Cosmological model discrimination from Weak Lensing

S. Pires, J.-L. Starck, A. Amara, A. RefregierDAPNIA/SEDI-SAP, Service d’Astrophysique, CEA-Saclay, F-91191 Gif-sur-Yvette Cedex, France

Abstract

Recent observations indicate that the Universe follows a Λ-CDM model. The comparison be-tween simulations and observations permits a much deeper understanding of the Large ScaleStructures (LSS) formation process and gives a unique tool for constraining theoretical modelsand associated parameters. If we use weak lensing data to constrain the cosmological model,using a classical two-point statistics, there is a degeneracy between two main parameters : σ8

and Ωm. The goal of this paper is to find a new approach that breaks this degeneracy. In thiscontext, we have analyzed realistic simulations of weak lensing mass maps for five cosmologicalmodels along this degeneracy. Then, we have tested several statistical tools based on severallinear transforms. We found that a wavelet denoising followed by a peak counting technique isthe most powerful to discriminate between different cosmological models.

Key words: Cosmology : Weak Lensing, Methods : Statistics, Data Analysis

1. Introduction

Visible matter represents less than 5% of the total matter of the Universe. The Uni-verse is composed essentially of invisible matter called dark matter. The gravitationaldeflection of light generated by mass concentrations (visible or not) along light pathsinduces magnification, multiplication and distortion of background galaxies. When thedistortions are not important this effect is called weak gravitational lensing. The proper-ties and the interpretation of this effect depend on the projected mass density integratedalong the line of sight and on the cosmological angular distance between the observer,the lens and the source (see Fig. 1). Today the weak gravitational lensing is widely usedto map dark matter of the Universe. Unlike other methods that probe the distribution oflight, weak lensing that measures the mass distribution can be directly used to constrain

Email address: sandrine.pires@cea.fr (S. Pires).

Preprint submitted to Elsevier 15 December 2006

Fig. 1. Gravitational Lensing effect

theoretical models of structure formation. Indeed, the measurement of distortions thatlensing induces in the images of background galaxies enables a direct measurement ofthe large-scale structures. An essential aspect of weak lensing is that measurements of itseffects are statistical. Indeed, galaxies have a shape of their own, and the change in theshape of an individual galaxy caused by weak lensing is too small to be useful. Generally,galaxies that are closen on the sky experience similar deflections. We can therefore av-erage over the shapes of many galaxies. Since the orientation of the intrinsic ellipticitiesof the galaxies is random, each observed image ellipticity provides an unbiased estimateof the lensing shear. Then, we can easily derive the mass that induced this deflection.

The Λ-CDM model is commonly accepted by many astrophysicists to describe thestructure formation and the distribution of the matter in the Universe. Ongoing effortsare made in the assessment of its parameters. The two-point correlation function is animportant measure of structure in the Universe. It is usually used in weak lensing forparameter extraction and is defined by : Rj =

∑

n(κn)(κ∗n−j), where κ is the density field.The Wiener-Khinchin theorem relates the autocorrelation (i.e. two-point correlation)function to the power spectral density via the Fourier transform. Accurate determinationof the two-point correlation function or of the power spectral density provides a testfor theoretical cosmological models. In particular, it can provide limits on dark matterdistribution, mean baryon density, and formation processes of galaxies and clusters. Ifthe two-point correlation function is used on weak lensing data, there is a degeneracybetween two main parameters, σ8 and Ωm. Ωm is the total density of the Universe andσ8 the amplitude of the fluctuations of the matter in a 8 Mpc radius. We can visualizethe (σ8,Ωm)-degeneracy in the space parameter Fig. 2 (upper left).

Many approaches can be investigated for parameter extraction. The two-point corre-lation function already has been studied [4,5]. Some groups applied the two-point cor-relation to the three-point correlation function on real mass maps [3]. But for the timebeing, no study has been done to compare the different statistics and to claim that onestatistic is better to discriminate between different models. In this paper, we investigatehow to characterize the non-gaussianities present in the mass maps due to the growthof structures using different statistical methods such as skewness and kurtosis of coeffi-cients obtained using different multiscale transforms : the ”a trous” wavelet transform[10], the bi-orthogonal wavelet transform [11], the ridgelet and the curvelet transform [8].We have also tested a wavelet denoising before parameter extraction in order to improvethe characterization.

