Source-lens clustering and intrinsic-alignment bias of weak-lensing estimators

# Source-lens clustering and intrinsic-alignment bias of weak-lensing estimators

Patrick Valageas Institut de Physique Théorique, CEA Saclay, F-91191 Gif-sur-Yvette, Cédex, France
CNRS, URA 2306, F-91191 Gif-sur-Yvette, Cédex, France
###### Key Words.:
weak gravitational lensing; cosmology: large-scale structure of Universe
###### Abstract

Context:

Aims:We estimate the amplitude of the source-lens clustering bias and of the intrinsic-alignment bias of weak lensing estimators of the two-point and three-point convergence and cosmic-shear correlation functions.

Methods:We use a linear galaxy bias model for the galaxy-density correlations, as well as a linear intrinsic-alignment model. For the three-point and four-point density correlations, we use analytical or semi-analytical models, based on a hierarchical ansatz or a combination of one-loop perturbation theory with a halo model.

Results:For two-point statistics, we find that the source-lens clustering bias is typically several orders of magnitude below the weak lensing signal, except when we correlate a very low-redshift galaxy () with a higher redshift galaxy (), where it can reach of the signal for the shear. For three-point statistics, the source-lens clustering bias is typically of order of the signal, as soon as the three galaxy source redshifts are not identical. The intrinsic-alignment bias is typically about of the signal for both two-point and three-point statistics. Thus, both source-lens clustering bias and intrinsic-alignment bias must be taken into account for three-point estimators aiming at a better than accuracy.

Conclusions:

## 1 Introduction

Weak gravitational lensing of background galaxies by foreground large-scale structures is an important probe of both the geometry of the Universe and the growth of these large-scale structures. This makes it a powerful tool when studying the distribution of dark matter and the nature of dark energy (Albrecht et al. 2006). This effect arises from the deflection of light rays from distant galaxies by the fluctuations of the gravitational potential along the line of sight (Bartelmann & Schneider 2001; Munshi et al. 2008). This yields both a deformation of the shape of the images of distant galaxies (associated with the “cosmic shear” , at lowest order) and a magnification of their flux (associated with the “convergence” ). Because we do not know a priori the shape or luminosity of individual background galaxies, cosmological studies use statistical averages over many galaxies to detect the coherent shear due to large-scale structures (typically on angular scales of a few arcmin), assuming that background galaxies are statistically isotropic. Thus, in practice we use the cosmic shear (through the coherent orientation of galaxies on arcmin scales that it induces) rather than the convergence as a probe of gravitational lensing (because it is difficult to predict the luminosity distribution of background galaxies with a good accuracy and we lack a standard candle). Moreover, we usually do not measure a shear or convergence map from a galaxy survey, that is, the fields or on some region of the sky, but the two-point correlation , by averaging over all galaxy pairs separated by an angular distance .

More precisely, weak gravitational lensing is measured from the ellipticities of galaxies, , which are related to the cosmological shear distortions by , where is the intrinsic galaxy ellipticity. Then, assuming that intrinsic galaxy ellipticities are independent and decorrelated from the shear, one measures the gravitational lensing signal by averaging over pairs of galaxies. This gives estimators (that we denote with a hat) of the form

 ^ξγγ∗(θ)=∑i,jwiwjϵ(\@vecθi)ϵ∗(\@vecθj)∑i,jwiwj, (1)

where we sum over all galaxy pairs in the survey with an angular distance that falls within some bin around . (In this example, we correlate with its complex conjugate to obtain nonzero results, as the shear and the ellipticity are spin-2 quantities.) The weights may be chosen to diminish the importance of noisy objects, to improve the signal-to-noise ratio. This provides an estimator of the real-space two-point shear correlation function . By summing over all pairs that are separated by a distance shorter than some angular radius , one also obtains the integral of within this scale or the variance of the smoothed shear or of the aperture mass (e.g., Bartelmann & Schneider (2001); Munshi et al. (2008); Kilbinger et al. (2013)).

In this fashion, the two-point correlation (Bacon et al. 2000; Van Waerbeke et al. 2000; Wittman et al. 2000; Hamana et al. 2003; Jarvis et al. 2006; Semboloni et al. 2006; Fu et al. 2008; Schrabback et al. 2010) and three-point correlation (Bernardeau et al. 2002b; Semboloni et al. 2011) of the cosmic shear have been detected and measured on scales of a few arcmin.

