Accurate Analytic Model for the Weak Lensing Convergence One-Point Probability Distribution Function and its Auto-Covariance
AAccurate Analytic Model for the Weak Lensing Convergence One-Point ProbabilityDistribution Function and its Auto-Covariance
Leander Thiele ∗ ,
1, 2
J. Colin Hill,
3, 4 and Kendrick M. Smith Perimeter Institute for Theoretical Physics, Waterloo ON N2L 2Y5, Canada Department of Physics, Princeton University, Princeton, NJ, USA 08544 Department of Physics, Columbia University, 538 West 120th Street, New York, NY, USA 10027 Center for Computational Astrophysics, Flatiron Institute, New York, NY, USA 10003
The one-point probability distribution function (PDF) is a powerful summary statistic for non-Gaussian cosmological fields, such as the weak lensing (WL) convergence reconstructed from galaxyshapes or cosmic microwave background maps. Thus far, no analytic model has been developed thatsuccessfully describes the high-convergence tail of the WL convergence PDF for small smoothingscales from first principles. Here, we present a halo-model formalism to compute the WL convergencePDF, building upon our previous results for the thermal Sunyaev-Zel’dovich field. Furthermore,we extend our formalism to analytically compute the covariance matrix of the convergence PDF.Comparisons to numerical simulations generally confirm the validity of our formalism in the non-Gaussian, positive tail of the WL convergence PDF, but also reveal the convergence PDF’s strongsensitivity to small-scale systematic effects in the simulations (e.g., due to finite resolution). Finally,we present a simple Fisher forecast for a Rubin Observatory-like survey, based on our new analyticmodel. Considering the { A s , Ω m , Σ m ν } parameter space and assuming a Planck CMB prior on A s only, we forecast a marginalized constraint σ (Σ m ν ) ≈ .
08 eV from the WL convergence PDF alone,even after marginalizing over parameters describing the halo concentration-mass relation. This errorbar on the neutrino mass sum is comparable to the minimum value allowed in the normal hierarchy,illustrating the strong constraining power of the WL convergence PDF. We make our code publiclyavailable at https://github.com/leanderthiele/hmpdf . I. INTRODUCTION
While the cosmic microwave background (CMB) his-torically has been the driving force in cosmological pa-rameter inference, we are now experiencing a prolifera-tion in high-quality data from the late-time matter dis-tribution. In contrast to the primary CMB, late-timefields are described by non-linear clustering of matter,rendering the distribution of many relevant observableshighly non-Gaussian. For such non-Gaussian fields, theproblem of extracting all information contained thereinis unsolved; while for Gaussian fields, such as the pri-mary CMB, the power spectrum is an optimal summarystatistic containing all the information, no such summarystatistic is known in the non-Gaussian case.One late-time field of interest is the weak lensing (WL)convergence. Weak gravitational lensing describes thedeflection of light by the matter distribution, impart-ing a shear and magnification on the images of observedbackground galaxies or CMB fluctuations. The WL con-vergence is a redshift-weighted measure of the integratedmatter density along the line of sight; thus, it is a pow-erful probe of the matter distribution.Since in the course of the non-linear gravitational clus-tering the matter distribution departs significantly fromGaussianity, appreciable amounts of information leakfrom the power spectrum into higher-order statistics.This motivated previous studies to consider parameter ∗ [email protected] inference from such measures of non-Gaussianity, for ex-ample the WL skewness and bispectrum [1–8]. An alter-native summary statistic is the one-point probability dis-tribution function (PDF), which simply constitutes thehistogram of WL convergence pixel values. Originallyconsidered in the context of peak statistics [9, 10], morerecent studies have demonstrated that the WL conver-gence PDF can add significant constraining power in pa-rameter inference, not only in the σ -Ω m plane, but alsoon the neutrino mass sum [e.g., 11–13].In some respects WL shares similarities with the ther-mal Sunyaev-Zel’dovich (tSZ) effect, which describes thescattering of CMB photons by hot electrons residingmostly in massive halos. Because the tSZ signal is ap-proximately proportional to M / , the halo model allowsan excellent description of the tSZ PDF. This fact was re-cently utilized in order to construct a semi-analytic modelfor the tSZ PDF [14, 15] (here “semi-” accounts for thefact that the model contains some functions that are mostaccurately fixed by fitting to numerical simulations).In this work, we demonstrate that the halo-model for-malism developed in the tSZ context can be applied tothe WL convergence PDF as well, with small modifica-tions due to two complications: (1) In constrast to tSZ,the WL convergence signal does not include the addi-tional M / temperature bias, which brings the distri-bution closer to Gaussian and renders the halo modelslightly less accurate; (2) Furthermore, while the tSZ sig-nal is strictly positive, the WL convergence receives neg-ative contributions from underdense regions (voids). Incontrast to previous works on the subject [6, 16–18], ourformalism is better suited to describe relatively large pos- a r X i v : . [ a s t r o - ph . C O ] S e p itive values of the convergence PDF (which are sourcedby massive halos), while performing less well in the onlymildly non-linear regime and especially at negative con-vergences. As we will show, our formalism is also notvery accurate when the convergence field is smoothed onlarge angular scales. Perturbative methods [1, 2, 19–23],and the large deviation statistics-formalism developed inRefs. [24, 25] are better suited in such a situation. Interms of physical input, our formalism is quite similar tothe stochastic numerical method developed in Refs. [26–28].Besides a theoretical model for the expected form ofthe WL PDF, in order to do parameter inference wealso require a prescription for its statistical distribution.While this distribution is non-Gaussian and difficult tocompute, a first step is the computation of the covari-ance matrix. In terms of practical applications, the co-variance matrix can be useful if the PDF is sufficientlydownsampled such that the likelihood can be computedin the Gaussian approximation, as was done in Ref. [14];alternatively, non-Gaussian inference methods such aslikelihood-free inference [29] can benefit from the covari-ance matrix as a starting point. In view of these potentialapplications, we generalize the halo-model formalism tocompute not only the one- but also the two-point PDF,the latter being sufficient for computation of the covari-ance matrix.The remainder of this paper is structured as follows:In Sec. II, we present the theoretical part of this work,starting from the general theory of weak gravitationallensing and proceeding to our halo-model formalism forthe one- and two-point PDFs. There, we will also dis-cuss the modifications to the formalism in comparisonto the tSZ case. In Sec. III, we present various resultsobtained with our formalism for the one-point PDF: Anumber of calculations intended to build up intuition onthe WL convergence PDF, and comparisons to two in-dependent sets of numerical simulations. In Sec. IV, weturn to the two-point PDF and the covariance matrixof the one-point PDF. We perform a null test and com-pare the analytic covariance matrix to a large N -bodysimulation. In Sec. V, we utilize the previous results toproduce a simple Fisher parameter forecast. We con-clude in Sec. VI. Further analytic calculations useful inbuilding up intuition are presented in Appendix A, somedetails on the numerical evaluation of the formalism arecollected in Appendix B, and in Appendix C we discussthe validity of several approximations. II. THEORYA. Background
Gravitational lensing distorts and magnifies the shapesof distant sources (e.g., galaxies or CMB fluctuations) asa result of the projected gravitational potential of matteralong the line-of-sight, including dark and baryonic mat- ter. In the weak lensing limit, these effects are encodedin the lensing convergence field, κ ( ˆ n ): κ ( ˆ n ) = (cid:90) ∞ dz δ ( x ( χ ( z ) ˆ n , z )) W κ ( z ) , (1)where χ ( z ) is the comoving distance to redshift z , δ isthe matter density fluctuation, δ ( x ) ≡ ( ρ ( x ) − ¯ ρ ) / ¯ ρ , andthe lensing projection kernel is given by W κ ( z ) = 32 Ω m H (1 + z ) H ( z ) χ ( z ) c (cid:90) ∞ z dz s dndz s ( χ ( z s ) − χ ( z )) χ ( z s ) , (2)where dn/dz s is the distribution of sources, normalizedsuch that (cid:82) dz dn/dz = 1. Note that for CMB lensing, dn/dz = δ D ( z − z ∗ ), where δ D is the Dirac- δ functionand z ∗ ≈ dn/dz is generally a more com-plicated function. Note that we have specialized to thecase of a flat universe in Eqs. (1) and (2). For reference,the lensing convergence is related to the lensing poten-tial, φ ( ˆ n ), via κ ( ˆ n ) = −∇ φ ( ˆ n ) / ∇ is the two-dimensional Laplacian on the sky), or κ (cid:96) = (cid:96) ( (cid:96) + 1) φ (cid:96) / κ ( θ , M, z ) for a halo of mass M at redshift z : κ ( θ , M, z ) = Σ − ( z ) (cid:90) LOS ρ (cid:18)(cid:113) l + d A | θ | , M, z (cid:19) dl , (3)where ρ ( r , M, z ) is the halo density profile, d A ( z ) is theangular diameter distance to redshift z , and Σ crit ( z )is the critical surface density (in physical units here)for lensing at redshift z assuming a source distribution dn/dz :Σ − ( z ) = 4 πGχ ( z ) c (1 + z ) (cid:90) ∞ z dz (cid:48) ( χ ( z (cid:48) ) − χ ( z )) χ ( z (cid:48) ) dndz (cid:48) (4)= 8 πG m H H ( z ) c (1 + z ) W κ ( z ) . (5)For a spherically symmetric density profile, the conver-gence profile is azimuthally symmetric, i.e., κ ( θ , M, z ) = κ ( θ, M, z ). For the NFW density profile, analytic formsexist for the convergence profile. One subtlety, however,is the non-convergence of the enclosed mass in the NFWprofile as r → ∞ , which thus necessitates a radial cutoffin calculations using this profile. B. WL PDF in the Halo Model
In Ref. [15], an analytic approach based on the halomodel was constructed to describe the one-point PDF ofthe tSZ field, building on a simpler model presented inRef. [14]. In particular, the effects of halo overlaps alongthe LOS and halo clustering, which were neglected in [14],were included in [15]. However, the expressions derivedin [15] are more broadly applicable to the one-point PDFof any (projected) cosmic field that can be modeled in ahalo-based approach. The halo model approach is veryaccurate for the tSZ field as this field is heavily dominatedby contributions from massive halos (e.g., [31–33]), due tothe temperature dependence of the tSZ signal. A primarygoal of this paper is to assess the accuracy of this modelfor other cosmic fields, in particular the WL convergencefield. Thus, as a first step to model the WL convergencePDF, we can simply use the expressions from [15], butwith the y (tSZ) profile replaced by the κ profile definedin Eq. 3. The rest of the formalism derived in that workthen goes through unchanged.For completeness and ease of reference in later sections,we include the derivation of the one-point κ PDF in thisformalism here. In some places, algebraic manipulationsare omitted for brevity; we refer the interested readerto [15] for full details.
