Issue 
A&A
Volume 527, March 2011



Article Number  A26  
Number of page(s)  36  
Section  Cosmology (including clusters of galaxies)  
DOI  https://doi.org/10.1051/00046361/201015621  
Published online  20 January 2011 
Online material
Appendix A: Threedimensional number densityintrinsic shear correlation function
In this appendix, we derive the threedimensional number densityintrinsic shear (gI) correlation function, detail the inclusion of photometric redshift scatter into the formalism, and establish an approximate relation between the threedimensional correlation function and the angular power spectrum.
A.1. Correlation function for exact redshifts
We define the threedimensional correlation function between the galaxy density contrast δ_{g} and the radial intrinsic shear γ_{I, + } as (A.1)for a given mean redshift z of the galaxy pairs correlated. Here we introduced a threedimensional comoving separation vector x which has a lineofsight component Π ≡ x_{ ∥ }. Its transverse components are denoted by x_{ ⊥ }, with modulus r_{p} ≡  x_{ ⊥ }  . The first argument of both δ_{g} and γ_{I, + } denotes the position on the sky, the second the position along the line of sight, and the third quantifies the epoch, given in terms of the redshift. Note that a lineofsight separation Π ≠ 0 implies that δ_{g} and γ_{I, + } are not measured at precisely the same epoch, contrary to what we have written in (A.1). However, as Π is small compared to the comoving distance χ(z) to the galaxies under consideration, this approximation holds to good accuracy.
Following Hirata & Seljak (2004), the radial component of the intrinsic shear is measured with respect to x_{ ⊥ }, and without loss of generality we can choose the coordinate system such that γ_{I, + } = γ_{I,1}. Note that in the majority of weak lensing studies γ_{ + } is defined as the tangential component of the shear. Measuring radial instead of tangential alignment implies a change of sign, so that e.g. the galaxygalaxy lensing signal which we consider is negative. Denoting Fourier variables by a tilde, one can construct in analogy to the matter density contrast a threedimensional intrinsic convergence δ_{I}^{12} via (A.2)where ϕ is the polar angle of k_{ ⊥ }, i.e. the projection of the wave vector onto the plane of the sky. We will denote the lineofsight component of k by k_{ ∥ }.
Then one can write the correlation function by Fourier transforming (A.1) as (A.3)Inserting the definition of the threedimensional gI power spectrum, (A.4)and subsequently integrating (A.3) over k^{′} yields (A.5)where in order to arrive at the third equality, the definition of the secondorder Bessel function of the first kind was used. In this derivation it was implicitly assumed that the intrinsic shear field does not feature Bmodes, as is for instance the case for the linear alignment paradigm.
One can now integrate over the line of sight, making use of the definition of the Dirac deltadistribution, to obtain the projected gI correlation function as employed by Mandelbaum et al. (2006), Hirata et al. (2007), and Mandelbaum et al. (2010), (A.6)Real data cannot provide the correlation function for arbitrarily large lineofsight separations, so that a truncation of the integral in (A.6) is necessary. This formula is still applicable if one can stack observations for all values of Π for which galaxy pairs carry a signal. While this can easily be achieved for spectroscopic observations, photometric redshift scatter smears the signal in Π such that a cutoff Π_{max} needs to be taken into account explicitly in the modelling. Of course it would be possible to compute the observed correlations out to very large Π_{max}, but this way many uncorrelated galaxy pairs would enter the correlation function, thereby decreasing the signaltonoise dramatically.
Instead, we proceed from (A.5) by assuming that ξ_{gI} is a real function, and write (A.7)As can be seen from this equation, ξ_{gI} is an even function in both r_{p} and Π, so that it is sufficient to compute just one quadrant. Note that by definition r_{p} ≥ 0, whereas Π can also attain negative values.
