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Appendix A: Two-point cosmic shear correlation

Appendix A.1: Density-shear two-point correlation

The product of two-point correlations in Eq. (29) reads as (A.1)where we introduced (A.2)This involves the spin-2 correlation defined in Eq. (31). Writing the power spectrum in terms of the two-point 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 = x₂ − x₁. 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 three-point 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 left-hand 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^-1Mpc). By symmetry, the first term vanishes, (A.7)Indeed, x₁ 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 Jacobi-Anger 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′,1ξ_2′,1′ that arises for the convergence, as in Eq. (15), the source-lens clustering bias of the cosmic shear is suppressed by the spin-2 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(k₁) in terms of the two-point 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 top-hat with obvious notations (i.e., unity when the conditions are satisfied and zero otherwise), we can integrate over k₁ and we obtain (A.16)Again, as compared with the factor ξ_1′,1   ξ_1′,2′ that arises for the convergence, the spin-2 factor e^2iα suppresses the source-lens clustering bias by replacing a factor ξ by ξ⁽²⁾ and introducing another subtraction.

Appendix B: Three-point 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 J₀(z) + J₂(z) = 2J₁(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: Three-point lensing–intrinsic shear correlations

To compute the lensing–intrinsic three-point 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 = (χ₁ + χ₂)θ / 2 is the radius of the circumcircle at radial distance (χ₁ + χ₂) / 2 (this three-point correlation is only relevant when the two redshifts z₁ and z₂ are very close). Next, using the hierarchical ansatz (37), we can split into three contributions. The first term, associated with the product P(k₁)P(k₂) 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 two-point 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(k₁)P(k₃), writes as (C.6)where we used Eq. (A.4) to integrate over k_2 ⊥. Next, writing the power spectra in terms of the two-point 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 z₁ ≃ z₂ 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 = (χ₁ + χ₂ + χ₃)θ / 3 is the radius of the circumcircle at radial distance (χ₁ + χ₂ + χ₃) / 3 (this three-point 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(k₁)P(k₂) 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 two-point 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 three-point functions

	Fig. D.1 Convergence and shear three point correlations and , as a function of the angular scale θ, for the redshift triplet z1 = 0.5,z2 = 1,z3 = 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 three-point weak lensing correlations from the model of Valageas et al. (2012a,b),

which combines one-loop perturbation theory with a halo model (Valageas & Nishimichi 2011b) and has been checked against ray-tracing 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 three-point correlation is typically smaller than the convergence one because of the spin-2 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 three-point 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 three-point 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 source-lens clustering bias (which itself is dominated by contributions that only depend on the two-point density correlation) or of other sources of noise. This provides significantly faster numerical computations.

© ESO, 2013