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Second Order Corrections to Large Scale Structure Weak Lensing: Background Source Clustering

We discuss the second order contributions to lensing statistics resulting from the clustering of background sources from which galaxy shape measurements are made in weak lensing experiments. In addition to a previously discussed contribution to the lensing skewness, background source clustering also contributes to the two-point correlation function, such as the angular power spectrum of convergence or shear. At arcminute scales or above, the second order contribution to the angular power spectrum of convergence due to source clustering is below the level of a few percent. The background clustering of sources also results in a non-Gaussian contribution to the power spectrum covariance of weak lensing convergence through a four-point correlation function or a trispectrum in Fourier space. The increase in variance is, at most, a few percent relative to the Gaussian contribution while the band powers are also correlated at the few percent level. The non-Gaussian contributions due to background source clustering is at least an order of magnitude smaller than those resulting from non-Gaussian aspects of the large scale structure due to the non-linear evolution of gravitational perturbations. We suggest that the background source clustering is unlikely to affect the precision measurements of cosmology from upcoming weak lensing surveys.


INTRODUCTION
Weak gravitational lensing of faint galaxies probes the distribution of matter along the line of sight. Lensing by largescale structure (LSS) induces correlation in the galaxy ellipticities at the few percent level and can be detected through challenging statistical studies of galaxy shapes in wide-field surveys (e.g., Blandford et al. 1991;Miralda-Escudé 1991;Kaiser 1992). An important aspect of weak lensing is that these ellipticity correlations, or associated statistics, provide cosmological information that is, in some cases, complementary to those supplied by the cosmic microwave background data at the same level of precision or better (e.g., Jain & Seljak 1997;Bernardeau et al. 1997;Kaiser 1998;Schneider et al. 1998;Hu & Tegmark 1999;Cooray 1999;van Waerbeke 1999;see Mellier 1999 andBartelmann &Schneider 2000 for recent reviews). Indeed, a wide number of recent studies have provided the clear evidence for weak lensing due to large scale structure (van Waerkebe et al. 2000;Bacon et al. 2000;Wittman et al. 2000;Kaiser et al. 2000), though more work is clearly needed to understand even the statistical errors and biases. While biases can result from certain aspects related to the selection of observational fields (e.g., , statistical errors include those that are fundementally present due to the non-Gaussian nature of the large scale structure (e.g., Cooray & Hu 2001b).
Current predictions on statistics related to large scale structure weak lensing and their ability to measure cosmological parameters are based on several assumptions; for example, it is implicitly assumed that background galaxies, from which galaxy shape measurements are made, are uniformly distributed. Another assumption is the use of socalled Born approximation where one integrates along unperturbed photon geodesics instead of perturbed photon paths. Here, we discuss the former assumption while Cooray & Hu (2002) presents a discussion of the corrections resulting from dropping the Born approximation and including the so-called lens-lens coupling between two lenses at two different redshifts.
The effect of background source clustering was first discussed with respect to the lensing three point statistics, such as convergence skewness (Bernardeau 1998). Hamana et al. (2001) includes an extended discussion of this contribution. Recently, the same effect was revisited by Schneider et al. (2002) as a possible source of curl-like modes in lensing statistics. Here, we present a general discussion of the corrections resulting from clustering of background sources and consider effects two, three and four-point correlation functions. We suggest that the contributions are negligible and are unlikely to affect the current and upcoming weak lensing observations as a probe of cosmological parameters by comparing to predictions that neglect background source clustering.

