Cosmological constraints with weaklensing peak counts and secondorder statistics in a largefield survey
^{1} Laboratoire AIM, UMR CEACNRSParis 7, Irfu, Service d’Astrophysique, CEA Saclay, 91191 GifsurYvette, France
email: austin.peel@cea.fr
^{2} McWilliams Center for Cosmology, Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USA
^{3} Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK
^{4} Institut d’Astrophysique de Paris, CNRS UMR 7095 & UPMC, 98bis boulevard Arago, 75014 Paris, France
Received: 18 October 2016
Accepted: 15 December 2016
Peak statistics in weaklensing maps access the nonGaussian information contained in the largescale distribution of matter in the Universe. They are therefore a promising complementary probe to twopoint and higherorder statistics to constrain our cosmological models. Nextgeneration galaxy surveys, with their advanced optics and large areas, will measure the cosmic weaklensing signal with unprecedented precision. To prepare for these anticipated data sets, we assess the constraining power of peak counts in a simulated Euclidlike survey on the cosmological parameters Ω_{m}, σ_{8}, and w_{0}^{de}. In particular, we study how Camelus, a fast stochastic model for predicting peaks, can be applied to such large surveys. The algorithm avoids the need for timecostly Nbody simulations, and its stochastic approach provides full PDF information of observables. Considering peaks with a signaltonoise ratio ≥ 1, we measure the abundance histogram in a mock shear catalogue of approximately 5000 deg^{2} using a multiscale massmap filtering technique. We constrain the parameters of the mock survey using Camelus combined with approximate Bayesian computation, a robust likelihoodfree inference algorithm. Peak statistics yield a tight but significantly biased constraint in the σ_{8}–Ω_{m} plane, as measured by the width ΔΣ_{8} of the 1σ contour. We find Σ_{8} = σ_{8}(Ω_{m}/ 0.27)^{α} = 0.77_{0.05}^{+0.06} with α = 0.75 for a flat ΛCDM model. The strong bias indicates the need to better understand and control the model systematics before applying it to a real survey of this size or larger. We perform a calibration of the model and compare results to those from the twopoint correlation functions ξ_{±} measured on the same field. We calibrate the ξ_{±} result as well, since its contours are also biased, although not as severely as for peaks. In this case, we find for peaks Σ_{8} = 0.76_{0.03}^{+0.02} with α = 0.65, while for the combined ξ_{+} and ξ_{−} statistics the values are Σ_{8} = 0.76_{0.01}^{+0.02} and α = 0.70. We conclude that the constraining power can therefore be comparable between the two weaklensing observables in largefield surveys. Furthermore, the tilt in the σ_{8}–Ω_{m} degeneracy direction for peaks with respect to that of ξ_{±} suggests that a combined analysis would yield tighter constraints than either measure alone. As expected, w_{0}^{de} cannot be well constrained without a tomographic analysis, but its degeneracy directions with the other two varied parameters are still clear for both peaks and ξ_{±}.
Key words: gravitational lensing: weak / largescale structure of Universe / cosmological parameters / methods: statistical
© ESO, 2017
1. Introduction
The observed statistical distribution of matter in the Universe serves as a powerful discriminator of cosmological models. Different relative contributions to the Universe’s massenergy content produce different expansion histories and different amplitudes of matter clustering. Weak gravitational lensing (WL), through its ability to probe the largescale structure, has become a primary tool to study the total matter distribution in the Universe, as well as the nature of dark matter and dark energy.
The WL signal consists of tiny coherent distortions of galaxy shapes whose light rays have been bent by gravitational fields of dense matter structures lying along the line of sight. In contrast to the dramatic distortions seen in stronglensing systems, the images of weakly lensed galaxies are sheared and magnified only at the percent level relative to their original shapes. Weak lensing is therefore an inherently statistical probe that, given a sufficient density of background sources, provides a means to map the projected matter distribution across the sky. Furthermore, such mass maps trace the total matter distribution in an unbiased way, as WL is insensitive to the dynamical relationship between dark matter halos and the luminous galaxies that occupy them.
Numerous analyses have now constrained cosmological parameters using WL data from both ground and spacebased galaxy surveys. Recent studies include the Hubble Space Telescope Cosmic Evolution Survey (COSMOS; Schrabback et al. 2010), the CanadaFranceHawaii Telescope Lensing Survey (CFHTLenS; Heymans et al. 2012, 2013; Kilbinger et al. 2013), the Sloan Digital Sky Survey (SDSS; Huff et al. 2014), the Dark Energy Survey (DES; Abbott et al. 2016b,a), and the Kilo Degree Survey (KiDS; Kuijken et al. 2015; Hildebrandt et al. 2017). Future surveys like Euclid (Laureijs et al. 2011) and the Large Synoptic Survey Telescope (LSST; LSST Science Collaboration 2009) will afford unprecedented precision in WL measurements with their large survey areas and advanced optics. Cosmic shear studies planned for missions with a deeper but smaller field of view include the Hyper SuprimeCam of the Subaru Telescope (HSC; Miyazaki et al. 2012) and the WideField Infrared Survey TelescopeAstrophysics Focused Telescope Assets (WFIRSTAFTA; Spergel et al. 2013).
The basic WL measurements are the shear twopoint correlation function (2PCF) and its associated power spectrum. These measures capture the secondorder statistics, but as the shear distribution is not merely Gaussian, they neglect the nonGaussian information encoded by the nonlinear formation of structure. Studying higherorder statistics of the shear and convergence fields has therefore also become common. Examples of these include threepoint correlation functions (Semboloni et al. 2011; Fu et al. 2014), as well as nth order moments of the convergence field and Minkowski functionals (Petri et al. 2013, 2015). The statistics of peaks, which are local maxima in WL maps, provide another way to access the nonGaussian part of the signal.
Large signaltonoise ratio (S/N) peaks trace highmass regions of the Universe and can often be associated with massive galaxy clusters. High peaks therefore provide a direct and robust probe of the halo mass function. However, the origin of a WL peak does not typically admit of a unique interpretation. Low S/N peaks can arise from projection effects of the largescale structure, or they can simply be spurious noise fluctuations. Studies aimed at detecting massive galaxy clusters with WL thus focus on high peaks with S/N higher than ~4, whereas cosmological studies include the low peaks, since they also contain significant cosmological information (Yang et al. 2013; Lin et al. 2016).
Jain & Van Waerbeke (2000) initiated the study of WL peaks in their own right as a cosmological probe, using simulations to show that peaks can discriminate models with different total matter density parameters. Since then, many other papers have explored the efficacy of peaks for constraining both standard and nonstandard cosmological models (Marian et al. 2009, 2011, 2012, 2013; Dietrich & Hartlap 2010; Maturi et al. 2010, 2011; Pires et al. 2012; Cardone et al. 2013; Lin & Kilbinger 2015a,b; Lin et al. 2016). Regarding recent surveys, peak analyses have also been used to derive constraints on (σ_{8},Ω_{m}) with CFHTLenS data (Liu et al. 2015a), CFHT Stripe82 data (Liu et al. 2015b), and DES Science Verification data (Kacprzak et al. 2016). Euclid will observe approximately 15 000 deg^{2} of the extragalactic sky, an area about two orders of magnitude larger than CFHTLenS and the currently available DES SV data. It is therefore important to study not only the statistical improvements that will be afforded by such a large survey, but also to anticipate potential challenges regarding systematics and biases in our modelling of observations.
Our goal in this paper is to assess the ability of peak counts, modelled by the fast stochastic algorithm Camelus, to constrain cosmological parameters in a largearea survey. Camelus was introduced in Lin & Kilbinger (2015a) and studied further in the context of parameter constraint strategies and massmap filtering methods in Lin & Kilbinger (2015b) and Lin et al. (2016), respectively. See also Zorrilla Matilla et al. (2016) for a recent comparison of Camelus predictions to Nbody simulations over a broad range of cosmologies. The code uses a forwardmodel approach to generate lensing catalogues in a way that avoids timecostly Nbody simulations and only requires a halo mass function and halo profile as input.
In this work, we apply the Camelus model to mock lensing data of area ~5000 deg^{2}. Implementing a waveletbased massmap filtering technique, we compute peak histograms of the mock catalogue as a function of S/N and filtering scale. The multiscale approach allows us to build a peaks summary statistic, which separates out the cosmological information contained at different scales. Using Camelus with approximate Bayesian computation (ABC) for inference, we derive credible contours in σ_{8}–Ω_{m} space and compute the derived parameter Σ_{8} = σ_{8}(Ω_{m}/ 0.27)^{α} for the mock survey. To compare with probes of Gaussian information, we compare contours and the uncertainty in Σ_{8} from peaks to that of the two components of the 2PCF ξ_{+} and ξ_{−} of the shear field.
The remainder of the paper is organized as follows. In Sect. 2 we present the basic theory of weak gravitational lensing relevant to this work. In Sect. 3 we describe our method, including our multiscale massmapping technique, the simulated galaxy catalogue used as observations, the Camelus model, and finally our parameter inferences using ABC. Parameter constraint contours, as well as the comparison between peaks and ξ_{±}, are presented in Sect. 4. We conclude in Sect. 5.
2. Theoretical background
2.1. Weak gravitational lensing
Throughout this work, we assume that the Universe can be described by a weakly perturbed FLRW model with Newtonian potential Φ. Local deviations from the average matter density are characterized by the density contrast, (1)where coordinates (t,θ,w) represent cosmic time, angular position, and comoving radial distance, respectively. Density perturbations are related to Φ via Poisson’s equation for a pressureless ideal fluid, (2)which can be written more conveniently in terms of cosmological parameters as (3)where H_{0} is the present Hubble parameter, Ω_{m} is the present matter density, and a(t) is the scale factor of the Universe.
The images of distant galaxies undergo distortions as their light propagates through regions of nonuniform potential on their way to us. The original unlensed positions β of light beams undergo a remapping to the positions θ at which we observe them, which we can quantify by the Jacobian matrix of the transformation. It can be shown that to linear order in Φ, is determined by a lineofsight integral of secondorder transverse derivatives of Φ; see Schneider et al. (1998), for example. This motivates the definition of the lensing potential (4)and the decomposition of in terms of the convergence κ(θ) and shear γ(θ) = γ_{1}(θ) + iγ_{2}(θ): (5)In the above, f_{K}(w) is the comoving angular diameter distance (6)and K is the curvature of space. Then κ and γ are expressible directly as secondorder derivatives of the lensing potential
where ∂_{i} denotes the partial derivative with respect to θ_{i}. In particular, we can now relate κ directly to the density fluctuations as(9)making the interpretation of convergence as the projected mass density more apparent.
The explicit time dependence of Φ in Eqs. (4) and (9) has been suppressed, since the integral over w′ is understood as being performed along our past light cone. Furthermore, we restrict our attention to the weaklensing regime of small deviations (i.e. where  κ ,  γ  ≪ 1), and the Born approximation permits us to integrate along the unperturbed light path.
Further details of weaklensing theory and formalism can be found in the reviews of Bartelmann & Schneider (2001), Hoekstra & Jain (2008), and Kilbinger (2015), for example.
2.2. Mass mapping
Mass maps refer to maps of the convergence κ, a quantity that directly represents the matter distribution projected along the line of sight. This is readily seen in the integral of Eq. (9). As they are scalar fields, mass maps are more convenient for many applications compared to the complex shear γ, as in, for example, crosscorrelating with other scalar fields like the galaxy distribution or cosmic microwave background (CMB) temperature. In this work, we use mass maps to facilitate the straightforward identification of weaklensing peaks.
In practice, we cannot directly measure κ or γ from a galaxy survey. What we observe instead are galaxy ellipticities, which are an additive combination of intrinsic ellipticity and the shear due to lensing. Assuming that the source galaxies are not intrinsically aligned, we can therefore average the observed ellipticities over many sources in a small region of the sky to estimate the shear signal γ. This approximation works in the weaklensing regime, since ϵ^{obs} ≈ ϵ^{int} + g, where g = γ/ (1−κ) is the reduced shear. The average of many observed ellipticities is thus an accurate measure of g ≈ γ.
The inversion method of Kaiser and Squires (Kaiser & Squires 1993) gives a parameterfree prescription for computing κ from γ via the Fourier transforms of Eqs. (7) and (8). The equations become (10)where and are the Fourier transforms of κ and γ, ℓ is the Fourier counterpart to angular position θ, denotes complex conjugation, and (11)Convergence can therefore be determined directly from measurements of galaxy shapes in the weaklensing regime, up to an additive constant. It is well known that the KaiserSquires inversion creates undesirable artefacts at the boundaries of finite fields. We address this issue by excluding a border of 8 pixels (=4 arcmin) around each 5 × 5 deg^{2} convergence map generated throughout our analysis.
2.3. Shear twopoint correlation functions
A standard approach to constraining cosmology with weaklensing analyses is to use Npoint correlation functions of the shear field. Primary among these are the twopoint functions ξ_{+}(θ) and ξ_{−}(θ). We present a brief overview of the ξ_{±} statistics, since we later compare their parameter constraint results to those of peak counts.
The shear γ at angular position θ can be decomposed into a tangential component γ_{t} and cross component γ_{×} with respect to a point at position θ_{0}. If the separation vector θ−θ_{0} has polar angle ϕ, then (12)The minus sign ensures that γ_{t}> 0 for tangential alignment with respect to the position θ_{0}, and γ_{t}< 0 for radial alignment.
The shear 2PCF ξ_{±} are defined as combinations of γ_{t} and γ_{×} as (MiraldaEscude 1991) (13)as a function of angular scale θ. Following Schneider et al. (2002), we can estimate ξ_{±} from pairs of measured galaxy ellipticities as (14)Here ϵ_{t} and ϵ_{×} are defined analogously as γ_{t} and γ_{×} in Eq. (12). Weights are denoted by w, and the indices i,j refer to galaxies at positions θ_{i} and θ_{j}. The summation is carried out over all galaxy pairs with  θ_{i}−θ_{j}  lying within an angular bin around θ.
3. Method
Our goal is to study the constraints on cosmological parameters attainable from peak count statistics in a widefield survey like Euclid. We describe in the following sections the simulated Euclidlike galaxy catalogue we have chosen for analysis, as well as the algorithm we use to simulate stochastic lensing catalogues as a function of cosmological parameters. We then describe the multiscale massmapping technique we use to construct peak abundance data vectors for both the mock catalogue and for those simulated by Camelus. Finally, we present the method of approximate Bayesian computation, which we use for parameter inference.
3.1. Mock observations
The Marenostrum Institut de Ciències de l’Espai Grand Challenge simulation (Fosalba et al. 2008, 2015a,b; Crocce et al. 2010, 2015; Carretero et al. 2015; Hoffmann et al. 2015) is a set of largevolume Nbody runs carried out at the Barcelona Supercomputing Center using the GADGET2 Nbody code (Springel 2005). It contains approximately 70 billion dark matter particles in a simulation box of comoving side length ~ 3 h^{1} Gpc, giving high mass resolution in a large and statistically independent (i.e. nonrepeated) volume. Using a technique combining halo occupation distribution (HOD) and halo abundance matching (HAM) to populate dark matter halos, a mock galaxy catalogue was generated with a light cone extending out to z = 1.4. The MICECATv2.0 catalogue^{1} (MICE hereafter) we have used in this work covers an area of about 5000 deg^{2}, a contiguous octant of the sky lying within 0° ≤ RA ≤ 90° and 0° ≤ Dec ≤ 90°, which is approximately onethird the size of what is expected for Euclid’s widefield survey. The catalogue contains lensing information for nearly 500 million magnitudelimited galaxies, with completeness depending on redshift and position. In the least complete areas, for example, the catalogue is complete to observation band H ≈ 24 (23) up to z ≈ 0.45 (1.4). In this work, we only make use of the galaxy positions, observed ellipticities, and true redshifts.
The MICE cosmology is a flat ΛCDM model with parameters Ω_{m} = 0.25, Ω_{Λ} = 0.75, Ω_{b} = 0.044, σ_{8} = 0.8, h = 0.7, and n_{s} = 0.95. We find that its galaxy redshift distribution is well approximated by the parameterized form(15)with bestfit parameters p = 0.88, q = 1.40, and z_{0} = 0.78 based on chisquare minimization. The normalization factor is determined by numerical integration. Figure 1 shows the agreement between n(z) and the histogram of true redshifts from MICE. We use this functional form as input to the Camelus algorithm in order to generate its simulated catalogues. With real Euclid data, we would instead use the estimated photoz distribution, as true galaxy redshifts are not known.
Fig. 1 Histogram of the true redshift distribution of the MICE mock catalogue. The solid (blue) line is the bestfit n(z) in the form of Eq. (15) with parameters (p,q,z_{0}) = (0.88,1.40,0.78). The normalization is computed numerically. 

