The integrated 3-point correlation function of cosmic shear
MMNRAS , 1–22 (2021) Preprint 23 February 2021 Compiled using MNRAS L A TEX style file v3.0
The integrated 3-point correlation function of cosmic shear
Anik Halder, , (cid:63) Oliver Friedrich, , Stella Seitz, , Tamas N. Varga , Universitäts-Sternwarte, Fakultät für Physik, Ludwig-Maximilians Universität München, Scheinerstr. 1, 81679 München, Germany Max Planck Institute for Extraterrestrial Physics, Giessenbachstrasse 1, 85748 Garching, Germany Kavli Institute for Cosmology, University of Cambridge, CB3 0HA Cambridge, United Kingdom Churchill College, University of Cambridge, CB3 0DS Cambridge, United Kingdom
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
We present the integrated 3-point shear correlation function i ζ ± — a higher-order statistic of the cosmic shear field — whichcan be directly estimated in wide-area weak lensing surveys without measuring the full 3-point shear correlation function,making this a practical and complementary tool to 2-point statistics for weak lensing cosmology. We define it as the 1-pointaperture mass statistic M ap measured at di ff erent locations on the shear field correlated with the corresponding local ξ ± . Building upon existing work on the integrated bispectrum of the weak lensing convergence field,we present a theoretical framework for computing the integrated 3-point function in real space for any projected field withinthe flat-sky approximation and apply it to cosmic shear. Using analytical formulae for the non-linear matter power spectrumand bispectrum, we model i ζ ± and validate it on N-body simulations within the uncertainties expected from the sixth yearcosmic shear data of the Dark Energy Survey. We also explore the Fisher information content of i ζ ± and perform a joint analysiswith ξ ± for two tomographic source redshift bins with realistic shape-noise to analyse its power in constraining cosmologicalparameters. We find that the joint analysis of ξ ± and i ζ ± has the potential to considerably improve parameter constraints from ξ ± alone, and can be particularly useful in improving the figure of merit of the dynamical dark energy equation of state parametersfrom cosmic shear data. Key words: gravitational lensing: weak – large-scale structure of Universe – cosmological parameters – methods: statistical
Weak gravitational lensing involves the study of the cosmic shearfield γ — coherent distortions imprinted in the shapes of backgroundsource galaxies by the gravitational lensing e ff ect of foregroundmatter distribution in the Universe (Bartelmann & Schneider 2001;Schneider 2006; Kilbinger 2015). Statistical analysis of the shearfield facilitates the inference of various cosmological model param-eters describing the foreground (late-time) matter field. The spatialdistribution of these late-time matter density fluctuations consists ofseveral, moderately underdense regions (e.g. voids) and relativelyfewer, but highly overdense regions (e.g. galaxies, galaxy clusters)which have emerged through the interplay of gravitational and bary-onic processes over billions of years. As a consequence, the late-timedensity fluctuations follow a positively skewed non-Gaussian distri-bution. However, most of the statistical analyses currently performedon cosmic shear data are focused on the evaluation of 2-point shearcorrelation functions ξ ± (Troxel et al. 2018; Hamana et al. 2020; As-gari et al. 2021) which are insensitive to the information contained inthe higher-order moments of the distribution. Therefore, the need forexploring methods beyond 2-point statistics in the vast amounts ofobserved shear data is of paramount importance. These higher-orderstatistics may not constrain cosmological parameters better than 2-point correlation functions. However, due to di ff erent dependence on (cid:63) E-mail: [email protected] the parameters, they hold the potential to break parameter degenera-cies which appear in 2-point analyses.The 3-point correlation function of cosmic shear ( γ -3PCF), gen-eralized third-order aperture mass statistics (Schneider & Lombardi2003; Schneider et al. 2005; Kilbinger & Schneider 2005), weaklensing convergence bispectrum — the Fourier space counterpartof γ -3PCF (Takada & Jain 2004; Kayo & Takada 2013; Sato &Nishimichi 2013) — are examples of third-order statistics which canprobe the full 3-point information of the observed weak lensing field.Cosmological constraints using the γ -3PCF were first reported bySemboloni et al. 2010 in the COSMOS survey and by Fu et al. 2014in the CFHTLS survey . However, in current weak lensing surveys(such as DES, KiDs, HSC ) which span thousand square degrees andlarger areas on the sky (much larger than COSMOS and CFHTLS),measuring and analysing the full γ -3PCF remains unexplored due toboth theoretical and observational challenges.Hence, in recent years, many alternate methods to probe parts ofthe higher-order information in the cosmic shear field have been pro- COSMOS - Cosmic Evolution Survey https://cosmos.astro.caltech.edu ; CFHTLS - Canada France Hawaii Telescope Legacy Survey . DES - Dark Energy Survey ;KiDS - Kilo Degree Survey http://kids.strw.leidenuniv.nl/index.php ; HSC - Hyper Suprime-Cam survey https://hsc.mtk.nao.ac.jp/ssp/ . © a r X i v : . [ a s t r o - ph . C O ] F e b A. Halder et al. posed and some even measured in data, which although do not cap-ture the full 3-point information, are easier to measure and modelthan γ -3PCF. Examples are shear peak statistics (Kacprzak et al.2016), density split statistics (Friedrich et al. 2018; Gruen et al.2018; Burger et al. 2020), lensing mass-map moments (Chang et al.2018a; Gatti et al. 2020), and joint analyses of shear peaks with ξ ± (Martinet et al. 2021; Harnois-Déraps et al. 2020) to name a few.Most of them show potential in putting tighter constraints on cos-mological parameters obtained from ξ ± alone.In this paper we propose another such statistic which can be mea-sured directly from cosmic shear data, namely , the integrated 3-point shear correlation function i ζ ± . We define the statistic as the aperture mass measured using a compensated filter at several loca-tions, and correlate them with the position-dependent shear 2-pointcorrelation function measured within top-hat patches at the corre-sponding locations. Some key aspects that we explore in this paperare the following: • This statistic is the real space counterpart of the recently intro-duced integrated bispectrum of the weak lensing convergence field κ as studied by Munshi et al. 2020b; Jung et al. 2021. In this paper,we build upon the existing work and formulate a theoretical modelfor our real space statistic on the shear field γ and validate it on sim-ulated cosmic shear maps. • The most desirable feature of i ζ ± is that it can be easily mea-sured from the observed shear field, a direct observable. This is pos-sible because we define i ζ ± using an aperture mass — a weightedmeasurement of the shear field at a given location using a compen-sated window which filters out a constant convergence mass sheet —and the position-dependent 2-point shear correlation function whichis intuitively the ξ ± measured within top-hat patches (with area of afew square degrees). Our definition is di ff erent from Munshi et al.2020b who work with the convergence field in Fourier space andaccordingly define the integrated convergence bispectrum iB κ usingthe local mean convergence measured within a top-hat patch insteadof using a compensated filter. If one would want to measure iB κ thenit would first be necessary to construct a convergence map from theobserved shear field. This map-making process is not at all straightforward in the presence of complicated survey geometry and masks. • We investigate the information content of i ζ ± for a DES-sizedtomographic survey in terms of Fisher constraints on cosmologicalparameters. This is the first work to perform such an analysis in thecontext of the integrated weak lensing bispectrum.We organise the paper in the following manner. In chapter 2 we for-mulate the integrated 3-point function statistic for any projected fieldwithin the flat-sky approximation and then apply it to the case for thecosmic shear field in chapter 3. In chapter 4 we describe the simu-lations and numerical methods we use in order to measure and the-oretically model the statistic. Finally, in chapter 5 we validate ourtheoretical model on the simulations and present Fisher constraintson cosmological parameters. Throughout this paper we assume flatcosmology i.e. Ω K =
0. As we mainly work with projected 2D quan-tities, we di ff erentiate them from 3D quantities by explicitly speci-fying the sub or super-script ‘3D’ for the latter. The i in i ζ ± stands for ‘integrated’ and should not be confused with thecomplex imaginary unit. In this chapter we formulate the general framework of equations re-quired for describing the integrated 3-point function (in real-space)and the integrated bispectrum (in Fourier space) of any projected 2Dfield within the flat-sky approximation. For this chapter and the next,we provide a summary of this technical part of the paper at the endof chapter 3. Readers may feel free to skip these theoretical detailsand directly refer to the summary in section 3.4.
Any cosmic field f (cid:2) χ , η (cid:3) that we observe on our past light-coneat 3D comoving position χ and corresponding conformal lookbacktime η = η − χ (where η is the conformal time today and χ the radialcomoving distance), can be projected onto the 2D celestial sphere toobtain the weighted line-of-sight 2D quantity f (ˆ n ) towards a radialunit direction ˆ n (Bartelmann & Schneider 2001), f (ˆ n ) = (cid:90) d χ q f ( χ ) f (cid:2) χ ˆ n , η − χ (cid:3) (1)where q f ( χ ) is a particular weighting kernel over which f is pro-jected. Examples are the projected galaxy number density or theweak lensing convergence field that we observe on the celestialsphere. Assuming that the angular extent of the field of view is small— spanning an area of a few square degrees — we can make the flat-sky approximation, where we denote the position on the sky as a 2Dplanar vector θ = ( θ x , θ y ) and express f as f ( θ ) = (cid:90) d χ q f ( χ ) f (cid:2) ( χ θ , χ ) , η − χ (cid:3) . (2) The 2D power spectrum ( P gh ) and bispectrum ( B fgh ) of projectedfields f , g , h are defined as (Bartelmann & Schneider 2001): (cid:10) g ( l ) h ( l ) (cid:11) ≡ (2 π ) δ D ( l + l ) P gh ( l ) (3) (cid:10) f ( l ) g ( l ) h ( l ) (cid:11) ≡ (2 π ) δ D ( l + l + l ) B fgh ( l , l , l ) (4)where (cid:104) ... (cid:105) denotes an ensemble average over di ff erent realizationsof the Universe and δ D denotes the Dirac delta function. f ( l ) cor-responds to the Fourier space representation of field f ( θ ) (see Ap-pendix A); and similarly for fields g , h . The bispectrum is definedfor closed triangle configurations l + l + l = P ( k , η ), and bispectrum B ( k , k , k , η )of the 3D fields f , g , h with k i corresponding to 3D Fourierwave-vectors. This can be computed using the Limber approxima-tion (Limber 1954; Kaiser 1992; Buchalter et al. 2000): P gh ( l ) = (cid:90) d χ q g ( χ ) q h ( χ ) χ P (cid:32) k = l χ , η − χ (cid:33) (5) B fgh ( l , l , l ) = (cid:90) d χ q f ( χ ) q g ( χ ) q h ( χ ) χ B (cid:32) l χ , l χ , l χ , η − χ (cid:33) (6) In this paper we do not use any distinguishing symbol (e.g. the commonlyused tilde) for separately denoting the Fourier space representation of thefunction f . The Fourier representation is left understood when f appearswith argument l or q (2D Fourier wave-vectors).MNRAS000
Any cosmic field f (cid:2) χ , η (cid:3) that we observe on our past light-coneat 3D comoving position χ and corresponding conformal lookbacktime η = η − χ (where η is the conformal time today and χ the radialcomoving distance), can be projected onto the 2D celestial sphere toobtain the weighted line-of-sight 2D quantity f (ˆ n ) towards a radialunit direction ˆ n (Bartelmann & Schneider 2001), f (ˆ n ) = (cid:90) d χ q f ( χ ) f (cid:2) χ ˆ n , η − χ (cid:3) (1)where q f ( χ ) is a particular weighting kernel over which f is pro-jected. Examples are the projected galaxy number density or theweak lensing convergence field that we observe on the celestialsphere. Assuming that the angular extent of the field of view is small— spanning an area of a few square degrees — we can make the flat-sky approximation, where we denote the position on the sky as a 2Dplanar vector θ = ( θ x , θ y ) and express f as f ( θ ) = (cid:90) d χ q f ( χ ) f (cid:2) ( χ θ , χ ) , η − χ (cid:3) . (2) The 2D power spectrum ( P gh ) and bispectrum ( B fgh ) of projectedfields f , g , h are defined as (Bartelmann & Schneider 2001): (cid:10) g ( l ) h ( l ) (cid:11) ≡ (2 π ) δ D ( l + l ) P gh ( l ) (3) (cid:10) f ( l ) g ( l ) h ( l ) (cid:11) ≡ (2 π ) δ D ( l + l + l ) B fgh ( l , l , l ) (4)where (cid:104) ... (cid:105) denotes an ensemble average over di ff erent realizationsof the Universe and δ D denotes the Dirac delta function. f ( l ) cor-responds to the Fourier space representation of field f ( θ ) (see Ap-pendix A); and similarly for fields g , h . The bispectrum is definedfor closed triangle configurations l + l + l = P ( k , η ), and bispectrum B ( k , k , k , η )of the 3D fields f , g , h with k i corresponding to 3D Fourierwave-vectors. This can be computed using the Limber approxima-tion (Limber 1954; Kaiser 1992; Buchalter et al. 2000): P gh ( l ) = (cid:90) d χ q g ( χ ) q h ( χ ) χ P (cid:32) k = l χ , η − χ (cid:33) (5) B fgh ( l , l , l ) = (cid:90) d χ q f ( χ ) q g ( χ ) q h ( χ ) χ B (cid:32) l χ , l χ , l χ , η − χ (cid:33) (6) In this paper we do not use any distinguishing symbol (e.g. the commonlyused tilde) for separately denoting the Fourier space representation of thefunction f . The Fourier representation is left understood when f appearswith argument l or q (2D Fourier wave-vectors).MNRAS000 , 1–22 (2021) ntegrated 3-point shear correlation function where q f ( χ ), q g ( χ ) and q h ( χ ) are the weighting kernels with which f , g , and h are projected, respectively. Under the assumptionsof an isotropic Universe, the power spectrum is independent of thedirection of the wave-vector and the bispectrum does not depend onthe orientation of the closed triangle of its wave-vectors. It shouldbe noted again that these expressions are written assuming that theUniverse is flat. However, it is straight forward to generalize theseequations to a universe with non-zero spatial curvature (see Bartel-mann & Schneider 2001; Schneider 2006). The integrated bispectrum iB ( k ) of the 3D matter density contrastfield was first studied by Chiang et al. 2014 who defined it as thecorrelation of the local mean density perturbation and the position-dependent power spectrum evaluated within 3D sub-volumes. Theyshowed that this correlation can be expressed as integrals over dif-ferent k -modes of the full 3D matter density contrast bispectrum B δ . Chiang et al. 2015 studied the real space counterpart of iB ( k ),namely the integrated 3-point function i ζ ( r ) which they showedto be the correlation of the local mean density perturbation andthe position-dependent 2-point correlation function within 3D sub-volumes and presented the first detection of i ζ ( r ) in the BOSSDR10 CMASS galaxy sample. Recently, Munshi & Coles 2017;Munshi et al. 2020a,b; Jung et al. 2020, 2021 extended the formalismto the integrated bispectrum iB ( l ) of projected 2D fields. In particu-lar, Munshi et al. 2020b, studied this in the context of the weak lens-ing convergence field and developed various theoretical models forthe same. In this section we build upon the mathematical formalismof the integrated bispectrum developed in these previous works andintroduce its real space counterpart the integrated 3-point function i ζ ( θ ) for any projected 2D field. The central quantity to our discussion will be the projected field f ( θ ; θ C ) at a given location θ on the flat-sky weighted by an az-imuthally symmetric 2D window function W (of a given size or char-acteristic scale) centred at θ C f ( θ ; θ C ) ≡ f ( θ ) W ( θ C − θ ) , (7)where W ( θ C − θ ) = W ( θ − θ C ) = W ( | θ C − θ | ). For example, if thewindow function centred at θ C is a top-hat of size θ T , then f ( θ ; θ C ) = f ( θ ) only when | θ C − θ | ≤ θ T , otherwise f ( θ ; θ C ) =