1. One-Halo Term
We refer to the (differential) κ one-point PDF as P ( κ ). Considering a bin spanning [ κ i , κ i +1 ], we definethe binned version of the PDF as p i = (cid:90) κ i +1 κ i dκ P ( κ ) . (6)The fundamental concept underlying the model devel-oped in [14, 15] is that p i quantifies the sky fraction sub-tended by κ values in the range [ κ i , κ i +1 ]. For an individ-ual spherically symmetric halo with an azimuthally sym-metric projected κ -profile κ ( θ ), this sky fraction is simplythe area in the annulus between θ ( κ i ) and θ ( κ i +1 ), where θ ( κ i ) is the angular distance from the center of the profileto the radius where κ ( θ ) = κ i . If we then assume thathalos are sufficiently rare that they never overlap on thesky, the one-point PDF is simply given by adding up theannular area contributions from all halos: p i = (cid:90) dz dM χ H dndM π (cid:0) θ ( κ i ) − θ ( κ i +1 ) (cid:1) + δ i (1 − F halos ) , (7)where dn ( M, z ) /dM is the halo mass function (i.e., thenumber of halos of mass M at redshift z per unit massand comoving volume), θ ( κ, M, z ) is the inverse functionof κ ( θ, M, z ), F halos is the total sky area subtended by allhalos (assuming some radial cutoff for the halo profiles), δ i is unity if κ = 0 lies in the bin and zero otherwise, andredshift and mass dependences have been suppressed inthe equation for compactness. Eq. 7 is only accurate inthe limit in which halos do not overlap on the sky; more-over, it neglects effects due to the clustering of halos.While these assumptions are (moderately) accurate forthe tSZ field, they are not accurate for the WL conver-gence field.We thus seek a more general approach, in which theselimiting assumptions are discarded. The basic ideas and results for our improved formalism were presented for thetSZ field in [15]; here, we adapt the formalism to the WLconvergence field and introduce a more compact notation.Our goal is to compute the one- and two-point PDFs, P ( κ a ) and P ( κ a , κ b ; φ ), where φ is the angular separationbetween two sky locations at which we measure the twoconvergence values κ a and κ b . It will be convenient towork in Fourier space, introducing P a ≡ (cid:90) dκ a e iλ a κ a P ( κ a ) , (8) P ab ≡ (cid:90) dκ a dκ b e i ( λ a κ a + λ b κ b ) P ( κ a , κ b ; φ ) , (9)where we have abbreviated the notation for conciseness.We will separate the PDFs into a one- and a two-haloterm, writing P = P P . In this section we computethe 1-halo term, i.e., we ignore the clustering of halos forthe moment. We introduce two further pieces of notation:we denote the projected halo mass function n ≡ n ( M, z ) = χ ( z ) H ( z ) dn ( M, z ) dM , (10)which gives the expected number of halos per unit massand redshift interval in unit solid angle; furthermore weintroduce the quantitiesˆ K ( θ ) a ≡ e iκ ( θ ) λ a − , (11) K ( (cid:96) ) a ≡ (cid:90) θ ˆ K ( θ ) a J ( (cid:96)θ ) , (12)where κ are the convergence profiles, (cid:82) θ ≡ (cid:82) πθdθ and we have again suppressed the mass- and redshift-dependences.First, we consider a narrow bin of width dM dz inmass-redshift space, such that halo overlaps can be ne-glected for the infinitesimal number of halos in this bin.For a given realization of the halo distribution, we haveat the arbitrarily chosen origin : e iλ a κ ( ) = 1 + (cid:88) h ˆ K ( θ h ) a , (13)where the sum runs over all halos in the given mass-redshift bin and θ h is their separation from the origin.Thus, we find for the 1-point PDF P a = (cid:68) e iλ a κ ( ) (cid:69) h = 1 + dM dz n (cid:90) ˆ n ˆ K ( ˆ n ) a = 1 + dM dz nK (0) a , (14)where the subscript h indicates that we are averagingover realizations of the halo distribution. Likewise, wefind for the 2-point PDF P ab = (cid:68) e iλ a κ ( ) e iλ b κ ( φ ) (cid:69) h = 1 + dM dz n (cid:20) K (0) a + K (0) b + (cid:90) ˆ n ˆ K ( ˆ n ) a ˆ K ( ˆ n − φ ) b (cid:21) = P a P b (cid:20) dM dz n (cid:90) (cid:96) K ( (cid:96) ) a K ( (cid:96) ) b J ( (cid:96)φ ) (cid:21) , (15)where we have introduced (cid:82) (cid:96) ≡ (cid:82) (cid:96)d(cid:96)/ π . Note that tothe order considered so far, terms of the form 1+ dM dz A can equally well be written as exp dM dz A . Underthe Born approximation, the convergence is an additivequantity. Thus, the complete PDFs can be obtainedby convolution, which is equivalent to multiplication inFourier space: P a = exp (cid:90) M,z K (0) a , (16) P ab P a P b = exp (cid:90) M,z,(cid:96) K ( (cid:96) ) a K ( (cid:96) ) b J ( (cid:96)φ ) , (17)where we have introduced (cid:82) M ≡ (cid:82) dM n , (cid:82) z ≡ (cid:82) dz forbrevity. As we have demonstrated in Ref. [15], expandingthe exponentials to first order leads to the approximatemodel from Ref. [14]. In this sense, terms of order n p in the Taylor expansion of P can be interpreted as de-scribing overlaps of p halos along the line of sight.
