear measurement method based on ashapelet decomposition is not optimal.

Many other methods have been developed to addressthe global problem of the shear estimation. In preparationfor the next generation of wide-field survey, a wide rangeof the shear estimation methods have been comparedblindly in the Shear Testing Program (STEP) in order toimprove the accuracy of the methods ([4], [5]). Several

methods have achieved an accuracy of a few percentsin the simulated STEP images. However, the accuracyrequired for future surveys will be greater (of the orderof 0.1%) and will require new insights. A new challengecalled GREAT08 ([10]) has been recently set outsidethe weak lensing community as an effort to spur furtherdevelopments.

B. Shear field estimation

After PSF correction, a catalogue of galaxies canbe built with the shape and position of each galaxypresent in the field. As stated above, the measured shapeafter PSF correction ε is a combination of the intrinsicellipticity of the galaxy εint and the gravitational lensingshear γ. Thus, the gravitational lensing shear γ canonly be estimated by averaging over a large number ofgalaxies.

Usually this shape catalogue is used to characterize thegravitational shear statistically using correlation func-tions or other statistics (see section V). It can also beused to map the shear field in order to derive a darkmatter mass map (see section IV).

In practice, the shear map is obtained by pixelisationof the field in such a way that several background galax-ies fall in each pixel. The shear map is then obtained byaveraging for each pixel the ellipticity of the galaxiesfalling into the pixel:

γ(i, j) =1

N(i, j)

N(i,j)∑k=1

ε(xk, yk), (13)

where γ corresponds to the estimated shear for thepixel (i, j) of the shear map, N(i, j) is the number ofgalaxies within the pixel (i, j) used to estimate the localshear γ(i, j) and ε(xk, yk) corresponds to the estimatedellipticity of a galaxy at position (xk, yk). The resultingshear map is subject to some observational effects suchas noise that arises both from the measurement errorof galaxy ellipticities σmeas and the residual of galaxyintrinsic ellipticities σint. Another observational effectis the masking out of bright stars from the field thatgives a complex geometry to the survey. Fig. 5 showsan example of mask applied to real data. The analysisof weak lensing data requires to account for theseobservational effects (see section IV-A and section IV-B).

IV. MAPPING THE DARK MATTER

The problem of mass reconstruction has become acentral topic in weak lensing since the very first mapsof dark matter have demonstrated that we could see thedark side of the Universe.

LIGHT ON DARK MATTER WITH WEAK GRAVITATIONAL LENSING 6

Fig. 5. Mask applied to the CFHTLS data (obtained with theMegacam in the D1 field). The mask covers a field of 1◦ x 1◦. (J.Berge et al, 2008)

The reconstruction of the dark matter mass map fromshear measurements is an ill-posed inverse problembecause of observational effects such as noise, complexgeometry of the field, etc. This inverse problem can bedecomposed in two sub-problems: the reconstruction ofthe dark matter mass map from the shear field (sectionIV-A) and the filtering of the dark matter mass map(section IV-B).

A. Weak lensing inversion

1) Local inversion: We first consider the local inver-sions that have the advantage to address two differentproblems : the problem of missing data and the finitesize of the field. A relation between the gradient ofK = log(1 − κ) and combinations of first derivativesof g = γ

1−κ have been derived by [9] :

∇K ≡ u−1

1− |g|2

(1− g1 −g2

−g2 1 + g1

)(g1,1 + g2,2

g2,1 − g1,2

)≡ u (14)

This equation can be solved by line integration and thereexists an infinite number of local inverse formulae whichare exact for ideal data on a finite-size field. But theydiffer in their sensitivity to observational effects such asnoise. The reason why different schemes yield differentresults can be seen by noting that the vector field u (theright-hand side of equation 14) has a rotational compo-nent due to noise because it comes from observationalestimates.

In [12], the authors have split the vector field u into agradient part and a rotational part and they derive the bestformula that minimizes the sensitivity to observationaleffects by convolving the gradient part of the vector fieldu with a given kernel.