0. Its local Fouriertransform (see Appendix A) is given by f ( l ; θ C ) ≡ F [ f ( θ ; θ C )] = (cid:90) d θ f ( θ ) W ( θ C − θ ) e − i l · θ = (cid:90) d l (2 π ) f ( l ) W ( l − l ) e i ( l − l ) · θ C (8)where f can be any complex / real 2D field defined in a tomographicbin with projection kernel q f e.g. projected galaxy density contrast,weak lensing convergence κ , weak lensing shear γ . f ( l ) and W ( l )are the Fourier space representations of f ( θ ) and W ( θ ), respectively.If f is a real field i.e. f ∗ ( − l ) = f ( l ) then we can easily see that f ∗ ( l ; θ C ) = f ( − l ; θ C ). We can now find the weighted mean of f ( θ ; θ C ) defined within the2D window W at θ C as¯ f ( θ C ) ≡ A (cid:90) d θ f ( θ ; θ C ) = A (cid:90) d θ f ( θ ) W ( θ C − θ ) = A (cid:90) d l (2 π ) f ( l ) W ( l ) e i l · θ C (9)where for the final equality we have used the convolution theoremand have defined the 1-point area normalisation term as A ≡ (cid:90) d θ W ( θ C − θ ) . (10)Note that this normalisation term is a purely geometric factor inde-pendent of the location θ C of the window (evaluating it at any θ C gives the same result and for simplicity we evaluate it at θ C = ;see also footnote 6). If we use a normalised window function i.e. A = f ( θ C ) = A f ( l = ; θ C ) . (11) The 2-point correlation (as a function of the separation 2D vector α )of projected fields g and h is defined as ξ gh ( α ) ≡ (cid:10) g ( θ ) h ( θ + α ) (cid:11) . (12)This is the Fourier space counterpart of the projected power spec-trum P gh ( l ): ξ gh ( α ) = F − [ P gh ( l )] = (cid:90) d l (2 π ) P gh ( l ) e i l · α . (13)Considering isotropic fields, this inverse 2D Fourier transformationbecomes an inverse Hankel transform (see Appendix A): ξ gh ( α ) = F − [ P gh ( l )] where the correlation function (power spectrum) is in-dependent of the direction of the separation vector α (Fourier mode l ).For ergodic fields, we can write the expression for this 2-pointcorrelation function evaluated within a finite region of area A asˆ ξ gh ( α ) ≡ A (cid:90) d θ g ( θ ) h ( θ + α ) (14)where the integrand for a given separation α , is defined only forthose points θ for which both θ and θ + α lie within the boundaryof the region under consideration. As A → ∞ , ˆ ξ gh ( α ) → ξ gh ( α ).However, if the region spans only a small area (e.g. a small 2D aper-ture on the sky), then this limit does not hold and instead the ex-pression ˆ ξ gh ( α ) evaluates to a value which depends on the locationof the aperture. Hence, we now formally define the expression forthe position-dependent 2-point correlation function ˆ ξ gh ( α ; θ C ) of the We use the subscript ‘1pt’ for the window function in the equation forthe weighted mean of a field inside the window W at a given location todistinguish it from the case when we compute the position dependent 2-pointfunction within a di ff erent window W at the same location (see equation (15).MNRAS , 1–22 (2021) A. Halder et al. projected fields g and h both defined within a 2D aperture W centredat θ C asˆ ξ gh ( α ; θ C ) ≡ A ( α ) (cid:90) d θ g ( θ ; θ C ) h ( θ + α ; θ C ) = A ( α ) (cid:90) d θ g ( θ ) W ( θ C − θ ) × h ( θ + α ) W ( θ C − θ − α ) = A ( α ) (cid:90) d l (2 π ) (cid:90) d l (2 π ) (cid:90) d q (2 π ) g ( l ) h ( l ) × W ( q ) W ( l + l − q ) e i ( l + l ) · θ C e i ( q − l ) · α (15)where l i , q are 2D Fourier wave-vectors. In the above equa-tion A ( α ) is the area normalisation for this projected position-dependent 2-point function and is given by A ( α ) ≡ (cid:90) d θ W ( θ C − θ ) W ( θ C − θ − α ) = (cid:90) d q (2 π ) W ( q ) W ( − q ) e i q · α (16)which for simplicity we evaluate (using the first equality) at θ C = as this term is independent of the window’s location θ C . However,it is important to note that this area normalisation depends on theseparation vector α under consideration, unlike A defined in equa-tion (10). For azimuthally symmetric window functions that we areinterested in, it follows from isotropy considerations that this nor-malisation term only depends on the magnitude α of the separationvector i.e. A ( α ) = A ( α ). Hence, one can evaluate this term forany polar angle φ α (e.g. defined with respect to the x-axis of the flat-sky coordinate system). For simplicity, we shall consider φ α = g with the complex-conjugated field h ∗ we haveˆ ξ gh ∗ ( α ; θ C ) ≡ A ( α ) (cid:90) d θ g ( θ ; θ C ) h ∗ ( θ + α ; θ C ) = A ( α ) (cid:90) d l (2 π ) (cid:90) d l (2 π ) (cid:90) d q (2 π ) g ( l ) h ∗ ( − l ) × W ( q ) W ( l + l − q ) e i ( l + l ) · θ C e i ( q − l ) · α . (17)In case the field h is real i.e. h ∗ ( − l ) = h ( l ), it follows from equation(15) that ˆ ξ gh ∗ ( α ; θ C ) = ˆ ξ gh ( α ; θ C ).The position-dependent correlation function gives an unbiased es-timate of the 2-point correlation function i.e. (cid:10) ˆ ξ gh ( α ; θ C ) (cid:11) = ξ gh ( α ).Also, when we consider the fields and the window functions to beisotropic, then the above expressions only depend on the magnitude α of the separation vector i.e. ˆ ξ gh ( α ; θ C ) = ˆ ξ gh ( α ; θ C ). The power spectrum is the forward Fourier transform of the 2-pointcorrelation function. Hence, we define the Fourier space counterpart Of course, this is only true when we do not consider holes and masks in thedata. To account for this, one may randomly throw away some points insidea window centred at θ C so as to have only those pairs of points { θ , θ + α } yielding the same e ff ective area of another window at θ (cid:48) C but which has masksand holes within its aperture. of ˆ ξ gh ( α ; θ C ) asˆ P gh ( l ; θ C ) ≡ F [ A ( α )ˆ ξ gh ( α ; θ C )] = (cid:90) d α A ( α )ˆ ξ gh ( α ; θ C ) e − i l · α = (cid:90) d l (2 π ) (cid:90) d l (2 π ) g ( l ) h ( l ) × W ( l + l ) W ( l − l ) e i ( l + l ) · θ C = g ( − l ; θ C ) h ( l ; θ C ) . (18)This is slightly di ff erent from the position-dependent power spec-trum definition of Chiang et al. 2014 who define it for the 3D mat-ter density contrast field in their equation 2.3 with a constant vol-ume normalisation term. On the other hand, we factor out the scale-dependent area normalisation term A ( α ) in our definition of ˆ P gh .Similarly, the Fourier space counterpart of ˆ ξ gh ∗ ( α ; θ C ) can be writ-ten asˆ P gh ∗ ( l ; θ C ) ≡ F [ A ( α )ˆ ξ gh ∗ ( α ; θ C )] = (cid:90) d l (2 π ) (cid:90) d l (2 π ) g ( l ) h ∗ ( − l ) × W ( l + l ) W ( l − l ) e i ( l + l ) · θ C = g ( − l ; θ C ) h ∗ ( − l ; θ C ) . (19)When the field h is real, ˆ P gh ∗ ( l ; θ C ) = ˆ P gh ( l ; θ C ). We now define the integrated 3-point function of projected fieldsanalogous to the 3D case (Chiang et al. 2015) — the ensemble av-erage (over di ff erent locations θ C ) of the product of the position-dependent weighted mean and the position-dependent 2-point func-tion of projected fields: i ζ ( α ) ≡ (cid:68) ¯ f ( θ C ) ˆ ξ gh ( α ; θ C ) (cid:69) = A A ( α ) (cid:90) d θ (cid:90) d θ (cid:68) f ( θ ) g ( θ ) h ( θ + α ) (cid:69) × W ( θ C − θ ) W ( θ C − θ ) W ( θ C − θ − α ) = A A ( α ) (cid:90) d l (2 π ) (cid:90) d l (2 π ) (cid:90) d l (2 π ) (cid:90) d q (2 π ) × (cid:68) f ( l ) g ( l ) h ( l ) (cid:69) e i ( l + l + l ) · θ C × W ( l ) W ( q ) W ( l + l − q ) e i ( q − l ) · α , (20)and for the case with complex-conjugated field h ∗ : i ζ ∗ ( α ) ≡ (cid:68) ¯ f ( θ C ) ˆ ξ gh ∗ ( α ; θ C ) (cid:69) = A A ( α ) (cid:90) d l (2 π ) (cid:90) d l (2 π ) (cid:90) d l (2 π ) (cid:90) d q (2 π ) × (cid:68) f ( l ) g ( l ) h ∗ ( − l ) (cid:69) e i ( l + l + l ) · θ C × W ( l ) W ( q ) W ( l + l − q ) e i ( q − l ) · α . (21)For a real field h , it follows that i ζ ∗ ( α ) = i ζ ( α ). MNRAS000
0. Its local Fouriertransform (see Appendix A) is given by f ( l ; θ C ) ≡ F [ f ( θ ; θ C )] = (cid:90) d θ f ( θ ) W ( θ C − θ ) e − i l · θ = (cid:90) d l (2 π ) f ( l ) W ( l − l ) e i ( l − l ) · θ C (8)where f can be any complex / real 2D field defined in a tomographicbin with projection kernel q f e.g. projected galaxy density contrast,weak lensing convergence κ , weak lensing shear γ . f ( l ) and W ( l )are the Fourier space representations of f ( θ ) and W ( θ ), respectively.If f is a real field i.e. f ∗ ( − l ) = f ( l ) then we can easily see that f ∗ ( l ; θ C ) = f ( − l ; θ C ). We can now find the weighted mean of f ( θ ; θ C ) defined within the2D window W at θ C as¯ f ( θ C ) ≡ A (cid:90) d θ f ( θ ; θ C ) = A (cid:90) d θ f ( θ ) W ( θ C − θ ) = A (cid:90) d l (2 π ) f ( l ) W ( l ) e i l · θ C (9)where for the final equality we have used the convolution theoremand have defined the 1-point area normalisation term as A ≡ (cid:90) d θ W ( θ C − θ ) . (10)Note that this normalisation term is a purely geometric factor inde-pendent of the location θ C of the window (evaluating it at any θ C gives the same result and for simplicity we evaluate it at θ C = ;see also footnote 6). If we use a normalised window function i.e. A = f ( θ C ) = A f ( l = ; θ C ) . (11) The 2-point correlation (as a function of the separation 2D vector α )of projected fields g and h is defined as ξ gh ( α ) ≡ (cid:10) g ( θ ) h ( θ + α ) (cid:11) . (12)This is the Fourier space counterpart of the projected power spec-trum P gh ( l ): ξ gh ( α ) = F − [ P gh ( l )] = (cid:90) d l (2 π ) P gh ( l ) e i l · α . (13)Considering isotropic fields, this inverse 2D Fourier transformationbecomes an inverse Hankel transform (see Appendix A): ξ gh ( α ) = F − [ P gh ( l )] where the correlation function (power spectrum) is in-dependent of the direction of the separation vector α (Fourier mode l ).For ergodic fields, we can write the expression for this 2-pointcorrelation function evaluated within a finite region of area A asˆ ξ gh ( α ) ≡ A (cid:90) d θ g ( θ ) h ( θ + α ) (14)where the integrand for a given separation α , is defined only forthose points θ for which both θ and θ + α lie within the boundaryof the region under consideration. As A → ∞ , ˆ ξ gh ( α ) → ξ gh ( α ).However, if the region spans only a small area (e.g. a small 2D aper-ture on the sky), then this limit does not hold and instead the ex-pression ˆ ξ gh ( α ) evaluates to a value which depends on the locationof the aperture. Hence, we now formally define the expression forthe position-dependent 2-point correlation function ˆ ξ gh ( α ; θ C ) of the We use the subscript ‘1pt’ for the window function in the equation forthe weighted mean of a field inside the window W at a given location todistinguish it from the case when we compute the position dependent 2-pointfunction within a di ff erent window W at the same location (see equation (15).MNRAS , 1–22 (2021) A. Halder et al. projected fields g and h both defined within a 2D aperture W centredat θ C asˆ ξ gh ( α ; θ C ) ≡ A ( α ) (cid:90) d θ g ( θ ; θ C ) h ( θ + α ; θ C ) = A ( α ) (cid:90) d θ g ( θ ) W ( θ C − θ ) × h ( θ + α ) W ( θ C − θ − α ) = A ( α ) (cid:90) d l (2 π ) (cid:90) d l (2 π ) (cid:90) d q (2 π ) g ( l ) h ( l ) × W ( q ) W ( l + l − q ) e i ( l + l ) · θ C e i ( q − l ) · α (15)where l i , q are 2D Fourier wave-vectors. In the above equa-tion A ( α ) is the area normalisation for this projected position-dependent 2-point function and is given by A ( α ) ≡ (cid:90) d θ W ( θ C − θ ) W ( θ C − θ − α ) = (cid:90) d q (2 π ) W ( q ) W ( − q ) e i q · α (16)which for simplicity we evaluate (using the first equality) at θ C = as this term is independent of the window’s location θ C . However,it is important to note that this area normalisation depends on theseparation vector α under consideration, unlike A defined in equa-tion (10). For azimuthally symmetric window functions that we areinterested in, it follows from isotropy considerations that this nor-malisation term only depends on the magnitude α of the separationvector i.e. A ( α ) = A ( α ). Hence, one can evaluate this term forany polar angle φ α (e.g. defined with respect to the x-axis of the flat-sky coordinate system). For simplicity, we shall consider φ α = g with the complex-conjugated field h ∗ we haveˆ ξ gh ∗ ( α ; θ C ) ≡ A ( α ) (cid:90) d θ g ( θ ; θ C ) h ∗ ( θ + α ; θ C ) = A ( α ) (cid:90) d l (2 π ) (cid:90) d l (2 π ) (cid:90) d q (2 π ) g ( l ) h ∗ ( − l ) × W ( q ) W ( l + l − q ) e i ( l + l ) · θ C e i ( q − l ) · α . (17)In case the field h is real i.e. h ∗ ( − l ) = h ( l ), it follows from equation(15) that ˆ ξ gh ∗ ( α ; θ C ) = ˆ ξ gh ( α ; θ C ).The position-dependent correlation function gives an unbiased es-timate of the 2-point correlation function i.e. (cid:10) ˆ ξ gh ( α ; θ C ) (cid:11) = ξ gh ( α ).Also, when we consider the fields and the window functions to beisotropic, then the above expressions only depend on the magnitude α of the separation vector i.e. ˆ ξ gh ( α ; θ C ) = ˆ ξ gh ( α ; θ C ). The power spectrum is the forward Fourier transform of the 2-pointcorrelation function. Hence, we define the Fourier space counterpart Of course, this is only true when we do not consider holes and masks in thedata. To account for this, one may randomly throw away some points insidea window centred at θ C so as to have only those pairs of points { θ , θ + α } yielding the same e ff ective area of another window at θ (cid:48) C but which has masksand holes within its aperture. of ˆ ξ gh ( α ; θ C ) asˆ P gh ( l ; θ C ) ≡ F [ A ( α )ˆ ξ gh ( α ; θ C )] = (cid:90) d α A ( α )ˆ ξ gh ( α ; θ C ) e − i l · α = (cid:90) d l (2 π ) (cid:90) d l (2 π ) g ( l ) h ( l ) × W ( l + l ) W ( l − l ) e i ( l + l ) · θ C = g ( − l ; θ C ) h ( l ; θ C ) . (18)This is slightly di ff erent from the position-dependent power spec-trum definition of Chiang et al. 2014 who define it for the 3D mat-ter density contrast field in their equation 2.3 with a constant vol-ume normalisation term. On the other hand, we factor out the scale-dependent area normalisation term A ( α ) in our definition of ˆ P gh .Similarly, the Fourier space counterpart of ˆ ξ gh ∗ ( α ; θ C ) can be writ-ten asˆ P gh ∗ ( l ; θ C ) ≡ F [ A ( α )ˆ ξ gh ∗ ( α ; θ C )] = (cid:90) d l (2 π ) (cid:90) d l (2 π ) g ( l ) h ∗ ( − l ) × W ( l + l ) W ( l − l ) e i ( l + l ) · θ C = g ( − l ; θ C ) h ∗ ( − l ; θ C ) . (19)When the field h is real, ˆ P gh ∗ ( l ; θ C ) = ˆ P gh ( l ; θ C ). We now define the integrated 3-point function of projected fieldsanalogous to the 3D case (Chiang et al. 2015) — the ensemble av-erage (over di ff erent locations θ C ) of the product of the position-dependent weighted mean and the position-dependent 2-point func-tion of projected fields: i ζ ( α ) ≡ (cid:68) ¯ f ( θ C ) ˆ ξ gh ( α ; θ C ) (cid:69) = A A ( α ) (cid:90) d θ (cid:90) d θ (cid:68) f ( θ ) g ( θ ) h ( θ + α ) (cid:69) × W ( θ C − θ ) W ( θ C − θ ) W ( θ C − θ − α ) = A A ( α ) (cid:90) d l (2 π ) (cid:90) d l (2 π ) (cid:90) d l (2 π ) (cid:90) d q (2 π ) × (cid:68) f ( l ) g ( l ) h ( l ) (cid:69) e i ( l + l + l ) · θ C × W ( l ) W ( q ) W ( l + l − q ) e i ( q − l ) · α , (20)and for the case with complex-conjugated field h ∗ : i ζ ∗ ( α ) ≡ (cid:68) ¯ f ( θ C ) ˆ ξ gh ∗ ( α ; θ C ) (cid:69) = A A ( α ) (cid:90) d l (2 π ) (cid:90) d l (2 π ) (cid:90) d l (2 π ) (cid:90) d q (2 π ) × (cid:68) f ( l ) g ( l ) h ∗ ( − l ) (cid:69) e i ( l + l + l ) · θ C × W ( l ) W ( q ) W ( l + l − q ) e i ( q − l ) · α . (21)For a real field h , it follows that i ζ ∗ ( α ) = i ζ ( α ). MNRAS000 , 1–22 (2021) ntegrated 3-point shear correlation function The Fourier space counterparts of the above equations can be writtenas iB ( l ) ≡ F [ A ( α ) i ζ ( α ; θ C )] = A (cid:90) d l (2 π ) (cid:90) d l (2 π ) (cid:90) d l (2 π ) (cid:68) f ( l ) g ( l ) h ( l ) (cid:69) × e i ( l + l + l ) · θ C W ( l ) W ( l + l ) W ( l − l ) = (cid:68) ¯ f ( θ C ) ˆ P gh ( l ; θ C ) (cid:69) , (22) iB ∗ ( l ) ≡ F [ A ( α ) i ζ ∗ ( α ; θ C )] = A (cid:90) d l (2 π ) (cid:90) d l (2 π ) (cid:90) d l (2 π ) (cid:68) f ( l ) g ( l ) h ∗ ( − l ) (cid:69) × e i ( l + l + l ) · θ C W ( l ) W ( l + l ) W ( l − l ) = (cid:68) ¯ f ( θ C ) ˆ P gh ∗ ( l ; θ C ) (cid:69) . (23)where the last lines of both these equations show that the inte-grated bispectrum is the ensemble average of the position-dependentweighted mean and the position-dependent power spectrum of theprojected fields.From isotropy considerations (of the fields and of the symmetricwindow functions) we have iB ( l ) = iB ( l ) and i ζ ( α ) = i ζ ( α ). Wecan thereby relate the integrated 3-point function to the integratedbispectrum through an inverse Hankel transform: i ζ ( α ) = A ( α ) F − [ iB ( l )] . (24)The formalism for the integrated bispectrum and integrated 3-pointfunction we have developed so far is very general and applicableto any projected field within the flat-sky approximation. For thecurved-sky formulation of the projected integrated bispectrum thereader is referred to the work by Jung et al. 2020.In this paper, we shall look into only one application of our for-malism for the integrated 3-point function — on the cosmic shearfield. Having developed the general framework of equations for comput-ing the integrated 3-point function for any projected field, we nowapply it to the weak lensing shear field and formulate the equationsfor the integrated 3-point shear correlation function . The light from background (source) galaxies is weakly deflectedby the foreground (lens) intervening total matter distribution. Thiscauses a coherent distortion pattern in the observed shapes of thesebackground galaxies and is known as the cosmic shear field. Thisfield can be interpreted as the shear caused by a weighted line-of-sight projection of the 3D matter density field — known as the weaklensing convergence field. Statistical analysis of this shear field (di-rectly observable) through the widely used 2-point shear correlationfunction allows one to infer about the projected power spectrum ofthe total matter distribution (theoretically predictable) and therebyconstrain cosmological parameters.Following equation (2), the weak lensing convergence field κ ( θ )acting on source galaxies situated at the radial comoving distance χ s can be written as a line-of-sight projection of the 3D matter densitycontrast field δ : κ ( θ ) = (cid:90) d χ q ( χ ) δ (cid:2) ( χ θ , χ ) , η − χ (cid:3) (25)with projection kernel q ( χ ) (also known as lensing e ffi ciency) writ-ten as (Kilbinger 2015) q ( χ ) = H Ω m c χ a ( χ ) χ s − χχ s ; with χ ≤ χ s (26)where Ω m is the total matter density parameter of the Universe today, H the Hubble parameter today, a the scale factor and c the speed oflight. The convergence and the associated complex shear field arerelated to each other through