Weak Lensing Observables in the Halo Model
WWeak Lensing Observables in the Halo Model
Kimmo Kainulainen ∗ and Valerio Marra † Department of Physics, University of Jyv¨askyl¨a, PL 35 (YFL), FIN-40014 Jyv¨askyl¨a, Finland andHelsinki Institute of Physics, University of Helsinki, PL 64, FIN-00014 Helsinki, Finland
The halo model (HM) describes the inhomogeneous universe as a collection of halos. The fullnonlinear power spectrum of the universe is well approximated by the HM, whose prediction canbe easily computed without lengthy numerical simulations. This makes the HM a useful tool incosmology. Here we explore the lensing properties of the HM by use of the stochastic gravitationallensing (sGL) method. We obtain for the case of point sources exact and simple integral expressionsfor the expected value and variance of the lensing convergence, which encode detailed informationabout the internal halo properties. In particular a wide array of observational biases can be easilyincorporated and the dependence of lensing on cosmology is properly taken into account. Thissimple setup should be useful for a quick calculation of the power spectrum and the related lensingobservables, which can play an important role in the extraction of cosmological parameters from,e.g., SNe observations. Finally, we discuss the probability distribution function of the HM whichencodes more information than the first two moments and can more strongly constrain the large-scale structures of the universe. To check the accuracy of our modelling we compare our predictionsto the results from the Millennium Simulation.
PACS numbers: 98.62.Sb, 98.65.Dx, 98.80.Es
I. INTRODUCTION
Gravitational lensing by large-scale inhomogeneitiesaffects the light from distant objects in a way which es-sentially depends on size and composition of the struc-tures through which the photons pass on their way fromsource to observer. The fundamental quantity describingthis statistical (de)magnification is the lensing probabil-ity distribution function (PDF), which has to be under-stood well if one is to use cosmological observations toaccurately map the expansion history and determine theprecise composition of the universe (see, for example, [1–5]). It is not currently possible to extract the lensingPDF from the observational data and we have to resortto theoretical models. Two possible alternatives havebeen followed in the literature. A first approach (e.g. [6–9]) relates a “universal” form of the lensing PDF to thevariance of the convergence, which in turn is fixed by theamplitude of the power spectrum, σ . Moreover the coef-ficients of the proposed PDF may be trained on some spe-cific N-body simulations. A second approach (e.g. [10–12]) is to build ab-initio a model for the inhomogeneousuniverse and directly compute the relative lensing PDF,usually through time-consuming ray-tracing techniques.The flexibility of this method is therefore penalized bythe increased computational time.In Refs. [13, 14] we introduced the stochastic gravita-tional lensing (sGL) method which combines the flexi-bility in modeling with a fast performance in obtainingthe lensing PDF. The sGL method is based on the weaklensing approximation and generating stochastic config-urations of inhomogeneities along the line of sight. A ∗ kimmo.kainulainen@jyu.fi † [email protected] numerical code based on sGL, as the publicly available turboGL package [15], can compute the lensing proba-bility distribution function for a given inhomogeneousmodel in a few seconds. The speed gain is actually a sine-qua-non for likelihood approaches, in which one needsto scan many thousands different models (see, e.g., [4]).This makes sGL a useful tool to study how lensing de-pends on cosmological parameters and how it impactsthe observations. The method can also be used to simu-late the effect of a wide array of systematic biases on theobservable PDF.Here we develop a self-consistent setup to easily calcu-late matter power spectrum and lensing properties of thedesired model universe. The basic idea is to apply thesGL method to the so-called “halo model” (HM) (see, forexample, [16–23]), where