CALCULATIONAL METHOD
We discuss our statistics in terms of lensing convergence, where ns(χ) is the normalized radial distribution of background sources such that dχns(χ) = 1 and g(χ ′ , χ) is the lensing weight function when a background source is at a radial distance of χ: Here, χ is the radial distance, or lookback time, from the observer, given by and the analogous angular diameter distance with the expansion rate for adiabatic CDM cosmological models with a cosmological constant given by Here, H0 can be written as the inverse Hubble distance today cH −1 0 = 2997.9h −1 Mpc. We follow the conventions that in units of the critical density 3H 2 0 /8πG, the contribution of each component is denoted Ωi, i = c for the CDM, b for the baryons, Λ for the cosmological constant. We also define the auxiliary quantities Ωm = Ωc + Ω b andΩK = 1 − i Ωi, which represent the matter density and the contribution of spatial curvature to the expansion rate respectively. Note that as ΩK → 0, dA(χ) → χ and we define χ(z = ∞) = χ0. Though we present a general derivation of second order contributions to weak lensing due to background source clustering, we show results for the currently favorable ΛCDM cosmology with Ω b = 0.05, Ωm = 0.35, ΩΛ = 0.65 and h = 0.65.
We assume that background sources are clustered in radial space such that ns(χ) ≈ns(χ)[1 + δns(χ)] where fluctuations in the number counts of background sources are related to that of the density field via a time, and possibly a scale dependent, bias Here we have considered, perturbatively, contributions up to the second order in density perturbations. It is important that we consider contributions up to the δ 2 term since these terms can contribute at the same second order level in the power spectrum. Though we discuss the effect in terms of lensing convergence, it should be noted that our calculation equally well apply for, correlations of shear, such as the gradient, or electric-like, modes (see, Schneider et al. 2002).
In Fourier space, we can write the first, second and third order contribution to convergence as and respectively. We define the associated angular power spectrum, bispectrum and trispectrum of weak lensing convergence, in flat-sky as appropriate for current and upcoming experiments as Here, ki...j = ki + . . . + kj and δD is the delta function not to be confused with the density perturbation. Note that the subscript c denotes the connected piece, i.e., the trispectrum is defined to be identically zero for a Gaussian field.
Here and throughout, we occasionally suppress the redshift dependence where no confusion will arise. We define the power spectrum of density fluctuations as where in linear perturbation theory. We use the fitting formulae of Eisenstein & Hu (1999) in evaluating the transfer function T (k) for CDM models. Here, δH is the amplitude of presentday density fluctuations at the Hubble scale; with n = 1, we adopt the COBE normalization for δH (Bunn & White 1997) of 4.2×10 −5 , consistent with galaxy cluster abundance (Viana & Liddle 1999), with σ8 = 0.86. To capture the nonlinear aspects of the power spectrum, we use the prescription by Peacock & Dodds (1996).
We will now discuss contributions to the angular power spectrum, bispectrum and trispectrum of convergence from clustering of background sources. To illustrate results, we take a redshift distribution for the background sources of the form where (α, β) denote the slope of the distribution at low and high z's, respectively around a mean-like parameter given by ∼ z0. For the purpose of this calculation we take α = β = 1.5 and take z0 to be 1.0 so as to mimic the expected background sources from current and upcoming lensing catalogs. In general, since clustering evolves to low redshifts, with a decrease in the background source redshift distribution to a lower redshift, we expect an increase in the importance of second order effects associated with clustering of background sources.