2. Two-Halo Term
The two-halo term arises from the dependence of halodensity on the underlying long-wavelength linear densityfield, which at sky location ˆ n and redshift z we denoteby δ ( ˆ n ) ≡ δ lin ( ˆ n , z ) . (18)The change in the halo density can to a first approxima-tion be written as n → n ( ˆ n ) = n [1 + bδ ( ˆ n )] , (19)with b ≡ b ( M, z ) the linear halo bias. In order to com-pute the 2-halo term in the PDFs, we proceed in twosteps: first, we compute the correction to the 1-halo termin a given realization δ , obtaining P δ , and then we per-form the average over realizations. We note that in con-trast to Ref. [15] we denote by P δ only the multiplicativecorrection factor to the PDF. Using the substitution inEq. (19), the required correction factors can be writtendown immediately: P δa ( ) = exp (cid:90) z α a δ ( ) , (20) P δab ( ) = exp (cid:90) z α a δ (cid:0) − φ (cid:1) + α b δ (cid:0) φ (cid:1) + β ab δ ( ) , (21) where we have introduced α a ≡ (cid:90) M bK (0) a , (22) β ab ≡ (cid:90) M,(cid:96) bK ( (cid:96) ) a K ( (cid:96) ) b J ( (cid:96)φ ) . (23)We take the opportunity to point out a subtlety here:because we are working with a fixed realization δ at themoment, isotropy is broken and the PDFs depend explic-itly on sky location. Thus, we need to assume that thelinear density field δ varies sufficiently slowly that thehalo model formalism we have been assuming still makessense. As we will explicitly demonstrate in Appendix A 1,this assumption is equivalent to the statement that thelinear matter correlation function vanishes on scales sim-ilar to typical halo radii. Now all that is left to do isto perform the average over realizations of δ , giving theclustering corrections P = (cid:104) P δ (cid:105) δ . (24)We remind the reader that for light-cone integrals f i ( ˆ n ) = (cid:82) z W i δ ( ˆ n ) the Limber approximation allows usto write (cid:104) f i ( ) f j ( φ ) (cid:105) δ = (cid:90) z Hζ ( φ ) W i W j , (25)where ζ ( φ ) is the redshift-dependent line-of-sight pro-jected matter correlation function, which in terms of thelinear matter power spectrum can be written as ζ ( φ ) = (cid:90) k dk π P lin ( k, z ) J ( kφχ ( z )) . (26)Utilizing the Limber approximation and the identity (cid:104) e x (cid:105) = e (cid:104) x (cid:105) / valid for Gaussian distributed x , we ob-tain: P a = exp (cid:90) z Hζ (0)2 α a , (27) P ab P a P b = exp (cid:90) z H (cid:2) α a α b ζ ( φ ) + 12 β ab ζ (0)+ β ab ( α a + α b ) ζ (cid:0) φ (cid:1)(cid:3) . (28)This concludes the main theoretical part of this work.We would like to point out two features of the formalismpresented here: (1) Although we have assumed that halosare only described by their mass and redshift, one couldconsider additional labels c (e.g., related to halo envi-ronment or formation history). This would introduce a c -dependence in the halo mass function n and add fur-ther integrations over c on equal footing with the massintegrations (the redshift integrations are special becauseof the simplification introduced by the Limber approxi-mation); (2) The general formalism applies to any N -point PDF. For example, for the 3-point PDF one wouldhave to compute three-index objects γ ijk that are analo-gous to α i and β ij . However, the “momentum” labels (cid:96) would turn into vectorial quantities now, which presum-ably complicates the required integrations considerably. C. WL PDF Contributions from Non-VirializedMatter and Voids
The expressions above only account for the contribu-tions to the WL κ PDF due to matter in halos (“virializedmatter”). For the Compton- y field, this approximationwas very accurate due to the temperature dependence ofthe tSZ signal, which strongly biases the y field towardelectrons in massive halos. For κ , this approximation isless accurate, as the WL convergence field is an unbiasedtracer of the matter distribution. Moreover, in contrastto Compton- y , there are negative-signal regions in the κ field (i.e., projected under-densities in the matter distri-bution). We thus require some method to treat both the“matter outside of halos” and “voids”.With regard to the matter outside of halos, one optionwould be the following. We assume that the rest of the κ map (not accounted for by the halo-based model) ispurely a Gaussian random field (GRF) that is uncorre-lated with the virialized-halo part of the κ map. We cancompute the variance of this GRF (call it the “residualvariance”) by simply using the halo model to computethe angular power spectrum C κκ(cid:96) and truncating the halomodel integrals at the M values above which the explicitprofile-based calculation is used (so that the variance con-tributed by those objects is not double-counted). Alter-natively, we could compare the variance of the halo-modelPDF with the variance obtained from the “Halofit” fit-ting function [34, 35], and extract the residual variancefrom the difference of these quantities. After computingthe residual variance, a Gaussian PDF of this width canbe convolved with the halo-based PDF to obtain the final κ PDF.We implement both approaches described above. Wefind the changes in the PDF with respect to the unmod-ified halo model-only result to be extremely minor. Thefirst approach suffers from the problem that the halomodel becomes ill-defined in the low-mass regime rele-vant to this calculation, and thus the small residual vari-ance is quite uncertain. On the other hand, we frequentlyfind the variance computed from Halofit to be smallerthan the variance deduced from the halo-model PDF,invalidating the basic assumption. Thus, in this workwe do not include either of these ideas for incorporatingconvergence contributions from matter outside of halos,instead using only the halo model described in the previ-ous section. If exact results are necessary, we recommendtesting stability with respect to the lower limit in massintegrations.With regard to the negative- κ voids, a simple prescrip-tion is based on the fact that the mean (cid:104) κ (cid:105) = 0 by con-struction. Thus, to a first approximation, we simply com-pute the halo-model PDF as described above and thenshift it such that the physical constraint is enforced. Thisidea clearly fails to provide an accurate description of thenegative- κ tail of the PDF. It also is not immediately ob-vious that it leads to good predictions for the positive- κ tail, primarily because it does not take into account void- halo correlations. Thus, comparison to numerical simu-lations will be crucial in assessing the accuracy of thissimple approximation. III. RESULTS: ONE-POINT PDF
Before discussing various results obtained with theformalism developed in the previous section, we men-tion several choices for fitting functions and numericalsettings. For the halo mass function dn/dM and thelinear bias b , we use the fitting functions of Ref. [36].We describe halos with an NFW profile, using theconcentration-mass relation of Ref. [37]. We use theColossus package [38] for calculations in the halo model,and CAMB [39] and CLASS [40, 41] [with Halofit cor-rections 34, 35] for matter and WL convergence powerspectra. As mentioned before, the NFW profile ne-cessitates a radial cutoff; we find the WL convergencePDF to be nearly independent of this cutoff and chooseit at r max = 1 . r vir , where r vir is computed accordingto Ref. [42] and the prefactor of 1.6 is chosen to ob-tain good agreement between the halo model-computedand Halofit-computed WL convergence power spectra.Unless otherwise stated, all halo masses are given interms of M ≡ M , and we choose integration limits11 ≤ log M/h − M (cid:12) ≤
16, so that the PDF is very wellconverged (as will be shown in Sec. III C). For simplicity,we specialize to a Dirac- δ distribution of source galaxies, dn/dz = δ D ( z − z s ), at a single source redshift z s . Inorder to incorporate pixelization effects, we convolve theconvergence profiles with a window function W pix1pt ,(cid:96) = 4 π (cid:90) π/ dϕ sinc(cos( ϕ ) (cid:96)a )sinc(sin( ϕ ) (cid:96)a ) , (29)with a half the pixel sidelength. This prescription is onlyapproximate; there is no precise method to incorporatequadratic pixels while keeping the convergence profilesazimuthally symmetric. However, the error incurred isnegligible for the purposes of this work. Note that itwould also be irrelevant in any real parameter inference,since for realistic shape noise levels the Wiener filter onewould apply to the map (as described in Sec. V) cutsoff harmonic-space modes before the pixelization effectbecomes relevant.Having developed the analytic formalism in the previ-ous section, we now proceed to discuss various results inthe following subsections. In Sec. III A, we examine theeffect of corrections (overlaps and clustering) on the WLone-point PDF. In Sec. III B, we compare our model’spredictions to results from two sets of cosmological N -body simulations. In Sec. III C, we disentangle the con-tributions from different halo mass and redshift intervalsto the PDF. In Sec. III D, we discuss the dependence ofthe WL PDF on cosmological and concentration modelparameters. P D F P z s = 1.0fiducialno overlaps/clustering( M min = 10 h M ) no clusteringno overlaps/clustering( M min = 10 h M )0.0 0.1 0.2 0.30.50.0 P / P f i d z s = 1.50.0 0.2 0.410 P D F P z s = 2.00.0 0.2 0.4 0.6convergence 0.50.0 P / P f i d z s = 2.50.0 0.2 0.4 0.6convergence FIG. 1: The effects of clustering and overlaps for four differ-ent source redshifts z s . The PDF neglecting clustering ( green )seems to show similar behavior to what was observed in thetSZ case [15], but the clustering effect is much more pro-nounced. Note that the shift to (cid:104) κ (cid:105) = 0 does not make sensein the no-overlaps formalism: by assumption, the sky is in-finitely large so the condition is automatically satisfied (thiscorresponds to the divergence at κ = 0). A. Impact of Overlaps and Clustering
As discussed before, our formalism utilizes a halomodel based framework similar to the tSZ PDF calcu-lation in Ref. [14], with the crucial difference that weincorporate corrections arising from halo clustering andoverlaps along the line of sight (as developed in [15]).Naturally, we should examine the size of these correc-tions. In Fig. 1, we plot the exact result from our for-malism in red, while the result neglecting halo cluster-ing is represented in green. As noted in Ref. [14], theresult neglecting overlaps is only applicable if the mini-mum halo mass contribution is relatively large, thus, weplot two versions of the PDF neglecting both overlaps andclustering with different minimum masses in solid/dashedblue. These results are shown for four different choicesof source redshift ranging from z s = 1 to 2.5.We see that both clustering and overlaps constitutesubstantial corrections, of order a few 10 %. This is inmarked contrast to the conclusion we drew in the tSZcase, where the clustering effect was subdominant anddid not exceed a few percent. As we shall see in Sec. III C,the convergence PDF receives larger contributions fromlow-mass halos at lower redshifts in comparison to the P D F p () z s = 1.0MassiveNuS analytic(NFW) analytic(smoothed)0.0 0.2 0.4 z s = 1.50.0 0.2 0.4 0.6convergence 10 P D F p () z s = 2.0 0.00 0.25 0.50 0.75convergence z s = 2.5 FIG. 2: Comparison between the MassiveNuS simulations re-sults [43, black ] and our analytic model ( red ) for the WL con-vergence one-point PDF, for four different source redshifts.