Fig. A.1
Threedimensional gI correlation function as a function of comoving lineofsight separation Π and comoving transverse separation r_{p} at z ≈ 0.5. Contours are logarithmically spaced between 10^{2} (yellow) and 10^{6} (black) with three lines per decade. Top panel: applying a Gaussian photometric redshift scatter of width 0.02. Bottom panel: assuming exact redshifts. Note the largely different scaling of the ordinate axes. The galaxy bias has been set to unity, and (6) with SuperCOSMOS normalisation has been used to model P_{δI} in both cases. Redshiftspace distortions have not been taken into account. 

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Equation (A.7) yields the threedimensional gI correlation function for exact or, to good approximation, spectroscopic redshifts. For the model described in Sect. 4 with SuperCOSMOS normalisation and b_{g} = 1, we plot ξ_{gI}(r_{p},Π,z) for z ≈ 0.5 in Fig. A.1, bottom panel. As expected, the correlation is strongest for small separation, in particular for Π close to zero. If spectroscopic data is available, essentially all information is captured when a cutoff Π_{max} = 60 h^{1} Mpc is used in the integration (A.6), as e.g. in Mandelbaum et al. (2010). Due to the definition (A.1), the gI correlation function measures the radial alignment of the galaxy shape with respect to the separation vector of the galaxy pair considered. Therefore the correlation function vanishes for all Π at r_{p} = 0 since then the separation vector points along the of sight. Note that the contours do not approach the Π = 0axis asymptotically, but cross this line at some value of r_{p}, as expected for a differentiable correlation function. Throughout these considerations we have not taken the effect of redshiftspace distortions into account.
A.2. Incorporating photometric redshifts
Photometric redshift errors cause the observed correlation function to be a “smeared” version of (A.7), introducing a spread especially along the line of sight but to a lesser extent also in transverse separation (because an uncertain redshift is used to convert angular separation to physical separation). If we denote quantities determined via photometric redshifts by a bar, the actually measured threedimensional correlation function reads (A.8)where z_{m} denotes the mean redshift of the galaxy samples used for the number density and the shape measurement. Here, p is the probability distribution of the true values of r_{p}, Π, and z_{m}, given photometric redshift estimates of these quantities. In words, (A.8) means that in order to obtain the observed correlation function, we integrate over ξ_{gI} as given in (A.7), weighted by the probability that the true values for separations and redshift actually correspond to the estimates based on photometric redshifts.
The direct observables for this measurement are the redshifts of the two galaxy samples under consideration, z_{1} and z_{2}, and their angular separation θ. The sets of variables (z_{1},z_{2},θ) and (r_{p},Π,z_{m}) are related via a bijective transformation. Writing (A.8) in terms of the other set of variables, one obtains (A.9)In the second step it was assumed that the probability distributions of z_{1}, z_{2}, and θ are mutually independent, and that θ is exactly known, i.e. . We have introduced different redshift probability distributions for the galaxy sample with number density information p_{n} and the one with shape information p_{ϵ}. All quantities related to photometric redshifts have been expressed in terms of the arguments of the correlation function on the lefthandside.
We make use of the following approximate relations between the two triples of variables, (A.10)where H(z) is the Hubble parameter. Note that the same transformations have been used to bin the observational data in terms of redshift, transverse and lineofsight separation. With this equation for Π, in combination with the assignment of probability distributions in (A.9), we have introduced the convention that Π > 0 means that the galaxy from the density sample is at lower redshift than the galaxy from the shape sample. If and only if the distributions for the density and the shape sample are identical, which we assume throughout this work, the correlation function remains symmetric with respect to Π, i.e. .
With these equations at hand, one can also write down the inverse transformation of (A.10), which is needed to evaluate (A.9), (A.11)Then (A.9) can be expressed as (A.12)Note that the absolute value for z_{2} − z_{1} has been introduced in the second argument of ξ_{gI}, which is possible since it is an even function in this argument. The integrals in (A.12) run over the full range of spectroscopic (exact) redshifts. As a consequence,  z_{2} − z_{1}  in the second argument of ξ_{gI} can obtain relatively large values, leading to very large Π ≫ 100 h^{1} Mpc. However, the spectroscopic ξ_{gI} becomes very small for large Π, so that the integrand in (A.12) can safely be set to zero in this case.