In Eq.(1), may be taken as the tangential or cross-component or as the component along a given axis. One can also consider different redshift bins for and , as in tomographic studies that make use of the redshift dependence of the lensing signal (Hu 1999; Heymans et al. 2013). Next, one may take the Fourier transform of to obtain an estimator of the weak lensing power spectrum. This is more convenient than first taking the Fourier transform of the ellipticity field and second taking its variance, because the masks and intricate boundaries of galaxy surveys make it difficult to compute the Fourier transform of the field.

In practice, different sources of noise can bias the estimator (1). In addition to the instrumental noise itself (that may be included in the statistical properties of ), intrinsic galaxy alignments must be taken care of. They come either through the correlation between nearby galaxies, (Heavens et al. 2000; Croft & Metzler 2000; Catelan et al. 2001; Brown et al. 2002), or the correlation between the ellipticity of a foreground galaxy and the local density field, which contributes to the shear of a background galaxy and gives rise to a correlation (Hirata & Seljak 2004; Hirata et al. 2007; Mandelbaum et al. 2011; Heymans et al. 2013).

Another source of bias arises from the fact that galaxies are not located at random in space. Indeed, they are correlated with the density field, and this gives rise to a “source-lens clustering” bias (Bernardeau 1998; Hamana et al. 2002). For instance, in terms of the convergence , if galaxies were only located behind overdense regions their luminosity would appear systematically enhanced. This effect is expected to be rather small for measures of the two-point shear correlation, as compared with the full gravitational lensing signal, because it is restricted to density fluctuations close to the observed galaxies whereas the full signal arises from density fluctuations along the whole line of sight. It is further suppressed by the fact that the lensing efficiency [the kernel in Eq.(9) below] vanishes at the source plane. Moreover, on large scales this bias scales as whereas the weak lensing signal scales as (where is the matter density correlation), so that this bias should be subdominant. Nevertheless, in view of the increasing accuracy of future surveys, it is interesting to have an estimate of the magnitude of this systematic effect, to check that it can indeed be neglected (as in all current studies).

On the other hand, for measures of the shear three-point correlation, both the signal and the source-lens clustering bias scale as and one can expect a significant contamination, especially for triplets of galaxies that are at different redshifts, so that the lensing kernel is nonzero. Moreover, at leading order the bias writes as a sum of product of two terms. The first term again involves the correlation between a foreground galaxy and nearby density fluctuations along another line of sight, but the second term now involves the correlations between density fluctuations along two full lines of sight as in the cosmological signal and is not suppressed by the ratio of the density correlation length to the Hubble length.

The source-lens clustering effect has already been investigated in Bernardeau (1998) and Hamana et al. (2002), using perturbation theory and numerical simulations, for the skewness of the smoothed convergence as derived from a convergence map. In this paper, to keep close to current observational procedures, we investigate the source-lens clustering effect on estimators of the form of Eq.(1), without assuming a convergence map is first measured in the data analysis. As we explain in Sect. 6 below, the source-lens clustering bias associated with such two-point and three-point estimators is rather different from the one associated with the one-point estimator studied in Bernardeau (1998) and Hamana et al. (2002). In particular, it has the opposite sign. Moreover, we consider both the convergence (this allows us to introduce our approach in a simple fashion) and the cosmic shear . Then, we consider the bias due to galaxy intrinsic alignments, both for two-point and three-point shear correlation estimators. This allows us to extend previous works that focused on the two-point shear correlation to the case of the three-point shear correlation and to compare with the source-lens clustering bias.

We develop an analytical formalism to estimate these weak lensing biases and for numerical computations we use a linear bias model for the galaxy distribution and a linear intrinsic alignment model. We use semi-analytic models for the matter density two-, three- and four-point correlations.

This paper is organized as follows. We first study the source-lens clustering bias for estimators of the two-point correlation of the convergence in Sect. 2 and of the cosmic shear in Sect. 3. Then, we consider estimators of the three-point correlation of the convergence in Sect. 4 and of the cosmic shear in Sect. 5. We compare our approach with some previous works in Sect. 6. Next, we investigate the intrinsic-alignment bias in Sect. 7, for both estimators of the two- and three-point cosmic shear correlations. We conclude in Sect. 8.

