Appendix: Relations between other correlation functions

In Sect. 4.1.2 we have derived the relations between the shear and the convergence 2- and 3-pt correlation functions. The method we used is based on the relation between the 2-pt correlation functions of the convergence κ and the deflection potential ψ (4.12). Thus the method can easily be generated to derive relations between the correlation function of κ and that of any weak lensing quantity g which can be expressed as derivatives of ψ. We denote g = Dgψ, and write these 2-pt relations in a general form

g(x₁)g^′(x₂) = 1 π²

Z

d²y hκκi (y) H (x₁− x₂− y) . (4.91) Listed below are some candidates for g and the corresponding operator D_g, where α is the deflection angle defined as α = ∂ψ.

g ψ κ γ α

Dg 1 ∇²/2 ∂²/2 ∂

Appendix: Relations between other correlation functions

Table 4.1: Forms of the convolution kernel H as defined in (4.91) for different weak lensing 2-pt statistics.

hXXi H (F ) integration form of H H (z)

hψψi F R

d²v ln |v| ln |z − v| (π/2) |z|²(ln z − 1) hγψi ¹₂∂²F −R

d²v ₍v^∗−¹z^∗)² ln |v| (π/2) z/z^∗

hααi −∂²F R

d²v v^1∗v^∗−1z^∗ −πz/z^∗ hκψi ¹₂∂∂^∗F πR

d²v δ⁽²⁾_D(z − v) ln |v| π ln z hαγi ¹₂∂³F R

d²v v^1∗2v^∗−¹z^∗ −πz/z^∗2 hακi ¹₂∂²∂^∗F πR

d²v z^∗−¹v^∗δ⁽²⁾_D (v) π/z^∗ hα^∗γi ¹₂∂²∂^∗F R

d²v v^1∗2v⁻¹z π/z^∗ hγγi ¹₄∂⁴F R

d²v v^1∗2(v^∗−1z^∗)² 2πz/z^∗3 hκγi ¹₄∂³∂^∗F −πR

d²v v^1∗2δ⁽²⁾_D (z − v) −π/z^∗2

hκκi ¹₄∂²∂^∗2F π²δ⁽²⁾_D (z)

hγγ^∗i ¹

4∂²∂^∗2F R

d²v v¹²(v^∗−1z^∗)² π²δ⁽²⁾_D (z)

Let DgDg^′ act on both sides of the relation between hψψi and hκκi (4.11), and use the statistical homogeneity of the κ field, one can obtain an integration form of the kernel H , in analogy to (4.8).

Let the same operator act on both sides of (4.12), one can express H as derivatives of the convolution kernel F in (4.12), as H (x1− x₂− y) = D_gDg^′F (x₁− x₂− y). Further inserting the explicit form of F (4.19) allows one to obtain the explicit form of H as a function of z = x₁− x₂− y. We summarize some of the 2-pt relations in Table. 4.1. Note that the form of H (F ) for hααi has a minus sign, which is due to the fact that ∂x1∂x2F (x₁− x₂− y) = −∂²F (z) with z = x₁− x₂− y.

We write the relations between hκκκi and the 3-pt correlation functions of the g’s also in a uniform convolutional form,

gg^′g^′′ (x₁, x₂) = 1 π³

Z d²y₁

Z

d²y₂ hκκκi (y₁, y₂) I(x₁− y₁, x₂− y₂) . (4.92) To find the explicit form of the convolution kernel I, we first write it into an integral form, in analogy to (4.5), and then split it into terms which can be expressed also as derivatives of the kernel F , like (4.20). Then with the explicit form of F one can reach the explicit form of I. We list the forms of the convolution kernel I for some 3-pt statistics in Table. 4.2.

Some of these relations, e.g. those for hκγi, hγγκi, and hγκκi, can find their application in galaxy-galaxy(-galaxy) lensing which corresponds to the cross-correlation of shear and galaxy number den-sity. Some other relations, e.g. those for hααi, hακi, and hαακi, are potentially of interest for studies of the lensing effects on the Cosmic Microwave Background and its cross-correlation with galaxy weak-lensing maps (Hu 2000).