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We assign intrinsic ellipticities to the MICE galaxies so that the distributions of the two components ϵ_{1} and ϵ_{2} match those observed in the COSMOS survey. These closely approximate Gaussian distributions with zero mean and a standard deviation of 0.3. We generate observed ellipticities, which are used for mass mapping, by the relations (Seitz & Schneider 1997) (16)where ϵ_{obs} and ϵ_{int} are the observed and intrinsic ellipticities, respectively, and the asterisk represents complex conjugation. Shear γ and convergence κ values that comprise g are provided in the MICE catalogue for each galaxy line of sight. Given that  g  > 1 for only a few of galaxies, the resulting standard deviations of the observed ellipticity components remain at 0.3.
The Camelus algorithm, which we describe in Sect. 3.2, produces shear catalogues of size 25 deg^{2} in Cartesian space. For consistency, we therefore also extract and analyze 25 deg^{2} patches from the full MICE catalogue. We aim for as many independent patches as possible to ensure the best statistical constraining power of the peak count information. To do this, we take the straightforward approach of dividing the MICE octant into strips along lines of constant declination (Dec) such that each strip spans at least 5 deg in Dec. We then divide each strip into rectangles along lines of constant right ascension (RA) such that after a gnomonic (flatsky) projection, we achieve the maximum number of 25 deg^{2} squares for the strip.
The transformation equations are (17)and (18)where (λ,φ) are the (RA, Dec) coordinates to be projected, (λ_{0},φ_{0}) is the projection center, and (x,y) are the tangent plane coordinates. The nonconformal gnomonic projection has the useful property of mapping great circles to straight lines, but it introduces shape and distance distortions in the tangent plane radially away from the origin. For the small size of the patches we consider, however, these effects are subpercent level. For example, the distortion error in both the area and the maximum angular separation within the 25 deg^{2} fields is <0.3%.
Because of the spherical geometry, the number of usable patches within each strip varies significantly with declination. The strip with its edge at the equator contains 17, and the strip nearest the north pole contains only one.
Figure 2 shows an example of the extraction and projection geometry for a patch centred on (RA,Dec) = (40.0,57.9) deg. In the upper panel, the shaded rectangular region is the area in RA/Dec space before projection. This area is projected about the origin, marked by a red X, into the region enclosed by the dashed line in the lower panel. The inner tan square in the tangent space represents the area we use for massmapping for this patch.
This cutting is clearly not optimal in the sense that some areas are ultimately excluded from our analysis, namely those between the dashed and solid lines of the lower panel. All of the galaxies in the shaded RA/Dec rectangle end up inside the dashed region, but only those that fall in the tan square are used. Despite the simple approach, we still achieve 186 × 25 = 4650 deg^{2} of total effective area for our analysis. This should be sufficient to guide our intuitions about the constraining power of peak statistics from a survey like Euclid.
Fig. 2 Example of our patch extraction and projection scheme for MICE. The upper panel shows the original extent of the patch, which lies within 34.9° ≤ RA ≤ 45.0°, 55.3° ≤ Dec ≤ 60.4°. This area is projected into the region enclosed by the dashed line in the lower panel of the tangent space. The bounded inner square represents the 25 deg^{2} area for which we generate the multiscale maps and compute peak histograms. 