Solid red : fiducial result of our model; dashed red : the re-sult obtained by smoothing the NFW density profiles witha k -space filter calibrated on the convergence power spectrameasured in MassiveNuS, as described in the text and illus-trated in Fig. 3. This filter captures the non-negligible effectsdue to the finite resolution of the simulation. Here, as wellas in the other plots in which we show simulation data, theerror bars would be invisible by eye. tSZ PDF, which explains the more pronounced clusteringcontribution. We note the unphysical divergence of theresults neglecting overlaps near κ = 0, which is removedby the improved model presented in this work. B. Comparison to Numerical Simulations
We compare results from our formalism to WL con-vergence PDFs extracted from two different sets of N -body simulations, namely MassiveNuS [43, Sec. III B 1]and Takahashi et al. (hereafter T17) [44, Sec. III B 2].Both of these simulations provide ray-traced WL conver-gence maps. We provide further details on each simula-tion analysis below. http://astronomy.nmsu.edu/aklypin/SUsimulations/MassiveNuS/ http://cosmo.phys.hirosaki-u.ac.jp/takahasi/allsky_raytracing/ ( + ) C / z s = 1.0MassiveNuS analytic(NFW) analytic(smoothed) Halofit z s = 1.510 multipole 10 ( + ) C / z s = 2.0 10 multipole z s = 2.5 FIG. 3: Convergence power spectra.
Blue is the Halofit re-sult, while solid red is our fiducial analytic result from thehalo model. The MassiveNuS power spectra ( black ) show adeficiency in power at (cid:96) (cid:38) , likely due to resolution effects.The dashed red lines are analytic power spectra obtained bysmoothing the NFW density profiles, as described in the text,in order to mimic the resolution effect.
1. Comparison to MassiveNuS
We analyze a set of 10 ray-traced weak lensing conver-gence maps from the MassiveNuS simulation suite, whichare derived from a set of N -body simulations that includedark matter and an approximate treatment of massiveneutrinos via a linear response method [43]. Each con-vergence map is 3 . × . with 512 square pixels,corresponding to pixel side-length 0 .
41 arcmin. The ef-fect of the pixel window is treated in our analytic calcu-lations via Eq. 29. The simulations include maps for awide range of cosmological parameters, but we consideronly the fiducial simulated cosmology, with parametersgiven by Ω m = 0 .
3, Ω b = 0 . h = 0 . A s = 2 . × − , n s = 0 .
97, and zero neutrino mass ( σ = 0 . δ -function source planes at various redshifts. We consider κ values ranging from [ − σ κ , σ κ ], where σ κ is the vari-ance of the κ field measured from the full simulation setfor each source redshift option. The bins are linearlyspaced with width σ κ /
5. In all PDF measurements, weenforce the constraint that (cid:104) κ (cid:105) = 0.To ensure that the simulation results are robust to cos-mic variance fluctuations resulting from the small map size, we also analyze a set of 10 convergence mapsthat were produced for covariance matrix estimation atthe fiducial cosmology, using additional, independent N -body simulations. We sub-divide this large set into 10subsets of 10 maps each, and verify that any fluctu-ations in the measured κ PDFs across the subsets arenegligible.In Fig. 2, we plot a comparison between the fiducial an-alytic one-point PDF (solid red) and the PDF measuredin MassiveNuS (black), for four different source redshifts, z s = 1 , . , , .
5. While the discrepancies at negative κ are entirely expected, the large differences in the positive- κ tail are not expected given our intuition that the halomodel should perform very well in this regime. In orderto explain these discrepancies, we plot WL convergencepower spectra in Fig. 3. We observe good agreementbetween the Halofit result (blue) and the fiducial result(solid red) computed using the standard halo model ex-pressions [e.g., 46]. On the other hand, MassiveNuS lackspower for (cid:96) (cid:38) . This is likely related to small-scaleresolution effects in the simulation, presumably a combi-nation of finite mass resolution and force softening [e.g.,47, 48]. As a simple test of whether these resolution ef-fects can explain the discrepancies seen in the one-pointPDFs, we calibrate a k -space filter with which we smooththe NFW density profiles such that the resulting conver-gence power spectra match the MassiveNuS results. Wefind the filter W ( k ) = [1 + ( kR ) ] − . , (30)where R = 0 . h − Mpc comoving, to yield relativelygood agreement. The resulting power spectra are plot-ted in dashed red in Fig. 3. Having calibrated the filter W ( k ) on the power spectra, we then compute the re-sulting one-point PDF, plotted in dashed red in Fig. 2.We observe much better agreement now. In the part ofthe PDF that is relatively close to Gaussian the analyticresult matches the simulations almost exactly. Small dis-crepancies remain in the high- κ tail, which is not surpris-ing since our smoothing filter was calibrated solely on thetwo-point correlation function, while the one-point PDFin the tail depends strongly on higher-order correlationfunctions. Thus, we conclude that the resolution effectsleading to lack of power at high (cid:96) are likely responsiblefor the high- κ discrepancy between our model and thesimulation result, rather than a deficiency of our halomodel formalism.
2. Comparison to T17
We analyze a set of 108 full-sky, ray-traced weak lens-ing convergence maps from T17 [44], which are derivedfrom a suite of large dark-matter-only N -body simula-tions. The maps are provided in HEALPix [45] formatat resolution N side = 8192, corresponding to an approx-imate pixel scale of 0 .
43 arcmin. The parameters usedin the simulations are Ω m = 0 . b = 0 . h = 0 . P D F p () FWHM = None T17analytic(NFW) analytic(smoothed)analytic(var matched)0 0.1 0.2convergence 10 P D F p () FIG. 4: Comparison between the T17 simulation results [44,black] and the analytic result (red) for the convergence one-point PDF, for four different Gaussian smoothing scales la-beled by their FWHM in the panels.
Solid red : fiducial resultof our model; dashed red : the result obtained by smooth-ing the NFW density profiles; dotted red : the result obtainedby convolving with a Gaussian such that the final variancematches the variance measured in the simulation. Note thatin the top and bottom left panels the analytic variance isslightly larger than the one measured in the simulations, thusno dotted line is plotted. σ = 0 . n s = 0 .
97, and zero neutrino mass. We matchthese parameters in all analytic calculations that com-pare to T17.In our comparison to the T17 simulations we focus onthe effect of smoothing the convergence maps. Thus, weonly produce results for a single source redshift, z s =1 . κ values ranging from [ − σ κ , σ κ ], where σ κ is the variance of the κ field measured from the fullsimulation set for each choice of smoothing filter. Thebins are linearly spaced with width σ κ /
5. In all PDFmeasurements, we enforce the constraint that (cid:104) κ (cid:105) = 0.The results are plotted in Fig. 4. Black curves are sim-ulation results, while red is the analytic result. Focusingon the upper panel (where no additional smoothing hasbeen applied to the maps apart from the inherent pix-elization, which is treated via Eq. 29), we again observea discrepancy between the fiducial analytic result (solidred) and the simulation. As a further test to our hy-pothesis that the discrepancy can be explained with sim-ulation resolution effects, we again construct a k -spacesmoothing filter for the NFW profiles. We choose the redshift-dependent filter W ( k ; z ) = 11 + kR ( z ) ; (31) R ( z ) = 0 . h − Mpc × [log(1 + z ) + 0 . . The function R ( z ) was chosen to give a good fit to thesoftening lengths employed in the T17 simulation, withsome adjustments of the prefactor (our R is about 10 %smaller than the softening length). We find the resultingone-point PDF to be largely independent of the precisefunctional form chosen for W ( k ; z ), as long as it decreasessteadily to ∼ . kR ∼
1. The natural correspon-dence between the smoothing scale R and the softeninglength is a further indication that simulation resolutioneffects are responsible for the observed discrepancies inthe one-point PDF.A second purpose of this section is to evaluate howwell our formalism can describe the PDF of convergencemaps smoothed with a Gaussian filter. As we explicitlyshow in Appendix A 2, as the smoothing scale increasesthe PDF becomes closer to Gaussian (as is physically ex-pected) and receives larger contributions from the two-halo term. These facts imply that our formalism, whichis most accurate for the non-Gaussian parts of the PDFthat are dominated by massive halos, is expected to per-form worse. Thus, comparison to simulated maps is auseful test of the domain of validity. This is plottedin the lower three panels of Fig. 4. We observe that,as should be expected, the difference between solid anddashed red becomes negligible as the smoothing scale in-creases. However, our formalism does not recover thesimulation PDFs very well. As the smoothing scale in-creases, the PDF receives more and more contributionsfrom non-virialized matter, which is not included in ourhalo-model formalism. One attempt to solve this prob-lem is to convolve the analytic PDF with a Gaussian suchthat the resulting variance (cid:104) κ (cid:105) is equal to the variancemeasured in the simulation. The results of this procedureare plotted as dotted red lines. Although the agreement(naturally) gets better, it is still far from perfect. Onepossible explanation is the fact that we do not describethe negative- κ regime accurately enough in our formal-ism, and upon convolution with a relatively broad Gaus-sian this inaccuracy leaks into the positive- κ part.Thus, we draw two conclusions from our comparisonwith the T17 simulations: 1) we have presented furtherevidence that convergence one-point PDFs measured insimulations are susceptible to large errors due to small-scale resolution effects. Thus, the discrepancies observedin Figs. 2 and 4 do not invalidate our analytic formalism.2) Our approach seems inadequate to generate accuratepredictions for the PDF of convergence maps smoothedover scales larger than a few arcminutes. Perturbativemethods are likely better suited to compute theoreticalpredictions in this regime. l o g M / h M z s = 1 z s = 2.5 l o g M / h M z l o g M / h M z p ( ) from redshifts z and masses M FIG. 5: Cumulative mass and redshift contributions to thePDF for source redshifts 1 and 2.5 (columns). The rows rep-resent different values of the convergence in units of the stan-dard deviation. Note the different vertical scales in differentrows.