second-order derivatives of the lensingpotential ψ ( θ ) in the 2D sky-plane (Schneider 2006): κ ( θ ) = (cid:16) ∂ x + ∂ y (cid:17) ψ ( θ ) , γ ( θ ) = (cid:16) ∂ x − ∂ y + i ∂ x ∂ y (cid:17) ψ ( θ ) (27)where ψ ( θ ) is the line-of-sight projection of the 3D Newtonian grav-itational potential Φ (cid:2) ( χ θ , χ ) , η − χ (cid:3) of the total matter distribution: ψ ( θ ) = c (cid:90) d χ χ s − χχ s χ Φ (cid:2) ( χ θ , χ ) , η − χ (cid:3) ; with χ s > χ . (28)The shear γ ( θ ) = γ ( θ ) + i γ ( θ ) at a given location θ is a complexquantity where the shear components γ and γ are specified in achosen Cartesian frame (in 2D flat-sky). However, one is free to ro-tate the coordinates by any arbitrary angle β . With respect to thisreference rotation angle β , one defines the tangential and cross com-ponents of the shear at position θ as (Schneider 2006) γ t ( θ , β ) + i γ × ( θ , β ) ≡ − e − i β (cid:2) γ ( θ ) + i γ ( θ ) (cid:3) . (29)Now, given a pair of points θ and θ on the field which are separatedby the 2D vector α ≡ θ − θ , one can write the tangential and crosscomponents of the shear for this particular pair of points along theseparation direction β = φ α (polar angle of α ) as γ t ( θ j , φ α ) + i γ × ( θ j , φ α ) ≡ − e − i φ α (cid:2) γ ( θ j ) + i γ ( θ j ) (cid:3) (30)where j = , γ ( l ) is related to κ ( l ) as (Schnei-der 2006; Kilbinger 2015) γ ( l ) = ( l x + i l y ) l κ ( l ) = e i φ l κ ( l ) ; for l (cid:44) l = (cid:113) l x + l y and φ l = arctan (cid:16) l y l x (cid:17) is the polar angle of l .The weak lensing convergence power spectrum P κ, gh can bedefined through equation (3) — (cid:10) κ g ( l ) κ h ( l ) (cid:11) ≡ (2 π ) δ D ( l + l ) P κ, gh ( l ) for the convergence fields κ g and κ h , each defined withprojection kernels q g ( χ ) and q h ( χ ) for two di ff erent redshift bins (seeequation (25)) with sources located at χ s , g and χ s , h , respectively. Itcan be further expressed through equation (5) as P κ, gh ( l ) = (cid:90) d χ q g ( χ ) q h ( χ ) χ P δ (cid:32) k = l χ , η − χ (cid:33) (32)where P δ ( k , η ) is the 3D matter density contrast power spectrum.Similarly, the weak lensing convergence bispectrum definedthrough equation (4) — (cid:10) κ f ( l ) κ g ( l ) κ h ( l ) (cid:11) ≡ (2 π ) δ D ( l + l + In this paper we only consider the case when all source galaxies are locatedin a Dirac- δ function like bin at χ s . However, it is straight forward to write q ( χ ) for a general distribution of source galaxies in a tomographic redshiftbin (e.g. see Schneider 2006). MNRAS , 1–22 (2021) A. Halder et al. l ) B κ, fgh ( l , l , l ) of the convergence fields κ f , κ g and κ h with pro-jection kernels q f ( χ ), q g ( χ ) and q h ( χ ) respectively can be expressedthrough equation (6) as B κ, fgh ( l , l , l ) = (cid:90) d χ q f ( χ ) q g ( χ ) q h ( χ ) χ B δ (cid:32) l χ , l χ , l χ , η − χ (cid:33) (33)where B δ ( k , k , k , η ) is the 3D bispectrum of the matter densitycontrast field and k i = l i χ . From the statistical isotropy of the densitycontrast field, both P δ and B δ are independent of the direction ofthe k i wave-vectors. A widely used statistic to investigate the shear field γ ( θ ) is the 2-point shear correlation function. Using the notation γ t , j ≡ γ t ( θ j , φ α )and γ × , j ≡ γ × ( θ j , φ α ), the 2-point shear correlations (as a functionof separation vector α ) are defined as (Schneider & Lombardi 2003;Jarvis et al. 2004): ξ + ( α ) ≡ (cid:10) γ t , γ t , (cid:11) + (cid:10) γ × , γ × , (cid:11) = (cid:10) γ ( θ ) γ ∗ ( θ ) (cid:11) ,ξ − ( α ) ≡ (cid:10) γ t , γ t , (cid:11) − (cid:10) γ × , γ × , (cid:11) = (cid:10) γ ( θ ) γ ( θ ) e − i φ α (cid:11) (34)where the ensemble averages are over all pairs of points { θ , θ } with θ = θ + α .Considering a pair of shear fields γ g , γ h with projection kernels q g ( χ ) and q h ( χ ) respectively, the shear 2-point cross-correlations ξ ± , gh between the two fields can then be written as ξ + , gh ( α ) ≡ (cid:10) γ g ( θ ) γ ∗ h ( θ + α ) (cid:11) ,ξ − , gh ( α ) ≡ (cid:10) γ g ( θ ) γ h ( θ + α ) e − i φ α (cid:11) . (35)In general, both the correlations are complex quantities but havevanishing imaginary parts only for the so-called E-mode shear fields(which we consider in this paper) (Schneider et al. 2002; Kilbinger2015). Moreover, from statistical isotropy of the fields it follows that ξ ± , gh ( α ) = ξ ± , gh ( α ). These shear correlations are related to the con-vergence power spectrum (equation (32)) through inverse Hankeltransforms (see Appendix A) (Schneider 2006; Kilbinger 2015): ξ + , gh ( α ) = F − [ P κ, gh ( l )] = (cid:90) d l l π P κ, gh ( l ) J ( l α ) ,ξ − , gh ( α ) = F − [ P κ, gh ( l ) e − i φ l ] = (cid:90) d l l π P κ, gh ( l ) J ( l α ) (36)where J ( x ), J ( x ) are the zeroth and fourth-order Bessel functionsof the first kind, respectively.We can now write the position-dependent 2-point correlationfunctions ˆ ξ ± , gh ( α ; θ C ) of the shear field within a 2D window W cen-tred at position θ C . Using equations (17), (31) and the first line ofequation (35), we can write the ˆ ξ + , gh ( α ; θ C ) correlation asˆ ξ + , gh ( α ; θ C ) ≡ A ( α ) (cid:90) d θ γ g ( θ ; θ C ) γ ∗ h ( θ + α ; θ C ) = A ( α ) (cid:90) d l (2 π ) (cid:90) d l (2 π ) (cid:90) d q (2 π ) κ g ( l ) κ h ( l ) × e i ( φ − φ ) W ( q ) W ( l + l − q ) e i ( l + l ) · θ C e i ( q − l ) · α (37)where φ and φ are the polar angles of the Fourier modes l and l respectively.Taking into account the phase factor e − i φ α present in the secondline of equation (35), we can write the ˆ ξ − , gh ( α ; θ C ) using equations (15) and (31) asˆ ξ − , gh ( α ; θ C ) ≡ A ( α ) (cid:90) d θ γ g ( θ ; θ C ) γ h ( θ + α ; θ C ) e − i φ α = A ( α ) (cid:90) d l (2 π ) (cid:90) d l (2 π ) (cid:90) d q (2 π ) κ g ( l ) κ h ( l ) × e i ( φ + φ ) W ( q ) W ( l + l − q ) e i ( l + l ) · θ C × e i ( q − l ) · α e − i φ α . (38)For isotropic window functions W , both the estimators are indepen-dent of the direction of α i.e. ˆ ξ ± , gh ( α ; θ C ) = ˆ ξ ± , gh ( α ; θ C ). Moreover,taking the ensemble average of the above equations we can see that (cid:104) ˆ ξ ± , gh ( α ; θ C ) (cid:105) = ξ ± , gh ( α ).Along these lines we can also define the position-dependent shearpower spectra expressions as the Fourier space counterparts of theabove equations. Using equations (19) and (18) respectively (withan extra phase factor e i φ α in the latter), we getˆ P + , gh ( l ; θ C ) ≡ F [ A ( α )ˆ ξ + , gh ( α ; θ C )] = γ g ( − l ; θ C ) γ ∗ h ( − l ; θ C ) = (cid:90) d l (2 π ) (cid:90) d l (2 π ) κ g ( l ) κ h ( l ) e i ( φ − φ ) × W ( l + l ) W ( l − l ) e i ( l + l ) · θ C (39)andˆ P − , gh ( l ; θ C ) ≡ F [ A ( α )ˆ ξ − , gh ( α ; θ C ) e i φ α ] = γ g ( − l ; θ C ) γ h ( l ; θ C ) = (cid:90) d l (2 π ) (cid:90) d l (2 π ) κ g ( l ) κ h ( l ) e i ( φ + φ ) × W ( l + l ) W ( l − l ) e i ( l + l ) · θ C . (40)In this paper, we shall use a top-hat (disc) window function W ofradius θ T inside which we shall evaluate the 2-point shear correla-tions: W ( θ ) = W ( θ ) = (cid:40) θ ≤ θ T , θ > θ T (41)and the Fourier transform of this window function reads W ( l ) = W ( l ) = (cid:90) d θ W ( θ ) e − i l · θ = πθ J ( l θ T ) l θ T (42)where J is the first-order ordinary Bessel function of the first kind.One should note that this form of the top-hat window function is notnormalised since (cid:82) d θ W ( θ ) = πθ .Another statistic used for investigating the convergence / shearfield is the aperture mass M ap ( θ C ) which measures the weighted κ — a projected surface mass — inside an aperture U located at agiven point θ C (Kaiser 1995; Schneider 1996, 2006): M ap ( θ C ) = (cid:90) d θ κ ( θ ) U ( θ C − θ ) = (cid:90) d l (2 π ) κ ( l ) U ( l ) e i l · θ C (43)where the azimuthally symmetric aperture U ( θ ) = U ( θ ) has a char-acteristic size scale θ ap and in the second line we have expanded theequation with Fourier space expressions (see equation (9)). Further-more, if U is a compensated window function i.e. its integral over itssupport vanishes (cid:82) d θ U ( θ C − θ ) = Q (of size θ ap ) located at θ C (Kaiser 1995; Schneider MNRAS000
0. Its local Fouriertransform (see Appendix A) is given by f ( l ; θ C ) ≡ F [ f ( θ ; θ C )] = (cid:90) d θ f ( θ ) W ( θ C − θ ) e − i l · θ = (cid:90) d l (2 π ) f ( l ) W ( l − l ) e i ( l − l ) · θ C (8)where f can be any complex / real 2D field defined in a tomographicbin with projection kernel q f e.g. projected galaxy density contrast,weak lensing convergence κ , weak lensing shear γ . f ( l ) and W ( l )are the Fourier space representations of f ( θ ) and W ( θ ), respectively.If f is a real field i.e. f ∗ ( − l ) = f ( l ) then we can easily see that f ∗ ( l ; θ C ) = f ( − l ; θ C ). We can now find the weighted mean of f ( θ ; θ C ) defined within the2D window W at θ C as¯ f ( θ C ) ≡ A (cid:90) d θ f ( θ ; θ C ) = A (cid:90) d θ f ( θ ) W ( θ C − θ ) = A (cid:90) d l (2 π ) f ( l ) W ( l ) e i l · θ C (9)where for the final equality we have used the convolution theoremand have defined the 1-point area normalisation term as A ≡ (cid:90) d θ W ( θ C − θ ) . (10)Note that this normalisation term is a purely geometric factor inde-pendent of the location θ C of the window (evaluating it at any θ C gives the same result and for simplicity we evaluate it at θ C = ;see also footnote 6). If we use a normalised window function i.e. A = f ( θ C ) = A f ( l = ; θ C ) . (11) The 2-point correlation (as a function of the separation 2D vector α )of projected fields g and h is defined as ξ gh ( α ) ≡ (cid:10) g ( θ ) h ( θ + α ) (cid:11) . (12)This is the Fourier space counterpart of the projected power spec-trum P gh ( l ): ξ gh ( α ) = F − [ P gh ( l )] = (cid:90) d l (2 π ) P gh ( l ) e i l · α . (13)Considering isotropic fields, this inverse 2D Fourier transformationbecomes an inverse Hankel transform (see Appendix A): ξ gh ( α ) = F − [ P gh ( l )] where the correlation function (power spectrum) is in-dependent of the direction of the separation vector α (Fourier mode l ).For ergodic fields, we can write the expression for this 2-pointcorrelation function evaluated within a finite region of area A asˆ ξ gh ( α ) ≡ A (cid:90) d θ g ( θ ) h ( θ + α ) (14)where the integrand for a given separation α , is defined only forthose points θ for which both θ and θ + α lie within the boundaryof the region under consideration. As A → ∞ , ˆ ξ gh ( α ) → ξ gh ( α ).However, if the region spans only a small area (e.g. a small 2D aper-ture on the sky), then this limit does not hold and instead the ex-pression ˆ ξ gh ( α ) evaluates to a value which depends on the locationof the aperture. Hence, we now formally define the expression forthe position-dependent 2-point correlation function ˆ ξ gh ( α ; θ C ) of the We use the subscript ‘1pt’ for the window function in the equation forthe weighted mean of a field inside the window W at a given location todistinguish it from the case when we compute the position dependent 2-pointfunction within a di ff erent window W at the same location (see equation (15).MNRAS , 1–22 (2021) A. Halder et al. projected fields g and h both defined within a 2D aperture W centredat θ C asˆ ξ gh ( α ; θ C ) ≡ A ( α ) (cid:90) d θ g ( θ ; θ C ) h ( θ + α ; θ C ) = A ( α ) (cid:90) d θ g ( θ ) W ( θ C − θ ) × h ( θ + α ) W ( θ C − θ − α ) = A ( α ) (cid:90) d l (2 π ) (cid:90) d l (2 π ) (cid:90) d q (2 π ) g ( l ) h ( l ) × W ( q ) W ( l + l − q ) e i ( l + l ) · θ C e i ( q − l ) · α (15)where l i , q are 2D Fourier wave-vectors. In the above equa-tion A ( α ) is the area normalisation for this projected position-dependent 2-point function and is given by A ( α ) ≡ (cid:90) d θ W ( θ C − θ ) W ( θ C − θ − α ) = (cid:90) d q (2 π ) W ( q ) W ( − q ) e i q · α (16)which for simplicity we evaluate (using the first equality) at θ C = as this term is independent of the window’s location θ C . However,it is important to note that this area normalisation depends on theseparation vector α under consideration, unlike A defined in equa-tion (10). For azimuthally symmetric window functions that we areinterested in, it follows from isotropy considerations that this nor-malisation term only depends on the magnitude α of the separationvector i.e. A ( α ) = A ( α ). Hence, one can evaluate this term forany polar angle φ α (e.g. defined with respect to the x-axis of the flat-sky coordinate system). For simplicity, we shall consider φ α = g with the complex-conjugated field h ∗ we haveˆ ξ gh ∗ ( α ; θ C ) ≡ A ( α ) (cid:90) d θ g ( θ ; θ C ) h ∗ ( θ + α ; θ C ) = A ( α ) (cid:90) d l (2 π ) (cid:90) d l (2 π ) (cid:90) d q (2 π ) g ( l ) h ∗ ( − l ) × W ( q ) W ( l + l − q ) e i ( l + l ) · θ C e i ( q − l ) · α . (17)In case the field h is real i.e. h ∗ ( − l ) = h ( l ), it follows from equation(15) that ˆ ξ gh ∗ ( α ; θ C ) = ˆ ξ gh ( α ; θ C ).The position-dependent correlation function gives an unbiased es-timate of the 2-point correlation function i.e. (cid:10) ˆ ξ gh ( α ; θ C ) (cid:11) = ξ gh ( α ).Also, when we consider the fields and the window functions to beisotropic, then the above expressions only depend on the magnitude α of the separation vector i.e. ˆ ξ gh ( α ; θ C ) = ˆ ξ gh ( α ; θ C ). The power spectrum is the forward Fourier transform of the 2-pointcorrelation function. Hence, we define the Fourier space counterpart Of course, this is only true when we do not consider holes and masks in thedata. To account for this, one may randomly throw away some points insidea window centred at θ C so as to have only those pairs of points { θ , θ + α } yielding the same e ff ective area of another window at θ (cid:48) C but which has masksand holes within its aperture. of ˆ ξ gh ( α ; θ C ) asˆ P gh ( l ; θ C ) ≡ F [ A ( α )ˆ ξ gh ( α ; θ C )] = (cid:90) d α A ( α )ˆ ξ gh ( α ; θ C ) e − i l · α = (cid:90) d l (2 π ) (cid:90) d l (2 π ) g ( l ) h ( l ) × W ( l + l ) W ( l − l ) e i ( l + l ) · θ C = g ( − l ; θ C ) h ( l ; θ C ) . (18)This is slightly di ff erent from the position-dependent power spec-trum definition of Chiang et al. 2014 who define it for the 3D mat-ter density contrast field in their equation 2.3 with a constant vol-ume normalisation term. On the other hand, we factor out the scale-dependent area normalisation term A ( α ) in our definition of ˆ P gh .Similarly, the Fourier space counterpart of ˆ ξ gh ∗ ( α ; θ C ) can be writ-ten asˆ P gh ∗ ( l ; θ C ) ≡ F [ A ( α )ˆ ξ gh ∗ ( α ; θ C )] = (cid:90) d l (2 π ) (cid:90) d l (2 π ) g ( l ) h ∗ ( − l ) × W ( l + l ) W ( l − l ) e i ( l + l ) · θ C = g ( − l ; θ C ) h ∗ ( − l ; θ C ) . (19)When the field h is real, ˆ P gh ∗ ( l ; θ C ) = ˆ P gh ( l ; θ C ). We now define the integrated 3-point function of projected fieldsanalogous to the 3D case (Chiang et al. 2015) — the ensemble av-erage (over di ff erent locations θ C ) of the product of the position-dependent weighted mean and the position-dependent 2-point func-tion of projected fields: i ζ ( α ) ≡ (cid:68) ¯ f ( θ C ) ˆ ξ gh ( α ; θ C ) (cid:69) = A A ( α ) (cid:90) d θ (cid:90) d θ (cid:68) f ( θ ) g ( θ ) h ( θ + α ) (cid:69) × W ( θ C − θ ) W ( θ C − θ ) W ( θ C − θ − α ) = A A ( α ) (cid:90) d l (2 π ) (cid:90) d l (2 π ) (cid:90) d l (2 π ) (cid:90) d q (2 π ) × (cid:68) f ( l ) g ( l ) h ( l ) (cid:69) e i ( l + l + l ) · θ C × W ( l ) W ( q ) W ( l + l − q ) e i ( q − l ) · α , (20)and for the case with complex-conjugated field h ∗ : i ζ ∗ ( α ) ≡ (cid:68) ¯ f ( θ C ) ˆ ξ gh ∗ ( α ; θ C ) (cid:69) = A A ( α ) (cid:90) d l (2 π ) (cid:90) d l (2 π ) (cid:90) d l (2 π ) (cid:90) d q (2 π ) × (cid:68) f ( l ) g ( l ) h ∗ ( − l ) (cid:69) e i ( l + l + l ) · θ C × W ( l ) W ( q ) W ( l + l − q ) e i ( q − l ) · α . (21)For a real field h , it follows that i ζ ∗ ( α ) = i ζ ( α ). MNRAS000 , 1–22 (2021) ntegrated 3-point shear correlation function The Fourier space counterparts of the above equations can be writtenas iB ( l ) ≡ F [ A ( α ) i ζ ( α ; θ C )] = A (cid:90) d l (2 π ) (cid:90) d l (2 π ) (cid:90) d l (2 π ) (cid:68) f ( l ) g ( l ) h ( l ) (cid:69) × e i ( l + l + l ) · θ C W ( l ) W ( l + l ) W ( l − l ) = (cid:68) ¯ f ( θ C ) ˆ P gh ( l ; θ C ) (cid:69) , (22) iB ∗ ( l ) ≡ F [ A ( α ) i ζ ∗ ( α ; θ C )] = A (cid:90) d l (2 π ) (cid:90) d l (2 π ) (cid:90) d l (2 π ) (cid:68) f ( l ) g ( l ) h ∗ ( − l ) (cid:69) × e i ( l + l + l ) · θ C W ( l ) W ( l + l ) W ( l − l ) = (cid:68) ¯ f ( θ C ) ˆ P gh ∗ ( l ; θ C ) (cid:69) . (23)where the last lines of both these equations show that the inte-grated bispectrum is the ensemble average of the position-dependentweighted mean and the position-dependent power spectrum of theprojected fields.From isotropy considerations (of the fields and of the symmetricwindow functions) we have iB ( l ) = iB ( l ) and i ζ ( α ) = i ζ ( α ). Wecan thereby relate the integrated 3-point function to the integratedbispectrum through an inverse Hankel transform: i ζ ( α ) = A ( α ) F − [ iB ( l )] . (24)The formalism for the integrated bispectrum and integrated 3-pointfunction we have developed so far is very general and applicableto any projected field within the flat-sky approximation. For thecurved-sky formulation of the projected integrated bispectrum thereader is referred to the work by Jung et al. 2020.In this paper, we shall look into only one application of our for-malism for the integrated 3-point function — on the cosmic shearfield. Having developed the general framework of equations for comput-ing the