Analysis of two-point statistics of cosmic shear

# Analysis of two-point statistics of cosmic shear

III. Covariances of shear measures made easy
B. Joachimi Argelander-Institut für Astronomie (AIfA), Universität Bonn, Auf dem Hügel 71, 53121 Bonn, Germany    P. Schneider Argelander-Institut für Astronomie (AIfA), Universität Bonn, Auf dem Hügel 71, 53121 Bonn, Germany    Argelander-Institut für Astronomie (AIfA), Universität Bonn, Auf dem Hügel 71, 53121 Bonn, Germany    T. Eifler Argelander-Institut für Astronomie (AIfA), Universität Bonn, Auf dem Hügel 71, 53121 Bonn, Germany
Received 2 August 2007 / Accepted 24 October 2007
###### Key Words.:
cosmology: theory – gravitational lensing – large-scale structure of the Universe – methods: statistical
offprints: B. Joachimi,
###### Abstract

Context:

Aims:In recent years cosmic shear, the weak gravitational lensing effect by the large-scale structure of the Universe, has proven to be one of the observational pillars on which the cosmological concordance model is founded. Several cosmic shear statistics have been developed in order to analyze data from surveys. For the covariances of the prevalent second-order measures we present simple and handy formulae, valid under the assumptions of Gaussian density fluctuations and a simple survey geometry. We also formulate these results in the context of shear tomography, i.e. the inclusion of redshift information, and generalize them to arbitrary data field geometries.

Methods:We define estimators for the E- and B-mode projected power spectra and show them to be unbiased in the case of Gaussianity and a simple survey geometry. From the covariance of these estimators we demonstrate how to derive covariances of arbitrary combinations of second-order cosmic shear measures. We then recalculate the power spectrum covariance for general survey geometries and examine the bias thereby introduced on the estimators for exemplary configurations.

Results:Our results for the covariances are considerably simpler than and analytically shown to be equivalent to the real-space approach presented in the first paper of this series. We find good agreement with other numerical evaluations and confirm the general properties of the covariance matrices. The studies of the specific survey configurations suggest that our simplified covariances may be employed for realistic survey geometries to good approximation.

Conclusions:

## 1 Introduction

Despite the fast progress in cosmology during the past years (e.g. Spergel et al. 2007, and references therein), still only little is known about those 95% of the energy content of the Universe which is non-baryonic. Especially for the dominant dark energy, astronomical observations provide the only source of information to reveal more about its properties. One of the most promising methods to shed light on the dark ingredients of the Universe is cosmic shear, the gravitational lensing of distant galaxies by the large-scale matter structure.

Observations of a cosmic shear signal are challenging since the shear, i.e. the distortion of galaxy images by gravitational lensing, is small compared to the intrinsic galaxy ellipticities, of order 1%. Besides, measurements of galaxy ellipticities are further hampered by atmospheric seeing, telescope optics and pixelization of the image. Therefore, although theoretical work on light propagation in an inhomogeneous universe exists for four decades (Gunn, 1967), the first detection of cosmic shear was only reported in 2000, independently by Bacon et al. (2000), Kaiser et al. (2000), van Waerbeke et al. (2000) and Wittman et al. (2000). Recently, cosmic shear observations have reached statistical errors on cosmological parameters compatible to those of other established methods, see e.g. Jarvis et al. (2006) with CTIO data, Hoekstra et al. (2006), Semboloni et al. (2006) and Fu et al. (2007) with samples from CFHTLS, and Hetterscheidt et al. (2007) with an analysis of the GaBoDS survey. In combination with CMB data, cosmic shear is able to break parameter degeneracies because it probes the matter distribution at lower redshifts and on smaller scales than the CMB fluctuations. Cosmic shear is also complementary to methods such as type Ia supernovae and galaxy redshift surveys, its strength being that the underlying theory is built on few physical assumptions; in particular there is no dependence on the relation between baryons and dark matter.