The local inversions reduce the unwanted boundaryeffects but whatever the formula is used, the

reconstructed field is more noisy than that obtainedwith a global inversion. Another drawback is that thereconstructed dark matter mass map has still a complexgeometry that will make the later analysis more difficult.

2) Global inversion: A global relation between κ andγ can be derived, from the relations (6) and (7). Indeed,it has been shown by [13] that the least square estimatorˆκn of the convergence κ in the Fourier domain is:

ˆκn(k1, k2) = P1(k1, k2)γ1n(k1, k2) +

P2(k1, k2)γ2n(k1, k2), (15)

where the hat symbols denotes Fourier transform and:

P1(k1, k2) =k2

1 − k22

k21 + k2

2

and P2(k1, k2) =2k1k2

k21 + k2

2

, (16)

with P1(k1, k2) ≡ 0 when k21 = k2

2 , and P2(k1, k2) ≡0 when k1 = 0 or k2 = 0. The most importantdrawback of this method is that it requires a convolutionof shears to be performed over the entire sky. As aresult, if the observed shear field has a finite size or acomplex geometry, then the method can produce artifactson the reconstructed convergence distribution near theboundaries of the observed field. A solution that has beenproposed by [15] to deal with missing data consists infilling-in judiciously the masked regions by performingan “inpainting” method simultaneously with a globalinversion. Inpainting techniques are an extrapolationof the missing information using some priors on thesolution. This new method uses a prior of sparsity inthe solution introduced by [16]. It assumes that thereexists a dictionary Φ (here the Discrete Cosine Trans-form) where the complete data are sparse and wherethe incomplete data are less sparse. The weak lensinginpainting problem consists of recovering a completeconvergence map κ from the incomplete measured shearfield γi. The solution is obtained by minimizing:

minκ‖ΦT κ‖0 subject to

∑i

‖ γi −M(Pi ∗ κ) ‖2≤ σ,

(17)where σ stands for the noise standard deviation and Mis the binary mask (i.e. Mi = 1 if we have informationat pixel i, Mi = 0 otherwise). This method enables toreconstruct a complete convergence map κ that can beused to do statistic estimation with a good accuracy (seesection V). A comparison with other probes of the matterdistribution can also be performed. This comparison isusually done after a filtering of the dark matter map (seesection IV-B) whose quality will be improved by theabsence of missing data.

LIGHT ON DARK MATTER WITH WEAK GRAVITATIONAL LENSING 7

B. Weak lensing filtering

The convergence map obtained by inversion of theshear field is very noisy even with a global inversion Thenoise comes from the shear measurement errors and theresidual intrinsic ellipticities present in the shear mapsthat propagate during the weak lensing inversion. Anefficient filtering is required to compare the dark matterdistribution with other probes.

1) Non-Bayesian methods:• Gaussian filter:

The standard method [13] consists in convolving thenoisy convergence map κ with a Gaussian windowG with standard deviation σG:

κG = G ∗ κn = G ∗ P1 ∗ γ1n + G ∗ P2 ∗ γ2n. (18)

The Gaussian filter is used to suppress the highfrequencies of the signal. However, a major problemis that the quality of the result depends strongly onthe value of σG that controls the level of smoothing.

• Wiener filter:An alternative to Gaussian filter is the Wiener filter([17], [18]) obtained by assigning the followingweight to each k-mode:

w(k) =|κ(k)|2

|κ(k)|2 + |N(k)|2. (19)

In theory, if the noise follows a Gaussiandistribution, the Wiener filtering provides theminimum variance estimator. However, it is notthe best approach, in particular on small scaleswhere non-linear features deviate significantlyfrom gaussianity. However, Wiener filter leads toreasonable results, generally better than the simpleGaussian filter.

2) Bayesian methods:• Bayesian filters

Some recent filters are based on the Bayesian the-ory that considers that some prior information canbe used to improve the solution. Bayesian filterssearch for a solution that maximizes the a posterioriprobability using the Bayes’ theorem :

P (κ|κn) =P (κn|κ)P (κ)

P (κn), (20)

where :– P (κn|κ) is the likelihood of obtaining the data

κn given a particular convergence distributionκ.