the inhomogeneous universe isapproximated as a collection of different types of haloswhose positions satisfy the linear power spectrum. Wewill give explicit integral expressions for mean and vari-ance of the lensing convergence, which will be directlyrelated to the relative matter power spectrum. Theseresults will be valid for point sources, while extendedsources will be treated in forthcoming work. Our dis-cussion applies, for example, to the narrow light bundlesemitted by distant Supernovae (SNe). We expect indeedthat our formulas will be useful in the analysis of SNeobservations as the convergence variance can be straight-forwardly included in the standard χ analyses. We alsodiscuss the PDF of the HM which encodes more informa-tion than the mean and variance and can more stronglyconstrain the large-scale structures of the universe.This paper is organized as follows. In Section II weintroduce the HM and the basic formalism to calculateits power spectrum, while in Section III we will discussits lensing properties. To check the accuracy of our mod-elling we compare in Section IV our predictions to the a r X i v : . [ a s t r o - ph . C O ] A ug results from the Millennium Simulation (MS) [24]. Fi-nally, we give our conclusions in Section V. II. THE HALO MODEL
The halo model assumes that on small scales (largewavenumbers k ) the statistics of matter correlations aredominated by the internal halo density profiles, while onlarge scales the halos are assumed to cluster according tolinear theory. In other words, the nonlinear evolution isassumed to produce only concentrated halos. The modeldoes not include intermediate low density structures suchas filaments and walls. The two components are thencombined together. We do it here by simple addition sothat the total power spectrum is P ( k, z ) = P L ( k, z ) + P H ( k, z ) . (1)The first term on the right-hand side is also called 2-halo component and the second term 1-halo component.The usefulness of the halo model stems from the fact thatboth terms in Eq. (1) can be computed without having toresort to numerical simulations. We would like to stressthat more sophisticated and accurate versions of the HMare available in the literature [25]. Here we use the sim-plest version of the HM as we merely wish to illustrateits good agreement with the full power spectrum. Linear Power Spectrum
The linear part of the power spectrum is as usual: k π P L ( k, z ) = δ H (cid:18) cka H (cid:19) n s T ( k/a ) D ( z ) , (2)where H and a are the present-day Hubble parameterand scale factor, n s is the spectral index and δ H is theamplitude of perturbations on the horizon scale today,which we fix by setting σ . For the transfer function T ( k ) one can use the fits provided, e.g., by Ref. [26]. Fitfunctions for the growth function D ( z ) could be found,e.g., in Refs. [27, 28], but it can also be easily obtainednumerically [14]. Halo Power Spectrum
The nonlinear or halo contribution has been obtainedin, e.g., [18]. We will briefly summarize the importantsteps here. First we introduce the halo mass function f ( M, z ), which gives the fraction of the total mass inhalos of mass M at the redshift z . The function f ( M, z )is related to the number density n ( M, z ) by:d n ( M, z ) ≡ n ( M, z )d M = ρ MC M f ( M, z )d M , (3) where the density ρ MC ≡ a ρ M is the constant matterdensity in a co-moving volume. The halo function is bydefinition normalized to unity: (cid:82) f ( M, z )d M = 1 andwe defined d n as the number density of halos in the massrange d M . The power spectrum due to randomly locatedhalos is then [21] P H ( k, z ) = (cid:90) ∞ d n ( M, z ) (cid:18) M W k ( M, z ) ρ MC (cid:19) , (4)where W k is the Fourier transform of the halo densityprofile: W k ( M, z ) = 1 M (cid:90) R ρ ( r, M, z ) sin krkr π r d r , (5)and R is the halo radius. For the halo profile ρ ( r, M, z ),we shall use the Navarro-Frenk-White (NFW) profile [29].We will now apply this setup to the weak lensing. III. LENSING