The statistical properties of the (line-of-sight projected) matter distribution measured by cosmic shear are usually characterized by the convergence power spectrum , where denotes the Fourier variable on the sky. If the matter density field is a realization of a Gaussian random field, the second-order measure fully describes its properties. This is the case if the density perturbations are linear. Equivalently, other second-order statistical measures as the correlation functions, the shear dispersion and the aperture mass (for definitions see Sect. 2) can be used instead of . Of all these second-order statistical methods, the shear correlation function is most straightforward to measure from data, being insensitive to gaps and the field geometry of the imaged sky region, and it provides the full second-order statistical information contained in the data.

In order to assess strengths and limitations of all these measures it is important to know errors and their correlations on different scales characterized by the covariance matrices. Under the assumption of Gaussianity, Schneider et al. (2002a), hereafter Paper I, calculated such covariance matrices analytically, starting from an estimator for the correlation functions . By further assuming a connected survey area, negligible boundary effects due to the finite field size and randomly placed galaxies in the field, the ensemble average over galaxy positions of the covariances was taken, and expressions for the covariances as a function of the theoretical were derived. In this work we aim at recalculating the covariance matrices, but now in a Fourier space approach, leading to significantly simpler expressions. These are then shown to be equivalent to the result of Paper I.

In Sect. 2 we give an overview on second-order measures of cosmic shear and their interrelations, using the notation of Bartelmann & Schneider (2001) and Schneider (2006). Moreover, we briefly summarize the E- and B-mode decomposition of the shear field. We derive estimators for E- and B-mode power spectra in Sect. 3 and show them to be unbiased under the assumptions made in Paper I. The covariance of these estimators and successively the covariances of other second-order measures are calculated in Sect. 4. We demonstrate the equivalence of our formulae to those given in Paper I in Sect. 5. In addition, we evaluate the covariances numerically and discuss our results. In Sect. 6 the calculations of the foregoing sections are generalized to shear tomography, i.e. the inclusion of photometric redshift information, as well as to arbitrary survey geometries. We discuss the behavior of our estimators for specific examples of survey geometries. Finally, we summarize our findings and conclude in Sect. 7.

## 2 Second-order measures of cosmic shear

In order to obtain information about cosmic shear from a given data set, galaxy ellipticities are measured at positions in the sky. The observed complex ellipticity contains contributions from the intrinsic source ellipticity and the shear induced by gravitational lensing, i.e. . By considering a pair of galaxies at positions and , one defines the tangential and cross component of the ellipticity at position for this particular pair as

 (1)

where is the polar angle of the separation vector (likewise for and ). Then one defines the shear correlation functions

 ξ±(θ):=⟨γtγt⟩(θ)±⟨γ×γ×⟩(θ). (2)

Further second-order statistical measures are the shear dispersion , defined as the average of the mean shear in circular apertures of radius , and the dispersion of the aperture mass in circular apertures of radius . For an aperture located at , the aperture measures are given by

 Map(θ)=∫d2ϑQ(ϑ)γt(\boldmathϑ);M⊥(θ)=∫d2ϑQ(ϑ)γ×(\boldmathϑ), (3)

when . The weight function is axially-symmetric, but otherwise arbitrary. Crittenden et al. (2002) suggested a weight function with exponential fall-off. Here is chosen to be the same as in Schneider et al. (2002b, hereafter S02):

 Q(ϑ)=6ϑ2πθ4(1−ϑ2θ2)H(θ−ϑ), (4)

where is the Heaviside step-function.

The convergence or dimensionless projected surface mass density is related to the shear via

 γ(\boldmathθ)=1π∫d2ϑ D(\boldmathθ−\boldmathϑ)κ(\boldmathϑ)   with   D(\boldmathθ)=θ22−θ21−2iθ1θ2|% \boldmathθ|4. (5)

In ordinary lens theory is a real quantity. If the complex can be calculated from , and are mutually dependent, which leads to a relation that ensures a shear field, generated purely by lensing, to be curl-free as demonstrated in Crittenden et al. (2002) and S02. As is a polar quantity, a general shear field can be decomposed in analogy to electromagnetic polarization into a curl-free E-mode and a divergence-free B-mode. Although not caused by gravitational lensing, B-modes can be present in cosmic shear data, for instance due to the underestimation of noise or uncorrected systematic errors. Further (small) sources of B-modes are source redshift clustering, see S02, and the limited validity of the Born approximation, see e.g. Jain et al. (2000).