PTER4.RELATIONSBETWEENTHREE-POINTCONFIGURATIONSPACESHEARCONVERGENCESTATISTICS

Table 4.2: Forms of the convolution kernel I as defined in (4.92) for different weak lensing 3-pt statistics.

hXXXi integration form of I split form of I I(a, b)

hαααi −R

Chapter 5

Bispectrum covariance in the flat-sky limit

As the matter density field evolves to be more and more non-Gaussian under non-linear gravi-tational clustering, information which is originally contained exclusively in the 2-pt statistics leaks into higher-order statistics. In cosmic shear studies, several authors have shown that the lowest order of them, i.e. the 3-pt statistics, already adds much information to the 2-pt one; in particular it can break the near degeneracy between the density parameter Ω_mand the power spectrum normalization σ₈(Bernardeau et al. 1997; Jain & Seljak 1997; van Waerbeke et al. 1999; Hui 1999). More recent studies by Takada & Jain (2004), TJ04 afterwards, and Berg´e et al. (2010), showed that including 3-pt statistics can improve parameter constraints significantly, typically by a factor of three.

In order to quantify the information content in lensing 3-pt statistics theoretically, one needs to have an expression for the covariance matrix of the 3-pt statistics. In this chapter we aim at deriving an expression for the bispectrum covariance hB(ℓ₁, ℓ₂, ℓ₃)B(ℓ₄, ℓ₅, ℓ₆)i for cosmic shear.

Previous work done within a flat-sky spherical harmonic formalism (Hu 2000) in the context of the CMB has been frequently referred to for such an expression. However several drawbacks exist in this approach. For instance, the expression given by Hu (2000) is valid only for integer arguments and does not allow a free binning choice, whereas it is desirable to evaluate the bispectrum and its covariance at real-valued angular frequencies and use e.g. a logarithmic binning. The other draw-backs are formal ones, e.g. the formula contains the Wigner symbol whose physical meaning within a flat-sky consideration remains obscure; the finite survey size is accounted for only by multiplying a factor, which lacks justification. There is also an unjustified assumption made in the coordinate transformation between the full sky and the 2D plane.

All these drawbacks are associated with the spherical harmonic formalism Hu (2000) adopted.

Thus we attempt a pure 2D Fourier-plane approach. We also work in the flat-sky limit since it greatly simplifies the mathematical form. Furthermore, the flat-sky limit is appropriate for practically all applications of weak lensing as the correlation of signals is only measured up to separations of a few degrees.

The major results of this chapter is published in Joachimi et al. (2009).

5.1 Bispectrum estimator

5.1.1 Estimator for B(ℓ₁, ℓ₂, ℓ₃)

The first and the most crucial step of deriving an expression for the bispectrum covariance is to find a proper expression for the estimator of the bispectrum B(ℓ1, ℓ2, ℓ3), where the ℓ’s are real-valued angular frequencies in our approach. We will use the convergence bispectrum (3.37) instead of the shear bispectrum due to formal simplicity. Since the two differ only by a phase factor, the result can easily be applied to the shear bispectrum.

To define an estimator of the bispectrum is to express it in terms of the convergence κ, i.e. to

‘invert’ the equation (3.37). There are three points to consider in doing so. First, the argument ℓ’s in the bispectrum are the absolute values of the vectorℓ’s, suggesting that angular averaging is needed.

Second, the bispectrum is defined only when the triangle condition is satisfied. If this condition is satisfied, the value of the Dirac delta function is infinity, nevertheless one needs to ‘invert’ it to obtain an estimator for the bispectrum. This seemingly unsolvable problem vanishes if one considers a finite survey size A. The Dirac Delta function can be expressed as

δ⁽²⁾_D(ℓ) = 1 (2π)²

Z

d²x e^iℓ·x. (5.1)

One can easily verify that, when the integral on the r.h.s. of (5.1) is confined to a region with size A, one has δ⁽²⁾_D(ℓ → 0) → A/(2π)²instead of infinity, which means the inversion of the delta function here should simply give a factor 1/A. Third, one still needs to specify the triangle condition. This can be done by adding a δ⁽²⁾_D(ℓ₁+ℓ₂+ℓ₃) to the estimator.