C. Halo Mass and Redshift Contributions
We now proceed to build up some physical intuitionon the dominant contributions to the convergence PDF.Since we label halos solely by their mass and redshift,we disentangle the contributions that different mass andredshift intervals give to the PDF. In Fig. 5, we plotheatmaps of the mass and redshift contributions to thePDF for source redshifts z s = 1 and 2 .
5. Each of thethree rows represents a different bin of the PDF at com-parable values of the convergence κ in units of the respec-tive standard deviation σ κ . Each pixel in the individualheatmaps corresponds to the fraction of the final value ofthe PDF if all masses and redshifts smaller than or equalto the one corresponding to the pixel are included (thus,the upper right corner has by definition a value of one ineach heatmap). Note that overlaps make the interpreta-tion of these heatmaps somewhat complicated, in partic-ular for low values of κ . In terms of redshift evolution,we can clearly see the growth of structure modulated bythe lensing kernel. In terms of mass contributions, theintuitive picture that higher values of κ are sourced bymore massive objects is confirmed. p () A s A smm mm variedcosmology AABBCC variedconcentration0.00 0.25 0.50 0.75convergence 0.500.751.001.251.501.75 p / p f i d m : ×0.25 0.00 0.25 0.50 0.75convergence A : ×0.50 10 20 30/ 0 10 20 30/ FIG. 6: Dependence of the convergence one-point PDF oncosmology ( left panel ) and concentration model ( right panel ).For an explanation of the parameters
A, B, C see the text.
Solid/dashed lines represent in/de-creases of the respectiveparameter by 10 %, except for the neutrino mass sum Σ m ν ,for which the fiducial model is 0 .
06 eV while solid/dashed rep-resent 0 .
12 and 0 eV respectively.
D. Parameter Dependence
In this section, we discuss the dependence of the con-vergence one-point PDF on the cosmological model aswell as the halo concentration-mass relation. The resultspresented here are a pre-requisite for the Fisher forecastin Sec. V, but are also useful as a means to build upintuition. We show results for a single source redshift z s = 1.We choose our fiducial cosmology as h = 0 .
7, Ω m =0 .
3, Ω b = 0 . A s = 2 . × − , n s = 0 .
97, andΣ m ν = 0 .
06 eV. We assume the normal hierarchy forthe neutrino masses. Following Ref. [37], we write theconcentration-mass relation as c ( M, z ) = A (cid:18) MM (cid:19) B (cid:18) z z (cid:19) C , (32)where we choose M = 10 . h − M (cid:12) , z = 0 .
35, so as tobreak the leading degeneracy between the three parame-ters
A, B, C (c.f. Fig. 5).Our results are shown in Fig. 6, with varied cosmologyin the left panel and varied concentration model in theright panel. The solid/dashed lines generally representparameter variations by ±
10 %, except for the neutrinomass sum, where solid corresponds to 0 .
12 eV and dashedto massless neutrinos. Note that the residual curves for0Ω m and A were shrunk by the stated factors to increasereadability. With regard to varied cosmological param-eters, with a fixed fractional change Ω m has by far thestrongest influence on the PDF. From the slight shapedifference in the residual curves, we can hope that thedegeneracy between neutrino mass sum and other pa-rameters is not too large. With regard to the concentra-tion model, the amplitude has by far the largest effect,while the variation with halo mass and redshift are ofminor importance. This finding rests on the fact thatthe mass- and redshift-integrands are relatively sharplypeaked around M , z for our simple Dirac- δ source dis-tribution (c.f. Fig. 5), a condition that may not be metin the case of a more realistic source distribution. IV. RESULTS: TWO-POINT PDF
As numerical evaluation of the two-point PDF is some-what involved, we collect some useful simplifications inAppendix B. In order to validate the analytic formalismfor the two-point PDF, and by extension the covariancematrix, we perform two tests, detailed in the following.
A. Two-point correlation function
In Fig. 7, we plot a comparison between the (isotropic)two-point correlation function obtained in our formalismwith the result predicted by the Halofit fitting function(for sources at z s = 1). Note that while the 1-halo termcan be directly computed in our formalism, the 2-haloterm is simply inferred as the difference of the other twocurves. The top x -axis in the figure indicates the comov-ing scale at the approximate redshift where the main con-tribution to the one-point PDF is sourced. The small dis-crepancies on large scales are due to the relatively largeminimum halo mass chosen in the computation. We con-firmed that the precise choice of this cut-off has no effecton the covariance matrix. Note that the relatively goodagreement on large angular scales is a good indicationthat we are treating the halo clustering effect correctly(in fact, this is the first direct validation, since the com-parison of the one-point PDF to numerical simualations iscomplicated by simulation resolution effects). However,as we show in Appendix A 1, the correlation function isnot sensitive to all terms in the two-point PDF. The dis-crepancies on small scales are likely stemming from theHalofit rather than the halo model side, since we do notexpect Halofit to be accurate on these rather non-linearscales. (Another possible explanation would be devia-tions from the NFW profile on small scales in the nu-merical simulations used to calibrate Halofit.) We pointout that this type of plot is useful to identify numericalinstability in a calculation of the covariance matrix. angular separation [arcmin]10 c o rr e l a t i o n f un c t i o n C () comoving scale at z = 0.35 [Mpc] FIG. 7: Comparison between the 2-point correlation func-tion reduced from our formalism for the 2-point PDF and theHalofit computation, for sources at z s = 1. Small discrepan-cies on large scales are related to the choice of radial cutoffor minimum halo mass. The discrepancies on small scalesare likely due to inaccuracies of Halofit in the very non-linearregime. B. Covariance matrix
Given the one- and two-point PDFs, the covariancematrix of the one-point PDF can be computed asCov ab = 1 N pix (cid:88) φ [ P ab ( φ ) − P a P b ] . (33)Here, the indices label the κ -bins, and the sum runs overall pixel separations in a given map. N pix is the numberof pixels in the map, i.e. related to the sky coverage. Inpractice, it is accurate enough to explicitly perform thesum over pixels for the smallest separations and approxi-mate the remaining summation by an integral. Note that P ab ( φ = 0) = P a δ ab .We measure the covariance matrix using the 108 full-sky T17 simulations and compare it to our analytic result.As discussed in Sec. III B 2, resolution issues appear tocause a discrepancy between the analytic result and theT17 (and also MassiveNuS) simulations. Thus, in orderto make the comparison as direct as possible, we computethe covariance matrix with the smoothed NFW profilesdescribed before, corresponding to the dashed red line inthe top panel of Fig. 4.In the bottom panel of Fig. 8, we show a comparison ofthe correlation matrices (i.e., Cov ab / √ Cov aa Cov bb ). We1 C o v aa / P a analyticT17 simulations0.0 0.1 0.2 0.3convergence 0.00.10.20.3 1.000.750.500.250.000.250.500.751.00 c o rr e l a t i o n FIG. 8: Comparison of the analytic and T17 covariance matri-ces.
Top panel : diagonal elements, rescaled by the respectiveone-point PDFs;
Bottom panel : correlation matrices (T17 inupper left triangle, analytic in lower right triangle; the colorcodes are as in the top panel). observe good agreement in the general structure. Theanalytic model is able to recover the transition to anti-correlation at low κ (albeit displaced by about one bin).The simulations appear to have more correlations in thehigh- κ bins, although these bins are noisy and dominatedby rare events in the simulations. There appears to bea step-like feature in the simulation covariance matrix at κ ∼ .