Still, any sizeable photometric redshift scatter leads to a considerable spread of the threedimensional correlation function in Π, as can be seen in Fig. A.1. Assuming a Gaussian photometric redshift scatter with width 0.02 around every true redshift, the strong signal concentrated at small Π and r_{p} ≲ 10 h^{1} Mpc in the spectroscopic case is scattered along the line of sight, so that the values of at Π > 200 h^{1} Mpc are still more than a per cent of those at Π = 0 for any r_{p}. In contrast, we find that the net scatter of signal between different transverse separations is negligible. Hence, in principle the projected correlation function (A.6) does not change when using photometric instead of spectroscopic redshift information as long as the complete range of Π for which a signal is measured enters the lineofsight integration. However, in practice the lineofsight integral has to be truncated for reasons of a good signaltonoise ratio, so that in the case of photometric redshifts part of the signal is lost. Therefore it is crucial to repeat the same steps applied to the data also to the model and use the same cutoff Π_{max} in (A.6).
A.3. Relation to angular power spectra
We now derive a relation between the threedimensional gI correlation function in the presence of photometric redshift scatter and the angular power spectrum, which proves most convenient to compute in practice. Inserting (A.5) into (A.12), one can write (A.13)where we employed z_{m} = (z_{1} + z_{2})/2 as a shorthand notation. Making use of , see (A.11), and defining the angular frequency , one obtains (A.14)We then transform the integration variables {z_{1},z_{2}} to {z_{m},Δz ≡ z_{2} − z_{1}}. Note that the determinant of the Jacobian of this transformation is unity. We apply Limber’s approximation, which in this case can be written as (A.15)Here we have assumed that the two redshift probability distributions are sufficiently broad and have similar forms, so that an evaluation at z_{m} instead of z_{m} ± Δz/2 does not change the results significantly. Since the photometric redshifts on which the distributions are conditional encapsulate the dependence of on the lineofsight separation Π, we do not extend this approximation to the second argument. Equation (A.14) thereby simplifies to (A.16)where in order to arrive at the second equality, we integrated over Δz. The resulting Dirac deltadistribution renders the k_{ ∥ } integration trivial. Making use of the expressions and p(z) = p(χ) dχ/dz, one obtains the result (A.17)where in the last step we implicitly defined the projected gI autocorrelation power spectrum C_{gI}. In addition to the angular frequency, we have written the photometric redshifts and , which characterise the redshift distributions entering C_{gI}, explicitly as arguments. Note that Limber equations, such as (5), in general hold only approximately, the range of validity being the more limited the narrower the kernels in the lineofsight integration (e.g. Simon 2007).
We have verified that the calculations of the threedimensional gI correlation function according to (A.12) and (A.17) agree within the numerical accuracy. The latter can be computed much more efficiently by computing the angular power spectrum via Limber’s equation and then using Hankel transformations to obtain the correlation function via (A.17), employing the transformation (A.10). One can proceed likewise to obtain analogous expressions for the gg signal. Galaxygalaxy lensing vanishes if the density field probed by the galaxy distribution and the source galaxies on whose images the gravitational shear is measured are located at exactly the same redshift. Thus one cannot proceed with the same formalism as used to derive the gI contribution (see the assumptions underlying the definition (A.1)), but must instead incorporate redshift probability distributions from the start, again arriving at an expression analogous to (A.17).
Appendix B: Redshift dependence of the linear alignment model
In this appendix we rederive the redshift dependence of the linear alignment model, obtaining a different result than Hirata & Seljak (2004), but being in full agreement with Hirata & Seljak (2010). Practically all attempts at constructing a physical description for intrinsic alignments are based on the linear alignment model originally suggested by Catelan et al. (2001). They assumed that the shape of the luminous part of a galaxy exactly follows the shape of its host halo, and that the ellipticity of the latter is determined by the local tidal gravitational field of the largescale structure.