Having taken care of all three points, our estimator of the bispectrum reads B(ℓˆ 1, ℓ2, ℓ3) = 1 where φℓi is the polar angle ofℓ, and we have put in a normalization function Γ to keep the estimator unbiased:

The next step is to express Γ in terms of the absolute values of the ℓ’s. This we achieve by writing the Dirac delta function in its integral form and exchanging the order of the integrals,

Γ =

The last expression in (5.4) was given in Gradshteyn et al. (2000), where Λ is defined to be

Λ(ℓ₁, ℓ₂, ℓ₃) ≡

Bispectrum estimator

5.1.2 Geometrical interpretation

It is interesting to note that Λ⁻¹is just the area of the triangle constructed byℓ₁,ℓ₂andℓ₃. This motivated us to find a geometrical interpretation for the angular averaging of δ⁽²⁾_D (ℓ₁+ℓ₂+ℓ₃) in (5.3). If one fixes the lengths ofℓ₁,ℓ₂andℓ₃and allows their polar angles to vary, in almost all cases the three of them do not form a triangle. Since the delta function specifies the triangle condition of the three vectors, it actually corresponds to the probability of the three vectors forming a triangle when their polar angles can be any value from 0 to 2π. Based on this idea, we consider a fixed vector ℓ₁, and allowℓ₂andℓ₃to vary within annuli with widths ∆ℓ₂and ∆ℓ₃, as sketched in Fig. 5.1.

Figure 5.1: Sketch of the annuli and their overlap for fixedℓ₁. The region of overlap is approximated by the shaded parallelograms. Figure from Joachimi et al. (2009).

The probability of the three vectors forming a triangle can be represented by the area of the overlap regions of the two annuli Akdivided by the areas of the annuli AR(ℓ2) and AR(ℓ3), in the limit of ∆ℓ₂, ∆ℓ₃ → 0. Thus

Z 2π 0

dφ_ℓ1 2π

Z 2π 0

dφ_ℓ2 2π

Z 2π 0

dφ_ℓ3

2π δ⁽²⁾_D (ℓ₁+ℓ₂+ℓ₃) = lim

∆ℓ2,∆ℓ3→0

2 Ak

A_R(ℓ₂) A_R(ℓ₃) . (5.6) With the triangle formed byℓ₁, ℓ₂, andℓ₃ being parametrized by ℓ2, ℓ3, and α, which is the internal angle opposite ℓ₁(see Fig. 5.1), Λ as defined in (5.5) can be written as Λ = 2ℓ⁻¹₂ ℓ⁻¹₃ / sin α.

Observing that Ak= ∆ℓ₂∆ℓ₃/ sin α, and

AR( ¯ℓi) = 2π ¯ℓi∆ℓi when ∆ℓi → 0 for i = 1, 2, 3 , (5.7) one reproduces (5.4).

5.1.3 Estimator for bin-averaged bispectrum B( ¯ℓ₁, ¯ℓ₂, ¯ℓ₃)

In practice, the bispectrum is estimated not at every angular frequency but in angular frequency bins. Thus we further average (5.2) over the bin-widths to obtain the bin-averaged bispectrum

esti-mator

which takes the average over the angular frequency annuli AR( ¯ℓi).

We demonstrate that (5.8) is an unbiased estimator by taking the ensemble average of the esti-mator,

In the first step the definition of the bispectrum (3.37) was inserted, whereas in the second step the identity δ⁽²⁾_D (ℓ → 0) → A/(2π)²has been used.

Further inserting (5.4) into (5.9), one obtains D ˆB( ¯ℓ₁, ¯ℓ₂, ¯ℓ₃)E

We take the approximation that the annuli are thin enough such that Λ (ℓ1, ℓ2, ℓ3) within the integral can be taken out of the integration and be replaced by Λ ¯ℓ₁, ¯ℓ₂, ¯ℓ₃

. Applying in addition (5.7), one arrives at where in the last step the definition of the bin-averaged bispectrum, which has a similar form as (5.8), was used. Hence, (5.8) defines an unbiased estimator of the bin-averaged bispectrum.