## 2 Two-point convergence correlation function

### 2.1 Weak lensing convergence κ

For simplicity, we first consider the estimator of the two-point correlation function of the convergence , which is a scalar rather than a spin-2 quantity. Then, Eq.(1) becomes

 ^ξκκ(θ)=∫dχ1dΩ1χ21n1∫dχ2d\@vecθ2χ22n2κ1κ2∫dχ1dΩ1χ21n1∫dχ2d\@vecθ2χ22n2, (2)

where is the comoving radial and angular distance (we assume a flat universe) and is the observed galaxy number density (a sum of Dirac peaks at the observed galaxy positions). We did not write weighting factors , which are not important for our purposes (and may be absorbed within the number densities ). In Eq.(2), we count all pairs by first counting all galaxies in the survey, of total angular area , and next integrating over all their neighbors at the angular distance . We did not explicitly write the boundaries of the redshift and angular bins in the integration signs.

The estimator (2) is somewhat academic, because in practice we do not measure the convergences but only the ellipticities (which boil down to if we discard intrinsic alignments and instrumental noise). However, it provides a simpler presentation of our approach. Moreover, it will be interesting to compare the results obtained for the convergence and the shear, to see whether the latter could be estimated from the former ones.

For a sufficiently wide survey, we can neglect the fluctuations of the denominator in Eq.(2). Indeed, defining

 D=∫dχ1dΩ1χ21n1∫dχ2d\@vecθ2χ22n2, (3)

we obtain (assuming for simplicity thin redshift and angular bins and )

 ⟨D⟩ = ∫dχ1dΩ1χ21¯n1∫dχ2d\@vecθ2χ22¯n2⟨(1+b1δ1)(1+b2δ2)⟩ (4) = (Δχ1)(ΔΩ)χ21¯n1(Δχ2)(2πθΔθ)χ22¯n2(1+b1b2ξ1,2).

Here, is the matter density contrast, , its two-point correlation function, and is the mean bias of galaxies at redshift , assuming a linear bias model, . The second order moment of reads as

 ⟨D2⟩ = ∫dχ1dΩ1χ21¯n1∫dχ2d\@vecθ2χ22¯n2∫dχ′1dΩ′1χ′21¯n′1 (5) ×∫dχ′2d\@vecθ′2χ′22¯n′2⟨(1+b1δ1)(1+b2δ2)(1+b′1δ′1)(1+b′2δ′2)⟩ = ⟨D⟩2+cross terms,

where the cross terms correspond to contributions that involve correlations between and . They are negligible if , where is the correlation length, and this restricts the integral over to . Therefore, in terms of the scaling with respect to the survey width, we have

 ⟨D⟩∝(ΔΩ),σ2D∝(ΔΩ),σD⟨D⟩∝1√(ΔΩ), (6)

where is the variance of the denominator of Eq.(2). Thus, the relative amplitude of the fluctuations of this denominator vanish as for large surveys.

Then, neglecting the fluctuations of the denominator111This is possible for a sufficiently wide survey because the denominator sums all pairs of separation over the survey. In contrast, if we consider the one-point estimator for a convergence map , as in the analysis of Bernardeau (1998), for each direction on the sky the numerator and denominator only sum the small number of galaxies included within the smoothing radius . Then, the numerator and denominator show significant correlated fluctuations that must be taken into account. See the discussion in Sect. 6 below. (there is no shot noise because of the nonzero angular separation), the expectation value of Eq.(2) reads as

 ⟨^ξκκ(θ)⟩=⟨(1+b1δ1)(1+b2δ2)κ1κ2⟩1+b1b2ξ1,2. (7)

Here, we have chosen infinitesimally thin redshift and angular bins, to avoid being too specific. Averaging over finite redshift bins for and , and a finite angular bin for , gives the appropriate results for a given survey strategy.

Next, the weak lensing convergence of a distant galaxy due to density fluctuations along the line of sight is given by (in the Born approximation, Bartelmann & Schneider (2001); Munshi et al. (2008))

 κi=∫χi0dχi′gi′,iδ(χi′), (8)

where the lensing kernel is given by

 g(χi′,χi)=3Ωm0H202c2χi′(χi−χi′)χi(1+zi′), (9)

and denotes the point along the line of sight to the galaxy . (Hereafter, primed indices or coordinates refer to points along the line of sight, which contribute to the weak lensing signal, whereas unprimed indices or coordinates refer to the background source galaxies.) Then, the average (7) can be split into four components,

 ⟨^ξκκ⟩=ξκκ+ξδκδκ+ζδκκ+ηδδκκ. (10)