For a pure lensing signal we will set the B-mode contribution to zero later on. Nevertheless we must account for noise in the B-mode channel as it is measured by the real-space estimator for the correlation functions used in Paper I, which does not discriminate between an E- or B-mode origin of galaxy ellipticities. In order to account for both E- and B-modes, we rewrite the originally purely real convergence as a complex quantity, which reads in Fourier space . In terms of and the following power spectra are defined via

 ⟨~κE(\boldmathℓ)~κ∗E(\boldmathℓ′)⟩ = (2π)2δ(2)(\boldmathℓ−\boldmathℓ′) PE(ℓ), ⟨~κB(\boldmathℓ)~κ∗B(\boldmathℓ′)⟩ = (2π)2δ(2)(\boldmathℓ−\boldmathℓ′) PB(ℓ), (6) ⟨~κE(\boldmathℓ)~κ∗B(\boldmathℓ′)⟩ = (2π)2δ(2)(\boldmathℓ−\boldmathℓ′) PEB(ℓ),

where is the two-dimensional Dirac delta-distribution. Note that all two-dimensional power spectra appearing in this work are power spectra of , which is why the index is dropped. To lowest order, i.e. in the Born approximation and without lens-lens coupling, the E-mode power spectrum is related to the three-dimensional power spectrum of the density fluctuations via Limber’s equation

 PE(ℓ)=9H40Ω2m4c4∫χh0dχg2(χ)a2(χ)Pδ(ℓχ,χ), (7)

where is the comoving distance and the comoving horizon distance. Hence, is the projection of along the line-of-sight, weighted with the lensing efficiency , where is the distribution of source galaxies in distance. For simplicity we have assumed a spatially flat geometry. From now on we set because the cross power is expected to vanish for a statistically parity-invariant shear field, as is the case for its corresponding correlation function .

The Fourier transform of (5) reads

 ~γ(\boldmathℓ)=e2iβ~κ(\boldmathℓ), (8)

where is the polar angle of , and where is now a complex quantity. By means of this relation and (6) one can derive the second-order shear measures as a function of the E- and B-mode power spectra, see e.g. S02,

 ξ+(θ) = ∫∞0dℓ ℓ2π J0(ℓθ){PE(ℓ)+PB(ℓ)};ξ−(θ)=∫∞0dℓ ℓ2π J4(ℓθ){PE(ℓ)−PB(ℓ)}, ⟨M2ap⟩(θ) = ∫∞0dℓ ℓ2π 576J24(ℓθ)(ℓθ)4PE(ℓ);⟨M2⊥⟩(θ)=∫∞0dℓ ℓ2π 576J24(ℓθ)(ℓθ)4PB(ℓ), (9) ⟨|¯γ|2⟩(θ) =

where is the Bessel function of the first kind of order .

So the above measures are all linear functionals of the power spectra, each employing a different filter. As a consequence they are interrelated, as shown by Crittenden et al. (2002). Thus, in practice only one measure has to be estimated from the data, the others can then be deduced. As real data contains gaps, masks and CCD defects, the placing of apertures needed for and is impractical, so that the correlation functions are the preferred primary observables. Due to the broad filter especially of the signal is strong, but mixes information about the power spectrum over a wide range of scales. On the contrary, the aperture mass statistics employ a narrow filter, which can be replaced by a Dirac -distribution with an error of only 10% (Bartelmann & Schneider, 1999). Thus the signal is small, but probes the power spectrum very locally, and measures on different scales quickly decorrelate (see Sect. 5). In addition to this, (9) shows that is sensitive only to E-modes, whereas , being sensitive only to B-modes, can be used to quantify systematic errors etc. The filters of the second-order measures also determine the form of their covariance matrices, as will be discussed in Sect. 5.