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3.2. Modelling peak counts with fast simulations
In this section we briefly describe the Camelus algorithm that we use to generate shear catalogues from dark matter halo simulations. The code has been developed and tested in Lin & Kilbinger (2015a,b) and Lin et al. (2016), and the software is available on GitHub^{2}. We refer to those papers for a more complete description.
The first step is to sample halo masses from a mass function, which is taken to have the form (Jenkins et al. 2001) (19)The parameter σ(z,M), which serves as a proxy for mass, is defined as the standard deviation of the linear density field, which has been smoothed with a spherical tophat filter of radius R such that . Halos are selected with masses within the range 5 × 10^{12}h^{1}M_{⊙} ≤ M ≤ 10^{17}h^{1}M_{⊙}.
Next, the sampled halos are distributed randomly in an observation field of 5 × 5 deg^{2}. The radial mass distributions of each halo are taken to be that of a truncated NavarroFrenkWhite (NFW) profile (Navarro et al. 1996, 1997) (20)where ρ_{s} is the characteristic overdensity related to the halo’s concentration, r_{s} is the scale radius, r_{vir} is the virial radius, α_{NFW} is the inner slope, and Θ is the Heaviside step function. The scale radius can be written as r_{s} = r_{vir}/c, the ratio of the physical virial radius to the concentration parameter. Following Takada & Jain (2002), we assume a parameterized form of c as a function of redshift and mass, (21)Here, the pivot mass M_{⋆} is defined such that σ(M_{⋆}) equals the critical density for spherical collapse at z = 0. We fix the parameters as (c_{0},β) = (9,0.13), as we find these to give good agreement with the MICE data in terms peak count histograms across different filtering scales. These, as well as other parameter settings adopted in this work, are summarized in Table 1.
Parameter settings for Camelus.
The final step is to generate a simulated catalogue of reduced shear values for source galaxies in the field. Sources are distributed randomly according to Eq. (15) and assigned ellipticities from a Gaussian distribution with dispersion . Orientations are random, meaning that we neglect intrinsic alignment in this work. The projected lensing quantities are then calculated, which can be done analytically for an NFW profile; see Takada & Jain (2003a,b), for example. Intrinsic source ellipticities are combined with the computed reduced shear g to give the final simulated lensing catalogue. We make S/N maps from this catalogue and compute peak count statistics following the procedure described in Sect. 3.3.
We note here some important assumptions implicit in the Camelus approach to simulating weaklensing maps. The first is that peak number counts arise primarily due to bound matter, and not from the diffuse matter that makes up, for example, cosmic filaments. The second is that the spatial correlation of dark matter halos does not have a significant impact on peaks. The results of Lin & Kilbinger (2015a) support the reasonableness of these assumptions for relatively small surveys. The present work, however, reveals limitations of Camelus in the context of larger survey areas; see Fig. 4 below and the parameter constraint results in Sect. 4.
Fig. 3 Starlet decomposition of an example noiseless convergence map κ(θ). The images are 150 × 150 with pixels of size 0.5 arcmin. The original map is shown along with the wavelet coefficient maps { w_{j} } up to j_{max} = 4. The final smoothed map c_{5}, i.e., the lowpass filtered version of κ, is not shown. The transform clearly picks out features of κ at successively larger scales as j increases. 