23, transitioning rather suddenly from high to lowcorrelation. The analytic model does not predict sucha feature, and since the halo model should work well atthis relatively large convergence, we are inclined to trustthe model more than the simulations.In the top panel of Fig. 8, we show a comparison be-tween the diagonal elements (we divide out the respectiveone-point PDFs). While the model recovers the diagonalelements to better than an order of magnitude (and towithin ∼
10 % for κ (cid:38) . κ are likely due to failure of the model. However,for κ (cid:38) .
1, we would expect the halo model to performwell, while, as we have already seen in Sec. III B 2, thesimulations are susceptible to resolution issues. Although we have tried to take these into account by smoothing theNFW density profiles with a filter calibrated on the one-point PDF, it is likely that the covariance matrix dependsdifferently on this filter and thus systematic discrepanciesare still to be expected. This also highlights the possibledangers in purely simulation-based inference procedures,which could be biased by such resolution effects.
V. FISHER FORECAST
In this section, we present a simple Fisher matrix pa-rameter forecast using the WL convergence PDF. We as-sume a Rubin Observatory-like survey with sky coverage20 ,
000 deg , pixel size Ω pix = (0 .
41 arcmin) and sourcedensity n gal = 45 arcmin − . The latter number is takenfrom Ref. [11]; in contrast to this work we make the sim-plifying assumption of a Dirac- δ source distribution at z s = 1. (Note that the inclusion of tomographic informa-tion would further improve the forecast error bars com-puted here.) Our fiducial cosmological model is the sameas in Sec. III D.In order to upweight the cosmological signal with re-spect to the shape noise, we convolve the convergenceprofiles with a Wiener filter, constructed as F (cid:96) = C κκ(cid:96) / ( C κκ(cid:96) + N (cid:96) ) , (34)where we compute the convergence power spectrum usingHalofit and the noise power spectrum N (cid:96) = 0 . /n gal isflat. Applying the Wiener filter is crucial in optimizingthe sensitivity of the one-point PDF, since it is a real-space statistic.The shape noise in the filtered maps has correlationfunction ζ κ, sn ( φ ) = (cid:88) (cid:96) (cid:96) + 1 / π N (cid:96) F (cid:96) W pix2pt ,(cid:96) P (cid:96) (cos φ ) , (35)where W pix2pt ,(cid:96) is the pixel window function, W pix2pt ,(cid:96) = 4 π (cid:90) π/ dϕ [sinc(cos( ϕ ) (cid:96)a )sinc(sin( ϕ ) (cid:96)a )] , (36)with a half the pixel sidelength. The “noisy” one-pointPDF and covariance matrix are computed as p sn ,a = ( G [ P i ]) a (37)Cov sn ,ab = (cid:88) φ (cid:2) ( G ( φ )2 [ P ij ( φ )]) ab − ( G [ P i ]) a ( G [ P j ]) b (cid:3) , (38)where G n [ · ] indicates convolution with an n -dimensionalGaussian. Here, G has variance ζ κ, sn (0) and G ( φ )2 hascovariance matrix C ( φ )sn = (cid:18) ζ κ, sn (0) ζ κ, sn ( φ ) ζ κ, sn ( φ ) ζ κ, sn (0) (cid:19) . (39)2 p s n () A s A smm mm variedcosmology AABBCC variedconcentration0.00 0.05 0.10 0.15convergence 201001020 ( p s n p f i d s n ) / d i a g ( C o v s n ) m : ×0.25 0.00 0.05 0.10 0.15convergence A : ×5 B : ×10 C : ×10 10 R u b i n O b s . p i x e l s FIG. 9: The convergence one-point PDFs, as a function ofparameter variations, including shape noise of σ κ, sn = 0 . κ -axis are not directlycomparable. As in Fig. 6, all parameter variations are 10 %,except for Σ m ν for which it is 100 %. The bottom row showsthe ratio of the residuals to the square-root of the diagonalelements of the covariance matrix. In Fig. 9, we plot the same PDFs already shown inFig. 6, this time including the shape noise contribution.We bin the PDFs into bins of width ∆ κ = 1 . × − ,and quote the number of pixels expected in the modelsurvey. Again, the parameter variations are 10 % for allparameters except the neutrino mass sum, for which thevariation is 100 % (dashed is massless neutrinos, solid isΣ m ν = 0 .
12 eV). Note that, for clarity, we rescaled someof the curves in the bottom row of Fig. 9 by the givenamounts. It is interesting to see that the Wiener filternot only serves the purpose of minimizing the noise con-tribution, but also increases the sensitivity of the PDFon cosmology in comparison to the concentration model(compare Fig. 6, which has the same parameter varia-tions). This can be understood as a consequence of thesuppression of small-scale power which makes the exactshape of the convergence profiles (and thus the concen-tration model) less relevant, while the main dependenceon the cosmological model comes through the halo massfunction. This will be crucial for the parameter forecast.We compute the Fisher matrix as F ab = ∂ p T ∂θ a Cov − ∂ p ∂θ b + 12 tr Cov − ∂ Cov ∂θ a Cov − ∂ Cov ∂θ b , (40)with p the binned one-point PDF and θ the parameter σ ( X ) /X fid [%] A s Ω m Σ m ν A B C no priors 11 2.1 490 9.3 30 58+Duffy2008 8.5 1.1 350 1.9 6.7 8.1+Planck2018 ( A s ) 1.4 1.9 230 8.8 27 58+Duffy2008+Planck2018 ( A s ) 1.4 1.0 132 1.7 6.6 8.0TABLE I: 1 σ constraints on cosmological and concentrationmodel parameters, for four different choices of priors. Thenumbers are relative to the fiducial value, in percent. vector (indexed by a, b ). Contrary to conventional wis-dom, the second term in the Fisher matrix is not alwaysnegligible and changes parameter constraints by a few10 %. In the data vector, we include the PDF in the in-terval κ ∈ [ − . , . κ part of thePDF, which is not well described by our model. How-ever, due to the Wiener filter and noise convolution it israther challenging to cleanly exclude this regime, and aswe will argue below we do not believe that keeping theuncertain values invalidates the major conclusions fromour forecast.We consider six free parameters, namely { A s , Ω m , Σ m ν } on the cosmology side and { A, B, C } in the parameterization of the concentration modelfrom Ref. [37]. For the concentration model, we choosethe mass- and redshift-pivots described at the end ofSec. III D (in the end we rescale A to the original value).In Fig. 10, we plot four different Fisher forecasts, dif-fering solely by the priors we place on the concentrationmodel and the scalar fluctuation amplitude A s . Blackincludes no priors at all, magenta includes the error barson A , B , C from Ref. [37] as diagonal Gaussian priors,green includes the CMB prior from Ref. [49] on A s , andorange includes both priors. For clarity, in Table I wequote the fractional 1 σ constraints (in percent) on thethree cosmological parameters as well as the concentra-tion model parametrization for the different choices ofpriors. We observe that if priors on the concentrationmodel as well as A s are included, our results suggest thatthe WL convergence PDF alone can provide an error baron Σ m ν comparable to the minimum possible neutrinomass sum. Including tomographic information and/orthe full WL convergence power spectrum would only im-prove these constraints.Our simple Fisher forecast has several limitations: • We are assuming a Gaussian likelihood even thoughwe know that the one-point PDF is a non-Gaussianstatistic. Unfortunately, the full likelihood for thisobservable is not yet known. Evidence of a smallbias when assuming a Gaussian likelihood was seenfor the tSZ PDF in Ref. [14]. However, a Gaussianlikelihood was used and shown to be unbiased forthe WL PDF in [11], albeit with the caveat thathigh- κ bins were removed, which Gaussianizes the3 no priors+Duffy2008 ( A , B , C )+Planck2018 ( A s )+Duffy2008 ( A , B , C )+Planck2018 ( A s ) m m [ e V ] A B A s [10 ] C m m [ eV ] A B C FIG. 10: Four Fisher forecasts forparameter constraints from the weaklensing one-point PDF. We are as-suming Gaussian shape noise as de-scribed in the text, Rubin Obser-vatory sky coverage of 20 ,
000 deg ,and a source density of 45 arcmin − in a Dirac- δ distribution at redshift z s = 1. The ellipses are 1 σ con-fidence intervals. Black : both cos-mology and concentration model arecompletely inferred from the data; magenta : including simulation priorsfrom Ref. [37] on the concentrationmodel; green : including CMB priorfrom Ref. [49] on A s ; orange : includ-ing both priors on the concentrationmodel and A s . statistic. The PDF observable could be a usefulopportunity to apply new methods in likelihood-free inference (e.g., [29, 50]). • We work in the Fisher approximation; however,MCMC results from Ref. [11] indicate that this isnot a bad approximation. • Our analytic covariance matrix is likely not ex-act for small values of κ ; however, our resultsfrom Sec. IV B indicate that the formalism tends tooverestimate the covariance matrix in this regime,which implies that in this respect our forecast pa-rameter constraints are conservative. • Our formalism is unable to make exact predictionsfor the negative- κ tail. However, this loss of detailalso is likely to imply that our forecast is conserva-tive. Including full information from the negative- κ region would only improve the constraints foundhere. • We make the simplifying assumption of a Dirac- δ source distribution in a single bin. Accounting fora realistic spread in the source distribution wouldlikely not significantly affect the constraints; more-over, Ref. [11] has demonstrated that tomography with multiple source bins has the potential to sig-nificantly improve the constraints. • It is not clear that the Wiener filter employed isoptimal. One possibility would be to consider thePDFs of maps smoothed on various scales simulta-neously. This approach should recover some of thescale-dependent information contained in the full N -point functions that is lost when compressing tothe zero-lag one-point PDF. The cross-covariancebetween the PDFs should be easy to compute in asimple extension of the formalism presented above(all terms in our model for the covariance matrixare symmetric under interchange of two conver-gence profiles, the modification should amount tobreaking that symmetry by filtering them with twodifferent kernels). • We do not consider any cosmology-dependence ofthe concentration model (although this is likelysmall), and our prior on the concentration modelparameters may be optimistic, particularly givenbaryonic effects on the small-scale matter distribu-tion.The last point implies that a more robust understand-ing of the concentration-mass relation and halo density4 min m a x f i g u r e o f m e r i t [ ] FIG. 11: Dependence of the normalized 3-parameter fig-ure of merit (FOM), as defined in Eq. (41), on the min-imum/maximum κ -cutoff. We observe that the negative- κ part of the PDF contains substantial information, and thatour fiducial choice of κ max = 0 .