## 3 Power spectrum estimators

Consider now galaxies at positions with measured ellipticities in a data field covering a solid angle . Repeating a calculation by Kaiser (1998), we write

 ~ϵα(\boldmathℓ)=N∑i=1ϵα,iei\boldmath\scriptsizeℓ⋅% \boldmath\scriptsizeθi=∫d2θei\boldmath\scriptsizeℓ⋅\boldmath\scriptsizeθn(\boldmathθ)γα(\boldmathθ)+∑iϵsα,iei\boldmath\scriptsizeℓ⋅\boldmath\scriptsizeθi, (10)

where in the second step was split up into shear and intrinsic ellipticity. Note that is the discrete Fourier transform of the real ellipticity component so that holds. The introduction of regains a continuous Fourier transform for the shear. Transforming back to Fourier space then yields

 ~ϵα(\boldmathℓ)=∫d2ℓ′(2π)2~γα(\boldmathℓ′)~n(\boldmathℓ−\boldmathℓ′)+∑iϵsα,iei\boldmath\scriptsizeℓ⋅\boldmath\scriptsizeθiwith~n(\boldmathℓ)=∫d2θei\boldmath\scriptsizeℓ⋅\boldmath% \scriptsizeθn(\boldmathθ). (11)

The explicit dependence of on the galaxy positions hampers analytical progress. To simplify this expression, we make the assumption that the galaxies in the field are uniformly sampled, which means that can be replaced by its ensemble average over all galaxy positions

where is the mean number density of galaxies in the field and is the aperture function, which is for within the field and else. In particular, . With the definition

 Δ(\boldmathℓ):=~n(\boldmathℓ)(2π)2¯n=∫d2θ(2π)2ei% \boldmath\scriptsizeℓ⋅\boldmath\scriptsizeθΠ(\boldmathθ) (12)

one can now write . Note that is the Fourier transform of the aperture. If one further assumes that the observed field is simply connected and that all relevant angles are considerably smaller than the extent of the field, i.e. , then the aperture function can be approximated by unity, so that consequently . Through this (11) takes on the form

 ~ϵα(\boldmathℓ)≈¯n~γα(\boldmathℓ)+∑iϵsα,iei\boldmath\scriptsizeℓ⋅\boldmath\scriptsizeθi. (13)

In the following calculations terms containing ellipticities to the square will appear. In this case the approximation for is executed as follows

 |Δ(\boldmathℓ)|2≈δ(2)(\boldmathℓ)Δ(\boldmathℓ)=δ(2)(\boldmathℓ)Δ(0)=δ(2)(\boldmathℓ)A(2π)2. (14)

With these considerations at hand we now define the following estimators for the E- and B-mode power spectra,

 ^PE(¯ℓ):=1¯n2AAR(¯ℓ)∫AR(¯ℓ)d2ℓ ∣∣  ~ϵ1(\boldmathℓ)cos2β+~ϵ2(\boldmathℓ)sin2β∣∣2−σ2ϵ2¯n, ^PB(¯ℓ):=1¯n2AAR(¯ℓ)∫AR(¯ℓ)d2ℓ ∣∣−~ϵ1(%\boldmath$ℓ$)sin2β+~ϵ2(\boldmathℓ)cos2β∣∣2−σ2ϵ2¯n. (15)

These estimators are band powers for disjunct bins in Fourier space, constructed by averaging over annuli with mean radius and area . denotes the dispersion of the complex source ellipticities. The Fourier ellipticities in (15) are calculated from the data via (10). In the following we demonstrate that these estimators are unbiased under the assumptions made above.

On calculating the expectation value, one splits up the image ellipticities into a shear and a source ellipticity part following (13). In order to process the latter terms, consider the definition of the source ellipticity dispersion . Moreover, as the Universe is isotropic, we expect the combination , which has a net orientation, to vanish. From these two complex correlators one concludes for the ellipticity components

 ⟨ϵsα,iϵsβ,j⟩=δijδαβσ2ϵ2;α,β={1,2}. (16)

The source ellipticity terms are processed via this relation and result in , so that they cancel with the last term in (15). In case of E-modes, the remaining shear terms yield

 ⟨^PE(¯ℓ)⟩=1AAR(¯ℓ)∫AR(¯ℓ)d2ℓ⟨|~γ1(\boldmathℓ)|2cos22β+|~γ2(\boldmath% ℓ)|2sin22β+2{~γ1(\boldmathℓ)~γ∗2(\boldmathℓ)+~γ∗1(% \boldmathℓ)~γ2(\boldmathℓ)}sin2βcos2β⟩. (17)