Power Spectrum
To the first order, using κ (1) (l)κ (1) (l ′ ) , we simplify with the Limber approximation (Limber 1954) in Fourier space following Kaiser (1992;Kaiser 1998) to obtain the well known result that The second order convergence power spectrum resulting from source clustering involves two terms: κ (2) κ (2) and κ (3) κ (1) . The latter includes an additional, and equal, term through a permutation. We simplify these contributions again using the Limber approximation (Limber 1954) such that and Note that the total contribution to the convergence power spectrum follows as κ (2) We denote the first contribution by C 22 l term and the latter two terms by C 31 l . Thus, the total contribution to the power spectrum due to source clustering is In figure 1, we show the second order correction to the angular power spectrum of convergence. The solid line is the well known first order result, while the dashed line is the contribution from the C 22 l term. Note that C 22 l ∝ b 2 1 , and we have taken the value of b1 = 1 for illustration purposes. The long-dashed line is the C 31 l contribution, where C 31 l ∝ b2 and we have taken b2 = 1 for simplicity. Since we do not have detailed information on the galaxy bias, to illustrate our results, we have taken the bias to be redshift and scale independent. Adding a redshift dependent bias of the from b(z) ∝ (1 + z) γ , however, did not lead to a significantly different result from the one suggested in figure 1 when |γ| < 2.
There is one aspect of bias that should be kept in mind when interpreting figure 1. In general, quadratic bias is expected to be negative, such that the two terms, C 22 l and C 31 l , added together will give a contribution which is lower than what one would naively expect if simply added together. In the IRAS PSCz catalog, Feldman et al. (2001) finds 1/b1 = 1.20 +0.18 −0.19 and b2/b 2 1 = −0.42 ± 0.19. Similar results our available from the 2dF survey by Verde et al. (2002): b1 = 1.04 ± 0.11 and b2 = −0.054 ± 0.08. Since there is no conclusive evidence for a non-zero value for b2, the first approximation that b2 = 0 and b1 = 1 leads to the dashed line with the conclusion that second order effects are generally below the few percent level for multipole less than 10 4 corresponding to angular scales less than few arcminutes.
On the other hand, if b2 = −0.5 and b1 = 1, we obtain the dot-dashed line as the total contribution to the angular power spectrum of convergence due to source clustering; the shape of the power spectrum is due to the fine cancellation of C 22 l and C 31 l terms. We suggest that, in addition to the linear bias, the extent to which background source clustering affects statistics such as power spectra or correlations depends on the detail aspects of galaxy biasing such as the quadratic bias. In any case, we find that source clustering effects are unlikely to be a strong contaminat for current lensing experiments. The angular power spectrum of convergence. The solid line shows the first order contribution while corrections due to background source clustering is shown with dashed (C 22 l ) and long-dashed (C 31 l ) lines. We have assumed b 1 = 1 and b 2 = 1 in these two cases, respectively. The dot-dashed line is the total second order power spectrum when b 1 = 1 and b 2 = −0.5 consistent with suggestions in the literature for galaxy bias (see text for details). The dotted line is the C 31 l contribution when galaxies trace the linear density field instead of the fully non-linear power spectrum.
There is also another important aspect related to C 31 l . The integral over l1 denotes the power spectrum traced by galaxies and C 31 l effectively scales with this integral as an overall normalization. If galaxies do not fully trace the nonlinear power spectrum, as predicted by the Peacock & Dodds (1996) formulae for the dark matter, then the contribution would be lower than what we have predicted. We can bracket the expected range of variation by replacing the power spectrum involved with the integral over l1 with that of the linear power spectrum, as it is generally expected that the galaxy power spectrum lies between the linear and non-linear cases of dark matter. Since the contribution to the integral here comes from all angular scales, the bahvior of either the linear or non-linear power spectrum at small angular scales becomes to some extent important for the calculation presented here. As we do not have a reliable method to predict the non-linear power spectrum at small scales, we safely cut off the calculation at k ∼ 10 6 h Mpc −1 .

Bispectrum
We can write the resuling contribution to the bispectrum by considering terms such as κ (2) (l1)κ (1) (l2)κ (1) (l3) and add the necessary permutations. We write one of these terms as so that the bispectrum is where the permutations are with respect to the ordering of (l1, l2, l3) and involves five additional terms. Following Cooray & Hu (2001a), we can write the third moment of convergence using the bispectrum as where the quantity within parantheses is the Wigner-3j symbol, which in the case of no angular dependence can be written as l1 l2 l3 for even L; it vanishes for odd L. We refer the reader to  for additional details on the Wigner 3-j symbol.
Using the third moment, we construct the skewness as where the all-sky expression for bispectrum, in terms of the flat-sky derivation, is Similarly, the second moment is defined as where W l are the multipole moments (or Fourier transform in a flat-sky approximation) of the window. For simplicity, we will choose a window which is a two-dimensional top hat in real space with a window function in multipole space of W l (σ) = 2J1(x)/x with x = lσ.  Cooray & Hu (2001). For comparison, we show results from N-body particle-mesh simulations by White & Hu (1999). The higher order correction to skewness resulting from background source clustering is shown with a dot-dashed line. The second order contribution is at the level of few tens of percent of the total expected from non-linear evolution of gravitational perturbations.
In figure 2, we summarize our results on the expected contribution to the convergence skewness. We refer the reader to Bernardeau (1998) and Hamana et al. (2002) for an extended discussion on the effects on background source clustering on skewness. As shown in figure 2 and discussed in prior publications, the contribution to skewness is at the level of few tens of percent and depends strongly on parameters such as the mean redshift of background sources and the width of the redshift distribution. A higher mean redshift and a smaller width result in a smaller contribution to skewness while a lower mean redshift and a broader distribution can contribute up to 30% or more of the skewness expected from non-linear evolution of gravitational perturbations.