The simplest possible form allowed by the assumptions made above is a linear relation between the intrinsic shear and the gravitational field, given by (Catelan et al. 2001) (B.1)where we wrote the normalisation in the notation of Hirata & Seljak (2004) in which C_{1} is an arbitrary constant. The partial derivatives are with respect to comoving coordinates, and Φ_{p}(x) ≡ Φ(x,z_{p}) denotes the “primordial” potential, i.e. the linear gravitational potential evaluated at the epoch of galaxy formation, at a redshift z_{p} well within the matterdominated era. For ease of notation we have omitted a smoothing of the gravitational potential on galactic scales in (B.1) which can be implemented by a simple cutoff of high wavenumbers in Fourier space, see Hirata & Seljak (2004). These authors used the relations (B.1) in their derivation of the intrinsic alignment power spectra for the linear alignment model, which we closely follow.
In a first step, the primordial gravitational potential is related to the matter density contrast via the Poisson equation (B.2)where denotes the comoving Laplacian. This expression is Fouriertransformed, yielding (B.3)The growth factor D(z) quantifies the dependence of the matter density contrast on redshift in the limit of linear structure formation, and is normalised to D(z) = (1 + z)^{1} during matter domination (Hirata & Seljak 2004)^{13}. Restricting (B.3) to linear scales, one obtains the ratio (B.4)where in the last step we made use of the fact that z_{p} lies in the matterdominated era. Inserting (B.4) into (B.3), and considering the linear regime, one arrives at the relation between primordial potential and linear matter density contrast, (B.5)This expression differs from the result given in Hirata & Seljak (2004), Eq. (14), by an additional factor (1 + z)^{2}. This discrepancy was also found in other rederivations (Hirata & Seljak 2010; Bean, Laszlo; priv. comm.).
Hirata & Seljak (2004) inserted their Eq. (14) into (B.1) and then computed the threedimensional intrinsic shear (II) power spectrum and the intrinsic shearmatter crosspower spectrum. Neglecting source clustering but otherwise following their steps in exact analogy, we obtain where in this work we use the full nonlinear matter power spectrum on the righthand side instead of the linear power spectrum, as written here in the original form of the linear alignment model. If Hirata & Seljak (2004), Eq. (14), were employed instead of (B.5), (B.6) would have an additional term (1 + z)^{4}, and (B.7) an additional term (1 + z)^{2}, in the numerator. These modifications would correspond to a shift by − 2 in η_{other} in our models (19) and (23).
Appendix C: Volume density and luminosities of red galaxies
To make realistic predictions for the intrinsic alignment contamination of cosmic shear surveys, we must specify, at each redshift, the distribution of galaxy luminosities that enter (19). Since this intrinsic alignment model only holds for red galaxies, we additionally must estimate the fraction of earlytype galaxies in the total weak lensing population as a function of redshift. In this paper, both quantities are determined using fits to the observed luminosity functions given in Faber et al. (2007). In this appendix, we present technical details about these calculations, assess the sensitivity of our results to this particular luminosity function choice, and provide data that can be used to forecast the intrinsic alignment contamination of other cosmic shear surveys (with different limiting magnitudes) besides that discussed in the main text. In all cases, we are extrapolating the luminosity functions to fainter magnitudes at a given redshift relative to the samples used to determine the luminosity function.
Fig. C.1
Red galaxy fraction f_{r} (top panel) and mean luminosity of red galaxies (bottom panel) as a function of redshift for a magnitude limit r_{lim} = 25. We compare the results for the sets of luminosity functions provided by Faber et al. (2007, black solid curves), Brown et al. (2007, dark grey solid curves), Willmer et al. (2006, light grey solid curves), Giallongo et al. (2005, black dotted curves), and Wolf et al. (2003, grey dotted curves). The results for Faber et al. (2007) luminosity functions used with the value of B − r expected at z = 1 are indicated by the cross in each panel. 