The shears are replaced by means of the expressions

 ~κE(\boldmathℓ)=~γ1(% \boldmathℓ)cos2β+~γ2(\boldmathℓ)sin2β;~κB(\boldmathℓ)=−~γ1(\boldmathℓ)sin2β+~γ2(%\boldmath$ℓ$)cos2β, (18)

derived from (8). Via (6) one then inserts the power spectrum. Formally, at this stage a Dirac delta-distribution with argument appears, but as terms quadratic in the ellipticities are involved, the approximation (14) applies, leading to

 ⟨^PE(¯ℓ)⟩=1AAR(¯ℓ)∫AR(¯ℓ)d2ℓ (2π)2Δ(0)PE(ℓ)=1AR(¯ℓ)∫AR(¯ℓ)d2ℓ PE(ℓ)=PE(¯ℓ). (19)

In complete analogy, one can show to be an unbiased estimator of the power spectrum, too. In Sect. 6 we will drop the assumptions on made here and examine how well the estimators (15) are suited for more realistic situations.

## 4 Covariances

Next we are going to calculate the covariance of the estimators defined above. Introducing the shorthand notation , the covariance reads

 Cov(PX;¯ℓ,¯ℓ′):=⟨ΔPX(¯ℓ)ΔPX(¯ℓ′)⟩=⟨^PX(¯ℓ)^PX(¯ℓ′)⟩−PX(¯ℓ)PX(¯ℓ′). (20)

For simplicity we keep to the case of the E-mode power spectrum in the following; the B-mode calculation is analogous. Plugging in the estimators yields for the correlator

 ⟨^PE(¯ℓ)^PE(¯ℓ′)⟩ = ∫AR(¯ℓ)d2ℓ¯n2AAR(¯ℓ)∫AR(¯ℓ′)d2ℓ′¯n2AAR(¯ℓ′)⟨∣∣~ϵ1(\boldmathℓ)cos2β+~ϵ2(\boldmathℓ)sin2β∣∣2∣∣~ϵ1(\boldmathℓ′)cos2β′+~ϵ2(\boldmathℓ′)sin2β′∣∣2⟩ − σ2ϵ2¯n∫AR(¯ℓ)d2ℓ¯n2AAR(¯ℓ)⟨∣∣~ϵ1(\boldmathℓ)cos2β+~ϵ2(\boldmathℓ)sin2β∣∣2⟩−σ2ϵ2¯n∫AR(¯ℓ′)d2ℓ′¯n2AAR(¯ℓ′)⟨∣∣~ϵ1(\boldmathℓ′)cos2β′+~ϵ2(\boldmathℓ′)sin2β′∣∣2⟩+σ4ϵ4¯n2.

By comparison with (15) the integrals of the second and third term amount to and , respectively. The expansion of the first term results in a number of four-point correlators of ellipticities. If the shear field is assumed to be Gaussian, which is realistic if one considers scales sufficiently large that effects of non-linear density evolution do not have an effect on the shear field, this can be written as a sum of products of two-point correlators. The four-point correlators of the intrinsic source ellipticities can be decomposed analogously. So terms of type remain, which can be further processed via (13) as follows:

 ⟨~ϵα(\boldmathℓ)~ϵ∗β(\boldmathℓ′)⟩=¯n2⟨~γα(\boldmathℓ)~γ∗β(\boldmathℓ′)⟩+∑ij⟨ϵsα,iϵsβ,j⟩ei(\boldmath\scriptsizeℓ⋅\boldmath\scriptsizeθi−\boldmath\scriptsizeℓ′⋅\boldmath\scriptsizeθj)=¯n2⟨~γα(\boldmathℓ)~γ∗β(\boldmathℓ′)⟩+δαβσ2ϵ2(∑iei(\boldmath% \scriptsizeℓ−\boldmath\scriptsizeℓ′)⋅%\boldmath$θ$i), (22)

where for the first step we assumed that intrinsic ellipticities and shear are not correlated, while in the second step we used (16). The term in brackets can be recognized as , whereby we can approximate the equation as done before, leading to

 ⟨~ϵα(\boldmathℓ)~ϵ∗β(\boldmathℓ′)⟩=¯n2⟨~γα(\boldmathℓ)~γ∗β(\boldmathℓ′)⟩+δαβσ2ϵ2(2π)2¯n δ(2)(\boldmathℓ−\boldmathℓ′). (23)

Correlators of the form and can be dealt with analogously by employing .