The lens convergence κ in the weak-lensing approxi-mation is given by the following integral evaluated alongthe unperturbed light path [30]: κ ( z s ) = (cid:90) r s dr ρ MC G ( r, r s ) δ M ( r, t ( r )) (6)where we defined the auxiliary function G ( r, r s ) = 4 πGc a f k ( r ) f k ( r s − r ) f k ( r s ) , (7)which gives the optical weight of an inhomogeneityat the comoving radius r . The functions a ( t ) and t ( r ) are the scale factor and geodesic time for thebackground FLRW model, and r s = r ( z s ) is the co-moving position of the source at redshift z s . Also, f k ( r ) = sin( r √ k ) / √ k, r, sinh( r √− k ) / √− k dependingon the curvature k >, = , <
0, respectively. Neglectingthe second-order contribution of the shear, the shift in thedistance modulus caused by lensing is expressed solely interms of the convergence:∆ m ( z ) (cid:39) (1 − κ ( z )) . (8)Eqs. (6) and (8) show that for a lower-than-FLRW col-umn density the light is demagnified (e.g., empty beam δ M = − δ M ( r, t ) is the local matter den-sity contrast and it is clear that an accurate statisticalmodelling of the magnification PDF requires a detaileddescription of the inhomogeneous mass distribution. Inthis work we will model δ M according to the halo model.See [14] for a more refined modelling which also includesfilamentary structures confining the halos. The followingresults will be valid for point sources (smoothing angle θ = 0). We will discuss extended sources in a forthcom-ing paper.Because in Eq. (6) the contributions to the total con-vergence are combined additively, it is useful to decom-pose the density field into a sum of Fourier modes: δ M ( r , t ) = (cid:90) ∞ d k (2 π ) e i k · r δ M ( k , t ) . (9)Eq. (9) can be used to separate the contributions tothe convergence due to small-scale inhomogeneities ( k (cid:38) k cut ) from the ones due to large-scale inhomogeneities( k (cid:46) k cut ) so that, similarly to Eq. (1), it is: κ ( z ) = κ L ( z ) + κ H ( z ) . (10)The quantity k cut separates the modes relevant to κ L from the ones relevant to κ H and can be defined as thescale at which the two power-spectrum components havethe same value, P L ( k cut , z ) = P H ( k cut , z ).We are in particular interested in the expected valueand variance of the lensing convergence. From Eq. (10)it follows that: (cid:104) κ (cid:105) = (cid:104) κ L (cid:105) + (cid:104) κ H (cid:105) , (11) σ κ = σ κ L + σ κ H . (12)We will now give directly computable expressions forEqs. (11-12). Linear Contribution
Because of photon conservation, if no lines of sight areobscured, the expected value of the weak lensing conver-gence is zero. This can be seen in Eq. (6) from the factthat (cid:104) δ M ( r, t ) (cid:105) = 0. Mechanisms that are able to obscurelines of sight involve small scales and so we have (cid:104) κ L (cid:105) = 0 . (13)The linear power spectrum, however, contributes to thevariance of the convergence according to [31]: σ κ L = (cid:90) r s drρ MC G ( r, r s ) (cid:90) k cut k dk π P L ( k, z ( r )) , (14)where the scale k cut can depend on the redshift z . Halo Contribution
The sGL method for computing the lensing conver-gence is based on generating stochastic configurations ofhalos and filaments along the line of sight and comput-ing the associated integral in Eq. (6) by binning into anumber of independent lens planes. Because the halos arerandomly placed their occupation numbers in parameter-space volume cells follow Poisson statistics. This allowsrewriting Eq. (6) as a sum over these cells characterized by the corresponding Poisson occupation numbers. Thevarious individual contributions to the convergence arethen additively combined. By generating many halo con-figurations one can easily sample the convergence PDF.Besides the approximations relative to Eqs. (6-8), corre-lations in the halo positions are neglected here, similarlyto Eq. (4). In the full sGL one can model also filamentarystructures confining the halos, thus accounting for someof the correlations among the halo positions. A detailedexplanation of the sGL method can be found in [13, 14]and a publicly-available numerical implementation, the turboGL package, in [15].In the present case the filamentary structures areturned off and the inhomogeneous matter distributionintroduced in [14] exactly corresponds to the halo model.The sGL method then predicts the following direct andexact results for mean and variance of the convergence: (cid:104) κ H (cid:105) = (cid:90) r s d r G ( r, r s ) (cid:90) ∞ d n ( M, z ( r )) × (15) × (cid:90) R ( M,z ( r ))0 d A ( b ) ( P sur −