integrated 3-point function for any projected field, we nowapply it to the weak lensing shear field and formulate the equationsfor the integrated 3-point shear correlation function . The light from background (source) galaxies is weakly deflectedby the foreground (lens) intervening total matter distribution. Thiscauses a coherent distortion pattern in the observed shapes of thesebackground galaxies and is known as the cosmic shear field. Thisfield can be interpreted as the shear caused by a weighted line-of-sight projection of the 3D matter density field — known as the weaklensing convergence field. Statistical analysis of this shear field (di-rectly observable) through the widely used 2-point shear correlationfunction allows one to infer about the projected power spectrum ofthe total matter distribution (theoretically predictable) and therebyconstrain cosmological parameters.Following equation (2), the weak lensing convergence field κ ( θ )acting on source galaxies situated at the radial comoving distance χ s can be written as a line-of-sight projection of the 3D matter densitycontrast field δ : κ ( θ ) = (cid:90) d χ q ( χ ) δ (cid:2) ( χ θ , χ ) , η − χ (cid:3) (25)with projection kernel q ( χ ) (also known as lensing e ffi ciency) writ-ten as (Kilbinger 2015) q ( χ ) = H Ω m c χ a ( χ ) χ s − χχ s ; with χ ≤ χ s (26)where Ω m is the total matter density parameter of the Universe today, H the Hubble parameter today, a the scale factor and c the speed oflight. The convergence and the associated complex shear field arerelated to each other through second-order derivatives of the lensingpotential ψ ( θ ) in the 2D sky-plane (Schneider 2006): κ ( θ ) = (cid:16) ∂ x + ∂ y (cid:17) ψ ( θ ) , γ ( θ ) = (cid:16) ∂ x − ∂ y + i ∂ x ∂ y (cid:17) ψ ( θ ) (27)where ψ ( θ ) is the line-of-sight projection of the 3D Newtonian grav-itational potential Φ (cid:2) ( χ θ , χ ) , η − χ (cid:3) of the total matter distribution: ψ ( θ ) = c (cid:90) d χ χ s − χχ s χ Φ (cid:2) ( χ θ , χ ) , η − χ (cid:3) ; with χ s > χ . (28)The shear γ ( θ ) = γ ( θ ) + i γ ( θ ) at a given location θ is a complexquantity where the shear components γ and γ are specified in achosen Cartesian frame (in 2D flat-sky). However, one is free to ro-tate the coordinates by any arbitrary angle β . With respect to thisreference rotation angle β , one defines the tangential and cross com-ponents of the shear at position θ as (Schneider 2006) γ t ( θ , β ) + i γ × ( θ , β ) ≡ − e − i β (cid:2) γ ( θ ) + i γ ( θ ) (cid:3) . (29)Now, given a pair of points θ and θ on the field which are separatedby the 2D vector α ≡ θ − θ , one can write the tangential and crosscomponents of the shear for this particular pair of points along theseparation direction β = φ α (polar angle of α ) as γ t ( θ j , φ α ) + i γ × ( θ j , φ α ) ≡ − e − i φ α (cid:2) γ ( θ j ) + i γ ( θ j ) (cid:3) (30)where j = , γ ( l ) is related to κ ( l ) as (Schnei-der 2006; Kilbinger 2015) γ ( l ) = ( l x + i l y ) l κ ( l ) = e i φ l κ ( l ) ; for l (cid:44) l = (cid:113) l x + l y and φ l = arctan (cid:16) l y l x (cid:17) is the polar angle of l .The weak lensing convergence power spectrum P κ, gh can bedefined through equation (3) — (cid:10) κ g ( l ) κ h ( l ) (cid:11) ≡ (2 π ) δ D ( l + l ) P κ, gh ( l ) for the convergence fields κ g and κ h , each defined withprojection kernels q g ( χ ) and q h ( χ ) for two di ff erent redshift bins (seeequation (25)) with sources located at χ s , g and χ s , h , respectively. Itcan be further expressed through equation (5) as P κ, gh ( l ) = (cid:90) d χ q g ( χ ) q h ( χ ) χ P δ (cid:32) k = l χ , η − χ (cid:33) (32)where P δ ( k , η ) is the 3D matter density contrast power spectrum.Similarly, the weak lensing convergence bispectrum definedthrough equation (4) — (cid:10) κ f ( l ) κ g ( l ) κ h ( l ) (cid:11) ≡ (2 π ) δ D ( l + l + In this paper we only consider the case when all source galaxies are locatedin a Dirac- δ function like bin at χ s . However, it is straight forward to write q ( χ ) for a general distribution of source galaxies in a tomographic redshiftbin (e.g. see Schneider 2006). MNRAS , 1–22 (2021) A. Halder et al. l ) B κ, fgh ( l , l , l ) of the convergence fields κ f , κ g and κ h with pro-jection kernels q f ( χ ), q g ( χ ) and q h ( χ ) respectively can be expressedthrough equation (6) as B κ, fgh ( l , l , l ) = (cid:90) d χ q f ( χ ) q g ( χ ) q h ( χ ) χ B δ (cid:32) l χ , l χ , l χ , η − χ (cid:33) (33)where B δ ( k , k , k , η ) is the 3D bispectrum of the matter densitycontrast field and k i = l i χ . From the statistical isotropy of the densitycontrast field, both P δ and B δ are independent of the direction ofthe k i wave-vectors. A widely used statistic to investigate the shear field γ ( θ ) is the 2-point shear correlation function. Using the notation γ t , j ≡ γ t ( θ j , φ α )and γ × , j ≡ γ × ( θ j , φ α ), the 2-point shear correlations (as a functionof separation vector α ) are defined as (Schneider & Lombardi 2003;Jarvis et al. 2004): ξ + ( α ) ≡ (cid:10) γ t , γ t , (cid:11) + (cid:10) γ × , γ × , (cid:11) = (cid:10) γ ( θ ) γ ∗ ( θ ) (cid:11) ,ξ − ( α ) ≡ (cid:10) γ t , γ t , (cid:11) − (cid:10) γ × , γ × , (cid:11) = (cid:10) γ ( θ ) γ ( θ ) e − i φ α (cid:11) (34)where the ensemble averages are over all pairs of points { θ , θ } with θ = θ + α .Considering a pair of shear fields γ g , γ h with projection kernels q g ( χ ) and q h ( χ ) respectively, the shear 2-point cross-correlations ξ ± , gh between the two fields can then be written as ξ + , gh ( α ) ≡ (cid:10) γ g ( θ ) γ ∗ h ( θ + α ) (cid:11) ,ξ − , gh ( α ) ≡ (cid:10) γ g ( θ ) γ h ( θ + α ) e − i φ α (cid:11) . (35)In general, both the correlations are complex quantities but havevanishing imaginary parts only for the so-called E-mode shear fields(which we consider in this paper) (Schneider et al. 2002; Kilbinger2015). Moreover, from statistical isotropy of the fields it follows that ξ ± , gh ( α ) = ξ ± , gh ( α ). These shear correlations are related to the con-vergence power spectrum (equation (32)) through inverse Hankeltransforms (see Appendix A) (Schneider 2006; Kilbinger 2015): ξ + , gh ( α ) = F − [ P κ, gh ( l )] = (cid:90) d l l π P κ, gh ( l ) J ( l α ) ,ξ − , gh ( α ) = F − [ P κ, gh ( l ) e − i φ l ] = (cid:90) d l l π P κ, gh ( l ) J ( l α ) (36)where J ( x ), J ( x ) are the zeroth and fourth-order Bessel functionsof the first kind, respectively.We can now write the position-dependent 2-point correlationfunctions ˆ ξ ± , gh ( α ; θ C ) of the shear field within a 2D window W cen-tred at position θ C . Using equations (17), (31) and the first line ofequation (35), we can write the ˆ ξ + , gh ( α ; θ C ) correlation asˆ ξ + , gh ( α ; θ C ) ≡ A ( α ) (cid:90) d θ γ g ( θ ; θ C ) γ ∗ h ( θ + α ; θ C ) = A ( α ) (cid:90) d l (2 π ) (cid:90) d l (2 π ) (cid:90) d q (2 π ) κ g ( l ) κ h ( l ) × e i ( φ − φ ) W ( q ) W ( l + l − q ) e i ( l + l ) · θ C e i ( q − l ) · α (37)where φ and φ are the polar angles of the Fourier modes l and l respectively.Taking into account the phase factor e − i φ α present in the secondline of equation (35), we can write the ˆ ξ − , gh ( α ; θ C ) using equations (15) and (31) asˆ ξ − , gh ( α ; θ C ) ≡ A ( α ) (cid:90) d θ γ g ( θ ; θ C ) γ h ( θ + α ; θ C ) e − i φ α = A ( α ) (cid:90) d l (2 π ) (cid:90) d l (2 π ) (cid:90) d q (2 π ) κ g ( l ) κ h ( l ) × e i ( φ + φ ) W ( q ) W ( l + l − q ) e i ( l + l ) · θ C × e i ( q − l ) · α e − i φ α . (38)For isotropic window functions W , both the estimators are indepen-dent of the direction of α i.e. ˆ ξ ± , gh ( α ; θ C ) = ˆ ξ ± , gh ( α ; θ C ). Moreover,taking the ensemble average of the above equations we can see that (cid:104) ˆ ξ ± , gh ( α ; θ C ) (cid:105) = ξ ± , gh ( α ).Along these lines we can also define the position-dependent shearpower spectra expressions as the Fourier space counterparts of theabove equations. Using equations (19) and (18) respectively (withan extra phase factor e i φ α in the latter), we getˆ P + , gh ( l ; θ C ) ≡ F [ A ( α )ˆ ξ + , gh ( α ; θ C )] = γ g ( − l ; θ C ) γ ∗ h ( − l ; θ C ) = (cid:90) d l (2 π ) (cid:90) d l (2 π ) κ g ( l ) κ h ( l ) e i ( φ − φ ) × W ( l + l ) W ( l − l ) e i ( l + l ) · θ C (39)andˆ P − , gh ( l ; θ C ) ≡ F [ A ( α )ˆ ξ − , gh ( α ; θ C ) e i φ α ] = γ g ( − l ; θ C ) γ h ( l ; θ C ) = (cid:90) d l (2 π ) (cid:90) d l (2 π ) κ g ( l ) κ h ( l ) e i ( φ + φ ) × W ( l + l ) W ( l − l ) e i ( l + l ) · θ C . (40)In this paper, we shall use a top-hat (disc) window function W ofradius θ T inside which we shall evaluate the 2-point shear correla-tions: W ( θ ) = W ( θ ) = (cid:40) θ ≤ θ T , θ > θ T (41)and the Fourier transform of this window function reads W ( l ) = W ( l ) = (cid:90) d θ W ( θ ) e − i l · θ = πθ J ( l θ T ) l θ T (42)where J is the first-order ordinary Bessel function of the first kind.One should note that this form of the top-hat window function is notnormalised since (cid:82) d θ W ( θ ) = πθ .Another statistic used for investigating the convergence / shearfield is the aperture mass M ap ( θ C ) which measures the weighted κ — a projected surface mass — inside an aperture U located at agiven point θ C (Kaiser 1995; Schneider 1996, 2006): M ap ( θ C ) = (cid:90) d θ κ ( θ ) U ( θ C − θ ) = (cid:90) d l (2 π ) κ ( l ) U ( l ) e i l · θ C (43)where the azimuthally symmetric aperture U ( θ ) = U ( θ ) has a char-acteristic size scale θ ap and in the second line we have expanded theequation with Fourier space expressions (see equation (9)). Further-more, if U is a compensated window function i.e. its integral over itssupport vanishes (cid:82) d θ U ( θ C − θ ) = Q (of size θ ap ) located at θ C (Kaiser 1995; Schneider MNRAS000 , 1–22 (2021) ntegrated 3-point shear correlation function M ap ( θ C ) = (cid:90) d θ γ t ( θ , φ θ C − θ ) Q ( θ C − θ ) (44)where the tangential shear γ t ( θ , φ θ C − θ ) at any given location θ is de-fined with respect to φ θ C − θ which is the polar angle of the separationvector between θ and the centre of the aperture θ C . The azimuthallysymmetric aperture Q has the form (Schneider 2006) Q ( θ ) = Q ( θ ) = − U ( θ ) + θ (cid:90) θ d θ (cid:48) θ (cid:48) U ( θ (cid:48) ) . (45)The aperture mass statistic can be interpreted as a position-dependent weighted mean of the shear / convergence field (see equa-tion (9)) with W = U . However, as we define it using a compen-sated filter, an area normalisation term for this statistic is irrelevant(see (10)).For the filter functions Q and U , several choices have been inves-tigated. In this paper we use the forms proposed by Crittenden et al.2002 (see also Kilbinger & Schneider 2005, Schneider et al. 2005): U ( θ ) = πθ (cid:32) − θ θ (cid:33) exp (cid:32) − θ θ (cid:33) Q ( θ ) = θ πθ exp (cid:32) − θ θ (cid:33) . (46)We shall also work closely with the Fourier space representationof U for our theoretical modelling: U ( l ) = U ( l ) = (cid:90) d θ U ( θ ) e − i l · θ = l θ − l θ . (47) We now have all the necessary ingredients to define the integrated3-point function (see Section 2.3.5) of the cosmic shear field as fol-lows: i ζ ± , fgh ( α ) ≡ (cid:68) M ap , f ( θ C ) ˆ ξ ± , gh ( α ; θ C ) (cid:69) (48)where M ap , f ( θ C ) is the aperture mass at location θ C (see equations(43), (44)) evaluated from the shear field γ f with projection ker-nel q f ( χ ) and ˆ ξ ± , gh ( α ; θ C ) are the position-dependent shear 2-pointshear correlation functions (see equations (37),(38)) computed in-side a top-hat patch centred at θ C from fields γ g , γ h with projectionkernels q g ( χ ) and q h ( χ ) respectively. Note again that each of theseprojection kernels indicate source redshifts corresponding to di ff er-ent comoving distances χ s , f , χ s , g , and χ s , h respectively.Using equations (21), (43) and (37), we can write the expressionfor the i ζ + correlation function as i ζ + , fgh ( α ) ≡ (cid:68) M ap , f ( θ C ) ˆ ξ + , gh ( α ; θ C ) (cid:69) = A ( α ) (cid:90) d θ (cid:90) d θ (cid:68) κ f ( θ ) γ g ( θ ) γ ∗ h ( θ + α ) (cid:69) × U ( θ C − θ ) W ( θ C − θ ) W ( θ C − θ − α ) = A ( α ) (cid:90) d l (2 π ) (cid:90) d l (2 π ) (cid:90) d q (2 π ) × B κ, fgh ( l , l , − l − l ) e i ( φ − φ − − ) × U ( l ) W ( q ) W ( − l − q ) e i ( q − l ) · α (49)where φ − − is the polar angle of the − l − l
2D Fourier-mode andin the last equality we have used the definition of the convergence bispectrum B κ which can be further expressed in terms of a line-of-sight projection of the 3D matter density bispectrum using equation(33) to obtain: i ζ + , fgh ( α ) = A ( α ) (cid:90) d χ q f ( χ ) q g ( χ ) q h ( χ ) χ (cid:90) d l (2 π ) (cid:90) d l (2 π ) × (cid:90) d q (2 π ) B δ (cid:32) l χ , l χ , − l − l χ , η − χ (cid:33) e i ( φ − φ − − ) × U ( l ) W ( q ) W ( − l − q ) e i ( q − l ) · α . (50)Similarly, using equation (38) the i ζ − correlation reads i ζ − , fgh ( α ) ≡ (cid:68) M ap , f ( θ C ) ˆ ξ − , gh ( α ; θ C ) (cid:69) = A ( α ) (cid:90) d θ (cid:90) d θ (cid:68) κ f ( θ ) γ g ( θ ) γ h ( θ + α ) (cid:69) × e − i φ α U ( θ C − θ ) W ( θ C − θ ) W ( θ C − θ − α ) = A ( α ) (cid:90) d χ q f ( χ ) q g ( χ ) q h ( χ ) χ (cid:90) d l (2 π ) (cid:90) d l (2 π ) × (cid:90) d q (2 π ) B δ (cid:32) l χ , l χ , − l − l χ , η − χ (cid:33) e i ( φ + φ − − ) × U ( l ) W ( q ) W ( − l − q ) e i ( q − l ) · α e − i φ α . (51)As stated before, for isotropic window functions, these correla-tions are independent of the direction of α i.e. i ζ ± , fgh ( α ) = i ζ ± , fgh ( α ).One thing to note is the similarity between the expressions of i ζ ± , fgh and the generalized third-order aperture mass statistics with di ff erentcompensated filter radii as proposed by of Schneider et al. 2005 (seetheir Section 6). Our expressions can be interpreted as a special-caseof these generalized aperture mass-statistics where we use 2 top-hatfilters of same radii and 1 compensated filter with a di ff erent sizeinstead of using 3 compensated filters.Computationally, it is more convenient to arrive at these expres-sions for the integrated 3-point shear correlation functions from theinverse Fourier transforms of the integrated shear bispectra whichwe define as iB + , fgh ( l ) ≡ F [ A ( α ) i ζ + , fgh ( α )] , iB − , fgh ( l ) ≡ F [ A ( α ) i ζ − , fgh ( α ) e i φ α ] . (52)Upon simplification, the expressions for these integrated bispectraread: iB ± , fgh ( l ) = (cid:90) d χ q f ( χ ) q g ( χ ) q h ( χ ) χ (cid:90) d l (2 π ) (cid:90) d l (2 π ) × (cid:90) d q (2 π ) B δ (cid:32) l χ , l χ , − l − l χ , η − χ (cid:33) e i ( φ ∓ φ − − ) × U ( l ) W ( l + l ) W ( − l − l − l ) = (cid:68) M ap , f ( θ C ) ˆ P ± , gh ( l ; θ C ) (cid:69) (53)where the last equality confirms our expectation (see equations (22)and (23)) that the integrated bispectrum of the shear field is the cor-relation of the aperture mass and the position-dependent shear powerspectrum.Due to the isotropy argument, we have iB ± , fgh ( l ) = iB ± , fgh ( l ) i.e.the above equation is true for any polar angle φ l and the integrated3-point functions are then inverse Hankel transforms of these inte- MNRAS , 1–22 (2021)
A. Halder et al. grated bispectra: i ζ + , fgh ( α ) = A ( α ) F − [ iB + , fgh ( l )] = A ( α ) (cid:90) d l l π iB + , fgh ( l ) J ( l α ) , i ζ − , fgh ( α ) = A ( α ) F − [ iB − , fgh ( l ) e − i φ l ] = A ( α ) (cid:90) d l l π iB − , fgh ( l ) J ( l α ) . (54)The J ( l α ) filter puts more weight on low- l values of the integratedbispectrum than the J ( l α ) filter at a given angular separation α .Hence, i ζ + ( α ) is more sensitive to large scale fluctuations (lower- l )than i ζ − ( α ) at the same angular separation α . So far, we have developed the following:(i) The integrated 3-point shear correlation function i ζ ± can be es-timated from the cosmic shear field by measuring the aperture massstatistic (with a compensated filter) at di ff erent locations and thencorrelating it with the position-dependent 2-point shear correlationfunction (evaluated inside top-hat apertures) located at the corre-sponding locations (see equation (48)).(ii) Given a prescription of the 3D matter density bispectrum B δ ( k , k , k , η ) for a set of cosmological parameters, we can the-oretically predict the i ζ ± through an inverse Hankel transform of theintegrated shear bispectrum iB ± — an integral of the convergencebispectrum (see equations (33), (53) and (54)). This is analogous tothe way in which one obtains the shear 2-point correlation function ξ ± from the convergence power spectrum which is in turn related tothe 3D matter density power spectrum P δ ( k , η ) through a line-of-sight projection (see equations (32) and (36)).(iii) In chapter 2 we provide a general framework of equationsfor the integrated 3-point function (equations (20), (21)) and the in-tegrated bispectrum (equations (22), (23)) which can be extended tothe analysis of any projected field within the flat-sky approximation.We shall now proceed to measure the ξ ± and i ζ ± statistics on sim-ulated cosmic shear data and also perform theoretical calculationsfor the same using the equations mentioned above. We will test theaccuracy of our models on the simulations and then investigate theirconstraining power on cosmological parameters. In this chapter we describe the simulations (sections 4.1 and 4.2)we use in order to measure our data vector and the data-covariancematrix (section 4.3). We will then discuss the methods we use inorder to theoretically model the data vector in section 4.4.