Section 2 describes our simulated data. In section 3, we present the different statisticalmethods to be compared. In this same section, a description of the filtering method is alsoprovided. In section 4, we compare the different methods for discriminating cosmologicalmodels. Conclusions are given in section 5.

2

2. Weak lensing mass maps simulations

To bring to fruition this study, we have run realistic simulated mass maps derived fromray-tracing through N-body cosmological simulations using the RAMSES code [2]. Thecosmological model is taken to be in concordance with the Λ-CDM model and we havechosen 5 models along the (σ8,Ωm)-degeneracy with the following parameters :

Cosmological Model Box Ωm ΩL h σ8

model 1 200 0.23 0.7 0.7 1

model 2 200 0.3 0.7 0.7 0.9

model 3 200 0.36 0.7 0.7 0.8

model 4 200 0.47 0.7 0.7 0.7

model 5 200 0.64 0.7 0.7 0.6

Table 1Characteristics of the five different cosmological models chosen along the σ8-ΩM degeneracy

where the box side is in Mpc/h. The simulations have 1283 particles.

Fig. 2. Upper left, All the models along the degeneracy, upper middle, one realization of model 1(σ8 = 1and ΩM = 0.23), upper right, one realization of model 2 (σ8 = 0.9 and ΩM = 0.3), bottom left, onerealization of model 3 (σ8 = 0.8 and ΩM = 0.36), Bottom middle, one realization of model 4 (σ8 = 0.8and ΩM = 0.47), and bottom right one realization of model 5 (σ8 = 0.7 and ΩM = 0.64)

We have run 18 different N-body 3D simulations for each model using the RAMSEScode. In N-body simulations, density distribution are represented using discrete particles.The simplest way to convert such a particle distribution into a density is to divide thesimulation into pixels. Next, we have derived three 2D maps per simulation by projection

3

of the density fluctuation along the 3 axis. Indeed in weak lensing, the effective 2Dmass distribution κe is calculated by integrating the density fluctuation along the lineof sight. In this approach we neglect the facts that the light rays don’t follow straightlines following the Born approximation. The mass distribution can be expressed by :

κe ≈ 3H20ΩmL2c2

∑

iχi(χ0−χi)

χ0a(χi)

(

npR2

Nts2 − ∆rfi

)

where H0 is the Hubble constant, Ωm is the

density of matter, c is the speed of light, L is the length of the box χ are co-movingdistances, with χ0 being the co-moving distance to the source galaxies. The summationis performed over the box i. np is the number of particles associated with a pixel of thesimulation, Nt is the total number of particles within a simulation, s = Lp/L where Lp isthe length of the plane doing the lensing. R is the size of the 2D maps and ∆rfi = r2−r1

Lwhere r1 and r2 are co-moving distances.

We have 54 realizations for 5 models along the degeneracy. Fig. 2 shows a realizationper model. In all cases, the field is 4.74 x 4.74, downsampled to 8102 pixels and weassume that the sources lie at exactly z=1. The overdensities correspond to halos ofgroups and clusters of galaxies. The typical values of κ are thus of the order of a few %.

3. Cosmological model discrimination apparatus

In this study, in order to break the degeneracy, we look at the maximum discrepancybetween the different models. The cosmological model is labeled k and r is the realizationnumber. Considering several transforms, we note W i(Y k

r ) the transformation on therealization Y k

r by the transform W i and considering several statistical tools Sj .

discrepancymaxSj ,W i

5∑

k,k′=1

54∑

r=1

|Sj(W i(Y kr )) − Sj(W i(Y k′

r ))| (1)

We are interested in comparing the sensitivity of different transforms (W i) and theefficiency of different statistical tools (Sj) to characterize the non-gaussianity in each ofthe realization Y k

r .

3.1. Different basis functions

The weak lensing is composed of clumpy structures such as clusters and filamentarystructures such as walls. Consequently, the weak lensing maps exhibit both isotropic andanisotropic features. These kind of objects can be better represented in some basis func-tions. Indeed, a transform is optimal to detect structures which have the same shape of itsbasis elements. We have then tested different basis function : the Direct space, the Fourierspace, an anisotropic Wavelet space, an isotropic Wavelet space, the Ridgelet space andthe Curvelet space. Follows a brief description of the different multiscale transforms.