1) Σ( b, M, z ( r )) , and σ κ H = (cid:90) r s d r G ( r, r s ) (cid:90) ∞ d n ( M, z ( r )) × (16) × (cid:90) R ( M,z ( r ))0 d A ( b ) P sur Σ ( b, M, z ( r )) , where the integral limits for the last two integrals are im-plicitly defined, d A ( b ) ≡ πb d b and Σ is the halo surfacedensity:Σ( b, M, z ) = a (cid:90) Rb r d r √ r − b ρ ( r, M, z ) . (17)Eqs. (15-16) can be directly integrated without having toconsider the full formalism of the sGL method. We alsowould like to point out that, in contrast to, e.g., [32], ourexpressions use the halo profiles in real space and notin Fourier space, thus including higher order correlationterms beyond the power spectrum. As a direct conse-quence the survival probability P sur can be included inthe way the sGL method predicts.The quantity P sur = P sur ( b, M, z, z s ) is a generic func-tion that describes the probability that a light ray, whichencounters a halo of mass M at the redshift z and im-pact parameter b for a source at redshift z s , does notfall below detection threshold. P sur can be used to simu-late the effect of a very wide array of systematic biases,such as any sources leading to obscuration of the lightbeam, either alone or in combination with restrictionsarising from imperfect search efficiencies and strategies.For example, selection by extinction effects or by out-lier rejection mainly relates to high-magnification eventswhich are clearly correlated with having high interveningmass concentrations (halos with large M and/or small b )along the light geodesic. Similarly, short duration eventsmight be missed by search telescopes or be rejected fromthe data due to poor quality of the light curve, for ex-ample in cases where a supernova is not separable fromthe image of a bright foreground galaxy. Probability ofsuch events would be correlated with the brightness ofthe source, with the density and redshift of the interven-ing matter and of course with the search efficiencies. Allthese effects could be modelled by P sur , carefully adjustedby the use of the astrophysical input. As a simple illus-tration, in the next Section we will model the survivalprobability by a step function in the impact parameter,such that the halo is opaque for radiuses smaller than theNFW scale radius R s = R/c : P sur ( b, M, z ) = (cid:26) b/R < c ( M, z ) − , (18)where c ( M, z ) is the mass- and redshift-dependent NFWconcentration parameter.Eqs. (15-16) allow to draw some general considerations.First, if P sur = 1, the expected value of the convergenceis correctly zero as demanded by photon flux conserva-tion, showing the “benevolent” nature of weak lensingcorrections for unbiased observations. If, however, thesurvival probability is not trivial, selection biases persisteven in very large datasets so that the average conver-gence approaches a nonvanishing value. Second, Eq. (16)is a product of positive quantities and so a nontrivialsurvival probability always reduces the observed variance. Thus, neglecting existing systematic biases could lead toan underestimation of the observationally inferred vari-ance, and hence to too strong constraints on additionalvariance caused by inhomogeneities.Finally, Eqs. (15-16) are integrated (see second d n -integral) over all halo masses 0 < M < ∞ . It is, how-ever, straightforward to generalize them by changing theintegration limits in order to have expected value andvariance relative to a specific halo mass bin ∆ M . Thismay be useful if one wants to connect the lensing given byhalos within a mass bin (see Fig. 1) to the correspondingamount of correlation. IV. COMPARISON WITH THE MILLENNIUMSIMULATION
We will now compare the halo model power spectrumand lensing properties to those from the Millennium Sim-ulation. Accordingly, we will fix the cosmological param-eters to h = 0 .
73, Ω M = 0 .