We use the publicly available simulated data sets from Takahashiet al. 2017 cosmological simulations (hereafter T17 simulations). The data products of the simulation are available at http://cosmo.phys.hirosaki-u.ac.jp/takahasi/allsky_raytracing/ . The simulations were generated primarily for the gravitational lens-ing studies for the HSC Survey. In this paper, we use the full-skylight cone weak lensing shear and convergence maps of the simula-tion suite.These data sets were obtained from a cold dark matter (CDM)only cosmological N-body simulation in periodic cubic boxes. Thesimulation setting consisted of 14 boxes of increasing side lengths L , L , L , ..., L (with L =
450 Mpc / h), nested around a commonvertex (see Figure 1 of Takahashi et al. 2017). Each box contained2048 particles (smaller boxes hence have better spatial and massresolution) and their initial conditions were set with second-orderLagrangian perturbation theory (Crocce et al. 2006) with an initialpower spectrum computed for a flat Λ CDM cosmology with the fol-lowing parameters : Ω cdm = . , Ω b = . , Ω m = Ω cdm + Ω b = . , Ω de = Ω Λ = . , h = . , σ = .
82 and n s = .
97. Weadopt this set of parameters as our fiducial cosmology. The particlesin each box were then made to evolve from the initial conditionsusing the the N-body gravity solver code
GADGET2 (Springel et al.2001; Springel 2005). The evolved particle distribution of the di ff er-ent nested boxes were combined in layers of shells, each 150 Mpc / hthick, to obtain full-sky light cone matter density contrast insidethe shells. The simulation boxes were also ray traced using themultiple-lens plane ray-tracing algorithm GRAYTRIX (Hamana et al.2015; Shirasaki et al. 2015) to obtain full-sky weak lensing con-vergence / shear maps (in Healpix format Górski et al. 2005; Zoncaet al. 2019) for several Dirac- δ like source redshift bins. Multiplesimulations were run to produce 108 realizations for each of theirdata products. The authors report that the average matter power spec-tra from their several realizations of the simulations agreed with thetheoretical revised Halofit power spectrum (Smith et al. 2003,later revised by Takahashi et al. 2012) to within 5 (10) per cent for k < / Mpc at z <
1. They also provide correction formulaefor their 3D and angular power spectra in order to account for thediscrepancies stemming from the finite shell thickness, angular res-olution and finite simulation box size e ff ect in their simulations. Werefer the reader to our Appendix B for a summary of those correc-tions.In this paper, for validating the 2-point and integrated 3-pointshear correlation functions (see section 4.3) we use the 108 full-skyweak lensing convergence and shear maps from the simulation suite.These maps come in the Healpix format (Górski et al. 2005; Zoncaet al. 2019) for various angular resolutions. We only use the mapswith
NSIDE = . (cid:48) ) at source redshifts z = . z = . In order to get an appropriate estimate for the covariance matrix ofour data vector which shall consist of 2-point and integrated 3-pointcorrelations of the shear field (see section 4.3), we need more thanthe 108 T17 maps from the simulation suite. For this purpose, wesimulate full-sky lognormal random fields which have been exten-sively studied and shown to be a very good approximation for the1-point probability density function (PDF) of the weak lensing con-vergence / shear (Hilbert et al. 2011; Xavier et al. 2016) or the late The density parameter for species X is defined at η = η i.e. Ω X ≡ Ω X , .MNRAS , 1–22 (2021) ntegrated 3-point shear correlation function time matter density contrast fields (Friedrich et al. 2018; Gruen et al.2018). This assumption has been confirmed from the DES ScienceVerification data for the convergence field (Clerkin et al. 2017) andmost recently been used to compute covariances for the 2-point shearcorrelations for the third year data analysis of the DES (Friedrichet al. 2020).In order to do so, we use the publicly available FLASK tool (Full-sky Lognormal Astro-fields Simulation Kit) (Xavier et al. 2016)which can be used to create realisations of correlated lognormalfields on the celestial sphere at di ff erent redshifts. Concisely, FLASK draws from a lognormal variable κ with the PDF (Xavier et al. 2016) p ( κ ) = exp (cid:18) − σ [ln( κ + λ ) − µ ] (cid:19) √ πσ ( κ + λ ) κ > − λ, µ and σ are the mean and variance of the associated normalvariable and λ is the lognormal shift parameter marking the lowerlimit for possible values that κ can realise. Using FLASK we createlognormal mocks of the T17 convergence / shear fields which approx-imately follow the 1-point PDFs of the T17 maps at redshifts z and z respectively. As input to FLASK , one needs to provide the conver-gence power spectra P κ, gh ( l ) and the lognormal-shift parameters λ i for the two redshifts (with g , h , i = P δ ( k , η ) along the line-of-sight as described in equation (32). Weuse the open-source Boltzmann solver code CLASS (Lesgourgues2011; Blas et al. 2011) for computing the non-linear P δ ( k , η ) in thefiducial T17 cosmology for which we use the revised halofit prescription for the non-linear matter power spectrum (Takahashiet al. 2012; Bird et al. 2012; Smith et al. 2003) which is included in CLASS . For obtaining the lognormal shift parameters we follow thestrategy of Hilbert et al. 2011 and fit the above form of the lognor-mal PDF to the 1-point PDF of the T17 maps at both redshifts andget the following values : λ = .
012 and λ = . FLASK pairs (each pair consistsof 2 maps at source redshifts z and z respectively) of full-sky shearmaps in Healpix format with
NSIDE = / convergence maps without any noise. In real data, theshear is obtained from the measured ellipticities of backgroundgalaxies which — besides the gravitational shear e ff ect — aresubject to di ff erent sources of noise such as non-circular intrinsicellipticities of the galaxies, measurement noise, noise from point-spread-function correction etc. In our covariance matrix we want toinclude the e ff ect of this shape-noise . This is important when wewant to forecast realistic constraints on cosmological parameters.In principle, this can be modelled by adding a complex noise term N ( θ ) = N ( θ ) + iN ( θ ) to the shear field γ θ ) = γ ( θ ) + i γ ( θ ) (Pireset al. 2020) where θ represents a pixel on the Healpix shear map.The noise components N , N can both be modelled as uncorrelated currently hosted at . currently hosted at http://class-code.net . Precisely, we only fit the PDF to the first of the 108 T17 maps at bothredshifts to obtain the quoted λ i values. We have also tested the fits on othermaps at each redshift and the values for the logshift parameters di ff er in onlythe third decimal place whose e ff ect on the summary statistics evaluated fromthe corresponding FLASK maps is insignificant. The values for the other fitparameters are: µ = − . , µ = − .
565 and σ = . , σ = . Gaussian variables with zero-mean and variance σ N = σ (cid:15) n g · A pix (56)where A pix is the area of the pixel at the given NSIDE , σ (cid:15) is thedispersion of intrinsic galaxy ellipticities which we set to be 0.3as found for weak lensing surveys (Leauthaud et al. 2007; Schrab-back et al. 2018), n g is the number of observed galaxies per squarearcminute for which we assume a value of 5 at each redshift bin.Note that for the two Dirac- δ source redshift bins we consider, thisadds up to give 10 galaxies per square arcminutes which is in ac-cordance with the expected number density of galaxies for the fullDES Year 6 cosmic shear data. To every pixel in a FLASK gener-ated shear map we add an independent draw of each Gaussian noiseterm. We then convert these noisy shear maps into noisy convergencemaps on the curved sky using a Kaiser-Squires (KS) mass map re-construction method as described in section 2.1 of Gatti et al. 2020(see also Chang et al. 2018a). This process of first adding noise tothe shear field and then converting it to a convergence map is moreaccurate than the usually prevalent way of adding independent Gaus-sian noise to the pixels of the noiseless convergence field. This isbecause convergence at a given pixel is a convolution of the shearin several pixels around the desired location. This makes the noisein the convergence at a given pixel be correlated with the noise inneighbouring pixels. Although the KS method ensures this, the ap-proach where uncorrelated Gaussian noise is added to the pixels ofa noiseless convergence map directly does not account for it and istherefore not entirely accurate.
We carry out measurements of the position-dependent 2-point shearcorrelations ˆ ξ ± , gh ( α ; θ C ) on the T17 and FLASK shear maps at sourceredshifts z and z (i.e. g , h = ,
2) within top-hat windows W with radius θ T = (cid:48) . Approximately, this results in a circularpatch of area 5 square degrees . We use the publicly available code TreeCorr (Jarvis et al. 2004) to measure these correlations in 20log-spaced bins with angular separations 5 (cid:48) < α < (cid:48) . To be pre-cise, we execute TreeCorr on those pixels of the map which liewithin a disc of radius θ T centred at a given location θ C in order toobtain ˆ ξ ± , gh ( α ; θ C ).For computing the aperture mass M ap , f ( θ C ) (with f = ,
2) we usea compensated window U with an aperture scale θ ap = (cid:48) . From aconvergence map at a given source redshift z f , we measure the aper-ture mass at location θ C through a convolution of the U filter withpixels in the neighbourhood of θ C (see equation (43)). Note that itis completely equivalent to compute the aperture mass from the cor-responding shear map at z f by convolving shear pixels with the Q filter (with the same aperture scale size as that of U , see equation(44)) and completely skip the KS convergence map making proce-dure (see Harnois-Déraps et al. 2020). Hence, the way in which wecompute the aperture mass using the convergence field is redundant.As we are working in a simulated setting and do not consider holesand masks in our data, the map making procedure is straight forward. which is small enough for the flat-sky approximation to hold. currently hosted at: https://rmjarvis.github.io/TreeCorr/_build/html/index.html . We found that for θ ap = (cid:48) the amplitude of the iB + signal was largerthan other aperture scales when measured in combination with the top-hatpatch of θ T = (cid:48) . Optimization of the filter sizes remains an interestingavenue to explore. MNRAS , 1–22 (2021) A. Halder et al.
However, this is not the case in real data and it is then practical toevaluate the aperture mass from the shear map directly.In the 108 T17 noiseless simulation maps, we evaluate the abovestatistics at locations distributed over the full-sky. We do not do thisfor every pixel in the
Healpix map but rather choose well separatedpixels (about 2 θ T apart — the diameter of W ) for which the top-hatpatches at those chosen pixels only slightly overlap with the patchescentred at neighbouring chosen pixels. The overlap is not a prob-lem and allows us to maximise the area over which we evaluate thestatistics. For computing the 2-point shear correlations ξ ± , gh ( α ) in agiven map we take the average of the position-dependent shear cor-relations evaluated at all chosen patches on the map (see the discus-sion after equation (38)). The integrated 3-point shear correlations i ζ ± , fgh ( α ) are evaluated by first taking the product of the aperturemass and the position-dependent shear correlation at a chosen loca-tion and then performing an average of this product evaluated at allother locations (see equation (48)) for a specific realization.We perform the same measurements on the FLASK maps (withshape-noise). Unlike the T17 maps, we do not distribute patches overthe whole sky but rather cut out two big circular footprints of 5000square degrees (approximately the size of the DES footprint) in eachhemisphere of a
FLASK map and restrict the distribution of patchesto within the extent of each footprint. In each
FLASK map, the twofootprints are widely separated which allows us to treat each regionas an independent survey realization. This helps to maximize theusage of our
FLASK simulations and allows us to have a total of 2000DES-like realizations (from 1000
FLASK maps) which we considersu ffi cient for the estimation of the covariance matrix of our data-vector for a DES-sized survey as the number of realizations is muchlarger than the maximum size of our data vector which we discussnext.For the two source redshifts z and z , our data vector D i evalu-ated from the i − th simulation realization (T17 or FLASK ) consists ofthe 2-point shear cross-correlations and the integrated 3-point shearcross-correlations (each correlation function evaluated at 20 angularseparations α ) as depicted below: D i ≡ (cid:0) ξ ± , , ξ ± , , ξ ± , , i ζ ± , , i ζ ± , , i ζ ± , , i ζ ± , (cid:1) T (57)where T stands for transpose. This gives a data-vector of size N d = × × =
280 elements. The mean data vector is obtained bytaking an average of the individual data vectors obtained from eachof the N r realizations: D = N r N r (cid:88) i = D i . (58)On the other hand, we evaluate our covariance matrix of the data-vector as ˆC = N r − N r (cid:88) j = (cid:16) D j − D (cid:17) (cid:16) D j − D (cid:17) T (59)thus resulting in an N d × N d = ×
280 matrix. For validatingour theoretical model for the data vector we compare it with themean data vector from the 108 T17 noiseless maps. For obtainingour DES-like data-covariance matrix (with impact of shape-noise)we evaluate it from the 2000 footprints cut out from the
FLASK sim-ulations.
