Modeling of weaklensing statistics
II. Configurationspace statistics
^{1} Institut de Physique Théorique, CEA Saclay, 91191 GifsurYvette, France
email: patrick.valageas@cea.fr
^{2} Department of Physics, Nagoya University, 4648602 Nagoya, Japan
^{3} Institute for the Physics and Mathematics of the Universe, University of Tokyo, Kashiwa, 2778568 Chiba, Japan
Received: 5 December 2011
Accepted: 30 January 2012
Aims. We investigate the performance of an analytic model of the 3D matter distribution, which combines perturbation theory with halo models, for weaklensing configurationspace statistics.
Methods. We compared our predictions for the weaklensing convergence twopoint and threepoint correlation functions with numerical simulations and fitting formulas proposed in previous works. We also considered the second and thirdorder moments of the smoothed convergence and of the aperturemass.
Results. As in our previous study of Fourierspace weaklensing statistics, we find that our model agrees better with simulations than previously published fitting formulas. Moreover, we recover the dependence on cosmology of these weaklensing statistics and we can describe multiscale moments. This approach allows us to obtain the quantitative relationship between these integrated weaklensing statistics and the various contributions to the underlying 3D density fluctuations, decomposed over perturbative, twohalo, or onehalo terms.
Key words: gravitational lensing: weak / largescale structure of Universe
© ESO, 2012
1. Introduction
The standard paradigm known as the ΛCDM cosmology includes dark components: dark matter and dark energy (Komatsu et al. 2011). Weak lensing of background galaxies by foreground largescale structures, the socalled “cosmic shear”, has been recognized as a potentially powerful tool for probing the distribution of dark matter as well as the nature of dark energy (Albrecht et al. 2006). Reports of significant detections of cosmic shear have been made by various groups (Bacon et al. 2000; Van Waerbeke et al. 2000; Wittman et al. 2000; Hamana et al. 2003; Jarvis et al. 2006; Semboloni et al. 2006; Fu et al. 2008; Schrabback et al. 2010).
By analyzing the cosmic shear data, one can directly measure the power spectrum of the matter density fluctuations on cosmological scales, which contain a wealth of cosmological information such as neutrino masses and dark energy equationofstate parameters (Jarvis et al. 2006; Semboloni et al. 2006; Ichiki et al. 2009; Schrabback et al. 2010). Consequently, it is a main goal of cosmology to infer and constrain these quantities from observations. To do this, a number of ambitious surveys are planned, such as the Hyper SuprimeCam Weak Lensing Survey (Miyazaki et al. 2006)^{1}, the Dark Energy Survey (DES)^{2}, the Large Synoptic Survey Telescope (LSST)^{3}, Euclid (Refregier et al. 2011)^{4}, and the WideField Infrared Survey Telescope (WFIRST)^{5}.
Most weaklensing information is contained in small angular scales and therefore weaklensing statistics are nonlinear and nonGaussian (Munshi et al. 2008; Takada & Jain 2009; Sato et al. 2009, 2011b). If we aim to exploit the full information, we have to treat the nonlinear effects to accurately model the weak lensing statistics. Furthermore, one has to use an appropriate likelihood function with given marginal distributions, otherwise the obtained results would be systematically biased (Sato et al. 2010, 2011a).
In a first companion paper (Valageas et al. 2012, hereafter Paper I), we studied the Fourierspace weaklensing statistics such as the weaklensing power spectrum and bispectrum, and found that our model proposed by Valageas & Nishimichi (2011a,b), which combines perturbation theory with halo models, agrees better with raytracing simulations than previously published fitting formulas and phenomenological models. In this second paper, we study the realspace weaklensing statistics, which are more often used for the statistical analysis of actual measurements than Fourierspace statistics, because observations are made in configuration space.
Previous works have already shown that on small scales the halo model provides a good description of the two, three and fourpoint correlations or smoothed moments of the cosmic shear (using some approximations) (Takada & Jain 2002, 2003; Benabed & Scoccimarro 2006), whereas a stochastic halo model can recover the probability distribution function of the unsmoothed convergence (Kainulainen & Marra 2011a,b). Here we include all “onehalo”, “twohalo” and “threehalo” terms, as well as oneloop perturbative results, and we compare these with largerscale simulations. This yields a greater accuracy and allows us to compare these different contributions, from very large to small scales. This should be useful for practical purpose because these different terms have different theoretical accuracies and probe different regimes of gravitational clustering, hence it is important to know their relative impact on weaklensing probes as a function of angular scale.
This paper is organized as follows. In Sect. 2 we briefly recall how configurationspace weaklensing statistics are computed from polyspectra of the 3D matter density field. We describe our numerical simulations and the data analysis in Sect. 3. Then, we present detailed comparisons between the simulation results, previous models, and our theoretical predictions for twopoint functions in Sect. 4 and threepoint functions in Sect. 5. We study the relative importance of the different contributions arising from “onehalo”, “twohalo”, or “threehalo” terms in Sect. 6. Then, we check the robustness of our model when we vary the cosmological parameters in Sect. 7 and we briefly study multiscale moments in Sect. 8. Finally, we conclude in Sect. 9.
2. From 3D statistics to weaklensing statistics
2.1. Lensing power spectrum and bispectrum
Using Born’s approximation, the weaklensing convergence κ(θ) can be written as the integral of the density contrast along the line of sight (Bartelmann & Schneider 2001; Munshi et al. 2008), (1)where χ and are the radial and angular comoving distances, (2)and z_{s} is the redshift of the source (in this article we only consider the case where all sources are located at a single redshift to simplify the comparisons with numerical simulations and the dependence on the source redshift). Then, using a flatsky approximation, which is valid for small angles below a few degrees (Valageas et al. 2011), we define its 2D Fourier transform through (3)As in Paper I, we define the 2D convergence power spectrum and bispectrum as (4)and (5)From Eq. (1) one obtains at once from Limber’s approximation (Limber 1953; Kaiser 1992; Bartelmann & Schneider 2001; Munshi et al. 2008) (6)(7)where P(k;z) and B(k_{1},k_{2},k_{3};z) are the 3D power spectrum and bispectrum of the matter density contrast at redshift z. As described in Paper I, this provides the weaklensing convergence power spectrum and bispectrum from the model we developed in Valageas & Nishimichi (2011a,b) for the 3D power spectrum and bispectrum through a simple integration over the radial coordinate up to the source plane.