The first component, which does not include cross-correlations between the galaxies and the density fluctuations along the lines of sight, is the weak lensing signal,

 ξκκ(θ)=⟨κ1κ2⟩=∫χ10dχ1′g1′,1∫χ20dχ2′g2′,2ξ1′,2′. (11)

The second component involves products of the two-point correlations between a galaxy and a line of sight,

 ξδκδκ = b1b21+b1b2ξ1,2[⟨δ1κ1⟩⟨δ2κ2⟩+⟨δ1κ2⟩⟨δ2κ1⟩] (13) = b1b21+b1b2ξ1,2∫dχ1′dχ2′g1′,1g2′,2 ×[ξ1,1′ξ2,2′+ξ1,2′ξ2,1′],

while the third and fourth components involve the three- and four-point density correlations and ,

 ζδκκ = b1⟨δ1κ1κ2⟩+b2⟨δ2κ1κ2⟩1+b1b2ξ1,2 (15) = 11+b1b2ξ1,2∫dχ1′dχ2′g1′,1g2′,2 ×[b1ζ1,1′,2′+b2ζ2,2′,1′],
 ηδδκκ = b1b21+b1b2ξ1,2⟨δ1δ2κ1κ2⟩c (16) = b1b21+b1b2ξ1,2∫dχ1′dχ2′g1′,1g2′,2η1,2,1′,2′. (17)

Thus, the last three terms in Eq.(10) bias the estimator (2) of cosmological gravitational lensing. Of course, they vanish when the galaxy bias goes to zero, that is, when the galaxy positions are uncorrelated with the density fluctuations along the lines of sight. This source-lens clustering bias does not depend on the size of the survey (we assumed a survey window that is large as compared with the angular scale at which we probe the gravitational lensing correlation), because it is due to the intrinsic correlations of the galaxy and matter distributions and not to shot noise effects.

### 2.2 Analytical approximations

To estimate the source-lens clustering bias for estimators of the two-point correlations, as in Eq.(10), we need the matter density three- and four-point correlation functions and . Because we are only interested in orders of magnitude estimates and do not require a or better accuracy, we use a simple hierarchical ansatz for these high-order density correlations (Groth & Peebles 1977; Peebles 1980). Thus, we write the three-point density correlation as a sum of products of two-point correlations,

 ζ1,2,3=S33[ξ1,2ξ1,3+ξ2,1ξ2,3+ξ3,1ξ3,2], (18)

and in a similar fashion for the four-point density correlation,

 η1,2,3,4=S416[ξ1,2ξ1,3ξ1,4+3cyc.+ξ1,2ξ2,3ξ3,4+11cyc.], (19)

where “3 cyc.” and “11 cyc.” stand for three and eleven terms that are obtained from the previous one by permutations over the labels “1,2,3,4” of the four points. For the normalization factors in Eqs.(18)-(19), we interpolate from the large-scale quasilinear limit (Bernardeau et al. 2002a) (where we take an angular average to neglect the angular dependence of that would arise in the exact leading-order perturbative result)

 SQL3=347−(n+3), (20)
 SQL4=607121323−623(n+3)+73(n+3)2, (21)

to the highly nonlinear HEPT approximation (Scoccimarro & Frieman 1999)

 SHEPT3=34−2n1+2n+1, (22)
 SHEPT4=854−27×2n+2×33n+6n1+6×2n+3×33n+6×66n, (23)

as

 Sn=SQLn+ξ21+ξ2(SHEPTn−SQLn). (24)

Here, is the two-point correlation at the scale of interest and the local slope of the linear matter power spectrum. Thus, in the quasilinear regime, where , we have , while in the highly nonlinear regime, where , we have . Since the density correlations only contribute on much smaller scales (Mpc) than the cosmological scales (Mpc), in the integrals such as (11) or (17) it is sufficient to use for the two-point correlations and the coefficients the mean redshift of the relevant points. [For , we also use the geometrical mean of the relevant scales to compute and in Eqs.(20)-(24).]

This ansatz is the simplest model that is in qualitative agreement with large-scale theoretical predictions [because and at leading order in perturbation theory, see Goroff et al. (1986); Bernardeau et al. (2002a)] and with numerical simulations on nonlinear scales (Colombi et al. 1996). It was already used to estimate the covariance matrices of galaxy surveys (Bernstein 1994; Szapudi & Colombi 1996) or X-ray cluster surveys (Valageas et al. 2011; Valageas & Clerc 2012). Its generalization to all-order density correlations was also used to compute the high-order cumulants and the probability distributions of the smoothed convergence and cosmic shear, providing a good agreement with results from ray-tracing in N-body simulations (Valageas et al. 2004; Barber et al. 2004; Munshi et al. 2004; Munshi & Valageas 2005).

