Issue 
A&A
Volume 561, January 2014



Article Number  A53  
Number of page(s)  26  
Section  Cosmology (including clusters of galaxies)  
DOI  https://doi.org/10.1051/00046361/201322146  
Published online  23 December 2013 
Online material
Appendix A: Twopoint cosmic shear correlation
Appendix A.1: Densityshear twopoint correlation
The product of twopoint correlations in Eq. (29) reads as (A.1)where we introduced (A.2)This involves the spin2 correlation defined in Eq. (31). Writing the power spectrum in terms of the twopoint correlation function, integrating over the longitudinal direction and the polar angles of r_{⊥} and k_{⊥}, Eq. (31) also reads as (A.3)(Here we note .) The integral over two Bessel functions satisfies
(A.4)where Θ is the Heaviside function. This yields Eq. (32) and Eq. (A.2) writes as (A.5)where we note x_{1,2} = x_{2} − x_{1}. Since we have α_{x1′,2} = α_{x1,2} and α_{x2′,1} = α_{x2,1} = α_{x1,2} + π, Eq. (A.5) gives Eq. (33).
Appendix A.2: Density–shear–shear threepoint correlation
Substituting the expression (37) of the bispectrum into Eq. (36), we are led to compute the quantity (A.5)where the superscripts in the three terms in the lefthand side refer to the arguments of the power spectra in the three terms in the bracket in the right hand side. Here and in the following, we use the flat sky approximation and the fact that the three points, 1, 1′, and 2′, are at the same redshift (i.e., within radial distances of order ~8 h^{1}Mpc). By symmetry, the first term vanishes, (A.7)Indeed, x_{1} and x_{1′} are along the same line of sight, hence their projected separation in the transverse plane is zero, x_{1′ ⊥} − x_{1 ⊥} = 0, and the angular integration over the polar angle α_{k1′} vanishes.
Using the exponential representation of the Dirac factor, the second term reads as (A.8)Using Eqs. (30) and (31) in Eq. (A.8) gives (A.9)Using the fact that x_{1′ ⊥} = x_{1 ⊥}, we make the change of variables . Then, using the JacobiAnger expansion, (A.10)and Eq. (A.4), we can perform the integration over k_{2′ ⊥}, which gives (A.11)As compared with the factor ξ_{2′,1}ξ_{2′,1′} that arises for the convergence, as in Eq. (15), the sourcelens clustering bias of the cosmic shear is suppressed by the spin2 factor e^{2iα}. It replaces one correlation ξ by a correlation ξ^{(2)}, which is smaller because of the subtraction in Eq. (32), and it yields a second subtraction in Eq. (A.11).
In a similar fashion, the third term of Eq. (A.5) also reads as (A.12)Then, making the change of variable and using the expansion (A.10) we can integrate over angles. Next, using the property (A.4) and the summation rule , we obtain (A.13)Then, expressing P(k_{1}) in terms of the twopoint correlation function, as in (A.14)and using the property (A.15)where a > 0,b > 0,c > 0, n is integer, and Θ is a unit tophat with obvious notations (i.e., unity when the conditions are satisfied and zero otherwise), we can integrate over k_{1} and we obtain (A.16)Again, as compared with the factor ξ_{1′,1} ξ_{1′,2′} that arises for the convergence, the spin2 factor e^{2iα} suppresses the sourcelens clustering bias by replacing a factor ξ by ξ^{(2)} and introducing another subtraction.
Appendix B: Threepoint cosmic shear correlation
Using the exponential representation of the Dirac distribution, Eq. (60) also writes as (B.1)Integrating one after the other over the longitudinal components, { x_{2′ ∥},x_{3′ ∥} }, { k_{2′ ∥},k_{3′ ∥} }, { r_{∥},k_{1′ ∥} }, the angles { α_{x1},α_{x2},α_{x3} }, { α_{k1′},α_{k2′},α_{k3′} }, and α_{r}, we obtain (B.2)where d = χ_{1′}θ is the radius of the circumcircle at radial distance χ_{1′}. Using the ansatz (37), this reads as (B.3)where we used Eq. (A.14) to write (B.4)Using the property (A.15) and (B.5)with the relation J_{0}(z) + J_{2}(z) = 2J_{1}(z) / z, we obtain (B.6)Then, using Eqs. (A.4) and (B.6), we can integrate Eq. (B.3) over wavenumbers, which yields (B.7)where the angles ϕ and ϕ′ are given by Eq. (65).