17 extracts essentially all ofthe information content (assuming κ min = − . profiles in general would be extremely beneficial for pa-rameter inference from the WL PDF, similar to the WLpower spectrum.Our forecast is similar in set-up to the simulation-basedone presented in Ref. [11]. However, there are a few keydifferences: (1) they do not make the simplifying assump-tion of a Dirac- δ source distribution; (2) they filter theconvergence maps with a different (cid:96) -space filter; (3) theyare able to use more of the negative- κ regime; (4) they(naturally) cannot marginalize over small-scale model-ing uncertainties; (5) they do not include noise correla-tions, which we find to have an appreciable effect. As aconsequence of these differences, we find the agreementbetween our results and theirs to be satisfactory. For arough qualitative comparison, we consider our result withthe concentration-model prior (i.e. the second row in Ta-ble I), where we have σ (10 A s ) = 0 . σ (Ω m ) = 0 . σ (Σ m ν ) = 0 .
21 eV. We find that including the effect ofnoise correlations gives about a factor of 2 degradationin constraints, which is approximately offset by increas-ing the source number density by a factor 4. Thus, forthe purposes of this rough comparison, we choose theturquoise contours in Fig. 3 of Ref. [11] as reference (thesedo not include noise correlations and have a source den-sity of 13 .
25 arcmin − ). Approximating the posterior asGaussian, we read off σ (10 A s ) = 0 . σ (Ω m ) = 0 . σ (Σ m ν ) = 0 .
12 eV. Thus, our halo model-only forecastreproduces these simulation-derived constraints to withina factor of two. Note, however, the difference in orienta-tion of ellipses involving Ω m .As a final result of this section, we explore the con- straints’ dependence on the choice of the minimum andmaximum convergence values in the data vector p . Weconsider the 3-parameter figure of merit,FOM = (cid:0) det F − (cid:1) − / , (41)where F cosmo is the sub-block of the Fisher matrix corre-sponding to the cosmological parameters { A s , Ω m , Σ m ν } .We work with the maximum set of priors, correspondingto the orange lines in Fig. 10. A heatmap of this quantityas a function of minimum and maximum cut-off is shownin Fig. 11. Again, we emphasize that the “convergence”values quoted there are related through the Wiener filterto the physical convergence, making the interpretationsomewhat difficult. The first conclusion from Fig. 11 isthat the minimum cut-off κ min is rather important forthe constraining power. Second, for the minimum cut-offchosen in our analysis above, κ min = − .
03, the informa-tion content of the positive- κ tail is essentially saturatedwith our choice of κ max = 0 .
17. In fact, using a somewhatsmaller value of κ max could serve to Gaussianize the like-lihood for the WL PDF, evidently without a significantloss in constraining power. VI. CONCLUSIONS
We have developed a halo model-based formalismfor the weak lensing convergence one-point and two-point PDFs (and, by extension, the covariance matrixof the one-point PDF). The strengths of our model liein its superior speed compared to simulations, the abil-ity to explicitly marginalize over small-scale uncertainties(parametrized through the concentration model), and itsinterpretability.As expected on physical grounds, the accuracy of ourmodel is highest in the positive-convergence tail. We haveshown that, in this regime, discrepancies in the one-pointPDF with respect to numerical simulations are explainedby simulation resolution issues and do not invalidate ourmethod. It may be argued that as soon as the conver-gence map is smoothed on a sufficient scale, the simula-tion resolution effects would be less severe. However, atthe wavenumbers at which the MassiveNuS simulationsstart to show appreciable power deficiencies, the Wienerfilter employed in this work still assumes values of or-der 0.1; thus, smoothing on a reasonable scale appearsnot sufficient to neutralize the small-scale issues in thesimulations.On the other hand, in the negative convergence regimeand for large smoothing scales our formalism is less accu-rate. Alternative approaches are likely better suited foraccurate theoretical predictions in these regimes.Validation of the covariance matrix was found to bechallenging, and discrepancies with respect to the simu-lations remain over a range of convergence values. How-ever, discrepancies at low κ are likely irrelevant in anyreal analysis due to the dominance of the shape noise5contributions there. The smaller discrepancies at high κ could simply be due to resolution effects in the simula-tions, as we already demonstrated for the one-point PDFitself.Using our formalism, we have performed a Fisher fore-cast in the { A s , Ω m , Σ m ν } parameter space for a RubinObservatory-like survey. We have found that the conver-gence PDF alone could provide a 1 σ error bar on Σ m ν that is comparable the minimum neutrino mass sum al-lowed from oscillation experiments, if a CMB prior on A s and simulation priors on the concentration-mass relationare included. Our results are in good agreement with pre-vious simulation-based forecasts, and could be generatedin a fraction of the time. We have also presented argu-ments why the several limitations of our simple forecastare likely to render it conservative, i.e., a more sophis-ticated analysis would probably find a further improve-ment in constraints (although this gain could be negatedby the inclusion and marginalization of systematic errorsin the measurement of convergence values). Our workdemonstrates that an analytic approach to non-GaussianWL statistics is feasible for upcoming surveys, at leastin terms of the statistical constraining power. Tests forbiases will necessitate end-to-end simulation analyses, be-yond the scope of this work.We believe that a comprehensive model for the weaklensing convergence one-point PDF and its covariancematrix would be most accurate if it combines differentapproaches. For example, one could imagine taking sim- ulation results for the negative-convergence and Gaus-sian part of the PDF, while the positive-convergence tailis generated with our formalism. A desirable side effectof this method could be that smaller simulation volumesare required in order to sample the quasi-linear part ofthe density field.Possible extensions of our model could involve the useof compensated density profiles [e.g., 51] (which couldhelp improve accuracy near the Gaussian peak of thePDF), the effective halo model approach from Ref. [52],and the inclusion of voids. Acknowledgments
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We can show quite easily that the 2-point function isgiven by C ( φ ) = (cid:2) ∂ ab log P ab (cid:3) λ a = λ b =0 , (A1)where ∂ a ≡ − i∂/∂λ a . Note that the 1-point factors in the2-point PDF give no contribution, since their logarithmis a sum of functions that depend only on λ a or λ b . Sincethe Fourier-space PDF P ab is a product of one- and two-halo term, the correlation function becomes a sum.The 1-halo term is given by Eq. (17):log P ab ⊃ (cid:90) M,z,(cid:96) K ( (cid:96) ) a K ( (cid:96) ) b J ( (cid:96)φ ) . (A2)Performing the differentiation, we obtain C ( φ ) = (cid:90) (cid:96) J ( (cid:96)φ ) (cid:90) M,z (cid:20)(cid:90) θ κ ( θ ) J ( (cid:96)θ ) (cid:21) , (A3)which is equivalent to the standard expression C ( (cid:96) ) = (cid:90) M,z | ˜ κ (cid:96) | . (A4)The 2-halo term is given by Eq. (28):log P ab ⊃ (cid:90) z H (cid:2) α a α b ζ ( φ ) + 12 β ab ζ (0)+ β ab ( α a + α b ) ζ (cid:0) φ (cid:1)(cid:3) . (A5)Since α a = 0, β ab = ∂ a β ab = 0 when λ a = λ b = 0, onlythe first term in the square brackets survives, and we get C ( φ ) = (cid:90) z Hζ ( φ ) (cid:20)(cid:90) M,θ bκ ( θ ) (cid:21) . (A6)Transforming to conjugate space, this gives C ( (cid:96) ) = (cid:90) z Hχ P lin ( (cid:96)/χ, z ) (cid:20)(cid:90) M,θ bκ ( θ ) (cid:21) . (A7)We see that this is not exactly equal to the usual halomodel calculation, which would have an extra Besselfunction in the square brackets. The difference, as re-marked above, arises from the fact that we approximatethe linear overdensity field as approximately constantover the typical size of a halo, so that the linear powerspectrum becomes negligible whenever the argument (cid:96)θ of the neglected Bessel function would become apprecia-ble.7