– P (κn) is the a priori probability of the dataκn. This terms, called evidence, is simply a

constant that ensures that the a posteriori prob-ability is correctly normalized.

– P (κ) is the a priori probability of the estimatedconvergence map κ. This term codifies ourexpectations about the convergence distributionbefore acquisition of the data κn.

– P (κ|κn) is called a posteriori probability.Searching for a solution that maximizes P (κ|κn)is the same that searching for a solution that mini-mizes the following quantity (Q) :

Q = − log(P (κ|κn)), (21)

Q = − log(P (κn|κ))− log(P (κ)). (22)

If the noise is uncorrelated and follows a Gaussiandistribution, the likelihood term P (κn|κ) can bewritten:

P (κn|κ) ∝ exp(−12χ2), (23)

with :

χ2 =∑x,y

(κn(x, y)− κ(x, y))2

σ2κn

. (24)

The equation (22) can be expressed as follows:

Q =12χ2 − log(P (κ)) =

12χ2 − βH, (25)

where β is a constant that can be seen as a parameterof regularization and H represents the prior that isadded to the solution.If we have no expectations about the convergencedistribution, the a priori probability P (κ) is uniformand the maximum a posteriori is equivalent to thewell-known maximum likelihood. This maximumlikelihood method has been used by [19], [20] toreconstruct the weak lensing field, but the solutionneeds to be regularized in some way to preventoverfitting the data. It has been done via the apriori probability of the convergence distribution.The choice of this prior is one of the most criticalaspects of the Bayesian analysis. An Entropic prioris frequently used but there exists many definitionsof the Entropy (see [21]). One that is currently usedis the Maximum Entropy Method (MEM) (see [22],[24]).Some authors ([19], [23]) have also suggested toreconstruct the gravitational potential Ψ insteadof the convergence distribution κ, still using aBayesian approach. But this is clearly better toreconstruct the mass distribution κ directly becauseit allows a more straightforward evaluation of theuncertainties in the reconstruction.

LIGHT ON DARK MATTER WITH WEAK GRAVITATIONAL LENSING 8

• Multiscale Bayesian filtersA multiscale maximum entropy prior has been pro-posed by [24] which uses the intrinsic correlationfunctions (ICF) with varying width. The multichan-nel MEM-ICF method consists in assuming thatthe visible-space image I is formed by a weightedsum of the visible-space image channels Ij , I =∑Nc

j=1 pjIj where Nc is the number of channels andIj is the result of the convolution between a hiddenimage hj with a low-pass filter Cj , called ICF(Intrinsic Correlation Function) (i.e. Ij = Cj ∗ oj).In practice, the ICF is a Gaussian. The MEM-ICFconstraint is:

HICF =Nc∑j=1

| oj | −mj− | oj | log

(| oj |mj

). (26)

Another approach, based on the sparse representa-tion of the data, has been used by [25] that consistsin replacing the standard Entropy prior by a waveletbased prior. Sparse representations of signals havereceived a considerable interest in recent year. Theproblem solved by the sparse representation is tosearch for the most compact representation of asignal in terms of linear combination of atoms inan overcomplete dictionary.The entropy is now defined as :

H(I) =J−1∑j=1

∑k,l

h(wj,k,l). (27)

In this approach, the information content of animage I is viewed as sum of information at differentscales wj . The function h defines the amount ofinformation relative to a given wavelet coefficient.Several functions have been proposed for h.In [26], the most appropriate entropy for the weaklensing reconstruction problem has been found to bethe NOISE-MSE entropy that presents a quadraticbehavior for small coefficients and is very close tothe l1 norm (i.e. absolute value of the wavelet coef-ficient) when the coefficient value is large, which isknown to produce good results for the analysis ofpiecewise smooth images. The proposed filter calledMRLens (Multi-Resolution for weak Lensing) hasshown to outperform other techniques (Gaussian,Wiener, MEM, MEM-ICF) in the reconstruction ofdark matter. It has been used to reconstruct thelargest weak lensing survey ever undertaken withthe Hubble Space Telescope. The result is shownFig. 6. This map is the most precise and detaileddark matter mass map, covering a large enough areato see extended filamentary structures.