(i) The anisotropic bi-orthogonal wavelet transformThe most commonly used wavelet transform is the bi-Orthogonal Wavelet Trans-form (OWT). The two-dimensional algorithm leads to a wavelet transform with 3wavelet functions. At each scale there are three wavelet coefficient sub-images Agiven wavelet band is therefore defined by its resolution level j (j = 1...J) and itsdirection number d (d = 1...3 corresponding respectively to the horizontal, vertical

4

and diagonal band). The OWT of a 2D signal can be defined as :

S(kx, ky) =∑

lx,ly

CJ,lx,yφJ,kx,y

(lx, ly) +

3∑

d=1

∑

lx,ly

J∑

j=1

ψdj,kx,y

(lx, ly)wdj,lx,y

(2)

with J the number of resolutions used in the decomposition, wdj the wavelet coef-

ficients at scale j and sub-band d, CJ is a smooth version of the original imageS.The indexing is such that j = 1 corresponds to the finest scale. We expect theOWT to be optimal for detecting mildly isotropic or anisotropic features.

(ii) The isotropic ”a trous” wavelet transformThe Undecimated Isotropic Wavelet Transform decomposes an n× n image S as :

S(kx, ky) = CJ (kx, ky) +J

∑

j=1

wJ(kx, ky), (3)

where CJ is a smooth version of the original image S and wj represents the detailsin S at scale 2−j (see [10] for details). Thus, the algorithm outputs J + 1 sub-bandarrays of size n × n. This wavelet transform is well adapted to the detection ofisotropic features such as the clumpy structures (clusters) of the weak lensing data.

(iii) The Ridgelet transformThe classical multiresolution ideas only address a portion of the whole range ofinteresting phenomena: the roughly isotropic one at all scales and all locations. Theridgelet transform have been proposed as alternatives to wavelet representation ofimage data. Given a function f(x1, x2), the ridgelet transform is the superpositionof elements of the form a−1/2ψ((x1 cos θ+sin θ−b)/a), ψ is a wavelet, a > 0 a scaleparameter, b a location parameter and θ an orientation parameter. The ridgelet isconstant along lines x1 cos θ+x2 sin θ = const, and transverse to these ridges it is awavelet. The ridgelet transform has been developed [9] to process images includingridges elements, and so provides a representation of perfectly straight edges.

(iv) The Curvelet transformRidgelets are essentially focused on dealing with straight lines rather than curves,ridgelets can be adapted to representing objects with curved edges using an appro-priate multiscale localization. If one uses a sufficiently fine scale to capture curvededges, such edges are almost straight. As a consequence the Curvelet Transformhas been introduced, in which Ridgelet are used in a localized manner. The ideaof the Curvelet transform [8] is to first decompose the image into a set of waveletplanes, then to decompose each plane in several blocks (the block size can changeat each scale level) and to analyse each block with a ridgelet transform. The finestthe scale is, the more sensitive to the curvature the analysis is. As a consequence,curved singularities can be well approximated with very few coefficients.

3.2. A set of statistical tools

In a first step, different transforms (Fourier, ”a trous” Wavelet, Orthogonal Wavelet,...)have been used to reveal non-Gaussianity on our mass maps Y k

r . Let X1, X2, ..., Xn betheir transform coefficients. In a second step, we have tried a set of statistical tools on

5

these transform coefficients in order to find a new statistic to break the degeneracy.Knowing that the 2 point statistics lead to the (σ8,ΩM )-degeneracy and the field isclearly non-gaussian, we have chosen to use statistics of order 3 and 4 and we opted forstatistics currently used to detect non-gaussianity. We have selected the skewness, thekurtosis and the higher criticism. We have also tested the peak counting statistic becausethe formalism exist to predict the peak count to a given cosmological model. Here after,a brief description of these statistical tools is provided.

3.2.1. Non-gaussian estimators

We have selected the third and the fourth central moment of a distribution : theskewness and the kurtosis, which are currently used to measure the non-gaussianity. Theskewness is a measure of the asymmetry of a distribution and the kurtosis is a measureof the peakedness of a distribution.