25, Ω B = 0 . Λ = 0 . w = − σ = 0 . n s = 1; see [24] for more details.Let us point out that in [14] we have already accuratelyreproduced the MS lensing PDF with an sGL modelling This should be approximately true [33] if strong lensing eventsleading to secondary images and caustics [34] do not play a signif-icant role as far as the full-sky average is concerned [10, 35, 36]. Σ Κ Convergence variance per mass bin M FIG. 1. Shown is the variance per mass bin for a sourceat redshift z = 1. The variance is computed using Eq. (16)restricted to the mass bin ∆ M =]10 n , n +1 ] h − M (cid:12) . In theplot n labels the halo mass according to M n = 10 n h − M (cid:12) .For halos of mass M (cid:46) h − M (cid:12) the contribution to thetotal variance is negligible. See Section IV for more details. which included not only the halos, but also the filamen-tary structures. Our goal here is different; we are deliber-ately using a simpler modelling of the inhomogeneities inorder to study the validity of the halo model for lensinganalyses.Because we are assuming that all (virialized) matter isconcentrated in halos, we need to use a halo mass func-tion whose integral is normalizable to unity. We thereforeadopt the halo function provided by Sheth & Tormen inRef. [37], which should be approximately valid for thefull mass range of the integrals in Eqs. (15-16). Halofunctions obtained through numerical simulations, albeitpossibly more precise, are valid only above a mass cut-off imposed by the numerical resolution of the simulationitself [38]. We point out, however, that halos with masssmaller than M cut ∼ h − M (cid:12) act effectively as a meanfield in weak lensing, and so mass functions from numeri-cal simulations which are valid down to M cut may be usedwithin the present setup by introducing explicitly M cut in Eqs. (15-16). This fact can be appreciated quantita-tively by computing the convergence variance (16) permass bin ∆ M as shown in Fig. 1. Clearly for M (cid:46) M cut the contribution to the total variance is negligible whichmeans that the corresponding halos behave effectively asa fine homogeneous dust as far as weak lensing is con-cerned [14].In Fig. 2 we have plotted the power spectrum ofEq. (1), together with the halo and linear components.Also plotted is the power spectrum of the MillenniumSimulation. The agreement is rather good, although thehalo model power spectrum is slightly underpowered onsmall scales. A better power spectrum could be obtainedwith a more sophisticated HM, but this would not be rel-evant for the main goal of this paper, which is to propose k h Mpc P k FIG. 2. Shown is the present-day power spectrum of the halomodel (orange solid line and Eq. (1)) for the ΛCDM universeof the Millennium Simulation, together with the halo (bluedashed line and Eq. (4)) and linear (green dot-dashed line andEq. (2)) components. Note how the halo component gives aconstant contribution at large scales (shot noise). Also shownfor comparison is the power spectrum (black dotted line) rel-ative to the Millennium Simulation [24]. Σ Κ FIG. 3. Shown is the redshift dependence of the dispersionsdue to the linear (green dot-dashed line and Eq. (14)) and halo(blue dashed line and Eq. (16)) components, for the ΛCDMmodel of the Millennium Simulation. Also plotted is the sum(orange solid line) of the variances and the dispersion (blackdotted line) relative to the Millennium Simulation [35, 36].The labeling is as in Fig. 2. using Eqs. (15-16) for lensing. The oscillations at largescales are absent in the HM spectrum because we used atransfer-function fit that reproduces the baryon-inducedsuppression on the intermediate scales but ignores theacoustic oscillations, which are again not relevant for ushere [26]. For the NFW concentration parameters weused the fit of Ref. [39]. Note also that, as said before, We increased the fit by a factor of 4 . / .