For completeness, we check the approximation (24) in Fig. 1, where we plot the coefficients as defined by Eq.(24) and the skewness and kurtosis of the density contrast within spherical cells of radius measured in numerical simulations (Nishimichi & Taruya 2011). The latter are defined from the cumulants of the density contrast as , where the superscript “T.H.” refers to the top-hat filter. These two definitions only coincide if we neglect the scale dependence of the two-point correlations and of the coefficients when we compute the cumulants from the three- and four-point correlations (18) and (19). However, this is sufficient for our purpose because in our numerical computations of the cosmic shear bias below, we also factor out the coefficients , using their value at the typical angular scale of interest, so that geometrical integrations over angles, including the typical spin-2 factor , can be done analytically. Thus, Fig. 1 shows that our approximation provides the correct order of magnitude for three- and four-point correlations. On these scales, the match is better than for the skewness and for the kurtosis. It might be possible to improve the agreement with the simulations by using another interpolation form, such as where and are best-fit parameters, but in this paper we keep the simple interpolation (24), which is sufficient for our purposes.

Since galaxies have a bias of order unity, and we are only interested in general-purpose estimates, we take in our numerical computations [and our results may be multiplied by the appropriate factors if needed, as in Eqs.(13)-(17)]. For the nonlinear density correlation function , we use the semi-analytic model developed in Valageas et al. (2013), which combines one-loop perturbation theory with a halo model to predict the matter density power spectrum and correlation function with a percent accuracy on quasilinear scales and a ten-percent accuracy on highly nonlinear scales. For cosmological parameters, we use the best fit CDM cosmology from Planck observations (Planck Collaboration et al. 2013).

In numerical computations, we keep the real-space expressions (11)-(17), rather than going to Fourier space. This avoids integrations over oscillatory kernels, such as Bessel functions, and the use of Limber’s approximation. Indeed, in configuration-space expressions such as Eq.(11), which involves the correlation between density fluctuations and along two lines of sight, Limber’s approximation corresponds to setting in cosmological kernels such as the lensing efficiency (Limber 1953; Munshi et al. 2008). This is because the density correlation is negligible beyond Mpc whereas cosmological kernels vary on much larger scales Mpc. However, if we use this approximation in Eqs.(13)-(17), we obtain , because at . This also means that the source-lens clustering contributions in Eq.(10) will be suppressed by a factor (which vanishes in Limber’s limit) and that the computation of this source-lens clustering bias requires going beyond Limber’s approximation. (This is no longer the case for the three-point estimators studied in Sects. 4 and 5, where we use Limber’s approximation because the source-lens clustering contributions do not vanish in this limit).

### 2.3 Numerical results

We show our results for cases where the two galaxy redshifts are the same in Fig. 2. [Throughout this paper, a positive (resp. negative) bias is shown by a solid (resp. dotted) line in the figures.]

On small angular scales, the total bias is dominated by the four-point correlation contribution (17), because it scales as and grows faster than the other terms in the nonlinear regime. This arises from correlations between the two nearby source galaxies and close density fluctuations on the two lines of sight. On very large scales, the bias is dominated by the first term in Eq.(13), which does not depend on the angular scale because it arises from the correlation between each galaxy and density fluctuations along its line of sight, whereas the other terms and the weak lensing signal itself decrease for larger angles as they involve correlations between the two lines of sight.

In all cases, the last three terms in Eq.(10) only give rise to a relative bias of the weak lensing estimator that is smaller than about . This can be safely neglected for all practical purposes. This is due to the fact that:

(a) this bias only arises from density fluctuations close to the source galaxies, whereas the weak lensing signal is generated by density fluctuations along the whole line of sight,

(b) the lensing efficiency of Eq.(9) vanishes at the source plane, , which further suppresses the bias by a factor of order , where Mpc is the typical correlation length, and

(c) the bias scales as whereas the signal scales as .