2.2. Configurationspace statistics
We focus here on configurationspace weaklensing statistics, which may be more convenient than Fourierspace quantities for practical purposes because of complex survey geometries. Indeed, observations of the shear field are made in configuration space, by measuring largescale correlations of galaxy ellipticities, and going to Fourier space (with further operations that are not local in real space) can be difficult because galaxy surveys do not cover the whole sky and shear maps can show irregular boundaries and internal holes due to observational constraints.
In particular, we consider the twopoint and threepoint correlation functions of the weaklensing convergence, defined as (8)(9)where we used statistical homogeneity and isotropy (hence ζ_{κ} only depends on the lengths of the three sides of the triangle defined by the summits { θ_{1},θ_{2},θ_{3} } and the bispectrum (7) only depends on the three lengths { ℓ_{1},ℓ_{2},ℓ_{3} } ).
These realspace correlations can be expressed in terms of the Fourierspace power spectrum and bispectrum as (10)where θ = θ_{2} − θ_{1} is the pair angular distance as in Eq. (8), and (11)where { ν_{1},ν_{2},ν_{3} } are the lengths of the three sides of the triangle { θ_{1},θ_{2},θ_{3} } as in Eq. (9), α_{3} is the inner angle at summit θ_{3}, we chose without loss of generality (12)and we noted in Eq. (11) the multipoles (13)In addition to the threepoint correlation ζ_{κ} of the convergence, it can be useful to consider the threepoint correlation of the cosmic shear γ. Because the latter is a spin2 quantity, one is led to consider several threepoint shear correlations (or “natural components”), depending on the choice of the reference direction (or of the projection procedure of the cosmic shear vectors), see Schneider & Lombardi (2003); Schneider et al. (2005). However, they can all be written as integrals over the convergence bispectrum, such as Eq. (11), or integrals over the convergence threepoint correlation (9) (Shi et al. 2011). Therefore, although we only consider the convergence threepoint correlation (9) in this paper, we can expect a similar level of agreement between our analytical model and simulations for these other threepoint correlations (in addition we also consider the thirdorder moment of the aperturemass, which can be related to both the convergence and the cosmic shear).
It is also common practice to study smoothed averages X_{s} of the convergence or shear field, defined by their filtering window through (14)For instance, the smoothed convergence κ_{s} is defined by a tophat filtering, (15)while the “aperturemass” M_{ap} is defined by a compensated filter (Schneider 1996; Van Waerbeke et al. 2001), such as (16)and if θ > θ_{s}. This also reads in Fourier space as (17)with (18)In particular, we have (19)We mainly focus here on onepoint moments of κ_{s} and M_{ap}, that is , and we do not consider multipoint statistics such as ⟨ X_{s}(θ_{1};θ_{s1})...X_{s}(θ_{p};θ_{sp}) ⟩ associated with p windows centered on p different directions and with p different angular radii. However, we will check the validity of our model for multiscale statistics, that is, for windows of different sizes centered on the same direction, in Sect. 8.
In the simpler case of onepoint moments the variance reads as (20)while the thirdorder moment reads as (21)where we used the symmetries of the bispectrum and we noted (22)In practice, to avoid the numerous oscillations and changes of sign brought by the Fourierspace filters given in Eq. (19), we found it convenient to express the thirdorder moment (21) in terms of the realspace threepoint correlation (9), although this yields a fivedimensional integral instead of the threedimensional integral (21), (23)with
3. Numerical simulations
We performed the raytracing simulations through highresolution Nbody simulations of cosmological structure formation (Jain et al. 2000; Hamana et al. 2001; Sato et al. 2009; Takahashi et al. 2011) to obtain accurate predictions of the configuration statistics for weak lensing. To run the Nbody simulations, we used a modified version of the Gadget2 code (Springel 2005) and employed 256^{3} particles for each simulation. The raytracing simulations were constructed from 2 × 200 realizations of Nbody simulations with cubic 240 and 480 h^{1} Mpc on a side, respectively, to cover a light cone of angular size 5° × 5° (see Fig. 1 in Sato et al. 2009). For our fiducial cosmology, we adopted the standard ΛCDM cosmology with matter fraction Ω_{m} = 0.238, baryon fraction Ω_{b} = 0.0416, dark energy fraction Ω_{de} = 0.762 with the equation of state parameters w_{0} = −1 and w_{a} = 0, spectral index n_{s} = 0.958, normalization A_{s} = 2.35 × 10^{9}, and Hubble parameter h = 0.732, which are consistent with the WMAP threeyear results (Spergel et al. 2007). This fiducial cosmology gives the normalization σ_{8} = 0.759 for the variance of the linear density fluctuations in a sphere of radius 8 h^{1} Mpc. We considered source redshifts at either z_{s} = 0.6, 1.0, or 1.5. Using raytracing simulations we generated 1000 realizations of convergence maps for each source redshift.
In addition to the fiducial cosmology case, we performed raytracing simulations for several slightly different cosmologies. We varied each of the following cosmological parameters: A_{s}, n_{s}, the cold dark matter density Ω_{c}h^{2}, Ω_{de}, and w_{0} by ± 10%, respectively. Therefore h, Ω_{m}, and Ω_{b} are dependent parameters, because we assumed that the Universe is flat and the baryon density Ω_{b}h^{2} is fixed. For each of these ten different cosmologies, we obtain 40 realizations of convergence fields for each of the three source redshifts. Details of the methods used for the raytracing simulations can be found in Sato et al. (2009). All convergence maps used in this paper are the same as those used in Paper I.
In Sects. 4–6, we use the maps for the fiducial cosmology, while in Sect. 7 we use those for the varied cosmologies to investigate the robustness of our model. In Sect. 7 we show the results for six cases, varying n_{s}, Ω_{c}h^{2}, and w_{0} by ± 10%. The exact values for these cosmological parameters are listed in Table A.1 in Paper I.
4. Lensing twopoint functions
Fig. 1 Upper row: convergence twopoint correlation function for sources at redshifts z_{s} = 0.6,1, and 1.5, as a function of the angular pair separation θ. The points are the results from numerical simulations with 3 − σ error bars. The low black dashed line “L” is the linear correlation, the middle blue dashdotted line “S” is the result from the “halofit” of Smith et al. (2003), and the upper red solid line “comb.” is the result from our model, which combines oneloop perturbation theory with a halo model. The vertical arrow shows the scale down to which the simulation result is valid within 5%. Middle row: variance of the smoothed convergence for the same cases, as a function of the smoothing angle θ_{s}. Lower row: variance of the aperture mass for the same cases, as a function of the smoothing angle θ_{s}. 