93 (see [39]) in order tomatch the results of [40] which are relative to the MS cosmology. it is straightforward to calculate the correlation per massbin by binning Eq. (4) in the same way Eq. (16) has beenbinned in Fig. 1.In Fig. 3 we have plotted the redshift dependence ofthe convergence dispersions due to the linear and halocomponents, computed with Eq. (14) and Eq. (16), andalso the sum of the variances which shows that the lin-ear contribution is practically negligible. We wish tostress that here we are studying the case of point sources,i.e., we are adopting a smoothing angle θ = 0. For ex-tended sources and/or finite smoothing angles the re-sults of Fig. 3 change, and for smoothing angles largerthan some value ¯ θ the linear contribution dominates overthe one due to the halos. At present we cannot treatnonzero smoothing angles within the sGL model, butwe can nonetheless estimate the order of magnitude of¯ θ . Assuming for simplicity a flat universe, the opticalweight function G of Eq. (7) peaks at half the comov-ing distance to the source and vanishes at the observerand source locations. The physics at r s / θ can be converted into a smoothing scale by means of λ = θ d A ( r s / d A is the angular diameter dis-tance. Let us now take, for example, a source at z = 1and a smoothing scale of ¯ λ ∼ θ ∼ (cid:48) . Also plotted in Fig. 3 is the dispersionrelative to the Millennium Simulation [35, 36], whichshows a good agreement with the theoretical result. Wewish to stress that it is very easy to evaluate the integralsof Eqs. (15-16) numerically, and their predictions can bestraightforwardly implemented in χ analyses based onGaussian likelihoods. However, it should be rememberedthat Gaussian analysis is justified if the lensing PDF isnot strongly skewed. When this is not the case the fullPDF has to be used for the likelihood analysis, as wasdone, e.g., in [4].In Fig. 4 we have plotted the lensing PDFs obtainedwith the sGL method (histograms). Only the (1-)halocontribution is included as this is the direct output ofthe sGL method. Indeed at present we cannot properlyinclude the linear contribution which, in any case, shouldbe subdominant as shown in Fig. 3. For z = 0 .
83 and1 .
08 the agreement with the MS results (dotted lines)from [35, 36] is remarkable, especially considered the sim-plicity of the setup. For z = 1 .
50 the halo model predictsa less skewed PDF. This could mean that the model weare using to describe the halos (Sheth & Tormen mass In order not to exceed the validity region of the weak-lensingapproximation, the dispersion relative to the MS has been calcu-lated in the weak-lensing regime κ (cid:28) − . Thisresulted in κ cut = 0 . , . , . , . , .
365 for the PDFsrelative to z = 0 . , . , . , . , .
50, respectively. Κ Lensing PDF
FIG. 4. Shown are the lensing PDFs for a source at (fromtaller to shorter graph) z = 0 . , .
08 and 1 .
50 for the ΛCDMmodel of the Millennium Simulation. The dotted lines arethe lensing PDFs generated by shooting rays through the MS[35, 36], while the histograms are the corresponding lensingPDFs obtained with the sGL method [13–15]. function and NFW concentration parameters from [39])is less accurate at high redshifts. For example, a betteragreement with [35, 36] is found if the concentration pa-rameters are increased by 40%, and similar results maybe found by slightly changing the shape of the mass func-tion. Alternatively the extra skewness could be causedby unvirialized low-density large-scale structures like fil-aments and walls. These could be particularly importantat high redshifts where lesser amounts of virialized struc-tures, which are the only ones accounted for by the halomodel, are present. Indeed, a very good agreement withMS was found in [14] where the halo model was extendedto include filamentary structures. Note, finally, that thedifferences in shape between the two PDFs at z = 1 . Selection effects
It is interesting to discuss the impact on lensing of thetoy survival probability of Eq. (18). In Fig. 5 we showthe redshift dependence of the convergence dispersion(16) with (dot-dashed line) and without (dashed line)the selection effect, which halves the resulting disper-sion. Also plotted is the (negative) mean convergence(15), which shows the possible bias present even in verylarge datasets.In Fig. 6 we show the full lensing PDF with selectioneffects included (taller PDF). The lensing PDF with uni-tary survival probability is also shown for comparison(shorter PDF). As expected the selection effects reducethe high magnification tail of the PDF, without sizeablychanging the tail at low magnifications. See [14] for an- FIG. 5. Shown is the redshift dependence of the conver-gence dispersion with (red dot-dashed line) and without (bluedashed line) the selection effects of Eq. (18), for the ΛCDMmodel of the Millennium Simulation. Also plotted is the meanconvergence (red solid line) with selection effects (without se-lection effects it is zero). Κ Lensing PDF