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We employ the Schechter luminosity function parameters for red galaxies from Faber et al. (2007), where φ^{∗} and M^{∗} are given as a function of redshift, and where the faintend slope is fixed at α = − 0.5. While we consistently use magnitudes in the r band, Faber et al. (2007) determine M^{∗} in the B band. Therefore we the estimate restframe B − r colour from the tables provided in Fukugita et al. (1995), finding B − r = 1.32 for ellipticals. This conversion from B to r takes into account that Faber et al. (2007) give B band magnitudes in the Vegabased system, whereas this work uses AB magnitudes throughout. Furthermore, we have assumed r ≈ r′, where r′ is the filter listed by Fukugita et al. (1995). This assumption should hold to good accuracy^{14} for typical colours of the galaxies in our samples, i.e. 0.2 ≲ r − i ≲ 0.6.
For earlytype galaxies, B − r shows little evolution between z = 0 and z ~ 1 (Bruzual & Charlot 2003), so we assume the restframe colour to be constant in this redshift range, which we check via the following procedure. Since the Sloan g filter covers a similar wavelength range to the B band (although the peaks of the transmission curves differ, see Fukugita et al. 1995 for details), we use the evolution of g − r as determined from the Wake et al. (2006) templates as an approximation for the redshift dependence of B − r. We find a shift of 0.15 mag from z = 0 to z = 1, which has significantly less effect on our results than employing different observational results for luminosity functions, see Fig. C.1 and the corresponding discussion below. Finally, we correct for the fact that Faber et al. (2007) have computed absolute magnitudes assuming a Hubble parameter h = 0.7 while we give absolute magnitudes in terms of h = 1.
Fig. C.2
Comoving volume number density of red galaxies n_{V,r} and mean luminosity ⟨L⟩/L_{0} of red galaxies as a function of limiting magnitude r_{lim} and redshift z. Left panel: comoving volume number density of red galaxies n_{V,r} in units of 10^{4} Mpc^{3}. Contour values range between 10^{3} in the upper left and 50 at the bottom. In the upper left corner of the panel n_{V,r} ≈ 0. Right panel: mean r band luminosity. Contour values range between 0.16 in the lower part to 5 in the upper left corner. In both panels the red dashed line marks the values for r_{lim} = 25, the limiting magnitude we employ in our calculations. Decades in contour values are indicated by the black solid lines. 

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With all of these caveats, the limiting absolute B band magnitude at redshift z for a given apparent magnitude limit in the r band is given by (C.1)where k_{r,red}(z) is the kcorrection of red galaxies for the r band (Wake et al. 2006). In line with our convention for absolute magnitudes, the luminosity distance D_{L} is computed with h = 1. If absolute magnitudes are given for other values of the Hubble parameter, like e.g. in Faber et al. (2007), we convert these accordingly. The limiting absolute magnitude from (C.1) can then be transformed into the minimum luminosity entering (27) and (28), (C.2)where M_{0}(z) denotes the restframe absolute magnitude − 22, evolutioncorrected to redshift z using the redshift dependence of M^{∗} from Faber et al. (2007), which is given by − 1.2z. Note that this dependence accounts for the redshift evolution in the B band, but since B − r is nearly constant as a function z, we can also apply the correction (to good approximation) to r band magnitudes. Denoting the luminosity corresponding to M^{∗} by L^{∗}, we obtain for (28) the expression (C.3)where the incomplete Gamma function was introduced. Analogously, we arrive at (C.4)for the comoving volume density of red galaxies entering (27).
In addition to the luminosity functions from Faber et al. (2007), we also consider fitted Schechter parameters presented in Giallongo et al. (2005), as well as the sets of luminosity functions published by Wolf et al. (2003), Willmer et al. (2006), and Brown et al. (2007). We determine fit functions to the redshift dependence of both M^{∗} and φ^{∗} for the latter three works because we have to extrapolate beyond the range of redshift analysed therein. We use linear functions for M^{∗} and various functional forms with two to three fit parameters for φ^{∗}, but note that since the fits rely on only five to six data points, the extrapolation has considerable uncertainty. All five references give B band luminosity functions, but the magnitude system and the convention for h vary, as well as the redshift ranges covered and the definition of red galaxies.