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3.3. Multiscale wavelet filtering
Weaklensing maps are dominated by galaxy shape noise and must be filtered to access their signal. Numerous schemes have been employed to denoise WL maps, the most common of which is filtering with a Gaussian kernel. For the reasons described below, we choose to filter the mass maps in our analysis using the isotropic undecimated wavelet, or starlet, transform (Starck et al. 2007).
The starlet has many properties that make it useful in astrophysical image processing. First, isotropy makes it wellsuited to extract features from astrophysical data containing objects that are roughly round, such as stars, galaxies, and clusters. Next, it is a multiscale transform in which the information contained at different scales in an image is separated out simultaneously. The filter functions associated with the starlet transform are localized in real space, that is, they go to zero within a finite radius. The first wavelet function acts as a highpass filter, while the remaining wavelets act as bandpass filters for their respective scales, since they are localized in Fourier space as well. Finally, the wavelet functions are compensated, meaning they integrate to zero over their domains. This is beneficial, as it was shown by Lin et al. (2016) that in the context of peakcount analyses, compensated filters are better at capturing cosmological information than noncompensated filters like the Gaussian kernel. We note that the wavelet transform of a convergence map at a particular scale is also formally equivalent to aperturemass filtering by a corresponding compensated filter (Leonard et al. 2012).
Fig. 4 Comparison of peak abundance histograms between Camelus (red circles) and MICE (blue line) for 4650 deg^{2}. Error bars for Camelus represent the standard deviation of 500 realizations for the associated S/N bin. There is good agreement between the mock observations and the Camelus predictions across all wavelet filtering scales, especially for ν lower than ~5. The shaded regions represent bins where the relative different between MICE and Camelus is larger than 50%. We apply a conservative cut and omit these higherbias bins in our analysis. 