Another non-gaussian estimator is the Higher Criticism (HC) [7]. HC is optimal todetect a tiny fraction of non-Gaussian data. It measures non-gaussianity by looking atthe maximum deviation comparing the sorted pvalues of a distribution with the sortedpvalues of a normal distribution. To define HC, we first convert the individual Xi into p-values. The probability value (p-value) measure the probability of a statistical hypothesistest. Let pi = PN(0, 1) > Xi be the ith p-value, and let p(i) denote the p-values sortedin increasing order. The HC is defined as:

HC∗

n = maxi

∣

∣

∣

∣

∣

√n[i/n− p(i)]

√

p(i)(1 − p(i))

∣

∣

∣

∣

∣

(4)

Or in a modified form :

HC+n = max

i:1/n6p(i)61−1/n

∣

∣

∣

∣

∣

√n[i/n− p(i)]

√

p(i)(1 − p(i))

∣

∣

∣

∣

∣

(5)

3.2.2. Peak counting estimator

Peaks in the weak lensing mass maps are galaxy clusters. Dark energy influences thenumber and the distribution of clusters and how they evolve with time. Consequently,peak counting provides an important measure of cluster distribution and evolution. Mo-tivated by the work of J. Berge et al, 2006 (in prep) who studied the peak countingstatistic in a theoritical framework, we have also tested this statistic. We have then usedthe SExtractor software [6] to extract the peaks of the weak lensing mass maps and wehave performed a cluster counting. First a global count per realization and second a countconsidering the size of the peaks to mimic a wavelet transform.

3.3. Filtering

The weak lensing mass maps are derived from the measurement of the deformations ofgalaxies background. We have added the expected experimental level of Gaussian noiseat our realistic simulations of weak lensing mass maps. As we can seen in Fig. 3 (middlepanel) the maps are very noisy. The noise scale-up the error in the measure of statistics.

6

Consequently, we need a very competitive filtering if we want to extract any informationof these mass maps.

Fig. 3. Left, Noiseless simulated mass map and middle, simulated mass map obtained in space observa-tions, right, filtered mass map by the FDR multiscale entropy

We used a method of non-linear filtering in wavelet domain : the Multi-scale Entropyfiltering [12]. This method uses a recent very competitive tool to set the threshold inan adaptive way called FDR (False Discovery Rate) and is based on the standard MEMprior (see [14]) derived from the wavelet decomposition of a signal.

3.3.1. Selecting significant Wavelet coefficients using the FDR:

The False Discovery Rate (FDR) is a recent statistic developed [13] to select a thresholdusing the data. The FDR procedure provides the means to adaptively control the fractionof false discoveries over total discoveries. This procedure guarantees control over the FDRin the sense that E(FDR) ≤ Ti

V .α ≤ α.

The unknown factor Ti

V is the proportion of truly inactive pixels. A complete descriptionof the FDR method can be found in [13]. In [12], it has been shown that the FDRoutperforms standard method for source detection. In this application, the FDR methodis used in a multiresolution framework (see [12]). We select a detection threshold Tj foreach scale. A wavelet coefficient wj(k, l) is considered significant if its absolute value islarger than Tj .

3.3.2. Multiscale Entropy filtering :

Assuming Gaussian noise, the Multiscale Entropy restoration method reduces to findthe image Sf that minimizes J(Sf ), given the map Sobs output of source separation with:

J(Sf ) =‖ Sobs − Sf ‖2

2σ2n

+ β

J∑

j=1

∑

k,l

hn((WSf )j,k,l) (6)

where σn is the noise standard deviation in Sobs, J is the number of Wavelet scales, Wis the Wavelet Transform operator and hn(wj,k,l) is the multiscale entropy only definedfor non significant coefficient (outside the support selected by the FDR thresholding).

7

3.3.3. Results :

It was shown in [12] that this method outperforms several standard techniques (Gaus-sian filtering, Wiener filtering, MEM filtering,...). Fig. 3, shows in the left panel theoriginal map without noise and in the right panel, the same mass map after the FDRmultiscale entropy filtering. The visual aspect indicates that many clusters are recon-structed and the intensity of the peaks are well recovered. Thanks to the FDR method,we controll the fraction of false detections over the total number of detections. Withoutfiltering no discrimination was possible whatever the statistic. We have then applied thisfiltering on all the weak lensing mass maps before any processes.