In this section we detail the numerical recipes that go into the theo-retical computation of the constituents of the model vector M which we evaluate for the fiducial T17 cosmology (see section 4.1). Formodelling the 2-point shear correlations ξ ± , gh ( α ) (see equation (36))we need to compute the convergence power spectrum P κ, gh ( l ). Asalready stated before, we use the public Boltzmann solver code CLASS to compute the nonlinear revised halofit
3D matterpower spectrum P δ ( k , η ) which we integrate along the line-of-sightto obtain P κ, gh ( l ). We use the 1-dimensional adaptive quadrature in-tegration routine from the GNU Scientific Library gsl to performthe integration. To partly correct for the flat-sky and the Limber ap-proximation that goes into the derivation for the expressions of theshear correlations, we multiply the convergence power spectrum byan l -dependent correction factor specified by Kitching et al. 2017: C κ, gh ( l ) ≡ ( l + l + l ( l − (cid:16) l + (cid:17) P κ, gh (cid:32) l + (cid:33) . (60)Moreover, instead of performing the inverse Hankel transform l -integrals (i.e. the F − [ ... ] operations in equation (36)) for convert-ing the Fourier space power spectra to shear correlations, we useexpressions with summation over l as given in Friedrich et al. 2020(see also Stebbins 1996): ξ ± , gh ( α ) = (cid:88) l > l + π (cid:16) G + l , (cos α ) ± G − l , (cos α ) (cid:17) l ( l + C κ, gh ( l ) (61)where the functions G ± l , ( x ) can be expressed in terms of 2nd orderassociated Legendre polynomials P l , ( x ) (Friedrich et al. 2020; Steb-bins 1996): G + l , ( x ) ± G − l , ( x ) = P l , ( x ) (cid:32) − l ± x ( l − − x − l ( l − (cid:33) + P l − , ( x ) ( l + x ∓ − x . (62)These equations are exact for a curved-sky treatment and more ac-curate than the inverse Hankel transforms; the latter resulting in in-creasing errors for larger angular separations (Kitching et al. 2017).The expressions can be easily evaluated using the gsl library.For computing the integrated 3-point functions i ζ ± , fgh ( α ), we firstneed to evaluate the integrated shear bispectra iB ± , fgh ( l ) (see equa-tion (53)). We use the fitting formula for the 3D dark matter bispec-trum B δ ( k , k , k , η ) by Gil-Marín et al. 2012 (hereafter GM, seemore in Appendix C) with the revised halofit non-linear powerspectrum implementation in CLASS which we then integrate overthe l i -modes and along the line-of-sight to obtain iB ± , fgh ( l ). For nu-merically computing the 5-dimensional integration in equation (53)we use the publicly available adaptive multi-dimensional integrationpackage cubature and evaluate each integrated bispectrum for157 l -modes log-spaced in the range 1 ≤ l ≤ iB ± , fgh ( l ) in equation (54) by the summation over l expressions inequation (61) to obtain i ζ ± , fgh ( α ). In order to do so, we first linearlyinterpolate the iB ± , fgh ( l ) between the 157 log-spaced l -modes to getthe iB ± , fgh ( l ) for every integer- l multipole within the range specifiedabove. We then use the interpolated value at every multipole to per-form the summation.In order to validate the theoretical model for the 2-point and in-tegrated 3-point shear correlations on the T17 simulations, we also To be precise, we use the c ++ wrapper of the code which can be ob-tained from the o ffi cial repository, currently hosted at: https://github.com/lesgourg/class_public . currently hosted at: . currently hosted at: https://github.com/stevengj/cubature .MNRAS , 1–22 (2021) ntegrated 3-point shear correlation function need to account for the e ff ects in the simulations due to limited an-gular resolution of the maps, finite simulation box size and finitethickness of the lens shells as reported by Takahashi et al. 2017. Weinclude these corrections in our theory power spectra as summarizedin Appendix B. We now present the results of our measurements and theory calcula-tions. In section 5.1, we test the accuracy of our model in describingthe T17 data vector within the uncertainties expected from the sixthyear cosmic shear data of the DES using the
FLASK covariance ma-trix. And in section 5.2, we explore the Fisher constraining poweron cosmological parameters which can be obtained on performing ajoint analysis of ξ ± and i ζ ± .The results of the theory computation of the convergence powerspectra P κ, gh ( l ) for source redshifts z = . z = . g , h = ,
2) are shown in Figure 1. It is clear from the Fig-ure that the convergence power spectrum for sources at higher red-shift i.e. P κ, is larger than the lower redshift power spectrum P κ, indicating the presence of more amount of deflecting material be-tween the observer and the source at larger redshifts; in other words,a larger lensing e ffi ciency for sources situated at a higher redshift(see equation (26)). Also, the spectra are smooth as features like thebaryonic acoustic oscillations which are prominent in the 3D matterpower spectrum are smeared out due to the mixing of 3D k -modesinto 2D l -modes through the line-of-sight projection (see equation(32)). In Figure 2 we show the integrated bispectra iB + , fgh ( l ) for thetwo source redshifts z and z (where f , g , h = , θ ap = (cid:48) and two top-hat windows of radii θ T = (cid:48) .Other cross-combinations besides the four cross-spectra shown inthe Figure, e.g. iB + , ( l ) and iB + , ( l ) are the same as iB + , ( l ) and iB + , ( l ), respectively (e.g. this can be easily verified from equa-tion (53)). Hence, they add no extra information and we only con-sider these four. The iB − , fgh spectra look similar to iB + , fgh and are notshown separately.It should be noted here that the high- l end of the integrated shearbispectra pick up significant contributions from squeezed configu-rations of the convergence bispectrum B κ since the high- l valuescorrespond to computing the position-dependent correlation func-tion in real space on angular scales much smaller than the size ofthe patch ( l (cid:29) π/ θ T ≈ l end of iB picks upcontribution from triangle configurations other than squeezed as theangular scales that the low- l correspond to are close to the diameterof the patch where the squeezed limit does not hold (see Figure D1and discussion in Appendix D for more details). In Figure 4 we show each component of the data vector D (blackdots) evaluated from the mean of 108 T17 simulated maps for thetwo source redshifts. The error bars on the data points indicate thestandard deviation over the 108 maps (note that these are noiselesssimulations). The grey shaded region is the 1-sigma standard devia-tion computed from the data-covariance matrix ˆC estimated from2000 DES Year 6 sized footprints in FLASK lognormal sky-maps
Figure 1.
The scaled convergence auto and cross power spectra P κ ( l ) for twotomographic source redshift bins z = . z = . Figure 2.
The scaled integrated bispectra iB + ( l ) for two tomographic sourceredshift bins z = . z = . θ ap = (cid:48) and top-hat window ofradius θ T = (cid:48) . which include realistic shape-noise (see Figure 3). The model vec-tor for each statistic is also shown in the plots (in blue) where wealso include the corrections proposed by Takahashi et al. 2017 to ac-count for the various resolution e ff ects of the T17 simulation (seeAppendix B). The ξ ± models are in good agreement with the T17measurements within both the scatter of the simulations and the DESerror bars. This is another confirmation of the result already reportedby Takahashi et al. 2017 that the convergence power spectrum (thatwe obtain using the revised halofit
3D matter power spectrum)matches with the T17 simulations after taking into account the reso-lution corrections (see Appendix B). Our model predictions for the i ζ + statistic also agrees well on all angular scales with the T17 sim-ulations not only within the grey DES error bars but also within thescatter of the T17 simulations (black error bars). However, this is notthe case for i ζ − models as they are seen to be in agreement with theT17 simulations only on larger angular scales but over predict thesimulations on smaller scales. This stems from an inaccuracy of theGM bispectrum fitting formula. At the small angular scales, the i ζ − with its fourth-order Bessel function J (see equation (54) and dis- MNRAS , 1–22 (2021) A. Halder et al.
Figure 3.
The 280 ×
280 data-correlation matrix (normalised version of ˆC , see equation (59)) estimated from 2000 DES Year 6 sized footprints inFLASK lognormal sky-maps which include realistic shape-noise for twotomographic source redshift bins z = . z = . ×
20 box around the diagonal indicates the correlation matrix for the20 separation bins α of each of the 14 components of the data vector D = (cid:0) ξ + , ( α ) , ξ − , ( α ) , ξ + , ( α ) , ξ − , ( α ) , ..., i ζ + , ( α ) , i ζ − , ( α ) (cid:1) T (see equation(57)). The o ff -diagonal boxes indicate the cross-correlations between the an-gular bins of di ff erent correlation functions. cussion after the equation) is most sensitive to the very high- l valuesof the integrated bispectrum. At these very high- l values, the inte-grated bispectrum signal is mostly due to the contributions from thehighly squeezed configurations of the convergence bispectrum (seediscussion in Appendix D). The GM formula on the other hand, isknown to overestimate these highly squeezed bispectrum configu-rations (Sato & Nishimichi 2013; Namikawa et al. 2019; Takahashiet al. 2020) and hence causes the overestimation of the i ζ − signalon the small angular scales. On the other hand, i ζ + has a zeroth-order Bessel function J weighting which is more sensitive to lower l values (for a given angular scale) of the integrated bispectrum com-pared to i ζ − . At low to moderate- l , the integrated bispectrum receivescontribution from not so highly squeezed and other bispectrum trian-gle configurations where the GM fitting function works reasonablywell.To compare how well the model vector M describes the data vec-tor D of a given statistic quantitatively, we compute the χ value as χ = ( D − M ) T C − ( D − M ) (63)where C − is an unbiased estimator of the inverse data covariancematrix ˆC − measured from N r realizations for a data vector contain-ing N d elements (Hartlap, J. et al. 2007): C − = N r − N d − N r − ˆC − . (64)Using the χ value, computed using the FLASK covariance matrix(see Figure 3), we make angular scale-cuts for every individualstatistic (see the red-dashed vertical lines in Figure 4). For making ascale-cut we impose two conditions. Firstly, the χ value of a givenstatistic using all angular bins larger than the scale-cut must be lowerthan a threshold value of 0.15. Secondly, the fractional change in the Table 1.
Signal-to-noise ratio S / N for the T17 simulation data vectors com-puted with the FLASK covariance matrix. The χ values for the theory modelwith respect to the data vector are also reported along with the length ofthe data vector. All reported quantities are evaluated after imposing angu-lar scale-cuts (see Figure 4). The data vector for each statistic includes allauto and cross-correlations for both tomographic bins z = . z = . ξ + = (cid:0) ξ + , , ξ + , , ξ + , (cid:1) T , i ζ + = (cid:0) i ζ + , , i ζ + , , i ζ + , , i ζ + , (cid:1) T etc. Data vector Length of data vector S / N χ ξ +
55 43.65 0.26 ξ −
22 36.77 0.30 ξ ±
77 47.26 0.57 i ζ +
74 8.06 0.16 i ζ −
31 7.91 0.26 i ζ ±
105 9.41 0.51 ξ ± and i ζ ±
182 48.40 1.08 χ value when ignoring the smallest bin right after the scale-cut,should be less than 15 per cent. For further analyses, this enables usto include only those parts of the model vectors which agree verywell with the simulations with respect to the DES-like uncertainties.In Table 1 we report the signal-to-noise ratio ( S / N ) of the variousstatistics after imposing the angular scale-cuts. The S / N is computedas (Chang et al. 2018b): S / N = (cid:113) D T C − D (65)and it indicates the statistical significance of the data vector. We alsoreport the corresponding χ values for the data vectors. Althoughwe require the χ for each individual statistic e.g. ξ + , etc. to bebelow 0.15, there is no such restriction for the joint data vectors. Thelow χ value of 1.08 for the entire data vector (after the scale-cuts)confirms that the model is in good agreement with the simulationswithin the DES uncertainties. We also check in Appendix E (seeFigure E1) whether any remaining systematic o ff set between the T17data vector and our model vector after imposing the scale-cuts cancause any large parameter biases in our Fisher forecasts (see nextsection). We verify that the systematic o ff set for each parameter fromthe corresponding fiducial parameter value is smaller than one-thirdof the 1-sigma constraints expected from the Fisher analysis of theentire data vector.Although the S / N of the i ζ ± is not as high as ξ ± for a DES-likesurvey, the non-zero signals measured from the simulations withouthaving had to compute the full 3-point correlation function showsthe ease of measurement and also the potential of the integrated 3-point shear correlation function to probe higher-order information ofthe highly non-Gaussian late-time matter density field. Having validated our theory model for the integrated 3-point shearcorrelations — i ζ + on all angular scales that we are interested in and i ζ − on large angular scales — we shall now address the Fisher infor-mation content of this statistic on cosmological parameters whenanalysed jointly with the 2-point shear correlation function. TheFisher information matrix F for a model vector M which dependson a set of parameters π reads (Dodelson & Schmidt 2020; Huterer2002) F ij = (cid:32) ∂ M ( π ) ∂π i (cid:33) T C − (cid:32) ∂ M ( π ) ∂π j (cid:33) . (66) MNRAS000
182 48.40 1.08 χ value when ignoring the smallest bin right after the scale-cut,should be less than 15 per cent. For further analyses, this enables usto include only those parts of the model vectors which agree verywell with the simulations with respect to the DES-like uncertainties.In Table 1 we report the signal-to-noise ratio ( S / N ) of the variousstatistics after imposing the angular scale-cuts. The S / N is computedas (Chang et al. 2018b): S / N = (cid:113) D T C − D (65)and it indicates the statistical significance of the data vector. We alsoreport the corresponding χ values for the data vectors. Althoughwe require the χ for each individual statistic e.g. ξ + , etc. to bebelow 0.15, there is no such restriction for the joint data vectors. Thelow χ value of 1.08 for the entire data vector (after the scale-cuts)confirms that the model is in good agreement with the simulationswithin the DES uncertainties. We also check in Appendix E (seeFigure E1) whether any remaining systematic o ff set between the T17data vector and our model vector after imposing the scale-cuts cancause any large parameter biases in our Fisher forecasts (see nextsection). We verify that the systematic o ff set for each parameter fromthe corresponding fiducial parameter value is smaller than one-thirdof the 1-sigma constraints expected from the Fisher analysis of theentire data vector.Although the S / N of the i ζ ± is not as high as ξ ± for a DES-likesurvey, the non-zero signals measured from the simulations withouthaving had to compute the full 3-point correlation function showsthe ease of measurement and also the potential of the integrated 3-point shear correlation function to probe higher-order information ofthe highly non-Gaussian late-time matter density field. Having validated our theory model for the integrated 3-point shearcorrelations — i ζ + on all angular scales that we are interested in and i ζ − on large angular scales — we shall now address the Fisher infor-mation content of this statistic on cosmological parameters whenanalysed jointly with the 2-point shear correlation function. TheFisher information matrix F for a model vector M which dependson a set of parameters π reads (Dodelson & Schmidt 2020; Huterer2002) F ij = (cid:32) ∂ M ( π ) ∂π i (cid:33) T C − (cid:32) ∂ M ( π ) ∂π j (cid:33) . (66) MNRAS000 , 1–22 (2021) ntegrated 3-point shear correlation function Figure 4.
The 2-point shear correlation functions ξ ± ( α ) and the integrated 3-point shear correlation functions i ζ ± ( α ) for two tomographic source redshift bins z = . z = . ff ects in theT17 simulations (see Appendix B). The integrated 3-point functions have been computed using a compensated filter of size θ ap = (cid:48) and top-hat window ofradius θ T = (cid:48) . The red-dashed lines denote the angular scale-cuts imposed on the data / model vectors using a χ criterion (see text). The angular bins smallerthan the scale-cuts are not included in further analyses. where F ij corresponds to an element of F for the model parame-ters π i and π j . The partial derivative of the model vector with re-spect to a model parameter π i can be computed using a 4-point cen- tral di ff erence quotient (also known as 5-point stencil derivative) We prefer to use 4-point to 2-point central di ff erence quotient for obtain-ing more accurate first derivatives (see also Yahia-Cherif et al. 2020).MNRAS , 1–22 (2021) A. Halder et al.
Figure 5.
The derivatives ∂ M ∂π i of 2 components of the model vector M — ξ + , (blue) and i ζ + , (red) shear correlation functions for source redshift bin z = . π = { Ω cdm , σ , n s , w , w a } and normalised by the corresponding 1-sigma standard deviation ∆ M ( α ) (from the FLASK covariance matrix) at a given α . The derivatives are negated (indicated with dotted lines) where ∂ M ∂π i < Figure 6.
Same as Figure 5 but for ξ + , (blue) and i ζ + , (red) shear correlation functions for source redshift bin z = . (Abramowitz & Stegun 1964): ∂ M ( π ) ∂π i = − M ( π i + δ i ) + M ( π i + δ i ) − M ( π i − δ i ) + M ( π i − δ i )12 δ i (67)where δ i is a small change of the parameter π i about its fiducialvalue, and M ( π i ± δ i ) means evaluating the model vector at thechanged parameters π i ± δ i while keeping all other parameters fixed.For our purpose we shall be interested in the cosmological param-eters π = { Ω cdm , σ , n s , w , w a } where w and w a indicate the dy-namical dark energy equation of state parameters in the CPL pa-rameterization (Chevallier & Polarski 2001; Linder 2003) adoptedby the Dark Energy Task Force (Albrecht et al. 2006) to com- pare di ff erent dark energy probes. The fiducial values for our cos-mological parameters are the same as that of the T17 simulationsi.e. π = { . , . , . , − , } . For the first four parameters wechoose the step sizes δ i to be 4, 2, 10 and 8 per cent of the corre-sponding fiducial values. For the w a parameter we adopt δ w a = . Ω b , h fixed to their fiducial (T17)values and keep the flatness of the Universe unchanged. This meanswhen varying Ω cdm , the amount of dark energy in the Universe isadjusted accordingly. These step sizes were motivated from Yahia-Cherif et al. 2020 who proposed optimal steps for the 5-point stencilderivative for Fisher analysis with the galaxy power spectrum. Forour analysis we use slightly larger steps than them but within theproposed range of steps for the parameters. Our steps were found as MNRAS000
Same as Figure 5 but for ξ + , (blue) and i ζ + , (red) shear correlation functions for source redshift bin z = . (Abramowitz & Stegun 1964): ∂ M ( π ) ∂π i = − M ( π i + δ i ) + M ( π i + δ i ) − M ( π i − δ i ) + M ( π i − δ i )12 δ i (67)where δ i is a small change of the parameter π i about its fiducialvalue, and M ( π i ± δ i ) means evaluating the model vector at thechanged parameters π i ± δ i while keeping all other parameters fixed.For our purpose we shall be interested in the cosmological param-eters π = { Ω cdm , σ , n s , w , w a } where w and w a indicate the dy-namical dark energy equation of state parameters in the CPL pa-rameterization (Chevallier & Polarski 2001; Linder 2003) adoptedby the Dark Energy Task Force (Albrecht et al. 2006) to com- pare di ff erent dark energy probes. The fiducial values for our cos-mological parameters are the same as that of the T17 simulationsi.e. π = { . , . , . , − , } . For the first four parameters wechoose the step sizes δ i to be 4, 2, 10 and 8 per cent of the corre-sponding fiducial values. For the w a parameter we adopt δ w a = . Ω b , h fixed to their fiducial (T17)values and keep the flatness of the Universe unchanged. This meanswhen varying Ω cdm , the amount of dark energy in the Universe isadjusted accordingly. These step sizes were motivated from Yahia-Cherif et al. 2020 who proposed optimal steps for the 5-point stencilderivative for Fisher analysis with the galaxy power spectrum. Forour analysis we use slightly larger steps than them but within theproposed range of steps for the parameters. Our steps were found as MNRAS000 , 1–22 (2021) ntegrated 3-point shear correlation function Figure 7. Left panel : Fisher contours for the 5 cosmological parameters π = { Ω cdm , σ , n s , w , w a } for the model vectors — ξ ± (blue), i ζ ± (green dashed) andtheir joint model vector (orange) using the the FLASK DES-like covariance matrix with realistic shape-noise in a two tomographic source redshift bin settingwith z = . z = . Right panel : Same as left panel but zoomed in and only showing the ξ ± (blue) and the joint contours(orange). a trade o ff between neither being too big (in order to obtain accu-rate derivatives i.e. have low truncation errors) nor being too small(such that the derivatives are not dominated by numerical noise i.e.have low rounding-o ff errors). We do not impose any priors on these5 cosmological parameters.The inverse of the Fisher matrix gives the parameter covariancematrix C π under the assumptions that the measured data vectors aredrawn from a multi-variate Gaussian distribution and that the de-pendence of M on the parameters π is close to linear (Trotta 2017;Uhlemann et al. 2020): C π = F − . (68)Hence, using the derivatives and the expected data-covariance ma-trix (for a DES-sized survey) we can compute this parameter co-variance matrix and forecast error contours on the cosmological pa-rameters that we are interested in. In Figures 5 and 6 we show thederivatives of some components of our model vector with respect tothe 5 cosmological parameters, normalised by the standard deviationfor each component obtained from the FLASK covariance matrix (inother words, dividing the derivative of a statistic by the correspond-ing grey shaded error in Figure 4 for a given separation bin α ). Thisgives a visual estimate of the shape and amplitude of the ingredientsof the Fisher matrix. It is clear that the way in which the amplitudesand shapes of i ζ + derivatives (as a function of α ) di ff er from oneparameter to another is di ff erent compared to ξ + which results inslightly altered orientations of the error contours of each statistic inthe parameter planes. This can be seen in Figure 7. The error con-tours from ξ ± are shown in blue, the contours from i ζ ± are shown in To ensure that the steps were not too large, we verified that the δ i weresmaller than one-third of the 1-sigma marginalized Fisher constraints on theparameters (see Table 2) for the joint model in our final analysis. green dashed ellipses and the joint contours of the two together inorange. For clarity, we also remove the integrated 3-point functioncontours and show only the ξ ± and the joint contours on the righthand panel of the Figure. Although the i ζ ± alone has larger contourscompared to ξ ± — due to the lower amplitudes of the derivatives(see Figures 5, 6) which partly stems from the low S / N of i ζ ± (seeTable 1) — the degeneracy directions are slightly di ff erent. A jointanalysis of ξ ± along with i ζ ± thus helps to alleviate some of the pa-rameter degeneracies present in ξ ± alone and result in a significantdecrease in the contour sizes. The contribution from i ζ ± to the jointcontours is significant with respect to the w , w a parameters. Thiscan be reasoned by investigating the derivatives (see Figures 5 and6) of the statistics with respect to the dark energy equation of stateparameters. The derivatives change more significantly for the di ff er-ent source redshifts with respect to w , w a for i ζ + compared to ξ + .Partly, this arises due to the di ff erent scaling of the matter bispec-trum with respect to the parameters as compared to the matter powerspectrum. But more importantly, this can be attributed to the fact thatthe 2-point shear correlation is a projection of the power spectrumalong the line-of-sight with a weighting of q ( χ ) χ (see equations (36)and (32)), whereas the integrated 3-point shear correlation functionhas a factor of q ( χ ) χ (see equation (50)) implying that the latter isweighted more heavily at lower redshifts (or smaller χ ), especiallyin the dark energy dominated era. This sensitivity of the integrated3-point function to w and w a shows potential in probing the dynam-ical dark energy equation of state from cosmic shear data. Quanti-tatively, this can also be seen from the marginalized 1-sigma con-straints σ ( π i ) = (cid:112) C π , ii of the w and w a parameters for our analysisreported in the second column of Table 2. The constraints obtainedfrom the joint analysis of ξ ± and i ζ ± i.e. σ ( w ) = . σ ( w a ) = . ξ ± i.e. σ ( w ) = . σ ( w a ) = .