FIG. 6. Shown is the lensing PDF for a source at z = 1 . other discussion of the impact of selection effects on lens-ing.Finally, we would like to point out that a finite smooth-ing angle reduces the skewness of the PDF as it reducesthe contrast in the matter surface density. If observa-tional biases are added, however, the overall effect on thePDF is more involved and depends strongly on the kindof selection effect considered. In the case of obscuration,for example, the bias on the mean convergence shouldremain basically unchanged as photons relative to un-seen lines of sight are lost no matter the smoothing angleadopted. Therefore for increasing θ the PDF should tendto a gaussian with decreasing dispersion peaked at themean biased convergence. V. CONCLUSIONS
By applying the sGL method to the halo model, wehave obtained a simple setup that allows to quickly com-pute power spectrum and lensing observables for a de-sired model universe. We wish to stress that there are noextra free parameters in the model, besides the ones rel-ative to the halo model derived from the literature (i.e.,halo mass and profile fitting functions). In particular,we have given exact and directly-computable expressionsfor the expected value (see Eq. (15)) and variance (seeEq. (16)) of the convergence. We stress that Eqs. (15-16) contain more information than analogous expressionsbased on the power spectrum, like Eq. (14). They are adirect consequence of the sGL method which models thehalo profiles in real space, and therefore include infor-mation of coherent structures described by higher ordercorrelation terms beyond the power spectrum. In par-ticular, a wide array of systematic biases, which couldbe relevant for the extraction of cosmological parametersfrom, e.g., SNe observations and which persist even invery large data sets, can be included in Eqs. (15-16). Wehave shown a quantitative example of the impact of se-lection effects on lensing with Figs. 5 and 6. Moreover,these equations have the crucial advantage of includingthe dependence of lensing on cosmology, which was shownin [4] to give sizeable corrections to the confidence levelcontours from SNe observations. This is to be contrastedwith the usual approach (e.g., [5, 41]) where lensing ef-fects are included by means of a cosmology-independent dispersion, which neglects the important dependence on,e.g., σ , Ω M, Λ and h .We have then checked the accuracy of this setupagainst the cosmology of the Millennium Simulation. Wehave found that the theoretical predictions of the vari-ance are in good agreement with the results relative tothe Millennium Simulation [35, 36]. This is most impor-tant as we wish here to propose to the community the useof Eqs. (15-16) for lensing, which can be easily includedin the standard χ analyses. We have also found that thecontribution of the linear (2-halo) component to the to- tal variance is practically negligible. Then we have shownthe lensing PDFs of the halo model as computed usingthe sGL method and the turboGL package [15] for theredshifts of z = 0 . , .
08 and 1 .
50. Given the simplic-ity of the setup, we found a remarkable agreement withthe results from the Millennium Simulation, especiallyat z = 0 .
83 and 1 .
08. One could conclude that, as faras the weak lensing magnification is concerned, the maincontribution comes from the smooth halo profiles, with asmaller correction due to correlations among the halo po-sitions and substructures within the halos (even thoughone should keep in mind the possibility that the MS itselfmight lack some interesting substructures due to its res-olution limit). At z = 1 .
50 we have found that the halomodel predicts a less-skewed PDF. This could be eitherdue to the fact that the model we are using to describethe halos (Sheth & Tormen mass function with NFW ha-los) is less accurate at high redshifts or due to unvirial-ized low-density large-scale structures like filaments andwalls, which could be important at higher redshifts whereless virialized structures are present. One could indeedimprove the modelling of the inhomogeneities, for exam-ple by explicitly introducing the filamentary structures asin [14], or by considering substructures within halos andfilaments. Indeed, any extra matter structures such as fil-aments and walls increase the skewness of the PDF, andit would be interesting to see to what extent one can al-ter the lensing PDF by introducing such structures with-out conflicting with the observational constraints on thematter power spectrum. We will develop these thoughtsfurther in a forthcoming paper.
ACKNOWLEDGMENTS
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