2. Large smoothing limit
In this section we briefly discuss how smoothing theconvergence field with a Gaussian filter of large apertureaffects the 1-point PDF. Denoting the smoothing scaleby σ , we have for the smoothed convergence profiles κ σ ( θ ) = Gaussian σ ( θ ) ∗ κ ( θ ) , (A8)which, after inserting the conjugate space expressions,leads to κ σ ( θ ) = (cid:90) θ (cid:48) ,(cid:96) κ ( θ (cid:48) ) J ( (cid:96)θ ) J ( (cid:96)θ (cid:48) ) e − (cid:96) σ / . (A9)In general, the ratio θ (cid:48) /σ will attain its maximum when θ (cid:48) is comparable to the projected halo radius. Of course,this varies with halo mass and redshift, but it is still rea-sonable to formally introduce a scale ˆ θ that characterizestypical halo extents. Since we assume σ to be large, wecan expand in ˆ θ/σ .The zero-order term in the expansion parameter ˆ θ/σ is given by κ (0) σ ( θ ) = σ − e − θ / σ ¯ κ , (A10)where we have introduced¯ κ ≡ (cid:90) θ κ ( θ ) . (A11)It simply measures the total amount of signal in a singlehalo, which is then smoothed into a Gaussian by the σ -filter.We now compute the n -th order cumulants k n . Simi-larly to the power-spectrum calculation performed in theprevious section, the cumulants are related to derivativesof log P , so that the cumulants naturally split into one-and two-halo terms: k n = (cid:2) ∂ na log P a (cid:3) λ a =0 = 2 πn σ − n (cid:90) M,z ¯ κ n ; (A12) k n = (cid:2) ∂ na log P a (cid:3) λ a =0 = σ − n (cid:90) z Hζ (0)2 n − (cid:88) m =1 (cid:18) nm (cid:19) πm πn − m × (cid:20)(cid:90) M b ¯ κ n − m (cid:21) (cid:20)(cid:90) M b ¯ κ m (cid:21) . (A13)Thus, we see that the scalings with the expansion param-eter are k n ∝ (ˆ θ/σ ) n − ; k n ∝ (ˆ θ/σ ) n − . (A14)From this, we can draw two conclusions:1. Increasing the order of a cumulant by one in-troduces two powers of the expansion parameter.Thus, in the limit where the expansion parameteris small, we converge at a Gaussian distribution. 2. The cumulants arising from the clustering term arelarger by two powers of the expansion parameter, sothat for large enough smoothing scales the two-haloterm eventually takes over (despite being only arelatively small correction in the unsmoothed case). Appendix B: Numerical evaluation
For efficient computation of the one- and especially thetwo-point PDF, two observations can be made. First,integrals of the form (cid:90)
M,θ e iκ ( θ ) λ (B1)can be transformed to (cid:90) dκ e iκλ (cid:90) M π dθ ( κ ) dκ , (B2)where θ ( κ ) is the inverse function of the convergence pro-file κ ( θ ). Thus, we can use the FFT. Note that this re-quires that the convergence profiles are invertible (mono-tonic). Application of (cid:96) -space filters (such as the pixelwindow function or the Wiener filter) can occasionallylead to non-monotonic profiles; in that case we split theintegral into segments in which the convergence profilesare monotonic.Second, in the two-point PDF it is not necessary tocompute the K ( (cid:96) ) a from Eq. (12) explicitly, since the in-tegral over (cid:96) reduces to (cid:90) (cid:96) J ( (cid:96)θ a ) J ( (cid:96)θ b ) J ( (cid:96)φ ) , (B3)which has the analytic form [53, pg. 411 Eq. (3)][4 π ∆( θ a , θ b , φ )] − . (B4)Here, ∆ denotes the area of the triangle with the argu-ments as its sides. If no triangle can be formed, the in-tegral vanishes. The case φ = 0 is relevant for this work;then the integral is a multiple of δ D ( θ a − θ b ) and this prop-agates through in such a way that the zero-separationtwo-point PDF simplifies to P ( κ a , κ b ; φ = 0) = P ( κ a ) δ D ( κ a − κ b ) , (B5)as it should. Appendix C: Validating Assumptions of the HaloModel Approach
There are two distinct classes of assumptions made inthis work. The first class comprises the basic underpin-nings of the halo model. In comparing to two differentsets of simulations, we have seen that we can describe theconvergence PDF accurately well into the non-Gaussian8tail, while our model has deficiencies for κ (cid:46)
1. Born approximation
The first of these is the Born approximation. Our for-malism crucially requires this approximation, since it re-lies on the additivity of the convergence signal. Whilewe do not examine the Born approximation in any detailhere, we note two reasons why we believe it to constituteonly a minor correction to the PDF: (1) we performeda simple numerical test in which we placed a single haloof mass 10 h − M (cid:12) at z = 0 .
2. Triaxialiaty
The second technical assumption is the neglect of halotriaxiality. Our formalism could in principle be adaptedquite easily to allow for triaxiality, introducing more inte-grations over a shape-distribution function and halo ori-entations. However, these additional integrations, com-bined with the fact that the projected convergence pro-files would no longer be azimuthally symmetric, rendersthe computation substantially more complex. Thus, weexplicitly test for the influence of triaxiality on the con-vergence PDF. To this end, we measure the distributionof principal axes ratios in the MassiveNuS halo catalog.This distribution is plotted in Fig. 12. We find that therelatively coarse binning in both mass and redshift is suf-ficient to capture the variation of the shape distribution.The extreme principal axes ratios for the lowest mass binare likely driven by non-virialized objects erroneously in-cluded in the halo catalog by the halo finder. However,this mass bin gives only a negligible contribution to thePDF (c.f. Sec. III C).Then, we perform simplified simulations as describedin more detail in Ref. [15]. In short, these simulationsrandomly populate maps with convergence profiles drawnfrom a given distribution and measure the PDF in theend. By construction, the simplified simulations do notinclude the clustering effect, but this is immaterial forthe purposes of the test we want to perform here. For [11, 12] [ , . ] logarithmic mass bin r e d s h i f t b i n [12, 13] [13, 14] [14, 15] 00.51 b / a c / a [ . , ] c / a c / a c / a b / a P ( b / a , c / a | M , z ) [rescaled] FIG. 12: Measured axes ratio distribution from the Mas-siveNuS halo catolog, used to construct the simplified sim-ulations that we utilize to assess the effect of ellipsoidal vsspherical halos (c.f. Fig. 13). Note that the lower triangularpart of these matrices is necessarily zero by the definition of a > b > c . We note the peaks near b = c = 0 in the lowestmass bin, which are due to imprecise axes ratio measurementsand spurious halo identifications close to the simulation’s res-olution limit. P () sim (ellipsoidal)sim (spherical)analytic (spherical)0.05 0.10 0.15 0.20 0.250.500.250.000.250.50 s i m / a n a l y t i c - FIG. 13: Here we explore the effect of including non-spherical halos on the PDF (here source redshift z s = 1).We see that the discrepancy between the complete ellipsoidalsimplified simulations and the analytic result is relativelysmall, in particular in comparison to the discrepancy betweenour model and the MassiveNuS and T17 simulations. Notethat in this exercise we have set a minumum mass cutoff atlog M vir /M (cid:12) = 12 . M /M (cid:12) = 11). This is done purely in order to keepruntime of the simplified simulations reasonable, but since itis consistently implemented on the analytic side no issues arisefrom this choice. b/a = c/a = 1 (i.e. allhalos are spherically symmetrical), the result from thistest is represented by the orange line. First, we observesome discrepancies between the green and orange lines,which in principle we should expect to coincide. How-ever, there are two reasons why perfect agreement is notreached: First, we remind the reader that our treatmentof the quadratic pixels is not entirely correct; and second,we expect some systematic errors in the numerical inte-gration through the deformed NFW profiles. Thus, we consider this code test as passed. Keeping this in mind,the discrepancies between the analytic result and the sim-plified simulations incorporating triaxial halos are rela-tively small. Thus, triaxiality can certainly not accountfor the discrepancies we observed between our model andthe MassiveNuS as well as the T17 simulations. We con-clude that more reliable modelling of small scale matterclustering is of much greater importance than incorpo-rating the small corrections from triaxiality.
3. Substructure3. Substructure