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We now compare our results for weaklensing twopoint functions with numerical simulations. As in Paper I, we also considered the predictions obtained from the popular “halofit” fitting function for the 3D density power spectrum given in Smith et al. (2003), to estimate the advantages of more systematic approaches like ours.
We show our results for the convergence twopoint correlation ξ_{κ}(θ), the variance of the smoothed convergence , and the variance of the aperture mass in Fig. 1. The numerical error bars increase on large scales because of the finite size of the simulation box. On small scales the numerical error is dominated by systematic effects, because of the finite resolution, which leads to an underestimation of the smallscale power. This underestimation was clearly apparent for the power spectrum P_{κ}(ℓ) shown in Paper I and can also be seen (especially at low redshift) for the variance of the aperture mass, which involves a compensated filter and probes a narrow range of scales (Schneider 1996). For each source redshift we estimated the angular scale down to which the simulations have an accuracy of better than 5% by comparing with higherresolution simulations (with 512^{3} particles instead of 256^{3}). This scale is shown by the vertical arrow in Fig. 1 and we can check that our model indeed agrees with the numerical simulations down to this angular scale.
The twopoint correlation and the smoothed convergence are not as sensitive to this lowresolution effect because they involve uncompensated filters, , which implies that at a given smoothing angular scale θ_{s} they receive greater contributions from larger scales (which are unaffected by the numerical resolution) than the aperture mass.
For the same reason, ξ_{κ}(θ) and remain adequately described by linear theory down to ~10 arcmin, whereas already shows significant deviations at ~100 arcmin. This also explains why the predictions from our model and the “halofit” are closer for the first two statistics than for M_{ap}. In agreement with our results for the convergence power spectrum (Paper I) and previous works (White & Vale 2004; Hilbert et al. 2009; Sato et al. 2009; Semboloni et al. 2011), the “halofit” formula underestimates the power on small scales. The discrepancy is again larger for the aperture mass. As in Paper I, our model provides a more accurate match to the numerical simulations down to ~1 arcmin. On smaller scales the results from simulations show a fast drop (especially for M_{ap}) that is not physical but due to the finite resolution. This prevents an accurate comparison with our predictions. However, since our model is built from a physical halo model and has been tested for the 3D density field down to highly nonlinear scales with higherresolution simulations (Valageas & Nishimichi 2011a), it should be more reliable than the numerical results shown in Fig. 1 below ~1 arcmin. This shows one advantage of analytic (or semianalytic) approaches as compared with numerical simulations: they can provide realistic predictions on a wider range of scales.
5. Lensing threepoint functions
Fig. 2 Upper row: convergence threepoint correlation function for equilateral triangles, as a function of the triangle side ν, for sources at redshifts z_{s} = 0.6,1, and 1.5. The points are the results from numerical simulations with 3 − σ error bars. The low black dashed line “tree_{L}”, the black dotted line “tree_{NL}” and the two blue dashed lines “F_{2,NL}” are obtained from the ansatz (27), using either the linear 3D power spectrum, or the “halofit” power, or an effective kernel F_{2,NL} with the “halofit” power (lower curve) or the power from our model (upper curve). The red solid line “comb.” is our combined model, described in Paper I. The vertical arrows are at the same angular scale as in Fig. 1. Middle row: thirdorder moment of the smoothed convergence, as a function of the smoothing angle θ_{s}. Lower row: thirdorder moment of the aperture mass, as a function of the smoothing angle θ_{s}. 