4. Analysis and results

4.1. Processing description

First, on each of the sets of filtered maps, we have run different transforms : the Fouriertransform, the ” trous” wavelet transform, the bi-orthogonal wavelet transform (7/9filter), the local ridgelet transform, the fast curvelet transform [1]. This processing hasbeen repeated on each of the 54 simulated mass maps per model. We have then calculatedon each individual resolution level different statistics (skewness, excess of kurtosis, HC∗

n

and HC+n ). Second, we have performed on each sets of filtered maps a global peak

counting (per map) and a peak counting considering the peak size (per map).The first point of the present study is to compare the sensitivity of each transform on

a set of simulated mass maps and the second one is to compare the discriminating powerof non-gaussian statistics to cluster counting statistic.

4.2. Discrimination in noisy and filtered mass maps

Each of the statistic measurement Sj on each resolution level l of a transform W i

lead to a distribution for each of the 5 models k : D54Sj(W i

l(Y k

NG))

. Each of the statistic

measurement applied on each resolution level l of a transform W i can be seen as anindividual statistic by itself. For example, the excess of kurtosis in the second scale ofthe ”a trous” wavelet transform is one of our statistics. Consequently, given a statistic,we obtained 5 distributions, each one corresponding to one of the 5 models.

Because of the tails of the distributions we can’t totally separate between two distribu-tions, given a statistic. For assessing discrimination between the different models, we needto define, for each distribution of a model k, a criterion threshold T above which the mapis classified not belonging to the model k. A simple test based on the ratio of maximumlikelihoods can not be applied because these distributions are not always Gaussian. Firstbecause we have a small number of realizations, and second because some statistics likeHC∗ don’t follow a Gaussian distribution. We have then used the statistical tool FDRto set a threshold T in an adaptive way. This procedure correspond to select a p-valuethreshold given a false detection ratio. Thus, for each transform, we have determined ap-value threshold per model among the l*54 p-values.

In order to find the discrimination power between the different models for a givenstatistic, we have first apply the FDR to select an adaptive threshold for each of the

8

5 distributions. Having a threshold Tk, the maps whose statistic measure is above thisthreshold are classified not belonging to this model k. Consequently, our discriminationpower is calculated looking at the percentage of maps belonging to a model k’ that areseparated that is to say whose statistic measure is above Tk. In Table 2, we can visualizethe discrimination power of the different statistical tools obtained with noisy maps usinga False Discovery rate equal to 0.05 to derive the thresholds.

Basis/Statistics Skewness Kurtosis HC∗

n HC+n Peak count

Direct space 48.9 25.4 4.07 28.7 3.14

Fourier space 3.0 3.1 4.6 3.0 x

”a trous” Wavelet Transform 52.4 49.2 31.3 47.6 4.07

Bi-orthogonal Wavelet Transform 48.7 37.6 25.6 33.0 x

Ridgelet Transform 30.9 31.5 23.5 30.6 x

Fast Curvelet Transform 6.8 41.3 26.5 39.4 x

Table 2Discrimination power achieved on noisy mass maps with α = 0.05 in FDR method

As we can seen in Table 2, without filtering no discrimination is possible. In this table,on the peak counting column, we have just presented the results obtained in direct andin an isotropic wavelet space because the extraction of peaks in the others space has nomeaning. We have then applied our filtering on the noisy simulated mass maps. Fig. 3shows the result obtained on the filtered mass maps.