In Fig. C.1 the red galaxy fraction f_{r} and the mean luminosity ⟨L⟩/L_{0} for r_{lim} = 25 are plotted as a function of redshift, making use of the different luminosity functions. We find fair agreement between the results based on Faber et al. (2007) and Brown et al. (2007), while the mean luminosities derived from Willmer et al. (2006) already deviate considerably at high z although Faber et al. (2007) and Willmer et al. (2006) partly use the same data. The Wolf et al. (2003) luminosity functions produce significantly lower f_{r} and higher ⟨L⟩ at low redshifts which is caused by the very different value for the faint end slope, α = + 0.52. We note that one of the three fields chosen by Wolf et al. (2003) contained two massive galaxy clusters, so that the largescale structure in this field could strongly influence the luminosity function in particular of earlytype galaxies. However, small red galaxy fractions can be compensated by higher luminosities in (19), so that even the Wolf et al. (2003) luminosity functions may yield intrinsic alignment signals of similar magnitude to the results of e.g. Faber et al. (2007).
Mean luminosity of red galaxies and comoving volume number density of red galaxies as a function of redshift and limiting r band magnitude r_{lim}.
Applying the formalism to luminosity functions from Giallongo et al. (2005), we obtain very high f_{r} at low redshift, which is clearly inconsistent with the other observations. The red galaxy sample used for the fits of Giallongo et al. (2005) is very small and contains only galaxies with z > 0.4. While the resulting fit function captures the pronounced decrease in number density for high redshifts that Giallongo et al. (2005) observe, it can obviously not be used at z ≲ 0.4. In conclusion, we find that the sets of luminosity functions by Faber et al. (2007) who jointly analyse galaxy samples from four different surveys produce reasonable red galaxy fractions and luminosities although the uncertainty in the values of f_{r} and ⟨L⟩ at any given redshift is still large.
In Fig. C.2, the comoving volume density of red galaxies n_{V,r} and the mean luminosity of red galaxies in terms of L_{0} are plotted as a function of both r_{lim} and redshift, using the default set of luminosity functions from Faber et al. (2007). For a fixed magnitude limit, ⟨L⟩ increases with redshift while n_{V,r} decreases strongly at high redshifts. Both tendencies are more pronounced if the apparent magnitude limit is brighter. At low redshift, e.g. for z ≲ 0.3 at r_{lim} = 25, a large number of faint blue galaxies are above the magnitude limit and cause f_{r} to diminish for z approaching zero (see Fig. C.1) although n_{V,r} continues to increase. This behaviour might change when explicitly taking into account the size cuts inherent to weak lensing surveys, but in any case, galaxies at these low redshifts constitute only a small fraction of the total survey volume and are expected to have a low luminosity and hence low intrinsic alignment signal on average, so they are unlikely to affect our results severely.
The data shown in Fig. C.2, with the corresponding numbers collected in Table C.1, can be used in combination with the intrinsic alignment model fits presented in Sect. 5.5 to predict the effect of intrinsic alignments on cosmic shear surveys. The mean luminosity as a function of redshift for a given magnitude limit r_{lim} can be inserted into the luminosity term in (19), which constitutes a fair approximation as long as values of β close to unity (which includes our bestfit value β = 1.13) are probed. Together with an overall redshift distribution p_{tot}(z) for the cosmic shear survey under consideration, n_{V,r} can be read off and used with (27) to compute the fraction of red galaxies as a function of redshift. Again, we emphasise the limitations of this approach which relies on a substantial amount of extrapolation in luminosity, especially for fainter limiting magnitudes, and which is subject to the large intrinsic uncertainty in the different sets of luminosity functions.
© ESO, 2011
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