We show our results for cases where the two galaxy redshifts are different in Fig. 3. Let us choose for instance . Then, the contribution is now always dominated by the first term in Eq.(13), because the correlation in the second term in Eq.(13) is very small since . Nevertheless, this gives an overall contribution to the relative bias that is again smaller than about , as for the similar coincident redshifts of Fig. 2. In contrast, the four-point correlation and its bias contribution (17) are now very small, several orders of magnitude below the corresponding contribution obtained for similar coincident redshifts, as expected because it involves correlations between density fields at different redshifts. The only significant contribution that is left is the first term in Eq.(15), which involves the three-point correlation , because it is still possible for the three points to be located at about the same redshift. Moreover, this contribution is now much greater than for coincident redshifts and can reach . This is because this contribution is dominated by configurations where the points and are at about the redshift , and while the lensing kernel is still suppressed by a factor of order , the kernel is now of the same order as its typical value along the line of sight to galaxy , because is now significantly different from since .

To see more clearly the redshift dependence of the source-lens clustering bias, we show our results as a function of the second galaxy redshift , for a fixed first galaxy redshift , in Fig. 4. In agreement with the discussion above, the relative bias is minimum for because of the vanishing of both lensing kernels and for . For , the lensing signal saturates because it is dominated by density fluctuations in the common redshift range, , whereas the bias is dominated by the three-point correlation (with a similar kernel ), so that the relative bias also saturates and remains small. For , the lensing signal decreases with the length of the common redshift range, , while the bias is dominated by the three-point correlation . This leads to a steep growth of the relative bias for very small (because the short line of sight diminishes the signal, which only arises from scales where the three-point correlation is significant and contributes to the bias). The comparison between the panels shows that the relative amplitude of the bias decreases on larger angular scales, because the three-point correlation is smaller.

Thus, most of the source-lens clustering bias arises from the three-point correlation between a low-redshift source galaxy, nearby density fluctuations on its line of sight, and density fluctuations at about the same redshift on the line of sight to a second higher redshift source galaxy. Figure 4 shows that this source-lens clustering bias is almost always negligible, except when we cross-correlate the gravitational lensing distortions of a low-redshift galaxy, , with a higher redshift galaxy, (the effect being larger for higher and smaller ). There, the bias can actually dominate the weak lensing signal.

## 3 Two-point cosmic shear correlation function

### 3.1 Weak lensing shear γ

The measure of the convergence from galaxy surveys is not easy, because galaxies do not have a unique luminosity that could serve as a standard candle. In practice, one rather measures the two-point correlation function of the cosmic shear from the galaxy ellipticities, as in Eq.(1). Using in the following the flat sky approximation (which is sufficient for small angular scales), the shear is given by

 γ=∫χi0dχi′gi′,i∫dki′eiki′⋅xi′e2iαki′~δ(ki′), (25)

where is the Fourier transform of the density contrast, is the position of point along the line of sight, and is the polar angle of the wave vector in the plane that is orthogonal to the line of sight. The shear is a complex quantity, , where and are the real components along the two axis and in the transverse plane. Because of the factor , it is also a spin-2 quantity (Bartelmann & Schneider 2001; Munshi et al. 2008). This additional factor makes the computations somewhat heavier than for the convergence . From the shear one may compute several correlation functions, such as , , , , , where and are the tangential and cross-components, with respect to the direction of the pair in the transverse plane. They can all be expressed in terms of the convergence two-point correlation (11) and have similar magnitudes (Kaiser 1992; Bartelmann & Schneider 2001; Munshi et al. 2008). For our purposes we focus on the full shear correlation, , as in Eq.(1). Then, in a fashion similar to Eq.(7), the average of this estimator writes as

 ⟨^ξγγ∗(θ)⟩=⟨(1+b1δ1)(1+b2δ2)γ1γ∗2⟩1+b1b2ξ1,2, (26)

where again the indices and refer to the two lines of sight separated by the angular distance . This average can be split into four components,

 ⟨^ξγγ∗⟩=ξγγ∗+ξδγδγ∗+ζδγγ∗+ηδδγγ∗. (27)

The first component is again the weak lensing signal and it is equal to the convergence correlation (11),

 ξγγ∗(θ)=ξκκ(θ)=∫χ10dχ1′g1′,1∫χ20dχ2′g2′,2ξ1′,2′. (28)

The second component involves products of the two-point correlations between the galaxies and the density fluctuations along the line of sight,

 ξδγδγ∗=b1b21+b1b2ξ1,2⟨δ1γ∗2⟩⟨δ2γ1⟩. (29)

As compared with Eq.(13), there is no term because it vanishes thanks to the spin-2 factor . This is not the case for the cross term , where each product breaks the rotational invariance as it connects two different lines of sight (which defines a prefered direction) and the final contribution is nonzero (this direction is the same for the two terms so that averaging over the direction does not yield a null result).