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We now compare our analytical results with numerical simulations for threepoint functions. As in Paper I, we also considered the predictions obtained from the following three simple models, which have been used in some previous works. (We did not consider the scale transformation introduced in Pan et al. (2007) because we have already shown in Paper I that it does not provide a sufficiently accurate model for the convergence bispectrum. We have checked that we obtain similar results for and .)
The first model, “tree_{L}”, is the lowestorder (“treeorder”) prediction from standard perturbation theory, which reads for the 3D bispectrum (Bernardeau et al. 2002) (27)where μ_{12} = (k_{1}·k_{2})/(k_{1}k_{2}) and (28)The second model, “tree_{NL}”, is given by Eq. (27) where we replace the linear 3D power P_{L}(k) by the nonlinear power P_{S}(k) from Smith et al. (2003). The third model, “F_{2,NL}”, makes the additional modification to replace the kernel F_{2} by an effective kernel F_{2,NL} that interpolates from the largescale perturbative result (28) to a smallscale ansatz where the angular dependence on μ_{12} vanishes, using the fitting formula from Scoccimarro & Couchman (2001). We considered two variants, using either the “halofit” nonlinear power spectrum P_{S}(k) from Smith et al. (2003) or the nonlinear power spectrum P_{tang}(k) of our model, see Paper I and Valageas & Nishimichi (2011b).
We show our results for the convergence threepoint correlation function ζ_{κ} (for equilateral triangles) and for the thirdorder moments and in Fig. 2. In agreement with the secondorder statistics shown in Fig. 1, the lowestorder perturbation theory prediction, “tree_{L}”, remains valid down to smaller angular scales for ζ_{κ} and κ_{s} than for M_{ap} because the former involve uncompensated filters instead of a compensated filter.
The simulations slightly underestimate the power on moderate and large angular scales because of the finite size of the simulation box (240 h^{1} and 480 h^{1} Mpc, which corresponds to 343 and 686 arcmin at z = 1), which cuts contributions from longer wavelengths. This discrepancy is not due to the analytic model because on the largest angular scales we can check that all theoretical predictions converge on the linear theory (as they must for CDM power spectra) while predicting somewhat more power than measured in the simulations (see the first row in Fig. 2). Therefore, this mismatch is not caused by higherorder perturbative corrections (e.g., twoloop terms), which are even smaller than the oneloop contributions that we included in our model. In agreement with this explanation, the discrepancy is smaller for M_{ap} than for ζ_{κ} and κ because, for a given smoothing radius θ_{s}, M_{ap} is less sensitive to larger scales thanks to its compensated filter . This shows that analytical models, such as the one we propose here, are competitive with numerical simulations if one needs to describe a broad range of scales.
In agreement with the results obtained in Paper I for the convergence bispectrum, using the nonlinear power within the “treelevel” expression (27) significantly increases the thirdorder moments on smaller scales and improves the general shape but is not sufficient to bridge the gap with the simulations. Modifying the kernel F_{2} by using the fitting formula of Scoccimarro & Couchman (2001) improves the predictions even more, especially when we use the 3D nonlinear power spectrum given by our model, which was shown earlier to be reasonably accurate (see Fig. 1; Paper I; Valageas & Nishimichi 2011b). Indeed, as for the variance, the “halofit” power spectrum of Smith et al. (2003) yields too little power on small scales, in agreement with previous works (Semboloni et al. 2011). However, this approach still underestimates the weaklensing signal.
The best agreement with the numerical simulations is provided by our model. As was seen for the convergence bispectrum in Paper I, it is interesting to note again the good match on the transition scales, θ_{s} ~ 5 arcmin, which a priori are the most difficult to reproduce since they are at the limit of validity of both perturbative approaches (which break down at shell crossing) and halo models (which assume virialized halos). On small angular scales we again predict more power than is measured in the simulations, but like for the secondorder moments this is at least partly caused by the lack of smallscale power in the simulations because of the finite resolution. Thus, we again plot in Fig. 2 the vertical arrows that were plotted in Fig. 1. Since the bispectrum typically scales as the square of the power spectrum, this should roughly correspond to an accuracy threshold of about 10% for the simulations. We can check that our model agrees with the numerical results down to this angular scale. As in Fig. 1, M_{ap} is much more sensitive than ζ_{κ} and κ_{s} to this finiteresolution effect.
6. Relative importance of the different contributions
Fig. 3 Convergence twopoint correlation function (left panel), smoothed convergence (middle panel), and aperturemass (right panel) for sources at redshift z_{s} = 1. The points are the results from numerical simulations with 3 − σ error bars. The low black dashed line “L” is the linear prediction, the middle black dashdotted line “1loop” is the twohalo contribution, for which we used a perturbative resummation that is complete up to oneloop order, the upper blue dashed line “1H” is the onehalo contribution, and the red solid line is our full model, as in Fig. 1. 