Appendix C: Threepoint lensing–intrinsic shear correlations
To compute the lensing–intrinsic threepoint correlations (95) and (98) we proceed as in Appendix B. Using the exponential representation of the Dirac distribution, Eq. (95) also writes as (C.1)Integrating one after the other over the longitudinal components, { x_{3′ ∥},k_{3′ ∥} }, the angles { α_{x1},α_{x2},α_{x3} }, { α_{k1},α_{k2},α_{k3′} }, and α_{r}, we obtain (C.2)where d = (χ_{1} + χ_{2})θ / 2 is the radius of the circumcircle at radial distance (χ_{1} + χ_{2}) / 2 (this threepoint correlation is only relevant when the two redshifts z_{1} and z_{2} are very close). Next, using the hierarchical ansatz (37), we can split into three contributions. The first term, associated with the product P(k_{1})P(k_{2}) in the bispectrum ansatz, writes as (C.3)where we used Eq. (A.4) to integrate over k_{3′ ⊥}. Next, writing the power spectra in terms of the twopoint correlation functions, we can integrate over { k_{1 ∥},k_{2 ∥} }. This yields (C.4)Using Eq. (B.6) we can integrate over { k_{1 ⊥},k_{2 ⊥} }, which yields (C.5)where the angles ϕ_{i} and are given by (C.6)as in Eq. (65). The second term in Eq. (C.2), associated with the product P(k_{1})P(k_{3}), writes as (C.6)where we used Eq. (A.4) to integrate over k_{2 ⊥}. Next, writing the power spectra in terms of the twopoint correlation functions, with Eq. (B.4) for P(k_{3′ ⊥}), we can integrate over { k_{2 ∥},r_{∥},k_{1 ∥} }. This yields (C.8)Using Eq. (B.6) we can integrate over { k_{1 ⊥},k_{3′ ⊥} }, which yields (C.9)where the angles ϕ_{i} and are given by Eq. (C.6). The third contribution is obtained in the same manner, and within our approximation z_{1} ≃ z_{2} we have .
In a similar fashion, integrating over the angles { α_{x1},α_{x2},α_{x3} }, { α_{k1},α_{k2},α_{k3} }, and α_{r}, Eq. (98) writes as (C.10)where d = (χ_{1} + χ_{2} + χ_{3})θ / 3 is the radius of the circumcircle at radial distance (χ_{1} + χ_{2} + χ_{3}) / 3 (this threepoint correlation is only relevant when the three redshifts are very close). Next, using again the hierarchical ansatz (37), we can split into three contributions. The first term, associated with the product P(k_{1})P(k_{2}) in the bispectrum ansatz, writes as (C.11)where we used Eq. (A.4) to integrate over k_{3 ⊥} and we also integrated over { k_{3 ∥},r_{∥} }. Next, writing the power spectra in terms of the twopoint correlation functions, we can integrate over wavenumbers by using Eq. (B.6). This yields (C.12)where x_{i,j ∥} = x_{i ∥} − x_{j ∥} and the angles ϕ_{i} and are given by Eq. (C.6). The second and third contributions and are also given by Eq. (C.12) through permutations over the indices { 1,2,3 }.
Appendix D: Comparison of models for the lensing threepoint functions
Fig. D.1
Convergence and shear three point correlations and , as a function of the angular scale θ, for the redshift triplet z_{1} = 0.5,z_{2} = 1,z_{3} = 2. We show the predictions from the model of Valageas et al. (2012a,b) (solid lines) and from the hierarchical ansatz (18) (dotted lines). Because is negative we plot . 

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We compare in Fig. D.1 the predictions for threepoint weak lensing correlations from the model of Valageas et al. (2012a,b),
which combines oneloop perturbation theory with a halo model (Valageas & Nishimichi 2011b) and has been checked against raytracing numerical simulations, with the predictions from the hierarchical ansatz (18). We consider the source redshift triplet { 0.5,1,2 } but other redshifts give similar results.
The shear threepoint correlation is typically smaller than the convergence one because of the spin2 factor e^{2iα}, which leads to some cancellations as seen from the counterterms in Eq. (61) for the circular average (51). This effect is larger on smaller scales where the slope of is lower.
Figure D.1 shows that the hierarchical ansatz (18) provides the correct order of magnitude for weak lensing threepoint functions on scales θ ≲ 50′. More precisely, both approximations agree to better than a factor 1.5 for θ < 10′ and a factor 3 for θ < 40′, for ; and to better than a factor 1.5 for θ < 30′ for . Because most of the cosmological information from weak lensing threepoint correlations measured in galaxy surveys comes from θ ≲ 10′, as the amplitude of the signal decreases on larger scales, the hierarchical ansatz (18) would be sufficient to estimate the relative importance of the sourcelens clustering bias (which itself is dominated by contributions that only depend on the twopoint density correlation) or of other sources of noise. This provides significantly faster numerical computations.
© ESO, 2013
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