Whereas scalar quantities like the convergence only involve the matter density two-point correlation function, , which is the the Fourier transform of the power spectrum,

 ξ(x)=∫dkeik⋅xP(k), (30)

for quantities that involve the spin-2 cosmic shear, we also need the integral with a spin-2 factor ,

 ∫dkeik⋅xe2iαkP(k)=e2iαxξ(2)(x∥,x⊥), (31)

where and are the longitudinal and transverse components of the separation vector with respect to the line of sight. As shown in App. A.1, this correlation function reads as

 ξ(2)(x)=ξ(x)−∫x⊥0dr⊥2r⊥x2⊥ξ(x∥,r⊥), (32)

and if . Then, Eq.(29) also reads as (see App. A.1)

 ξδγδγ∗=b1b21+b1b2ξ1,2∫dχ1′dχ2′g1′,1g2′,2ξ(2)1,2′ξ(2)2,1′. (33)

As compared with Eq.(13), the first product vanishes because of the spin-2 factor , as explained above, whereas in the second product the scalar correlation is replaced by the “spin-2 correlation” . Because of the subtraction in Eq.(32), associated with the constraint if , is usually smaller than . Therefore, the spin-2 factor decreases the amplitude of the source-lens clustering bias of the cosmic shear, as compared with the convergence.

The third and fourth components involve the three- and four-point density correlations and read as

 ζδγγ∗=b1⟨δ1γ1γ∗2⟩+b2⟨δ2γ1γ∗2⟩1+b1b2ξ1,2, (34)
 ηδδγγ∗=b1b21+b1b2ξ1,2⟨δ1δ2γ1γ∗2⟩c. (35)

For instance, using Eq.(25), the first average that enters the numerator in Eq.(34) reads as

 ⟨δ1γ1γ∗2⟩ = ∫dχ1′dχ2′g1′,1g2′,2∫dk1dk1′dk2′ (36) ×ei(k1⋅x1+k1′⋅x1′+k2′⋅x2′)e2i(αk1′−αk2′) ×δD(k1+k1′+k2′)B(k1,k1′,k2′),

where is the matter density bispectrum. Because this contribution is dominated by configurations where the points are nearby and at almost the same redshift (otherwise the three-point correlation is negligible), the bispectrum can be taken at the mean redshift of these three points.

### 3.2 Analytical approximations

For the shear, computations are not as straightforward because of the spin-2 factor . As we have seen in Sect. 2.3, the source-lens clustering bias is only important when we correlate a high-redshift galaxy, , with a low-redshift galaxy, . Then, the bias is dominated by the first term in Eq.(34), which involves the three-point correlation between the low-redshift galaxy with density fluctuations at almost the same redshift on the two lines of sight. Therefore, we neglect the four-point contribution (35) and we only consider the contributions (33) and (34). In Fourier space, neglecting the scale dependence of the coefficient , the ansatz (18) yields the factorized bispectrum

 B(k1,k2,k3) = S33[P(k2)P(k3)+P(k1)P(k3) (37) +P(k1)P(k2)].

As described in App. A.2, substituting the ansatz (37) into Eq.(36) gives

 ⟨δ1γ1γ∗2⟩=∫dχ1′dχ2′g1′,1g2′,2S33[ζ(1,1′)1,1′,2′+ζ(1,2′)1,1′,2′], (38)

where and are given by Eqs.(115) and (120) (and the contribution vanishes because of the spin-2 factor ). A symmetric expression gives and this yields the three-point contribution (34).

### 3.3 Numerical results

We show our results for galaxy pairs at the same redshift in Fig. 5. As compared with the case of the convergence shown in Fig. 2, the three-point contribution is somewhat smaller while the two-point contribution is several orders of magnitude smaller. This is because of the spin-2 factor that replaces the density correlation by the smaller correlation in Eq.(33) and removes the contribution . This implies that the contribution now decreases at large angles so that the relative bias does not show the faster growth found in Fig. 2. In any case, Fig. 5 shows that as for the convergence the source-lens clustering bias is negligible for same-redshift sources.