56 or in i ζ ± i.e. σ ( w ) = . MNRAS , 1–22 (2021) A. Halder et al.
Table 2.
Comparison of our work in real space using shear 2-point and integrated ξ ± , i ζ ± against previous works in Fourier space by Takada& Jain 2004; Kayo & Takada 2013; Sato & Nishimichi 2013 who used the convergence power spectrum P κ and the full convergence bispectrum B κ . Some ofthe symbols used in the table that have not been defined in the text earlier are: the total number of source galaxies over all tomographic bins n g = (cid:82) d z p ( z )where p ( z ) is the entire source galaxy distribution, α s is the spectral running index parameter and A s = δ ζ is the normalization parameter of the primordialpower spectrum. We present the marginalized 1-sigma constraints on the cosmological parameters σ ( π i ) along with the dark energy figure-of-merit (FoM) forour work using ξ ± , i ζ ± and the combined (joint) data vector of the two, respectively (with scale-cuts on the data vector). The step sizes δ i (for computing thederivatives of the model with respect to the parameters) when specified in per cent are relative to the fiducial parameter values. We also show correspondingvalues reported in the other works. The ‘—’ indicates values which are not explicitly reported or inapplicable to the other works. Our work Takada & Jain 2004 Kayo & Takada 2013 Sato & Nishimichi 2013Total n g (per arcmin ) 10 100; p ( z ) following Huterer 2002 20 25 σ (cid:15) z i z = . , z = . ≤ z ≤ . , z > . z = . , z = . , z = . p i ( z ) Dirac- δ function Equal n g , i in each bin from n ( z ) Top-hat function Dirac- δ functionField cosmic shear γ convergence κ convergence κ convergence κ Analysis in real or Fourier space real Fourier Fourier FourierData vectors (DVs) ξ ± ( α ) , i ζ ± ( α ) , joint P κ ( l ) , B κ ( l , l , l ) , joint P κ ( l ) , B κ ( l , l , l ) , joint P κ ( l ) , B κ ( l , l , l ) , jointMinimum and maximum scales 5 (cid:48) < α < (cid:48) ≤ l i ≤ ≤ l i ≤ ≤ l i ≤ Ω cdm , σ , n s , w , w a Ω de , Ω b , h , n s , σ , w , w a Ω de , Ω m h , Ω b h , n s , α s , δ ζ , w , w a Ω de , Ω cdm h , n s , A s , w , w a Derivative step sizes δ i δ w a = .
1; 5% for other parameters — δ w a = .
5; 10% for other parametersAnalysis with flat or non-flat Universe flat flat non-flat flatPriors in analysis none Planck priors on Ω b , h , n s Planck priors on all parameters noneMarginalized σ ( Ω cdm ) 0.08 , 0.16 , 0.04 — — —Marginalized σ ( σ ) 0.09 , 0.24, 0.06 — — —Marginalized σ ( n s ) 0.20 , 0.29 , 0.10 — — —Marginalized σ ( w ) 1.55 , 1.36 , 0.62 0.34 , 0.32 , 0.11 0.51 , 0.62 , 0.38 —Marginalized σ ( w a ) 4.56 , 4.01 , 2.11 0.93 , 0.91 , 0.36 1.30 , 1.60 , 0.94 —Dark energy FoM 0.78 , 0.19 , 2.28 — 11 , 7.2 , 20 5 , 15 , 25 σ ( w a ) = .
01. The same is true for the other cosmological parame-ters. Alternatively, one often quotes the dark energy figure of merit(FoM) defined as (Albrecht et al. 2006; Sato & Nishimichi 2013):FoM ≡ √ det ( C π [ w , w a ]) (69)to characterize the power of a survey to constrain these two param-eters. The higher the FoM, the stronger are the constraints in the w − w a plane. For a DES-like survey, our joint analysis has a FoM = = ξ ± shear correlations alone; visually, this is reflected fromthe smaller size of the orange contours in the right-hand panel ofFigure 7 compared to the blue contours. The above quoted numbersare with the scale-cuts assumed in our analysis. We expect that in-cluding smaller angular scales will show more improvement on themarginalized constraints and also on the FoM . However this needsthe development of more accurate models down to small angularscales. We also show for comparison, the marginalized constraintsand FoM from previous works by Takada & Jain 2004; Kayo &Takada 2013; Sato & Nishimichi 2013 who investigated the conver-gence power spectrum and the full convergence bispectrum. Theirreported constraints are significantly better than ours which we as-sociate to several di ff erences in their analysis settings to ours e.g.higher n g , no assumed scale-cuts, and for Kayo & Takada 2013 theyassumed priors on the parameters of their Fisher analysis (see theirFigure 1) whereas we do not impose any priors. Most importantly,these works investigate the constraining power of the full conver-gence bispectrum whereas we study only an integrated quantity ofthe bispectrum. The full bispectrum can be targeted to probe generalbispectrum configurations thereby probing more information than For example, assuming that our model is correct on all angular scalesand without imposing any scale-cuts, we find that the FoM for the joint datavector improves by over a factor of 2. integrated quantities of the bispectrum. Of course, this is also true inreal space for the full 3-point shear correlation function γ -3PCF orthe generalized third-order aperture mass statistics (Schneider et al.2005). All these statistics should ideally be able to constrain the darkenergy equation of state parameters better than the integrated 3-pointshear correlation function. However, all of them rely on the accu-rate measurement of the full γ -3PCF (or the bispectrum) from datawhich is still unexplored in current wide-area weak lensing surveys.The integrated 3-point shear correlation function is much easier tomeasure and holds potential to improve upon the parameter con-straints obtained from 2-point shear analyses alone. On the theoryside, we expect that including other e ff ects such as galaxy intrin-sic alignments, baryonic feedback, impact of massive neutrinos etc.should be easier to tune into the i ζ ± model compared to includingthem for the full shear 3-point correlation function. From both ob-servational and theoretical aspects, this makes the integrated 3-pointshear correlation function a promising statistic to explore in currentand future cosmic shear data. In this paper we propose a higher-order statistic — the integrated 3-point shear correlation function — which can be measured directly from the cosmic shear field observed in current wide-area weak-lensing surveys such as DES, KiDS, HSC and future surveys likeEUCLID . The following are the key results of this work: • The integrated 3-point shear correlation function i ζ ± can bemeasured by dividing a large survey area into several top-hat patches(each having an area of a few square degrees) and correlating the position-dependent (local) 2-point shear correlation function insideeach patch with the aperture mass statistic evaluated at the centre of .MNRAS000
01. The same is true for the other cosmological parame-ters. Alternatively, one often quotes the dark energy figure of merit(FoM) defined as (Albrecht et al. 2006; Sato & Nishimichi 2013):FoM ≡ √ det ( C π [ w , w a ]) (69)to characterize the power of a survey to constrain these two param-eters. The higher the FoM, the stronger are the constraints in the w − w a plane. For a DES-like survey, our joint analysis has a FoM = = ξ ± shear correlations alone; visually, this is reflected fromthe smaller size of the orange contours in the right-hand panel ofFigure 7 compared to the blue contours. The above quoted numbersare with the scale-cuts assumed in our analysis. We expect that in-cluding smaller angular scales will show more improvement on themarginalized constraints and also on the FoM . However this needsthe development of more accurate models down to small angularscales. We also show for comparison, the marginalized constraintsand FoM from previous works by Takada & Jain 2004; Kayo &Takada 2013; Sato & Nishimichi 2013 who investigated the conver-gence power spectrum and the full convergence bispectrum. Theirreported constraints are significantly better than ours which we as-sociate to several di ff erences in their analysis settings to ours e.g.higher n g , no assumed scale-cuts, and for Kayo & Takada 2013 theyassumed priors on the parameters of their Fisher analysis (see theirFigure 1) whereas we do not impose any priors. Most importantly,these works investigate the constraining power of the full conver-gence bispectrum whereas we study only an integrated quantity ofthe bispectrum. The full bispectrum can be targeted to probe generalbispectrum configurations thereby probing more information than For example, assuming that our model is correct on all angular scalesand without imposing any scale-cuts, we find that the FoM for the joint datavector improves by over a factor of 2. integrated quantities of the bispectrum. Of course, this is also true inreal space for the full 3-point shear correlation function γ -3PCF orthe generalized third-order aperture mass statistics (Schneider et al.2005). All these statistics should ideally be able to constrain the darkenergy equation of state parameters better than the integrated 3-pointshear correlation function. However, all of them rely on the accu-rate measurement of the full γ -3PCF (or the bispectrum) from datawhich is still unexplored in current wide-area weak lensing surveys.The integrated 3-point shear correlation function is much easier tomeasure and holds potential to improve upon the parameter con-straints obtained from 2-point shear analyses alone. On the theoryside, we expect that including other e ff ects such as galaxy intrin-sic alignments, baryonic feedback, impact of massive neutrinos etc.should be easier to tune into the i ζ ± model compared to includingthem for the full shear 3-point correlation function. From both ob-servational and theoretical aspects, this makes the integrated 3-pointshear correlation function a promising statistic to explore in currentand future cosmic shear data. In this paper we propose a higher-order statistic — the integrated 3-point shear correlation function — which can be measured directly from the cosmic shear field observed in current wide-area weak-lensing surveys such as DES, KiDS, HSC and future surveys likeEUCLID . The following are the key results of this work: • The integrated 3-point shear correlation function i ζ ± can bemeasured by dividing a large survey area into several top-hat patches(each having an area of a few square degrees) and correlating the position-dependent (local) 2-point shear correlation function insideeach patch with the aperture mass statistic evaluated at the centre of .MNRAS000 , 1–22 (2021) ntegrated 3-point shear correlation function the corresponding patch using a compensated filter. For fixed filtersizes, the i ζ ± ( α ) is a function of a single variable — the separationscale α at which the local 2-point shear correlation function is mea-sured. This makes it analogous to the full shear 2-point correlationfunction ξ ± ( α ) which is widely measured in weak lensing surveys(see Figure 4). • We develop a theoretical model for i ζ ± which is the real spacecounterpart of the integrated convergence bispectrum as introducedby Munshi et al. 2020b in Fourier space . The authors however, for-mulated the integrated bispectrum using equal-sized top-hat patcheson the convergence field. Working in real space with cosmic shear,we instead propose the usage of a combination of compensated (forthe aperture mass statistic) and top-hat filters (for the local 2-pointshear correlation) of di ff erent sizes allowing for the evaluation of thestatistic directly from cosmic shear data without any need for con-structing a convergence map. We compute our theoretical modelsusing the Gil-Marín et al. 2012 bispectrum fitting formula with the revised halofit non-linear matter power spectrum (Takahashiet al. 2012) implementation in the CLASS software (Lesgourgues2011; Blas et al. 2011). • We validate our model for the integrated 3-point function usingthe weak lensing shear simulations from Takahashi et al. 2017. Wefind that our theoretical predictions are in excellent agreement for themeasured ‘ + ’ integrated 3-point functions i ζ + (analogous to the ξ + shear 2-point correlation function) within the scatter of the simula-tions for multiple source redshifts and the cross-correlations thereof(see Figure 4). However, our model for the ‘ − ’ integrated 3-pointshear correlation functions i ζ − (analogous to the ξ − shear correla-tion function) agree with the simulations on large angular scales butover predict the simulation results on small scales. We associate thiswith the over estimation of the bispectrum by the Gil-Marín et al.2012 fitting formula for the highly squeezed configurations of thebispectrum which the i ζ − correlation function is mainly sensitive to.A more theoretically motivated formalism e.g. using the responsefunction approach to modelling the squeezed lensing bispectrum asrecently studied by Barreira et al. 2019 — who also formulated thee ff ect of baryons on the squeezed bispectrum — may help to ac-curately model the i ζ − correlation functions down to smaller angularscales. This also shows the potential in encoding e ff ects of non-linearprocesses (e.g. baryonic feedback) in the integrated 3-point functionwhich we expect to be easier compared to modelling them for thefull 3-point shear correlation function. Alternatively, using a moreaccurate 3D matter bispectrum fitting function such as bihalofit ,recently introduced by Takahashi et al. 2020, can be considered formore accurate predictions. These are left as directions for futureworks. • Making appropriate scale-cuts on the model vectors, we usethe Fisher matrix formalism to forecast constraints on cosmologi-cal parameters for a DES Year 6 sized survey with realistic shape-noise in a 2-redshift bin tomographic setting (see Figure 7). For thedata-covariance matrix we use a set of lognormal simulations usingthe
FLASK tool (Xavier et al. 2016). We find that the joint analysisof the integrated 3-point function and the 2-point shear correlationfunctions can allow for a significant improvement in the parameterconstraints compared to those obtained from 2-point shear corre-lation functions alone (see Table 2). This is because the responsesof the integrated 3-point shear correlations to the cosmological pa-rameters are di ff erent from that of 2-point shear correlations therebyresulting in slightly di ff erent degeneracy directions in the parameterplanes (see Figures 5 and 6). In particular, we find that the integrated3-point function has the potential to significantly improve the darkenergy figure of merit on a combined analysis with 2-point shear cor- relation functions. This arises due to the derivatives of the integrated3-point function (with respect to the dark energy equation of stateparameters) varying considerably in shape and amplitude comparedto the derivatives of the 2-point shear correlation. This can partly beattributed to the fact that the line-of-sight projection kernel in theexpression for the convergence bispectrum is weighted considerablymore heavily down to low redshifts (in the late-time dark-energydominated era) compared to the convergence power spectrum (seeequations (33) and (32)). This can be very useful for probing thedark energy equation of state parameters from cosmic shear dataalone and makes the integrated 3-point shear correlation function apromising method to probe higher-order information content of theshear field and thereby complement 2-point shear analysis.Theoretically, the integrated 3-point function (or the integrated bis-pectrum) of the lensing convergence field should be easier to workwith than the i ζ ± shear correlation function that we investigate inthis paper. However, observationally, the former requires one to gothrough the convergence map making process from the cosmic shearfield. This process becomes challenging when the observed shearfield has complicated masks and survey geometry. Although ouranalysis involves a simulated setup with simplifying assumptionssuch as a circular survey footprint without masks and holes, account-ing for the masking e ff ects is straight forward as our statistic is de-signed to be measured directly from the cosmic shear data (wherethe masking e ff ects are inherent) without the need for any map mak-ing. The integrated 3-point shear correlation function with its easeof measurement through the 2-point position-dependent shear corre-lation function and the 1-point aperture mass statistic is tailor-madefor application to real data.Although we have concentrated on the integrated 3-point functionof the cosmic shear field, we provide a general framework of equa-tions in chapter 2 which can be used for computing the integrated3-point function for any projected field e.g. galaxy counts field andits cross-correlations with the shear field. This will be explored infuture works. ACKNOWLEDGEMENTS
We sincerely thank Alexandre Barreira, Daniel Gruen and EiichiroKomatsu for helpful discussions and suggestions at various stagesof the project. We remain grateful to Ryuichi Takahashi for makinghis simulation suite publicly available and for clarifying our queries.This research was supported by the Excellence Cluster ORIGINSwhich is funded by the Deutsche Forschungsgemeinschaft (DFG,German Research Foundation) under Germany’s Excellence Strat-egy - EXC-2094-390783311. Some of the numerical calculationshave been carried out on the computing facilities of the Computa-tional Center for Particle and Astrophysics (C2PAP).