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The starlet transform of an N × N image I amounts to successive convolutions of the image with a set of discrete filters corresponding to different resolution scales j = 1,...,j_{max}. The result is a set of J = j_{max} + 1 maps { w_{1},...,w_{jmax},c_{J} }, each of size N × N, where the “detail” maps { w_{j} } represent I filtered at a scale of 2^{j} pixels. The final map c_{J} represents a smoothed version of I, and the original image is exactly recoverable from the decomposition . Further details, including explicit expressions for the discrete filter bank associated with the starlet, can be found in Starck et al. (2007).
As an illustration, we show the starlet transform of a convergence map of 150 × 150 pixels in Fig. 3. The original image is a noiseless map κ(θ) with pixels of size 0.5 arcmin, and the decomposition is shown for j_{max} = 4 wavelet scales increasing to the right. Brighter pixels indicate higher values. The final smoothed map c_{5}, which is the lowpass filtered version of κ, is not shown. Progressively lower frequency features are picked out from κ as j increases.
After filtering, the noise level is different at different scales, and these levels are related by the ratio of the filter norms. If we denote the wavelet convolution kernel at scale j by W_{j}, then the noise at this scale is (22)where   ·  _{2} denotes the ℓ_{2} norm, and σ_{ref} is the known noise level at a particular reference scale.
Instead of using the analytic expression corresponding to the input galaxy ellipticity dispersion, we estimate σ_{ref} from the data as follows. Let κ_{j} denote the map filtered at scale j (analogous to w_{j} above). At the finest wavelet scale, j = 1, the noise is dominant, and we expect the wavelet coefficients to follow the noise distribution. We therefore take σ_{ref} as the dispersion of κ_{1}, the smallest resolution of the convergence map decomposition.
The S/N at scale j of a given coefficient at location θ can then be written as (23)where we only consider positive values of κ_{j}. For all maps used in our cosmological analysis, we bin the galaxies into pixels of size 0.5 arcmin, resulting in maps that are 600 × 600 pixels. We transform κ with j_{max} = 5 wavelet scales and identify peaks in these maps as local maxima of ν, where a peak is simply a pixel with a higher ν value than its eight neighbouring pixels. Since our filtering smooths the maps on scales larger than the pixel size, and we also treat the model and data identically, identifying peaks this way should be robust in terms of the final cosmological constraints.
3.4. Choice of data vector
To compare the predictions of Camelus to MICE, and ultimately to constrain parameters, we need a summary statistic that encapsulates the peak information contained at different wavelet scales. For this, we choose peaks of ν ≥ 1 with bin boundaries defined by [1.0,1.2,...,4.8,5.0] ∪ [5.3,5.6,5.9, + ∞). That is, the bin spacing is Δν = 0.2 for ν ∈ [1.0,5.0] and Δν = 0.3 for ν ∈ [5.0,5.9].
Lin et al. (2016) have shown that keeping the multiscale peak histograms separate, rather than combining them into a single data vector, yields tighter constraints on cosmological parameters. Following this result, we adopt a summary statistic that is the concatenation of peakcount histograms at five consecutive wavelet scales. The scales are arranged in order of decreasing resolution, analogous to the four scales in the example of Fig. 3.
In Fig. 4 we show a comparison of the MICE peak histograms at each scale with the Camelus predictions for the 4650 deg^{2} field. MICE results are shown as solid blue lines, Camelus as red circles. The data points for both align with the left edges of their associated S/N bins. The Camelus data were obtained by averaging 500 realizations of the code with the MICE cosmology as input, and error bars represent the standard deviations of these runs.
The plots reveal overall good agreement between the mock observations and the Camelus predictions across all scales, especially for ν lower than ~5. The highest S/N bins exhibit the largest bias at each scale, and the shaded regions indicate bins where the relative difference between MICE and the Camelus prediction exceeds 50%. Scales 2 and 5, corresponding to angular filtering scales of 2 and 16 arcmin, respectively, provide the best agreement in the high S/N range. Furthermore, Camelus predicts systematically fewer peaks than MICE at the first two scales, while the reverse is true for scales 3 and 4. This suggests that there could be an optimal filtering scale lying between scales 2 and 3; however, Camelus underpredicts the MICE peaks again at scale 5 over most of the range, making the choice of an optimal scale ambiguous.
There are numerous competing factors that contribute to the difference between Camelus peak predictions and MICE. The first is related to differences in the halo mass functions between the model and the data. Halo sampling by Camelus agrees well with the Jenkins model it is based on (Lin & Kilbinger 2015a). On the other hand, the Jenkins model underpredicts the measured halo abundance of the MICE simulation substantially for M> 10^{14}h^{1}M_{⊙} (Crocce et al. 2010, 2015). The proportion of highmass halos in a given field area is therefore expected to be smaller for Camelus than for MICE. Furthermore, as most high S/N peaks are due to single massive halos (Yang et al. 2011), the peak counts in the high S/N bins should be larger for MICE than for Camelus. We see in Fig. 4 that this is only true for the first two wavelet scales.
Another reason for the differences is the lack of halo clustering in the Camelus model. Lin & Kilbinger (2015a) showed that the effect of randomizing the angular positions of halos is to reduce the peak count number density by between 10% and 50%. The reason is that decorrelating halos leads to less overlap between them on the sky and therefore to a decrease in the number of high peaks. This effect is offset, however, by the replacement of Nbody halos by spherical NFW profiles, as well as ignoring the contribution of unbound matter to the lensing signal. A third consideration is that the S/N of a halo’s WL signal will be maximized when the filter size approximately matches that of the structure, which depends on its specific parameters such as mass, concentration, and redshift.
Interpreting the precise nature of the differences between MICE and Camelus in Fig. 4 is therefore difficult, and we leave this question for a followup study. In any case, we seek to minimize the effect of the highbias shaded bins on our cosmological analysis, and so we exclude them from the peak abundance summary statistic. This gives a data vector for our ABC analysis that contains 93 total bins. This also has the effect of ensuring that the included bins all contain at least 10 peaks (horizontal dashed line), which is desirable for statistical purposes.
3.5. Parameter inference with approximate Bayesian computation
Approximate Bayesian computation (ABC) is an approach to constraining model parameters that avoids the evaluation of a likelihood function. With ABC in general, statistics of the observed data set are compared with the corresponding statistics derived from simulations assumed to model the process that generated the observations. For our purpose, the observed data set is the MICE catalogue, and the simulations are generated by Camelus. Parameter space is then probed by acceptreject sampling, giving a fast and accurate estimate of the true parameter posterior distributions.
Lin & Kilbinger (2015b) and Lin et al. (2016) showed that ABC is an efficient and successful parameter inference strategy for WL peak count analyses. The method has also been used recently in several other astrophysical and cosmological applications (Cameron & Pettitt 2012; Weyant et al. 2013; Robin et al. 2014; Killedar et al. 2015; Ishida et al. 2015; Akeret et al. 2015). We constrain the parameter set in this work, where is the equation of state parameter of dark energy. corresponds to a pure cosmological constant.
We implement a population Monte Carlo (PMC) ABC algorithm to iteratively converge on the posterior distribution of parameters. We refer to algorithm 1 and the surrounding text in Lin & Kilbinger (2015b) for further details of PMC ABC as we implement it here. The algorithm proceeds by the following steps.

Draw an initial set of samples (called particles) from the priordistributions of the parameters. We assume flat priors of [0.1,0.9] for Ω_{m}, [0.3,1.6] for σ_{8},and [−1.8,0] for .

For each particle, generate a data vector from one simulation of Camelus with its parameters set to those of the particle’s location in parameter space.

Compute the distance, defined as (24)between the proposed parameter set and the observed MICE data vectors. Here, x is the model prediction data vector, x^{obs} is the MICE data vector, and C is the covariance matrix. We note that because of the bias found from modelling (see Fig. 6), we also perform constraints with the calibrated data vector. In this case, we add to the MICE data vector the difference between it and the model prediction computed with the MICE input parameters. See further details in Sect. 4.1.

Discard the particles whose distance from MICE is larger than a prescribed tolerance ϵ.