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Fig. 4 Convergence threepoint correlation ζ_{κ} for equilateral configurations (left panel) and thirdorder moments (middle panel) and (right panel), for sources at redshift z_{s} = 1. The points are the results from numerical simulations with 3 − σ error bars. The low black dashed line “L” is the lowestorder perturbative prediction (27), the “1loop” dashdotted black line is the prediction of oneloop standard perturbation theory, which is identified with our threehalo term, the blue dotted line “2H” is the twohalo contribution and the upper blue dashed line “1H” is the onehalo contribution. The red solid line is our full model. 

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As in Paper I, we have seen in the previous section that our model, which is based on a combination of perturbation theories and halo models, provides a good match to numerical simulations. Therefore, it can be used to predict weaklensing statistics for a variety of cosmologies, which is an important goal for observational and practical purposes. A second use of our approach is to compare the different contributions that eventually add up to the signal that can be measured in weaklensing surveys. Thus, we can distinguish the various perturbative terms as well as the nonperturbative contributions associated for instance with onehalo or twohalo terms. This is a second advantage of these models, as compared with fitting formulas or direct raytracing simulations. This enables a deeper understanding of which properties of the matter distribution, and of the overall cosmological setting, can be probed by a given gravitational lensing measure. It can also help to estimate the accuracy that can be aimed at in weaklensing statistics as a function of scales, because different contributions suffer from different theoretical uncertainties.
6.1. Twopoint statistics
We plot our results for ξ_{κ}, , and in Fig. 3 at redshift z_{s} = 1. In addition to the full model prediction that was already shown in Fig. 1, we show the underlying 2halo and 1halo contributions. Because of their uncompensated filters, which make statistics at a given angular scale receive contributions from 3D fluctuations on a wide range of scales, taking the oneloop perturbative term into account only yields a small increase of ξ_{κ} and over a wide range of angular scales, as compared with the linear prediction. In contrast, for this oneloop contribution peaks on a narrow range of angular scales around 40 arcmin and has a significant impact that improves the match to the numerical results. On small scales the twohalo contribution decreases close to the linear prediction thanks to the partial resummation of higher perturbative orders. As explained in Valageas & Nishimichi (2011a) and Paper I, this is a useful improvement over the standard oneloop perturbation theory because it ensures that the twohalo term does not give significant contributions on very small scales, in agreement with physical expectations.
Then, these twopoint weaklensing quantities become dominated on small scales by the onehalo term but like for the 3D and 2D power spectra, there remains a significant intermediate range. Again, our model provides a satisfactory interpolation on these scales, but it would be interesting to build a more systematic procedure, for instance by including higher orders of perturbation theory or by building a more refined matching between the twohalo and onehalo regimes. In any case, Fig. 3 clearly shows how ξ_{κ}, , and depend on largescale perturbative density fluctuations or on smallscale halo properties, as the angular scale varies.
We checked that we obtain similar results at redshifts z_{s} = 0.6 and 1.5.
6.2. Threepoint statistics
We now study the various contributions to the lensing threepoint functions, associated with the theehalo, twohalo, and onehalo terms.
We plot our results for ζ_{κ} (for equilateral configurations), , and in Fig. 4, at redshift z_{s} = 1. As in Paper I, the threehalo term is identified with the perturbative prediction and we used the standard perturbation theory at oneloop order. Like for the 3D bispectrum (Valageas & Nishimichi 2011b), and contrary to the power spectrum, this gives a contribution that becomes negligible on small scales, so that it is not necessary to use a resummation scheme or to add a nonperturbative cutoff to ensure a good smallscale behavior. On the other hand, contrary to the twopoint statistics shown in Fig. 3, going to oneloop order now provides a great improvement over the treeorder result, even for ζ_{κ} and . This feature was already noticed for the 3D bispectrum (Sefusatti et al. 2010; Valageas & Nishimichi 2011b) and the convergence bispectrum (Paper I). Thus, combining this oneloop perturbative contribution with the twohalo and onehalo terms is sufficient to obtain a good match to the simulations, from the quasilinear to the highly nonlinear scales. This suggests that higher orders of perturbation theory do not significantly contribute to the bispectrum and that we already have a reasonably successful model. The twohalo term is also subdominant on all scales (by a factor ~10 at least). This is a nice property since because it is a mixed term, which involves both largescale halo correlations and internal halo structures, it may be more difficult to predict than the threehalo term (which is derived from systematic perturbation theories) and the onehalo term (which only depends on internal halo profiles and mass function). These various features were also observed for the 3D bispectrum (Valageas & Nishimichi 2011b) and the convergence bispectrum (Paper I).
Again, we checked that we obtain similar results at redshifts z_{s} = 0.6 and 1.5.
7. Dependence on cosmology
Fig. 5 Convergence twopoint correlation function (left panel), smoothed convergence (middle panel), and aperturemass (right panel) for sources at redshift z_{s} = 1 for six cosmologies. The points are the results from numerical simulations and the lines are the predictions of our model. 