DATA AVAILABILITY
The data for the N-body simulations used in the articlewere accessed from the public domain: http://cosmo.phys.hirosaki-u.ac.jp/takahasi/allsky_raytracing/ . REFERENCES
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E., et al., 2003, Monthly Notices of the Royal Astronomical Soci-ety, 341, 1311Springel V., 2005, MNRAS, 364, 1105Springel V., Yoshida N., White S. D. M., 2001, New Astron., 6, 79Stebbins A., 1996, arXiv e-prints, pp astro–ph / arXiv:1701.01467 )Troxel M., et al., 2018, Physical Review D, 98Uhlemann C., Friedrich O., Villaescusa-Navarro F., Banerjee A., CodisS., 2020, Monthly Notices of the Royal Astronomical Society, 495,4006–4027Xavier H. S., Abdalla F. B., Joachimi B., 2016, Mon. Not. Roy. Astron. Soc.,459, 3693Yahia-Cherif S., et al., 2020, Validating the Fisher approach for stage IVspectroscopic surveys ( arXiv:2007.01812 )Zonca A., Singer L., Lenz D., Reinecke M., Rosset C., Hivon E., Gorski K.,2019, Journal of Open Source Software, 4, 1298MNRAS000 , 1–22 (2021) ntegrated 3-point shear correlation function APPENDIX A: FOURIER AND HANKEL TRANSFORMS
The forward and inverse Fourier transforms of a field f in the 2Dsky-plane can be written as f ( l ) = F [ f ( θ )] ≡ (cid:90) d θ f ( θ ) e − i l · θ (forward FT) f ( θ ) = F − [ f ( l )] ≡ (cid:90) d l (2 π ) f ( l ) e i l · θ (inverse FT) (A1)where l = ( l x , l y ) is the 2D Fourier wave-vector. If the field f isreal i.e. f ∗ ( θ ) = f ( θ ), then it follows from the above equation that f ∗ ( l ) = f ( − l ).If a function (e.g. correlation function) ξ ( α ) defined in the 2Dsky plane is independent of the direction of the vector α i.e. ξ ( α ) = ξ ( α ), then it follows from the Fourier transformation equation (A1)and from the properties of ordinary Bessel functions that (Schneider2006; Dodelson & Schmidt 2020): P ( l ) ≡ F [ ξ ( α )] = (cid:90) d α ξ ( α ) e − i l · α = π (cid:90) d α α ξ ( α ) J ( l α ) ,ξ ( α ) ≡ F − [ P ( l )] = (cid:90) d l (2 π ) P ( l ) e i l · α = (cid:90) d l l π P ( l ) J ( l α ) (A2)where J ( x ) is the zeroth-order Bessel function of the first kind.On the other hand, the 2D Fourier transform of ξ ( α ) with a com-plex phase factor e i φ α and its inverse transform reads P ( l ) ≡ F [ ξ ( α ) e i φ α ] = (cid:90) d α ξ ( α ) e − i l · α e i φ α = π (cid:90) d α α ξ ( α ) J ( l α ) ,ξ ( α ) ≡ F − [ P ( l ) e − i φ l ] = (cid:90) d l (2 π ) P ( l ) e i l · α e − i φ l = (cid:90) d l l π P ( l ) J ( l α ) (A3)where φ α is the polar angle of α , φ l is the polar angle of l and J ( x )is the fourth-order Bessel function of the first kind. These equationsare Hankel transformations. APPENDIX B: T17 SIMULATIONS POWER SPECTRACORRECTION FORMULAE
Takahashi et al. 2017 found that the convergence power spectra thatwas measured from the mean of the 108 simulated sky-maps intheir simulation suite, slightly underestimated the theoretical powerspectrum (using revised halofit ). They associated 3 e ff ects thatcaused the underestimation and provided correction factors to thetheory formulae to take them into account:(i) Finite simulation-box-size e ff ect : In Appendix B of Taka-hashi et al. 2017, the authors report that in order to consider thee ff ect of density fluctuations larger than the simulation-box-size L on the angular power spectrum, one needs to impose the conditionthat for k < π/ L , the matter power spectrum P δ ( k , η ) ! =
0, as thebox does not include fluctuations larger than L .(ii) Finite lens-shell e ff ect : The T17 lensing maps were pro-duced by ray tracing through lens shells of finite thickness in thesimulation boxes (see section 4.1). The finite thickness a ff ects the angular power spectrum of surface density fluctuations on a shell.To account for this, Takahashi et al. 2017 suggest to convolve thematter power spectrum with the window function of the shell (seetheir Appendix B). They provide a fitting formula for the convolvedpower spectrum: P δ ( k , η ) −→ (1 + c k − α ) α (1 + c k − α ) α P δ ( k , η ) (B1)with c = . × − , c = . × − , α = . α = . α = . z < . Finite angular resolution of sky-maps : For a given
NSIDE of a
Healpix map, the angular power spectrum C ( l ) measured fromthe sky-map is underestimated compared to the theoretical powerspectrum at large l due to lack of angular resolution. To account forthis in the theory spectrum, Takahashi et al. 2017 suggest a dampingfactor at small scales (high- l ) given by C ( l ) −→ C ( l )1 + ( l / l res ) (B2)where l res = . · NSIDE . APPENDIX C: 3D MATTER BISPECTRUM
The 3D matter bispectrum at leading order (tree-level) in densityperturbations as computed with standard Eulerian perturbation the-ory (PT) for Gaussian initial conditions is written as (Bernardeauet al. 2002; Dodelson & Schmidt 2020): B δ, tree ( k , k , k , η ) = F ( k , k , η ) P δ, L ( k , η ) P δ, L ( k , η ) + cyclic permutations (C1)where P δ, L ( k , η ) = D + ( η ) P δ, L ( k , η ) (C2)is the 3D linear matter power spectrum today evolved to time η usingthe linear growth factor D + ( η ) which is normalised to unity today i.e. D + ( η ) = F ( k i , k j , η ) is a symmetrized two-point mode couplingkernel which in a general Λ CDM universe takes the form (Friedrichet al. 2018): F ( k i , k j , η ) = µ ( η ) +
12 cos( φ ij ) (cid:32) k i k j + k j k i (cid:33) + [1 − µ ( η )] cos ( φ ij ) (C3)where φ ij is the angle between the two wave-vectors k i and k j . Inan Einsten-de Sitter (EdS) universe, the function µ ( η ) is a constantand takes the value µ ( η ) = . However, this form of the bispectrumonly works in the linear regime (large physical scales) and fails inthe non-linear regime. To improve upon this, one can go on to in-clude higher-order PT corrections but calculating the higher orderterms are cumbersome. Another way of predicting the non-linearmatter bispectrum is to propose a fitting formula for the bispectrumand calibrate the function’s parameters using the bispectra measuredfrom cold dark matter N-body simulations. This approach was firsttaken by Scoccimarro & Frieman 1999 and later improved by Scoc-cimarro & Couchman 2001 and Gil-Marín et al. 2012. In this paperwe use the bispectrum fitting formula of Gil-Marín et al. 2012: B δ ( k , k , k , η ) = F e ff ( k , k , η ) P δ ( k , η ) P δ ( k , η ) + cyclic permutations (C4)where P δ ( k , η ) is the 3D non-linear matter power spectrum (e.g.obtained using revised halofit (Takahashi et al. 2012)) and the MNRAS , 1–22 (2021) A. Halder et al. e ff ective mode coupling kernel F e ff ( k , k , η ) is a modified versionof the EdS F kernel and reads F e ff ( k i , k j , η ) = a ( k i , η ) a ( k j , η ) +
12 cos( φ ij ) (cid:32) k i k j + k j k i (cid:33) b ( k i , η ) b ( k j , η ) +
27 cos ( φ ij ) c ( k i , η ) c ( k j , η ) . (C5)The functions a ( k , η ), b ( k , η ) and c ( k , η ) are fitting formulae cali-brated with N-body simulations to interpolate the results betweenthe linear (tree-level bispectrum) and the non-linear regime bispec-trum measured from the simulations: a ( k , η ) = + σ a ( η )[0 . Q ( n e ff )] / ( qa ) n e ff + a + ( qa ) n e ff + a b ( k , η ) = + . a ( n e ff + qa ) n e ff + + a + ( qa ) n e ff + . + a b ( k , η ) = + . a / [1 . + ( n e ff + ]( qa ) n e ff + + a + ( qa ) n e ff + . + a . (C6)Although these functions have been expressed in terms of conformaltime, it is completely equivalent to replace η with the correspondingredshift z as the time argument in the above expressions. σ ( η ) isthe standard deviation of matter density fluctuations today linearlyevolved to time η i.e. σ ( η ) = D + ( η ) σ ( η ). The e ff ective logarith-mic slope of the linear matter power spectrum today n e ff ( k ) reads n e ff ( k ) = d log P δ, L ( k , η )d log k . (C7) q ≡ k / k nl is defined with the scale k nl ( η ) at which non-linearities startto become important and is defined as k P δ, L ( k nl , η )2 π ≡ Q ( n e ff ) is defined as Q ( n e ff ) ≡ − n e ff + n e ff + . (C9)The values for the parameters calibrated using simulations as foundby Gil-Marín et al. 2012 are: a = . , a = . , a = − . , a = . , a = . , a = − . , a = . , a = − . , a = − . . As reported by Gil-Marín et al. 2012, the fitting formula withthese parameter values works reasonably well for z < . k < . − h in Λ CDM cosmologies. However, in this paperwe use this fitting function for non- Λ CDM cosmologies, in partic-ular to predict the bispectrum for cosmologies with varying dark-energy equation of state parameters by encoding the informationof the latter into the fitting formula through the linear and non-linear ( revised halofit ) power spectra and σ ( η ) obtained using CLASS . Our approach is similar to what has previously been done bySato & Nishimichi 2013 who verified that the GM formula worksreasonably well in modelling the lensing bispectrum measured in N-body simulations with dynamical dark energy (see their Figure 8).Another approach, for w CDM cosmologies (i.e. w = constant and w a =
0) can be taken by using the recently introduced bihalofit fitting formula for the matter bispectrum by Takahashi et al. 2020which is more accurate than the GM formula especially in predictingthe highly squeezed configurations of the matter bispectrum whichthe GM formula overestimates.
Figure D1.
The scaled integrated shear bispectrum iB + , ( l ) for source red-shift bin z = . l are numerical artefacts arising fromthe integration routine being forced to exclude sampled points in the integra-tion volume for non-squeezed configurations. The computations for iB + wereperformed using a compensated filter of size θ ap = (cid:48) and top-hat windowof radius θ T = (cid:48) . APPENDIX D: INTEGRATED SHEAR BISPECTRUMUSING DIFFERENT APPROXIMATIONS
Here we compare the results of computing the integrated bispectrum iB + , ( l ) and correspondingly the integrated 3-point shear correla-tion function i ζ + , ( α ) for the source redshift z = . ff erent 3D matter bispectrum approximations in equations(53) and (50). In Figure D1 we plot the integrated bispectrum whencomputed with the GM bispectrum fitting formula (as already shownin Figure 2) along with the prediction when using the tree-levelbispectrum (see equation (C1)). We also plot the integrated bis-pectrum with the GM bispectrum but only when considering elon-gated / squeezed configurations i.e. when two modes of the bispec-trum are at least larger than two times the smallest mode. From theFigure, it is clear that the tree-level bispectrum and the GM formulamatch on low- l ( l (cid:28) ff er significantly on small scales which correspond to the non-linear regime (high- l ). On the other hand, when computing the iB + with only the squeezed configurations of the GM bispectrum, wefind that the result matches with the full GM result only in the high- l end, indicating that most of the iB + signal is dominated by squeezedconfigurations for l much larger than the characteristic mode cor-responding to the diameter of the patch within which the position-dependent shear correlation is measured i.e. l (cid:29) π/ (2 θ T ) ≈ l modes corresponding to scales approximately thesize of the patch or larger, the squeezed configuration result under-estimates the full GM result. This shows that our statistic probes notonly the squeezed but partly also other bispectrum configurations.This also explains the non-smoothed behaviour of the squeezed iB + result on low- l as the integration routine is forced to exclude sampledpoints in the integration volume for the non-squeezed configurationswhich contribute mostly at low- l . The non-smoothness is insignifi-cant and does not a ff ect the computation of the i ζ + signal which wediscuss next.In Figure D2 we show the corresponding real space i ζ + , ( α ) pre- MNRAS000
Here we compare the results of computing the integrated bispectrum iB + , ( l ) and correspondingly the integrated 3-point shear correla-tion function i ζ + , ( α ) for the source redshift z = . ff erent 3D matter bispectrum approximations in equations(53) and (50). In Figure D1 we plot the integrated bispectrum whencomputed with the GM bispectrum fitting formula (as already shownin Figure 2) along with the prediction when using the tree-levelbispectrum (see equation (C1)). We also plot the integrated bis-pectrum with the GM bispectrum but only when considering elon-gated / squeezed configurations i.e. when two modes of the bispec-trum are at least larger than two times the smallest mode. From theFigure, it is clear that the tree-level bispectrum and the GM formulamatch on low- l ( l (cid:28) ff er significantly on small scales which correspond to the non-linear regime (high- l ). On the other hand, when computing the iB + with only the squeezed configurations of the GM bispectrum, wefind that the result matches with the full GM result only in the high- l end, indicating that most of the iB + signal is dominated by squeezedconfigurations for l much larger than the characteristic mode cor-responding to the diameter of the patch within which the position-dependent shear correlation is measured i.e. l (cid:29) π/ (2 θ T ) ≈ l modes corresponding to scales approximately thesize of the patch or larger, the squeezed configuration result under-estimates the full GM result. This shows that our statistic probes notonly the squeezed but partly also other bispectrum configurations.This also explains the non-smoothed behaviour of the squeezed iB + result on low- l as the integration routine is forced to exclude sampledpoints in the integration volume for the non-squeezed configurationswhich contribute mostly at low- l . The non-smoothness is insignifi-cant and does not a ff ect the computation of the i ζ + signal which wediscuss next.In Figure D2 we show the corresponding real space i ζ + , ( α ) pre- MNRAS000 , 1–22 (2021) ntegrated 3-point shear correlation function Figure D2.
The integrated 3-point function i ζ + , ( α ) for source redshift bin z = . l > ff ects in the T17simulations. The computations use a compensated filter of size θ ap = (cid:48) andtop-hat window of radius θ T = (cid:48) . dictions by Hankel transforming (actually using equation (61)) theintegrated shear bispectra computed above and compare them withthe result of the T17 simulations. The GM bispectrum computedprediction matches well with the simulations as already seen in Fig-ure 4. The tree-level bispectrum computed i ζ + ( α ) signal only cap-tures the result on the largest angular scales but heavily deviatesin the non-linear regime. The squeezed configuration calculation ofthe GM bispectrum follows the trend of the simulation on smallscales while slightly underestimating the measured signal. This canbe attributed to the fact that at a given small angular separation α , i ζ + ( α ) receives contributions not only from the high- l end of iB + ( l )but also from the low- l end that correspond to scales larger thanthe separation scale (see equation (61)). As seen in Figure D1, thesqueezed bispectrum iB + ( l ) underestimates the full GM bispectrumresult in the low- l end thereby explaining the slight deficit. On largerscales, the squeezed bispectrum fails to describe the simulation re-sults showing that the squeezed-limit approximation does not holdas α approaches the size of the patch. We also show the result ofthe i ζ + ( α ) signal computation using the GM bispectrum but restrict-ing the Hankel summation of iB + ( l ) to include only l >
150 i.e.modes corresponding to scales much smaller than the size of thepatch. Although the result does not describe the simulations, the sig-nal matches the squeezed bispectrum results on the small scales con-firming that the i ζ + signal is indeed described by the squeezed limitbispectrum on these scales. However, it is worth noting that on verysmall scales (smaller than 5 (cid:48) ) the i ζ + prediction with the full GM bis-pectrum will eventually fail to describe the T17 simulations as theintegrated bispectrum result at extremely small scales (very high- l )receives most contribution from highly squeezed bispectrum config-urations which are known to be overestimated by the GM formula(Namikawa et al. 2019; Takahashi et al. 2020). This was apparent forthe i ζ − signals (see Figure 4) which are already sensitive to the very Figure E1. O ff sets between fiducial parameters (black dotted lines) and best-fit parameters (blue stars) in the parameter planes. The Fisher contours ex-pected from the analysis of the entire model vector (after imposing scale-cuts) are shown in orange centred around the fiducial parameters. high- l values of the integrated bispectrum for angular separationsaround 30 (cid:48) (due to the J Bessel function weighting).
APPENDIX E: IMPACT OF SYSTEMATIC OFFSETBETWEEN MODEL AND DATA VECTORS ONPARAMETER CONSTRAINTS
Here we discuss the impact of the remaining systematic o ff set be-tween the model M and data vector D (see Figure 4) after impos-ing the angular scale-cuts in our analysis. A systematic o ff set wouldamount to a bias in our parameter constraints which would causethe Fisher contours in Figure 7 to be centred around the wrong cos-mological values π — in our case the fiducial parameters. In otherwords, we want to explore how much the best-fit parameters π MP of the model describing the data vector is o ff from π . In order todo so, we need to minimize the χ ( π ) as a function of the param-eters (see equation (63)) between the data and model. We alreadysaw in section 5.1 that the χ ( π = π ) between D and M ( π ) has avalue of 1.08. We now want to find the parameters π MP which de-scribe the data vector with the lowest χ . We adopt the approachof Friedrich et al. 2020 (see their section 5.1) and study a linearizedapproximation of the model vector as a function of the parameters M ( π ) around the fiducial parameters π . This allows us to write thebest-fit parameters as (see equation 32 of Friedrich et al. 2020): π MP = π + F − x (E1)where we have assumed no priors on the parameters. F is the Fishermatrix (see equation (66)) of the model vector and x is another vector MP stands for maximum posterior in the notation of Friedrich et al. 2020.MNRAS , 1–22 (2021) A. Halder et al. with components: x i = (cid:16) D − M ( π ) (cid:17) T C − (cid:32) ∂ M ( π ) ∂π i (cid:33) (E2)where C is the data-covariance matrix (see equation (59)) and ∂ M ( π ) ∂π i are the derivatives of the model with respect to the parameters, eval-uated at the fiducial values π = π . We show our best-fit parametersfor the model describing the entire T17 data vector (after impos-ing the scale-cuts) in Figure E1 which can be seen to scatter veryclosely around the fiducial parameters. We also plot the orange con-tours (see Figure 7) of the parameters from the Fisher analysis forthe entire data vector (with ξ ± and i ζ ± including the assumed scale-cuts). The absolute o ff sets of the best-fit parameters from the fiducialvalues in units of the marginalized 1-sigma Fisher constraints for the5 parameters Ω cdm , σ , n s , w , w a are 0.12, 0.22, 0.25, 0.02, and 0.02respectively. As these o ff sets are smaller than one-third the marginal-ized 1-sigma constraints in the parameter planes, we conclude thatour fiducial model after imposing the angular scale-cuts describesthe T17 data-vector very well and there is no significant bias in ourresults. Ideally, one should include these o ff sets as a systematic errorbut as they are not significant we we deem it safe to ignore this forour analysis. This paper has been typeset from a TEX / L A TEX file prepared by the author.MNRAS000