Trispectrum
We can write the resuling contribution to the trispectrum by considering terms such as κ (2) (l1)κ (2) (l2)κ (1) (l3)κ (1) (l4) and add the necessary permutations. We write one of these terms as The ratio of non-Gaussian to Gaussian error. The solid line is the calculation for non-linear evolution of gravitational perturbations using dark matter halos following the prescription in Cooray & Hu (2001b), while the dotted line is contribution arising from source clustering effects. We use the same binning scheme in multipole space as Cooray & Hu (2001b).
so that the trispectrum is where the permutations are with respect to the ordering of (l1, l2, l3, l4).
For the purpose of this calculation, we assume that upcoming weak lensing convergence power spectrum will measure binned logarithmic band powers at several li's in multipole space with bins of thickness δli.
where As(li) = d 2 l is the area of 2D shell in multipole and can be written as As(li) = 2πliδli + π(δli) 2 .
We can now write the signal covariance matrix as where A is the area of the survey in steradians. Again the first term is the Gaussian contribution to the sample variance and the second the non-Gaussian contribution. A realistic survey will also have shot noise variance due to the finite number of source galaxies in the survey. For a comparison of previous calculations, we take the same binning scheme as the one used in Cooray & Hu (2001b) and used in White & Hu (1999).
In figure 3, we show the ratio of R ≡ Cii/C Gaus ii where C Gaus ii is the contribution with simply the Gaussian variance. This ratio can also be written as and we plot R − 1 to highlight the difference between source clustering and non-Gaussian aspect of large scale structure. As shown, the clustering only leads to a few percent contribution, at l ∼ 10 3 , beyond the Gaussian variance while the non-Gaussianities due to large scale structure clustering contributes at the level of 10% or more. One can safely ignore the relative increase in the variance of power spectrum measurements due to background source clustering. An additional aspect of the covariance resulting from non-Gaussianities is that band power estimates are correlated. These correlations can be written aŝ In figure 4, we show the behavior of the correlation coefficient between a fixed lj as a function of three li's. When li = lj the coefficient is 1 by definition. Due to the presence of the dominant Gaussian contribution at li = lj, the coefficient has an apparent discontinuity between li = lj and li = lj−1 that decreases as lj increases and non-Gaussian effects dominate. As shown, however, the correlation coefficients due to the non-Gaussian nature of the large scale structure is over an order of magnitude larger than than the correlations resulting from the clustering of background sources. The results related to the covariance suggests that non-Gaussian effects resulting from the clustering of background sources is not expected to strongly influence the abilities of weak lensing experiment to obtain precision measurements of cosmology.

SUMMARY & CONCLUSIONS
We have discussed the second order contributions to weak gravitational lensing convergence resulting from the clustering of background sources from which galaxy shape measurements are made in weak lensing experiments. The clustering of source galaxies induce a second order contribution to the two point statistics such as weak lensing convergence angular power spectrum. For the angular scales of interest, we have shown that this contribution is at the level of few percent; to some extent, however, the exact contribution is uncertain due to unknown aspects associated with galaxy biasing such as the quadractic bias. Our calculations related to the skewness generated by background source clustering is consistent with previous calculations by Bernardeau (1998) andHamana et al. (2001). We have discussed a new non-Gaussian aspect of the background source clustering involving a contribution to the four point correlation function of shear or the trispectrum in Fourier space. The non-Gaussian four-point function is of interest since it determines the covariance of power spectrum measurements. The background source clustering increases the Gaussian covariance at the level of few percent when l ∼ 10 3 . This increase, however, is an order of magnitude or more below the increase resulting from the intrinsic non-Gaussian nature of the large scale structure due to the non-linear evolution of gravitational perturbations. The trispectrum contribution to the covariance also leads to correlations between band power estimates, though, these are again at the few percent level or below and are unlikely to be a significant source of error for current and upcoming weak lensing experiments.

ACKNOWLEDGMENTS
We thank support from the Sherman Fairchild foundation and the Department of Energy grant DE-FG03-92-ER40701.