Reduce ϵ and iterate the process until the particle system representing the posterior distribution converges.
Computing the distance requires the covariance matrix for the peak abundance data vector, which we estimate from 500 Camelus runs. We assume that C does not vary with cosmology and estimate it under . We compute an unbiased inverse covariance estimator with a correction factor of (n−p−2)/(n−1) = 0.81, where n is the number of realizations, and p the number of bins (Hartlap et al. 2007). Sellentin & Heavens (2016) have shown that although this scaling indeed debiases C^{1}, retaining a Gaussian likelihood is not correct. However, our high n value ensures that is a good approximation to what should properly be a modified multivariate tdistribution.
The associated correlation coefficients for our 93 S/N bins are shown in Fig. 5. Blocks delineated by dotted lines represent the correlation between the two corresponding wavelet scales. We see that S/N bins are not strongly correlated, and neither are different scales, reflecting the efficient multiscale separation of information by the starlet filter.
Fig. 5 Correlation coefficients for our multiscale peak abundance data vector, computed from 500 Camelus runs with . Dotted lines delineate blocks corresponding to different wavelet scales. The correlation is weak both among bins within a particular scale and across the different scales themselves. The latter reflects the efficient separation of multiscale information by the starlet filter. 

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4. Results
We present in this section the cosmological parameter constraints attained from peak counts modelled by Camelus with ABC on the MICE field. We compare results with those of the 2PCF statistics ξ_{±}.
4.1. Parameter constraints from peak counts
Fig. 6 Parameter constraints in the σ_{8}–Ω_{m} plane from peaks as modelled by Camelus. The 1σ and 2σ contours represent the 68.3% and 95.4% most probable regions, respectively, after marginalizing over . The full Ω_{m} prior range is shown, and the star marks the MICE input cosmology. The constraint is biased significantly toward high σ_{8} and low Ω_{m} along the degeneracy direction. 

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Fig. 7 Credible parameter contours from calibrated peak counts data vector for the three 2D combinations of . In each figure, the third parameter has been marginalized over, and the star indicates the MICE cosmology. The left plot of σ_{8}–Ω_{m} space shows a right constraint with 1σ width measuring ΔΣ_{8} = 0.05. The middle and right plots show that we cannot constrain by our analysis, but the degeneracy directions are still identifiable. 

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The results of an ABC analysis are not point estimates of maximum likelihood, but rather credible regions of parameter space. We show the 1σ and 2σ credible contours in the σ_{8}–Ω_{m} plane after 12 iterations of ABC in Fig. 6. The areas within these contours represent the 68.3% and 95.4% most probable regions, respectively, after marginalizing over . With the PMC iteration scheme, ABC converges asymptotically to the true posterior. In practice, the contours stabilize quickly, meaning they do not continue to shrink appreciably after only a few iterations. In this case, we stop after iteration 12, since the size and width of the contours are not significantly different compared to iteration 11.
Kernel density estimation (KDE) was used on the final ABC particle system to derive the contours. The expression for the multivariate KDE posterior is(25)where N = 800 is the number of particles, x^{k} is the kth sample, and W_{H} is the multivariate normal kernel with bandwidth H satisfying Silverman’s rule (Silverman 1986).
It is well known that WL alone does not well constrain σ_{8} and Ω_{m} separately due to the strong degeneracy between the parameters, and so should be combined with other cosmological probes with orthogonal contours. What we look at therefore is the thickness the contours in the direction of the degeneracy. This is frequently expressed by the derived quantity (26)along with ΔΣ_{8}, the 1σ uncertainty. The parameter α represents the bestfit slope in log space. We find with α = 0.75, giving a 1σ width of ΔΣ_{8} = 0.11.
It is clear that despite having excluded the highbias bins, the residual Camelus systematics cause a significant bias in the parameter estimations. The peak contours are shifted along the degeneracy line so that the biases on Ω_{m} and σ_{8} are large, while that of Σ_{8} is much reduced. Excluding the highbias bins also weakens the constraining power, thereby broadening the contours. Whereas Zorrilla Matilla et al. (2016) have found a good agreement between the Camelus algorithm and Nbody runs on small fields, this test suggests that the systematics of Camelus need to be carefully studied in order to achieve accurate results for largefield analyses.
There are many possible sources of bias for peak predictions with Camelus. Zorrilla Matilla et al. (2016) have already identified some concerning the fast halo modelling. These include the lack of halo clustering, inaccurate modelling of the halo concentration, and the use of NFW profiles. Kacprzak et al. (2016) have also argued that ignoring source clustering results is a boost factor to peak counts. See also the discussion in Sect. 3.4 above.
As we are still interested in the constraining power of peaks with largefield statistics, we explore this further by assuming that the Camelus model can be improved and/or calibrated to correct the bias. One way to do this would be to first include effects like halo clustering in the model, and then to tune the model parameters (e.g. of the mass function, NFW profile, or massconcentration relation) if necessary so that the peak abundance data vector produced by Camelus under the MICE cosmology matches the observation as closely as possible. A simpler variation, although less well physically motivated, would be to tune the model parameters without otherwise modifying the algorithm. Alternatively, we can calibrate directly at the level of the data vector, matching the peak count histograms of Camelus to MICE across all wavelet scales. We choose the latter approach here and leave a study of the former to a future publication.
In effect, calibrating in this way is equivalent to pretending that the Camelus prediction under the MICE cosmology already matches the MICE observation. We note that in a more sophisticated treatment, and indeed in calibrating Camelus to use on real data, we would need to use many simulations with different cosmologies and interpolate by assuming, for example, that the calibration varies smoothly in parameter space.
We show the contours after calibration for the three combinations of in Fig. 7. The three panels represent results after 13 iterations of ABC, and the star again indicates the MICE input cosmology. In the left plot are the credible contours in the σ_{8}–Ω_{m} plane (note the different axis ranges compared to Fig. 6). We find now with α = 0.65. The 1σ width is therefore reduced to ΔΣ_{8} = 0.05, which agrees with the expectation that WL is better able to distinguish σ_{8} for higher Ω_{m} values. The posterior distribution of Σ_{8} is shown as the blue curve in Fig. 8. While the value of Σ_{8} is not very meaningful in an absolute sense, it serves as a useful basis for comparing the tightness of constraints attainable from different methods on the same data. We use ΔΣ_{8} to compare the amount of cosmological information contained in peaks to Gaussian probes of the WL signal in the following section.
The middle and right plots of Fig. 7 show the parameter spaces. The contours reveal that we cannot place constraints on even with largefield surveys. This is consistent with the understanding that tomography is necessary to probe the latetime evolution of the Universe, for which plays an important role. Nevertheless, the degeneracy between and the other parameters is clear.
4.2. Comparison with ξ_{±}
We compute ξ_{±} as defined by Eq. (14) on the same MICE subfields that were used for the peaks analysis with the publicly available Athena^{3} 2D tree code. We take the data vector in this case to be the concatenation of ξ_{+} and ξ_{−} computed in 10 logspaced angular bins each. The covariance matrix is computed using the 186 extracted patches from MICE, which we take to be independent realizations of 25 deg^{2} lensing fields. That is, our estimator essentially measures the subsample covariances and then rescales the result for the full survey (Norberg et al. 2009; Friedrich et al. 2016).
Fig. 8 Probability density functions (PDFs) of the derived parameter Σ_{8} from the peak analysis and the twopoint correlation functions ξ_{±} (2PCF). For peaks, the mode is Σ_{8} = 0.76 with ΔΣ_{8} = 0.05, while for ξ_{±}, these are 0.76 and 0.03, respectively. The comparable values of ΔΣ_{8} indicate the similar constraining power of the two WL statistics in a largefield survey. 