By expressing the correlators in terms of correlators (for X,Y), using (18), (4) can be written in the form

 ⟨^PE(¯ℓ)^PE(¯ℓ′)⟩=∫AR(¯ℓ)d2ℓ¯n2AAR(¯ℓ)∫AR(¯ℓ′)d2ℓ′¯n2AAR(¯ℓ′){AE+BE+CE}−σ2ϵ2¯n(PE(¯ℓ)+PE(¯ℓ′))−σ4ϵ4¯n2, (24)

with

 AE = ¯n4(⟨~κE(\boldmathℓ)~κ∗E(\boldmathℓ)⟩⟨~κE(\boldmathℓ′)~κ∗E(\boldmathℓ′)⟩+2⟨~κE(% \boldmathℓ)~κ∗E(\boldmathℓ′)⟩2)=¯n4(A2PE(ℓ)PE(ℓ′)+2 δ(2)(\boldmathℓ−\boldmathℓ′) A(2π)2P2E(ℓ)), BE = σ2ϵ2(2π)2¯n3(A(2π)2⟨~κE(\boldmathℓ)~κ∗E(\boldmathℓ)⟩+A(2π)2⟨~κE(\boldmathℓ′)~κ∗E(\boldmathℓ′)⟩+4 δ(2)(\boldmathℓ−\boldmathℓ′)⟨~κE(% \boldmathℓ)~κ∗E(\boldmathℓ)⟩) = σ2ϵ2(2π)2¯n3(A2(2π)2PE(ℓ)+A2(2π)2PE(ℓ′)+4 δ(2)(\boldmathℓ−\boldmathℓ′) APE(ℓ)), CE = σ4ϵ4(2π)4¯n2(A2(2π)4+2 δ(2)(\boldmathℓ−\boldmathℓ′)A(2π)2),

where in the second steps, respectively, we have made use of (6) together with (14). Here and in the following, denotes cosmic variance terms, is the shot noise contribution proportional to and stands for the mixed term. Shot noise is caused by the intrinsic ellipticity dispersion of the source galaxies and usually dominates on small angular scales (in the Gaussian approximation). The fact that a data field of finite extent is used to estimate statistical measures leads to another noise component called cosmic variance, which prevails on scales larger than about 5 or , respectively, see e.g. Hu & White (2001).

By subtracting the product of the mean of the estimators, (24) turns into

 ⟨ΔPE(¯ℓ)ΔPE(¯ℓ′)⟩=2⋅(2π)2A¯n2∫AR(¯ℓ)d2ℓ¯n2AAR(¯ℓ)∫AR(¯ℓ′)d2ℓ′¯n2AAR(¯ℓ′)δ(2)(\boldmathℓ−\boldmathℓ′)(¯nPE(ℓ)+σ2ϵ2)2. (25)

Note that the Dirac delta-distribution can only have a non-vanishing result if the two integration areas and overlap, i.e. if . Performing the integrations and repeating the above considerations in the B-mode case, one arrives at

 ⟨ΔPX(¯ℓ)ΔPX(¯ℓ′)⟩=4πA¯ℓΔℓ(PX(¯ℓ)+σ2ϵ2¯n)2δ¯ℓ¯ℓ′forX={E,B}, (26)

where the area of an annulus with thickness , , has already been inserted. Equation (26) for E-modes is in agreement with the results presented in Kaiser (1998). A similar calculation yields

 ⟨ΔPE(¯ℓ)ΔPB(¯ℓ′)⟩=0, (27)

i.e. as expected the E- and B-mode power spectra are not correlated.

