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Appendix A: Three-dimensional number density-intrinsic shear correlation function

In this appendix, we derive the three-dimensional number density-intrinsic shear (gI) correlation function, detail the inclusion of photometric redshift scatter into the formalism, and establish an approximate relation between the three-dimensional correlation function and the angular power spectrum.

A.1. Correlation function for exact redshifts

We define the three-dimensional 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 three-dimensional comoving separation vector x which has a line-of-sight 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 line-of-sight 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 galaxy-galaxy lensing signal which we consider is negative. Denoting Fourier variables by a tilde, one can construct in analogy to the matter density contrast a three-dimensional intrinsic convergence δ_I¹² 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 line-of-sight 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 three-dimensional 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 second-order Bessel function of the first kind was used. In this derivation it was implicitly assumed that the intrinsic shear field does not feature B-modes, 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 delta-distribution, 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 line-of-sight 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 cut-off 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 signal-to-noise 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 Three-dimensional gI correlation function as a function of comoving line-of-sight separation Π and comoving transverse separation rp 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 PδI in both cases. Redshift-space distortions have not been taken into account.

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Equation (A.7) yields the three-dimensional 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 cut-off 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 = 0-axis 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 redshift-space 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 three-dimensional 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, z₁ and z₂, and their angular separation θ. The sets of variables (z₁,z₂,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 z₁, z₂, 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 left-hand-side.

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 line-of-sight 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 z₂ − z₁ 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,  | z₂ − z₁ |  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 three-dimensional 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 line-of-sight integration. However, in practice the line-of-sight integral has to be truncated for reasons of a good signal-to-noise 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 cut-off Π_max in (A.6).

A.3. Relation to angular power spectra

We now derive a relation between the three-dimensional 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 = (z₁ + z₂)/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 {z₁,z₂} to {z_m,Δz ≡ z₂ − z₁}. 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 line-of-sight 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 delta-distribution 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 auto-correlation 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 line-of-sight integration (e.g. Simon 2007).

We have verified that the calculations of the three-dimensional 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. Galaxy-galaxy 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 re-derive 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 large-scale 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 C₁ 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 matter-dominated 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 cut-off 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 Fourier-transformed, 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 matter-dominated 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 re-derivations (Hirata & Seljak 2010; Bean, Laszlo; priv. comm.).

Hirata & Seljak (2004) inserted their Eq. (14) into (B.1) and then computed the three-dimensional intrinsic shear (II) power spectrum and the intrinsic shear-matter cross-power 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 right-hand 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 early-type 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 fr (top panel) and mean luminosity of red galaxies (bottom panel) as a function of redshift for a magnitude limit rlim = 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 faint-end 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 rest-frame 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 Vega-based 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 accuracy¹⁴ for typical colours of the galaxies in our samples, i.e. 0.2 ≲ r − i ≲ 0.6.

For early-type galaxies, B − r shows little evolution between z = 0 and z ~ 1 (Bruzual & Charlot 2003), so we assume the rest-frame 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 nV,r and mean luminosity ⟨L⟩/L0 of red galaxies as a function of limiting magnitude rlim and redshift z. Left panel: comoving volume number density of red galaxies nV,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 nV,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 rlim = 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 k-correction 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 M₀(z) denotes the rest-frame absolute magnitude  − 22, evolution-corrected 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⟩/L₀ 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 large-scale structure in this field could strongly influence the luminosity function in particular of early-type 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).

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 L₀ 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 best-fit 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