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Estimating the covariance matrix in this way ignores correlations on scales comparable to the size of the patches, as well as smallerscale separations spanning the boundaries between patches. With the large number of patches and a maximum angular scale of 3 deg, we do not expect these caveats to significantly affect the results. It is likely, however, that we underestimate the covariance to some degree because the patches are not perfectly independent.
We show the correlation coefficients associated with ξ_{±} in Fig. 9. Bins 1–10 correspond to ξ_{+}, and bins 11–20 correspond to ξ_{−}. The angular bins are evenly spaced logarithmically between (θ_{min},θ_{max}) = (1,180) arcmin. There is strong correlation among the ξ_{+} bins and little to moderate correlation among the ξ_{−} bins and their cross correlations with ξ_{+}.
Fig. 9 Correlation coefficients for ξ_{±}. Bins 1–10 correspond to ξ_{+}, and 11–20 correspond to ξ_{−}. The covariance matrix of the full survey is computed as a rescaling of the subsample covariance of the 186 MICE patches. 

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As we do not have an analogous stochastic forwardmodel to predict ξ_{±} as we do for peak counts, we cannot use ABC for parameter inference. Instead, we use the population Monte Carlo software package CosmoPMC^{4} (Wraith et al. 2009; Kilbinger et al. 2010). It is a Bayesian algorithm that uses adaptiveimportance sampling to improve its estimation of the posterior distribution iteratively. To compute cosmology, the fitting formula used for the nonlinear matter power spectrum is the halofit model of Smith et al. (2003). We use KDE in the same way as for the ABC result to analyse the resulting posterior particle system.
Fig. 10 Credible parameter contours from ξ_{±} (calibrated) with CosmoPMC for the three combinations of . The width ΔΣ_{8} of the σ_{8}–Ω_{m} contour is comparable to, but in fact larger than the value from peaks, despite the distributions seen in Fig. 8. As with peaks, cannot be constrained without a tomographic analysis. The star indicates the MICE cosmology. 

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The credible parameter contours for the calibrated 2PCF are shown in Fig. 10, analogous to Fig. 7 for peaks. In the σ_{8}–Ω_{m} plane, the 1 and 2σ contours are wider but less elongated than the peaks result. The degeneracy direction is also slightly offset from that of peaks. The middle and right panels show that despite the increased statistical power of the large survey area, as with peaks, cannot be constrained without tomography.
The Σ_{8} posterior distribution for 2PCF is shown as the red curve in Fig. 8, with and α = 0.70. At first glance, it appears that 2PCF yields a tighter constraint on Σ_{8}. However, ΔΣ_{8} here does not reflect the real contour width from the two left panels of Figs. 7 and 10. In fact, the contour width from peaks is clearly smaller than from 2PCF. The discrepancy is due to the definition of Σ_{8}. Since it defines a powerlaw curve, it is a poor fit to the contour of Fig. 7, which is elongated and barely bent. If the synergy between WL and other probes is to be optimized to lift degeneracy, Σ_{8} could be misleading. We also point out that Zorrilla Matilla et al. (2016) have found that the peak count covariance of Camelus is underestimated. This means that the contours in the left panel of Fig. 7 and the distribution of Σ_{8} in Fig. 8 should both be wider.
In the literature, similar comparisons have been reported by Dietrich & Hartlap (2010) using Nbody simulations and by Liu et al. (2015a) using CFHTLenS data. In both studies, the authors found that peakonly constraints are tighter than 2PCFonly constraints. The reason that our study does not show the same tendency is due to the conservative cut of highbias bins (see Fig. 4), which also removes cosmological information. A similar problem exists for the choice of scale cutting for 2PCF. The real capacity to extract cosmological information depends strongly on the accuracy of modelling, especially at nonlinear scales. Our study indicates that the tilt between the respective degeneracy lines of the two observables can be significant. Peaks and 2PCF are therefore complementary, as their combination facilitates the breaking of parameter degeneracies.
5. Summary and conclusions
Peak count statistics in weaklensing maps provide an important probe of the largescale structure of the Universe. Peaks access the nonGaussian information in the weaklensing signal, which is not captured by twopoint statistics and the related power spectrum. The upcoming Euclid mission will survey nearly onethird of the extragalactic sky and measure galaxy shapes with unprecedented precision, making it ideal for weaklensing analyses.
To prepare for such nextgeneration data, we have studied the ability of peak counts to constrain the parameter set in a largearea simulation with Euclidlike settings. We used the 5000 deg^{2} MICECATv2.0 lensing catalogue as mock observations, which we divided into 186 square 25 deg^{2} patches under gnomonic projection. Based on our cutting of the MICE octant, we extracted 4650 deg^{2} of effective area for the analysis. A more sophisticated scheme could be explored to make maximum use of the data, perhaps by computing quantities directly on the sphere. The area was already large enough, however, to see the effect of the increased statistical power on parameter constraints that we can expect from future surveys.
We used the stochastic forwardmodel code package Camelus to predict peak abundances as a function of cosmology. The model is semianalytic, requiring only a halo profile and a halo mass function as input to generate a 25 deg^{2} lensing catalogue in a few seconds. We implemented a waveletbased multiscale massmap filtering scheme in order to take advantage of cosmological information encoded at different scales. We showed that the Camelus model provides good agreement with the MICE peak histograms across five smoothing scales for S/N values lower than ~5. On the other hand, for higher S/N values, we found an increasing systematic deviation from MICE with increasing S/N.
Applying a conservative cut to exclude the highest bias S/N bins, we used approximate Bayesian computation to constrain the parameters of the MICE mock survey with peaks. In the σ_{8}–Ω_{m} plane, we found tight 1 and 2σ contours oriented along the typical degeneracy direction seen in weaklensing analyses. We measured the derived parameter Σ_{8} = σ_{8}(Ω_{m}/ 0.27)^{α} to be with a bestfit powerlaw slope α = 0.75 for a flat ΛCDM model. The uncertainty ΔΣ_{8} is representative of the width of the 1σ contour.
Although we omitted the highbias S/N bins from the peaks data vector, the residual systematics in Camelus resulted in a large bias in the contours from the MICE input cosmology. The contours were shifted towards higher σ_{8} and lower Ω_{m} along the degeneracy direction. This indicates that the systematics of Camelus need to be carefully studied before the method can be applied to real largefield data. We expect contributing systematics to include the lack of halo clustering, inaccurate modelling of the halo concentration, and the limitations of NFW profiles. It has recently been shown that these are a minor issue for relatively small fields (Zorrilla Matilla et al. 2016), but the assumptions of Camelus clearly break down for largearea surveys.
To compare peak count results to those from Gaussian probes of WL, we computed the shear twopoint correlation functions (2PCF) ξ_{±} on the MICE field. Lacking a similar fast stochastic prediction algorithm for 2PCF as we have for peaks, we used the population Monte Carlo software CosmoPMC instead of ABC for parameter inference. For the comparison, we corrected for the Camelus bias by calibrating the peaks data vector. We also corrected for a smaller bias found in the 2PCF result before comparing it to peaks.
The comparison revealed comparable constraints in the σ_{8}–Ω_{m} plane for the two WL observables, as measured by the width of their 1σ contours. We found ΔΣ_{8} = 0.05 for peaks and ΔΣ_{8} = 0.03 for 2PCF, although the actual width of the contours was larger for 2PCF than for peaks. The reason for the apparent discrepancy stems from the definition of Σ_{8}, as neither set of contours was particularly well fit by a power law. Nevertheless, the constraints of the two observables followed different degeneracy directions, indicating the benefit of a combined analysis with the two probes. Neither peak counts nor 2PCF were able to constrain without tomography.
We leave a tomographic study of peaks with Camelus to future work. We also did not consider here certain realistic observational effects, such as intrinsic alignment and masks. A primary benefit of the Camelus approach lies in its ability to probe the true underlying PDF of observables. Its flexibility as a forward model will make it straightforward to include such effects after other systematics have been addressed.
Acknowledgments
This work is supported in part by Enhanced Eurotalents, a Marie SkłodowskaCurie Actions Programme cofunded by the European Commission and Commissariat à l’énergie atomique et aux énergies alternatives (CEA). The authors acknowledge the Euclid Collaboration, the European Space Agency, and the support of the Centre National d’Etudes Spatiales (CNES). This work is also funded by the DEDALE project, contract No. 665044, within the H2020 Framework Program of the European Commission. The MICE simulations have been developed at the MareNostrum supercomputer (BSCCNS) thanks to grants AECT200620011 through AECT201510013. Data products have been stored at the Port d’Informació Científica (PIC), and distributed through the CosmoHub webportal (cosmohub.pic.es). Funding for this project was partially provided by the Spanish Ministerio de Ciencia e Innovacion (MICINN), projects 200850I176, AYA200913936, AYA201239620, AYA201344327, ESP201348274, ESP201458384, ConsoliderIngenio CSD200700060, research project 2009SGR1398 from Generalitat de Catalunya, and the Ramon y Cajal MICINN program. Finally, the authors would like to thank Peter Schneider, Sandrine Pires, and Samuel Farrens for useful comments and discussions.
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All Tables
All Figures
Fig. 1 Histogram of the true redshift distribution of the MICE mock catalogue. The solid (blue) line is the bestfit n(z) in the form of Eq. (15) with parameters (p,q,z_{0}) = (0.88,1.40,0.78). The normalization is computed numerically. 