Basis/Statistics Skewness Kurtosis HC∗

n HC+n Peak count

Direct space 48.9 38.1 38.0 22.0 80.2

Fourier space 0.9 11.5 4.8 31.1 x

”a trous” Wavelet Transform 75.9 66.7 67.4 18.3 76.5

Bi-orthogonal Wavelet Transform 48.1 77.6 62.8 63.0 x

Ridgelet Transform 44.4 35.5 40.4 46.1 x

Fast Curvelet Transform 5.6 53.9 47.2 67.0 x

Table 3Discrimination power achieved on filtered mass maps with α = 0.05 in FDR method

Among the different transforms tested the Bi-orthogonal Wavelet transform seems togive the best results. This is the one being optimal to detect the mildly anisotropicfeatures. The results obtain by the isotropic Wavelet Transform are quite good likewise.Among all the statistics the kurtosis on the Bi-Orthogonal Wavelet Transform gives goodresults. But the best results are achieved by the peak counting statistic in the direct spaceor in the isotropic wavelet space after wavelet denoising.

5. Conclusions and Discussions

We have used simulated weak lensing data with different statistics on different multi-scale transforms as a probe of dark matter distribution in order to constrain cosmologicalparameters. Our final goal is to try to break the degeneracy between σ8 and ΩM .

9

All the results must be balanced by the number of simulations, but it’s not really alimitation because we look for characterizing non-Gaussianity in a field that is clearlynon-Gaussian and the degeneracy that we try to break is important. In a future paper,we plan to have more simulations per model. Thus, we could check the quality of ourseparation using a part of the maps to test the completeness of our hypothesis test.

We have shown that in realistic simulated weak lensing data, no discrimination ispossible without filtering. We have applied a non-linear filtering in the wavelet domainusing the FDR method to set automatically thresholds at each resolution. This filteringhas been proven to outperform other methods (Gaussian filtering, Wiener filtering, MEMfiltering,...).

Among all the statistics based on multiscale transforms, kurtosis of the bi-orthogonalwavelet transform gives good results. Our study provides a strong motivation for the useof the peak counting after a wavelet filtering from weak lensing surveys for parameterextraction first, because of its simplicity and second, because the link with theory is clear.The results obtained by the peak counting could be besides improved considering the sizeof the peaks and theirs positions. It could also be interesting to test the bi-spectrum andthe tri-spectrum. This will be done in a future work.

References

[1] Candes, E. & Demanet, L. & Donoho, D. & Lexing, Y., Fast Discrete Curvelet Transforms, 2006, toappear in SIAM Multiscale Model. Simul.

[2] Teyssier, R., Cosmological hydrodynamics with adaptive mesh refinement. A new high resolutioncode called RAMSES, Apr 2002, AA, 385, 337-364

[3] Bernardeau, F. & van Waerbeke, L. & Mellier, Y., Patterns in the weak shear 3-point correlationfunction, Jan. 2003, AA, 397 405-414

[4] Bacon, D. J. & Massey, R. J. & Refregier, A. R. & Ellis, R. S., Joint cosmic shear measurements withthe Keck and William Herschel Telescopes, Sep 2003, MNRAS, 344, 673-685

[5] Massey, R. & Refregier, A. & Bacon, D. J. & Ellis, R. & Brown, M. L., An enlarged cosmic shearsurvey with the William Herschel Telescope, Jun 2005, MNRAS, 359, 1277-1286

[6] Emmanuel Bertin, 2003, http://terapix.iap.fr/[7] Donoho, D. & Jin, J., Higher criticism for detecting sparse heterogeneous mixtures, 2004, AS, 32,

962-994[8] E. Candes & D. Donoho, Curvelets: A Surprisingly Effective Nonadaptive Representation of Objects

with Edges, 1999[9] Candes, E & Donoho, D., ridgelets: a key to higher-dimensional intermittency, 1999, PTRS, 357,

2495-2509[10] J.-L. Starck & F. Murtagh, Astronomical Image and Data Analysis, 2002, Springer-Verlag[11] S. Mallat, A Wavelet Tour of Signal Processing, 1999, Academic press[12] Starck, J.-L. & Pires, S. & Refregier, A., Weak lensing mass reconstruction using wavelets”, Jun

2006, AA, 451, 1139-1150[13] Benjamini, Y. & Hochberg, Y, Controlling the false discovery rate - a practical and powerful approach

to multiple testing, 1995, J. R. Stat. Soc. B, 57, 289[14] Bridle, S. L. & Hobson, M. P. & Lasenby, A. N. & Saunders, R., A maximum-entropy method

for reconstructing the projected mass distribution of gravitational lenses, Sep 1998, MNRAS, 299,895-903

10