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Fig. 6 Convergence threepoint correlation ζ_{κ} for equilateral configurations (left panel) and thirdorder moments (middle panel) and (right panel) for sources at redshift z_{s} = 1 for six cosmologies. The points are the results from numerical simulations and the lines are the predictions of our model. 

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In this section we check the robustness of our model when we vary the cosmological parameters. As in Paper I, we considered six alternative cosmologies, where n_{s}, Ω_{c}h^{2}, and w_{0} are modified by ± 10% with respect to the fiducial cosmology used in the previous sections. The values of the associated cosmological parameters are given in Table I in Appendix A of Paper I. We compare the predictions of our model with numerical simulations for these six alternative cosmologies in Figs. 5 and 6 for the twopoint and threepoint statistics at z_{s} = 1. To avoid overcrowding the figures we did not plot the error bars of the numerical simulations. Each pair n_{s}, Ω_{c}h^{2}, and w_{0} gives two curves that are roughly symmetric around the fiducial cosmology result, because we consider small deviations of ± 10%. The deviations are largest for the n_{s} case, which changes the shape of the initial power spectrum as well as the normalization σ_{8}. These six cases roughly cover the range that is allowed by current data, and the n_{s} case is already somewhat beyond the observational bounds (Komatsu et al. 2011). Therefore, they provide a good check of the robustness of our model for realistic scenarios.
Like for the Fourierspace statistics studied in Paper I, the dependence on cosmology of the twopoint statistics is well reproduced by our model. For the threepoint statistics it is not easy to make a very precise comparison because the numerical results show a greater level of noise and are sensitive to finite resolution and finite size effects. However, where the simulations are reliable, we also obtain a good match with our predictions. We obtained similar results for z_{s} = 0.6 and z_{s} = 1.5, as well as for other cosmologies where we vary A_{s} or Ω_{de} by ± 10%. This shows that our model and, more generally, models based on combinations of perturbation theory and halo models provide a good modeling of the matter distribution and of weak gravitational lensing effects and capture their dependence on cosmology. Moreover, Figs. 5 and 6 clearly show that this analytical modeling is competitive with current raytracing simulations, because it provides reliable predictions over a greater range of scales. In particular, Figs. 5 and 6 show that the accuracy of our model is sufficient to constrain n_{s}, Ω_{c}h^{2}, and w_{0} to better than 10%.
8. Multiscale moments
Fig. 7 Twoscale secondorder moments ⟨ κ_{s}(θ_{s})κ_{s}(αθ_{s}) ⟩ (upper panel) and ⟨ M_{ap}(θ_{s})M_{ap}(αθ_{s}) ⟩ (lower panel) as a function of θ_{s}, at z_{s} = 1. We show the cases α = 2,5, and 10 from top to bottom. The symbols are the same as in Fig. 1. 

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Fig. 8 Threescale thirdorder moments at z_{s} = 1 as a function of θ_{s}. Upper panel: ⟨ κ_{s}(θ_{s})κ_{s}(αθ_{s})κ_{s}(βθ_{s}) ⟩ . Lower panel: ⟨ M_{ap}(θ_{s})M_{ap}(αθ_{s})M_{ap}(βθ_{s}) ⟩ . We show the cases { α,β } = { 2,5 } , { 3,9 } , and { 5,10 } from top to bottom. The blue dashed line is the “F_{2,NL}” ansatz (27), using the effective kernel F_{2,NL} from Scoccimarro & Couchman (2001) and the power from our model (as in the upper blue dashed line in Fig. 2), while the red solid line is our model. 