Open with DEXTER  
In the text 
Fig. 2 Example of our patch extraction and projection scheme for MICE. The upper panel shows the original extent of the patch, which lies within 34.9° ≤ RA ≤ 45.0°, 55.3° ≤ Dec ≤ 60.4°. This area is projected into the region enclosed by the dashed line in the lower panel of the tangent space. The bounded inner square represents the 25 deg^{2} area for which we generate the multiscale maps and compute peak histograms. 

Open with DEXTER  
In the text 
Fig. 3 Starlet decomposition of an example noiseless convergence map κ(θ). The images are 150 × 150 with pixels of size 0.5 arcmin. The original map is shown along with the wavelet coefficient maps { w_{j} } up to j_{max} = 4. The final smoothed map c_{5}, i.e., the lowpass filtered version of κ, is not shown. The transform clearly picks out features of κ at successively larger scales as j increases. 

Open with DEXTER  
In the text 
Fig. 4 Comparison of peak abundance histograms between Camelus (red circles) and MICE (blue line) for 4650 deg^{2}. Error bars for Camelus represent the standard deviation of 500 realizations for the associated S/N bin. There is good agreement between the mock observations and the Camelus predictions across all wavelet filtering scales, especially for ν lower than ~5. The shaded regions represent bins where the relative different between MICE and Camelus is larger than 50%. We apply a conservative cut and omit these higherbias bins in our analysis. 

Open with DEXTER  
In the text 
Fig. 5 Correlation coefficients for our multiscale peak abundance data vector, computed from 500 Camelus runs with . Dotted lines delineate blocks corresponding to different wavelet scales. The correlation is weak both among bins within a particular scale and across the different scales themselves. The latter reflects the efficient separation of multiscale information by the starlet filter. 

Open with DEXTER  
In the text 
Fig. 6 Parameter constraints in the σ_{8}–Ω_{m} plane from peaks as modelled by Camelus. The 1σ and 2σ contours represent the 68.3% and 95.4% most probable regions, respectively, after marginalizing over . The full Ω_{m} prior range is shown, and the star marks the MICE input cosmology. The constraint is biased significantly toward high σ_{8} and low Ω_{m} along the degeneracy direction. 

Open with DEXTER  
In the text 
Fig. 7 Credible parameter contours from calibrated peak counts data vector for the three 2D combinations of . In each figure, the third parameter has been marginalized over, and the star indicates the MICE cosmology. The left plot of σ_{8}–Ω_{m} space shows a right constraint with 1σ width measuring ΔΣ_{8} = 0.05. The middle and right plots show that we cannot constrain by our analysis, but the degeneracy directions are still identifiable. 

Open with DEXTER  
In the text 
Fig. 8 Probability density functions (PDFs) of the derived parameter Σ_{8} from the peak analysis and the twopoint correlation functions ξ_{±} (2PCF). For peaks, the mode is Σ_{8} = 0.76 with ΔΣ_{8} = 0.05, while for ξ_{±}, these are 0.76 and 0.03, respectively. The comparable values of ΔΣ_{8} indicate the similar constraining power of the two WL statistics in a largefield survey. 

Open with DEXTER  
In the text 
Fig. 9 Correlation coefficients for ξ_{±}. Bins 1–10 correspond to ξ_{+}, and 11–20 correspond to ξ_{−}. The covariance matrix of the full survey is computed as a rescaling of the subsample covariance of the 186 MICE patches. 

Open with DEXTER  
In the text 
Fig. 10 Credible parameter contours from ξ_{±} (calibrated) with CosmoPMC for the three combinations of . The width ΔΣ_{8} of the σ_{8}–Ω_{m} contour is comparable to, but in fact larger than the value from peaks, despite the distributions seen in Fig. 8. As with peaks, cannot be constrained without a tomographic analysis. The star indicates the MICE cosmology. 

Open with DEXTER  
In the text 