Fig. 6. Map of the dark matter distribution in the 2-square degreeCOSMOS field by [27]: the linear blue scale shows the convergencefield κ, which is proportional to the projected mass along the line ofsight. Contours begin at 0.4 % and are spaced by 0.5% in κ.

V. COSMOLOGICAL MODEL CONSTRAINTS

Image distortion measurements of background galax-ies caused by large-scale structures provides a directway to study the statistical properties of the growth ofstructures in the Universe. Weak gravitational lensingmeasures the mass and can thus be directly compared totheoretical models of structure formation. But because,we have only one realization of our Universe, a statisticalanalysis is required to do the comparison. The estimationof the cosmological parameters from weak lensing datacan be seen as an inverse problem. The direct problemthat consists of deriving weak lensing data from cosmo-logical parameters can be solved using numerical simula-tions. But the inverse problem cannot be solved so easilybecause the N-body equations used by the numericalsimulations can not be inverted. A statistical analysis isthen required to constrain the cosmological parameters.The statistical characteristics of the weak lensing fieldcan be quantified using a variety of measures estimatedeither in the shear field or in the convergence field. Mostlensing studies do the statistical analysis in the shearfield to avoid the inversion. But most of the followingstatistics can also be estimated in the convergence fieldif the missing data are carefully accounted.

A. Second-order statistics

The most common method for constraining cosmo-logical parameters uses second-order statistics of theshear field calculated either in real or Fourier space (orSpherical Harmonic space).

The most popular Fourier space second-order statisticis the power spectrum Pγ because it can be easily

LIGHT ON DARK MATTER WITH WEAK GRAVITATIONAL LENSING 9

related to the theoretical 3D matter power spectrumP (k, χ) to estimate cosmological parameters. The cor-relation properties are more convenient in Fourier space,but for surveys with complicated geometry due to theremoval of bright stars, the spatial stationarity is notsatisfied and the missing data need proper handling.Consequently, real space statistics are easier to estimate,although statistical error bars are harder to estimate.An example of real space second-order statistic is theshear variance < γ2 >, defined as the variance of theaverage shear γ evaluated in circular patches of varyingradius θs. The shear variance < γ2 > can be relatedto the underlying 3D matter power spectrum via the 2Dconvergence power spectrum Pγ . This shear variance hasbeen used in many weak lensing analysis to constraincosmological parameters. Another real space statistic isthe shear two-point correlation function ξi,j(θ) that iscurrently used because it is easy to implement and canbe estimated even for complex geometry. It is defined asfollows :

ξi,j(θ) =< γi(~θ′)γj(~θ′ + ~θ) >, (28)

where i, j = 1, 2 and the averaging is done over pairsof galaxies separated by angle θ = |~θ|. By parityξ1,2 = ξ2,1 = 0 and by isotropy ξ1,1 and ξ2,2 arefunctions only of θ. The shear two-point correlationfunctions can also be related to the underlying 3Dmatter power spectrum via the 2D convergence powerspectrum Pγ . These two-point correlation functions arethe most popular statistical tools used in weak lensinganalysis. The variance of the aperture mass Map

[29] that corresponds to an average shear two-pointcorrelation has been also used in many weak lensinganalyses. This statistic is the result of the convolutionof the shear two-point correlation with a compensatedfilter. Several forms of filters have been suggested whichtrade locality in real space with locality in Fourier space.

Second-order statistics measure the Gaussian proper-ties of the field. This is a limited amount of informationsince it is known that the low redshift Universe is highlynon-Gaussian on small scales. Indeed, gravitational clus-tering is a non linear process and in particular at smallscales the mass distribution is highly non-Gaussian.Consequently, if only second-order statistics are used toset constraints on the cosmological model, degenerateconstraints are obtained between some important cos-mological parameters.