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In the previous sections we considered singlescale moments, ⟨ X_{s}(θ_{s})^{p} ⟩ , associated with one smoothing window with a single angular radius θ_{s}. One can also use multipoint statistics such as ⟨ X_{s}(θ_{1};θ_{s1})...X_{s}(θ_{p};θ_{sp}) ⟩ associated with p windows centered on p different directions θ_{i} and with p different radii θ_{si}. In this section, we briefly check the validity of our model for the case of centered multiscale moments. To do this, all quantities X_{si} are centered on the same direction on the sky but we allow the angular radii θ_{si} to be different. Then, Eqs. (20) and (23) generalize as (29)and (30)We show in Fig. 7 the twoscale secondorder moments ⟨ κ_{s}(θ_{s})κ_{s}(αθ_{s}) ⟩ and ⟨ M_{ap}(θ_{s})M_{ap}(αθ_{s}) ⟩ for a scaleratio α = 2,5, and 10, at z_{s} = 1. A higher α yields a smaller moment because it corresponds to a larger second angular radius αθ_{s}. We obtain the same level of agreement as for the singlescale variances shown in Fig. 1. In particular, we obtain a good match on small angular scales where the “halofit” formula somewhat underestimates the weaklensing power. On large scales our analytical results are somewhat larger than the data obtained from the numerical simulations. This is due to the missing of largescale modes in the simulations because of the finite size of the simulation boxes (240 h^{1} and 480 h^{1} Mpc) that correspond to 343 and 686 arcmin at z = 1). This is more apparent than in the singlescale plots of Fig. 1 because we probe larger scales since the factor α is greater than unity. This again shows the advantage of analytical models such as ours, which are competitive with current raytracing numerical simulations to describe a broad range of scales.
We show in Fig. 8 the threescale thirdorder moments ⟨ κ_{s}(θ_{s})κ_{s}(αθ_{s})κ_{s}(βθ_{s}) ⟩ and ⟨ M_{ap}(θ_{s})M_{ap}(αθ_{s})M_{ap}(βθ_{s}) ⟩ for the scaleratios { α,β } = { 2,5 } , { 3,9 } , and { 5,10 } , at z_{s} = 1. Higher values of α and β give a smaller moment since they correspond to larger second and third angular radii αθ_{s} and βθ_{s}. We obtain the same level of agreement as for the singlescale thirdorder moments shown in Fig. 2. In particular, our model recovers the dependence on the ratios { α,β } and on the scale θ_{s} and performs better than the other models studied in this paper. To simplifiy the figures, we show in Fig. 8 only the secondbest model, i.e. the “F_{2,NL}” ansatz (27) using the effective kernel F_{2,NL} from Scoccimarro & Couchman (2001) and the power from our model (as in the upper blue dashed line in Fig. 2). Other models show similar behaviors to those found in Fig. 2 for singlescale moments (i.e., a lack of power on moderate and small angular scales). As for the secondorder statistics, the underestimate of the weaklensing signal by the simulations appears at a smaller angle θ_{s} than in the singlescale case shown in Fig. 2 because the factors α and β are larger than unity and increase the sensitivity to larger scales at fixed θ_{s}.
9. Conclusion
We have investigated the performance of current theoretical modeling of the 3D matter density distribution with respect to weaklensing statistics, focusing on configurationspace statistics, specifically the convergence twopoint and threepoint correlation functions and the second and thirdorder moments of the smoothed convergence and of the aperture mass. As in Paper I, where we studied Fourierspace statistics, we found that a model introduced in previous works (Valageas & Nishimichi 2011a,b), which combines the (resummed) oneloop perturbation theory with a halo model, fares better than some other recipes based on fitting formulae to numerical simulations or more phenomenological approaches. It yields a reasonable agreement with numerical simulations and provides a competitive approach, because it remains difficult and timeconsuming to describe a range of scales that spans three orders of magnitude or more by raytracing simulations.
One advantage of our approach compared with numerical simulations or fitting formulas is that it allows us to decompose the integrated weaklensing signal over several contributions that are associated with specific properties of the underlying 3D density field. Thus, we can distinguish perturbative terms, which can be derived from perturbation theory, from nonperturbative terms that are associated for instance with onehalo contributions, which depend on the density profile and mass function of virialized halos. This is useful because i) these different terms suffer from different theoretical uncertainties and ii) it allows one to understand which aspects of the matter distribution are probed by weaklensing statistics, while angular scales vary.
Like for the Fourierspace statistics studied in Paper I and the 3D statistics studied in Valageas & Nishimichi (2011a,b), we found that including oneloop terms in the perturbative contribution brings a more significant improvement compared with the lowestorder perturbation theory for threepoint statistics than for twopoint statistics. Then, while large scales are described by these perturbative contributions and small scales by onehalo contributions, the nonperturbative twohalo term that gives an additional contribution to threepoint statistics is always subdominant. This is a nice property because this mixed term is more difficult to model and may be less accurate than other contributions (see also Paper I).
Consequently, our model provides reliable predictions for weaklensing statistics, from small to large scales, and for a variety of cosmologies. It could still be improved in various manners. First, the accuracy of the perturbative contribution may be increased by including higher orders beyond oneloop or by using alternative resummation schemes. Second, the underlying halo model could be refined to include substructures (Sheth 2003; Giocoli et al. 2010), deviations from spherical profiles (Jing & Suto 2002; Smith et al. 2006), or the effect of baryons (Guillet et al. 2010). Next, the model could be generalized to nonGaussian initial conditions, which yield distinctive signatures in the bispectrum (Sefusatti et al. 2010).
Acknowledgments
We would like to thank Takashi Hamana for helpful discussions. M.S. and T.N. are supported by a GrantinAid for the Japan Society for Promotion of Science (JSPS) fellows. This work is supported in part by the French “Programme National de Cosmologie et Galaxies” and the FrenchJapanese “Programme Hubert Curien/Sakura, projet 25727TL”, the JSPS CoretoCore Program “International Research Network for Dark Energy”, a GrantinAid for Scientific Research on Priority Areas No. 467 “Probing the Dark Energy through an Extremely Wide and Deep Survey with Subaru Telescope”, a GrantinAid for Nagoya University Global COE Program, “Quest for Fundamental Principles in the Universe: from Particles to the Solar System and the Cosmos”, and World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan. We acknowledge KobayashiMaskawa Institute for the Origin of Particles and the Universe, Nagoya University for providing computing resources. Numerical calculations for the present work have been in part carried out under the “Interdisciplinary Computational Science Program” in Center for Computational Sciences, University of Tsukuba, and also on Cray XT4 at Center for Computational Astrophysics, CfCA, of National Astronomical Observatory of Japan.
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All Figures
Fig. 1 Upper row: convergence twopoint correlation function for sources at redshifts z_{s} = 0.6,1, and 1.5, as a function of the angular pair separation θ. The points are the results from numerical simulations with 3 − σ error bars. The low black dashed line “L” is the linear correlation, the middle blue dashdotted line “S” is the result from the “halofit” of Smith et al. (2003), and the upper red solid line “comb.” is the result from our model, which combines oneloop perturbation theory with a halo model. The vertical arrow shows the scale down to which the simulation result is valid within 5%. Middle row: variance of the smoothed convergence for the same cases, as a function of the smoothing angle θ_{s}. Lower row: variance of the aperture mass for the same cases, as a function of the smoothing angle θ_{s}. 