By means of (9) the covariances of all possible combinations of second-order cosmic shear measures can now easily be obtained. For instance, one gets for the covariance of the correlation function

 ⟨Δξ+(θ1) Δξ+(θ2)⟩ ≈ 14π2∑¯ℓ,¯ℓ′¯ℓ ¯ℓ′Δℓ2J0(¯ℓθ1)J0(¯ℓ′θ2)⟨(ΔPE(¯ℓ)+ΔPB(¯ℓ))(ΔPE(¯ℓ′)+ΔPB(¯ℓ′))⟩ (28) = 14π2∑¯ℓ,¯ℓ′¯ℓ ¯ℓ′Δℓ2J0(¯ℓθ1)J0(¯ℓ′θ2)(⟨ΔPE(¯ℓ)ΔPE(¯ℓ′)⟩+⟨ΔPB(¯ℓ)ΔPB(¯ℓ′)⟩),

where we have used a discretized version of the transformation because the power spectrum covariance is given for -bins. After insertion of (26) the covariance depends linearly on ; therefore, considering the limit , the sum transforms back into an integral,

 (29)

Similarly the results for other combinations of second-order measures are derived, so that for the case of correlation functions one arrives at

 ⟨Δξ−(θ1) Δξ−(θ2)⟩ = 1πA∫∞0dℓ ℓ J4(ℓθ1)J4(ℓθ2){(PE(ℓ)+σ2ϵ2¯n)2+(PB(ℓ)+σ2ϵ2¯n)2}, (30) ⟨Δξ+(θ1) Δξ−(θ2)⟩ = 1πA∫∞0dℓ ℓ J0(ℓθ1)J4(ℓθ2){(PE(ℓ)+σ2ϵ2¯n)2−(PB(ℓ)+σ2ϵ2¯n)2}, (31)

whereas for the aperture mass we get

 ⟨Δ⟨M2ap⟩(θ1) Δ⟨M2ap⟩(θ2)⟩=5762πAθ41θ42∫∞0dℓ ℓ−7 J24(ℓθ1)J24(ℓθ2)(PE(ℓ)+σ2ϵ2¯n)2. (32)

Up to now the covariances are continuous functions of although in reality angular scales are binned. If the binning is sufficiently small and the integrands in the above equations are smooth functions of and , one can simply replace the angles by the central values of the bins. However, the shot noise term of (29) and (30) is not smooth, but can be further processed by means of the orthogonality relation of the Bessel functions

 ∫∞0dℓ ℓ Jμ(ℓθ1)Jμ(ℓθ2)=1θ1 δ(θ1−θ2). (33)

The transition to a discrete set of angular bins is done by the relation between Kronecker and Dirac ’s: for . Introducing the number of galaxy pairs in an annulus of bin radius and thickness , , one can write (29) and (30) in the directly applicable form

 ⟨Δξ±(θ1) Δξ±(θ2)⟩=1πA∫∞0dℓ ℓ J0/4(ℓθ1)J0/4(ℓθ2){P2E(ℓ)+P2B(ℓ)+σ2ϵ¯n(PE(ℓ)+PB(ℓ))}+σ4ϵNp(θ1) δθ1θ2. (34)

The shot noise term in (31) cancels, so that in this case no further processing is needed.

## 5 Equivalence to real space covariances

In Paper I covariances of the correlation functions and other second-order measures were obtained by starting directly with an estimator for and by taking an ensemble average over all galaxy positions of the resulting covariances. These calculations were based on the same assumptions as this work; therefore our results and those from Paper I are equivalent, which is proven in the following.

### 5.1 Correlation function covariances

Starting from (34), the covariance for can again be split up into cosmic variance, shot noise and mixed term as follows

 A++ = 1πA∫∞0dℓ ℓ {P2E(ℓ)+P2B(ℓ)}J0(ℓθ1)J0(ℓθ2), B++ = σ2ϵπA¯n∫∞0dℓ ℓ {PE(ℓ)+PB(ℓ)}J0(ℓθ1)J0(ℓθ2), (35) C++ = σ4ϵNp(θ1) δθ1θ2,

where here and in the following, we drop the arguments , of the partial correlators etc. The quantity is already in the desired form of Paper I, (27). Inverting the first two equations of (9) by means of the orthogonality relation (33) results in (see also S02)

 (36)

from which one concludes . If one inserts this expression and the definition of the Bessel function in the form into (35), the mixed term yields

 B++=σ2ϵ8π4A¯n∫d2ℓ∫d2θ∫2π0dφ1∫2π0dφ2 ξ+(θ)ei\boldmath\scriptsizeℓ⋅(\boldmath\scriptsizeθ−\boldmath\scriptsizeθ1+\boldmath\scriptsizeθ2)=2σ2ϵπA¯n∫π0dφ ξ+(√θ21+θ