B. Higher-order statistics

An alternative procedure is to consider higher-orderstatistics of the weak lensing shear field enabling a

characterization of the non-Gaussian signal.The three-point correlation function ξi,j,k is the

lowest-order statistics which can be used to detect non-Gaussianity.

ξi,j,k(θ) =< γ(~θ1)γ(~θ2)γ(~θ3) > . (29)

In Fourier space it is called bispectrum and onlydepends on distances |~l1|, |~l2| and |~l3|:

B(|~l1|, |~l2|, |~l3|) ∝ < γ(|~l1|)γ(|~l2|)γ∗(|~l3|) > .(30)

It has been shown that tighter constraints can beobtained with the three-point correlation function [30].

A simpler quantity than the three-point correlationfunction is provided by measuring the third-order mo-ment (skewness) of the convergence κ that measures theasymmetry of the distribution. The convergence skew-ness is primarily due to rare and massive dark matterhalos. The distribution will be more or less skewedpositively depending on the abundance of rare andmassive halos. We can also estimate the fourth-ordermoment (kurtosis) of the convergence that measures thepeakedness of a distribution. A high kurtosis distributionhas a sharper “peak” and flatter “tails”, while a lowkurtosis distribution has a more rounded peak.

C. Other non-Gaussian statistics

The weak lensing field is highly non-Gaussian: onsmall scales, we can observe structures like galaxies,groups and clusters and on larger scales, we observesome filament structures. Another approach to lookfor non-Gaussianity is to perform a statistical analysisdirectly on the non-Gaussian structures present in theconvergence field. For example, the galaxy clusters thatare the largest virialized cosmological structures in theUniverse can provide a unique way to focus on non-Gaussianities present at small scales. One interestingstatistic is the peak counting that searches the numberof peaks detected on the convergence field correspondingroughly to the cluster abundance.

D. Statistical approach based on sparsity

It has been proposed by [14] to do a statistical analysisbased on the sparse representation of the weak lens-ing data. Several representations have been compared:Fourier, wavelet, ridgelet and curvelet representations.The comparison shows that the wavelet transform is themost sensitive to non-Gaussian cosmological structures.Indeed, by minimizing the number of large coefficient,the wavelet transform makes statistics be more sensitiveto the non-Gaussianities present in the weak lensingfield. In the same paper, several non-Gaussian statistics

LIGHT ON DARK MATTER WITH WEAK GRAVITATIONAL LENSING 10

have been compared and the peak counting estimated in awavelet representation, called Wavelet Peak Counting,has been found to be the best non-Gaussian statistic toconstrain cosmological parameters.

VI. CONCLUSION

The weak gravitational lensing effect that is directlysensitive to the gravitational potential provides a uniquemethod to map the dark matter. This can be used toset tighter constraints on cosmological models and tounderstand better the nature of dark matter and dark en-ergy. But the constraints derived from this weak lensingeffect depend on the techniques used to analyze the weaklensing signal which is very weak.

The field of weak gravitational lensing has recentlyseen great success in mapping the distribution of darkmatter (Fig. 6). But new methods are now necessary toreach the accuracy required by future wide-field surveysand ongoing efforts are done to improve the currentanalyses. This paper attempt to give an overview of thetechniques of signal processing that are currently usedto analysis the weak lensing signal along with futuredirections. It shows that the weak lensing is a dynamicresearch area in constant progress.

In this paper, we have detailed the different steps ofthe weak lensing data analysis thus presenting variousaspects of signal processing. For each problem, we havesystematically presented a range of methods currentlyused from earliest to up-to-date methods. This papershows that a milestone in weak lensing data analysisprogress has been the introduction of Bayesian ideasthat have provided a way to incorporate prior knowledgein data analysis. The next one could possibly be theintroduction of sparsity. Indeed, we have presented newmethods based on sparse representations of the data thathave already had some success.

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