We show our results for cases where the two galaxy redshifts are different in Fig. 6. We find again that the three-point contribution is somewhat smaller than for the case of the convergence shown in Fig. 3, especially on small scales, where it only reaches instead of . The two-point contribution is several orders of magnitude smaller than for the convergence. This is because it only involves the cross correlation , which correlates the high-redshift galaxy with low-redshift density fluctuations with , as the term is zero by symmetry.

We show the dependence of the source-lens clustering bias on the second galaxy redshift , for a fixed first galaxy redshift , in Fig. 7. We obtain behaviors that are similar to those found in Fig. 4 for the convergence, with a minimum at the coincident redshift , a saturation at high redshift , and a steep increase for (but the bias is no longer always positive). The amplitude of the bias is somewhat smaller than in Fig. 4, especially for small angular scales. This leads to an even smaller range of redshifts at where the bias reaches of the signal or more.

Therefore, as for the convergence, we find that the source-lens clustering bias of estimators of the cosmic shear two-point correlation function is almost always negligible. It is only relevant when we cross-correlate the shear of a low-redshift galaxy, , with the shear of a higher redshift galaxy, (the effect being larger for higher and smaller ). For , the bias can actually dominate the weak lensing signal. In practice, it would be sufficient to remove such pairs from the data analysis, because they are a very small fraction of the pairs measured in a survey and their cosmological information is highly redundant with other pairs (where both galaxies are at the same low redshift or at possibly different redshifts above ).

## 4 Three-point convergence correlation function

### 4.1 Source-lens clustering bias

We now consider the impact of the source-lens clustering bias on estimators of the three-point weak lensing correlation functions. As for the two-point correlation, we first investigate the simpler case of the weak lensing convergence . Then, the generalization of Eqs.(2)-(7) to three-point statistics, obtained by measuring triplets of galaxies, gives

 ⟨^ζκκκ⟩=⟨(1+b1δ1)(1+b2δ2)(1+b3δ3)κ1κ2κ3⟩⟨(1+b1δ1)(1+b2δ2)(1+b3δ3)⟩, (39)

for the estimator of the three-point convergence correlation. As seen in Sect. 2, because the lensing kernel vanishes for , the source-lens clustering bias is only significant when a foreground galaxy correlates with the density fluctuations along the line of sight to a background galaxy , with . Therefore, to simplify the analysis, we neglect correlations that correspond to a vanishing lensing efficiency kernel [that at next order give a damping factor instead of zero] or that involve different redshifts. This allows us to use Limber’s approximation (which gives zero for the discarded terms). Then, assuming without loss of generality , can only be correlated with , with , and with . Then, the average (39) reads as

 z1≤z2≤z3:⟨^ζκκκ⟩≃ζκκκ+ζδ, (40)

where the source-lens clustering contribution (that we denote with the superscript ) writes as

 ζδ = [b1b2⟨δ1δ2κ3⟩⟨κ1κ2⟩+(1+b1b3ξ1,3)b2⟨δ2κ3⟩⟨κ1κ2⟩ (41) +(1+b2b3ξ2,3)b1(⟨δ1κ2⟩⟨κ1κ3⟩+⟨δ1κ3⟩⟨κ1κ2⟩)] ×[1+b1b2ξ1,2+b2b3ξ2,3+b1b3ξ1,3+b1b2b3ζ1,2,3]−1.

Here we note again the density-density correlation (which arises from the galaxy-galaxy correlations).

In contrast with the case of the two-point estimator (10), the source-lens clustering bias is no longer dominated by contributions that involve the density three-point correlation, but by contributions that involve products of the density two-point correlation. In particular, for the generic case of three different source redshifts, the galaxy-galaxy correlations are negligible and Eq.(41) simplifies as

 z1

Therefore, the source-lens clustering bias is much easier to evaluate for the three-point convergence correlation functions than for the two-point statistics studied in Sects. 2 and 3. Nevertheless, in the following we use Eq.(41) to include the case where galaxy redshifts coincide.

On large scales, whereas in Eq.(10) the two-point weak lensing signal scales as the linear density correlation and the bias obeys the higher-order scaling , in Eq.(42) the three-point weak lensing signal and its bias show the same scaling . Therefore, the impact of the source-lens clustering bias is expected to be greater for measures of three-point lensing correlations than for two-point lensing correlations (see also Bernardeau (1998)).

The first contribution in Eq.(40) is the weak lensing signal,

 ζκκκ=⟨κ1κ2κ3