Open with DEXTER  
In the text 
Fig. 2 Upper row: convergence threepoint correlation function for equilateral triangles, as a function of the triangle side ν, for sources at redshifts z_{s} = 0.6,1, and 1.5. The points are the results from numerical simulations with 3 − σ error bars. The low black dashed line “tree_{L}”, the black dotted line “tree_{NL}” and the two blue dashed lines “F_{2,NL}” are obtained from the ansatz (27), using either the linear 3D power spectrum, or the “halofit” power, or an effective kernel F_{2,NL} with the “halofit” power (lower curve) or the power from our model (upper curve). The red solid line “comb.” is our combined model, described in Paper I. The vertical arrows are at the same angular scale as in Fig. 1. Middle row: thirdorder moment of the smoothed convergence, as a function of the smoothing angle θ_{s}. Lower row: thirdorder moment of the aperture mass, as a function of the smoothing angle θ_{s}. 

Open with DEXTER  
In the text 
Fig. 3 Convergence twopoint correlation function (left panel), smoothed convergence (middle panel), and aperturemass (right panel) for sources at redshift z_{s} = 1. The points are the results from numerical simulations with 3 − σ error bars. The low black dashed line “L” is the linear prediction, the middle black dashdotted line “1loop” is the twohalo contribution, for which we used a perturbative resummation that is complete up to oneloop order, the upper blue dashed line “1H” is the onehalo contribution, and the red solid line is our full model, as in Fig. 1. 

Open with DEXTER  
In the text 
Fig. 4 Convergence threepoint correlation ζ_{κ} for equilateral configurations (left panel) and thirdorder moments (middle panel) and (right panel), for sources at redshift z_{s} = 1. The points are the results from numerical simulations with 3 − σ error bars. The low black dashed line “L” is the lowestorder perturbative prediction (27), the “1loop” dashdotted black line is the prediction of oneloop standard perturbation theory, which is identified with our threehalo term, the blue dotted line “2H” is the twohalo contribution and the upper blue dashed line “1H” is the onehalo contribution. The red solid line is our full model. 

Open with DEXTER  
In the text 
Fig. 5 Convergence twopoint correlation function (left panel), smoothed convergence (middle panel), and aperturemass (right panel) for sources at redshift z_{s} = 1 for six cosmologies. The points are the results from numerical simulations and the lines are the predictions of our model. 

Open with DEXTER  
In the text 
Fig. 6 Convergence threepoint correlation ζ_{κ} for equilateral configurations (left panel) and thirdorder moments (middle panel) and (right panel) for sources at redshift z_{s} = 1 for six cosmologies. The points are the results from numerical simulations and the lines are the predictions of our model. 

Open with DEXTER  
In the text 
Fig. 7 Twoscale secondorder moments ⟨ κ_{s}(θ_{s})κ_{s}(αθ_{s}) ⟩ (upper panel) and ⟨ M_{ap}(θ_{s})M_{ap}(αθ_{s}) ⟩ (lower panel) as a function of θ_{s}, at z_{s} = 1. We show the cases α = 2,5, and 10 from top to bottom. The symbols are the same as in Fig. 1. 

Open with DEXTER  
In the text 
Fig. 8 Threescale thirdorder moments at z_{s} = 1 as a function of θ_{s}. Upper panel: ⟨ κ_{s}(θ_{s})κ_{s}(αθ_{s})κ_{s}(βθ_{s}) ⟩ . Lower panel: ⟨ M_{ap}(θ_{s})M_{ap}(αθ_{s})M_{ap}(βθ_{s}) ⟩ . We show the cases { α,β } = { 2,5 } , { 3,9 } , and { 5,10 } from top to bottom. The blue dashed line is the “F_{2,NL}” ansatz (27), using the effective kernel F_{2,NL} from Scoccimarro & Couchman (2001) and the power from our model (as in the upper blue dashed line in Fig. 2), while the red solid line is our model. 

Open with DEXTER  
In the text 