5.4 Two-point clustering statistics on a light-cone in cosmological redshift space

In order to explore the relation between the two-point statistics on a constant-time hypersurface in real space and that on a light-cone hypersurface in cosmological redshift space, we simply consider the case of the deterministic, linear, and scale-independent bias:
δ(x,z) = b(z) δm(x, z). (163 )
In what follows, we explicitly use the subscript ‘mass’ to indicate the quantities related to the mass density field, while those without the subscript correspond to objects satisfying Equation (163View Equation).

Using Equation (157View Equation), the two-point correlation function in the cosmological redshift space, ξ(CRD )(x ,x ;z) s⊥ s∥, is computed as

1 ∫ ξ(CRD )(xs;z ) = ------ P (CRD)(ks;z)exp (− iks ⋅ xs)d3ks (164 ) (2π)3∫ --1--- (S) 3 = (2π)3 P (k;z )exp(− ik ⋅ x)d k (165 ) = ξ (S)(c⊥xs⊥,c∥xs∥;z), (166 )
where (S) ξ (x⊥, x∥;z) is the redshift-space correlation function defined through Equation (131View Equation).

Since P (CRD)(ks;z) l and ξ(CRD )(xs;z) l are defined in redshift space, the proper weight should be

3 (CRD) CRD 2 3 com 2 2 d s [ϕ(z)n0 (z)] = d x [ϕ (z )n 0 (z)]c⊥ (z )c∥(z), (167 )
where nCRD (z) 0 and ncom(z) 0 denote the number densities of the objects in cosmological redshift space and comoving space, respectively, and ϕ(z) is the selection function determined by the observational target selection and the luminosity function of the objects. Then, the final expressions [84Jump To The Next Citation Point] reduce to
∫ zmax dVc- com 2 2 (CRD) dz dz [ϕ (z)n0 (z)]c⊥ (z)c∥(z)P l (ks;z) P (lLC,CRD )(ks) = -zmin--∫-zmax-------------------------------------, (168 ) dz dVc[ϕ(z)ncom(z)]2c⊥(z)2c∥(z ) zmin dz 0 ∫ z max dVc- com 2 2 CRD (LC,CRD ) z dz dz [ϕ (z)n0 (z)]c⊥ (z)c∥(z)ξl (xs;z) ξl (xs) = --min-∫-zmax------------------------------------, (169 ) dz dVc[ϕ(z)nc0om(z)]2c⊥(z)2c∥(z) zmin dz
where zmin and zmax denote the redshift range of the survey, 2 dVc∕dz = dC(z)∕H (z) is the comoving volume element per unit solid angle.

Note that ks and xs, defined in (CRD ) Pl (ks;z ) and ξClRD(xs;z), are related to their comoving counterparts at z through Equations (158View Equation) and (154View Equation), while those in P(LC,CRD)(ks) l and ξ(LC,CRD )(xs) l are not specifically related to any comoving wavenumber and separation. Rather, they correspond to the quantities averaged over the range of z satisfying the observable conditions √ ------------ xs = (c∕H0 ) δz2 + z2δ𝜃2 and k = 2π∕x s s.

Let us show specific examples of the two-point clustering statistics on a light-cone in cosmological redshift space. We consider SCDM and LCDM models, and take into account the selection functions relevant to the upcoming SDSS spectroscopic samples of galaxies and quasars by adopting the B-band limiting magnitudes of 19 and 20, respectively.

Figure 18View Image compares the predictions for the angle-averaged (monopole) power spectra under various approximations. The upper and lower panels adopt the selection functions appropriate for galaxies in 0 < z < zmax = 0.2 and QSOs in 0 < z < zmax = 5, respectively. The left and right panels present the results in SCDM and LCDM models. For simplicity we adopt a scale-independent linear bias model [23]:

--1-- b(z) = 1 + D (z)[b(k, z = 0) − 1 ], (170 )
with b(k,z = 0) = 1 and 1.5 for galaxies and quasars, respectively.
View Image

Figure 18: Light-cone and cosmological redshift-space distortion effects on angle-averaged power spectra. (Figure taken from [84Jump To The Next Citation Point].)

The upper and lower panels correspond to magnitude-limited samples of galaxies (B < 19 in 0 < z < zmax = 0.2; no bias model) and QSOs (B < 20 in 0 < z < zmax = 5; Fry’s linear bias model), respectively. We present the results normalized by the real-space power spectrum in linear theory P (R,lin)(k;z) [4], and P (S)(k;z = 0) 0, P (S)(k;z = zmax ) 0, P (CRD )(ks;z = zmax) 0, and P(LC,CRD)(ks) 0 are computed using the nonlinear power spectrum [67].

Consider first the results for the galaxy sample (upper panels). On linear scales (k < 0.1h Mpc − 1), P (0S)(k;z = 0) plotted in dashed lines is enhanced relative to that in real space, mainly due to a linear redshift-space distortion (the Kaiser factor in Equation (131View Equation)). For nonlinear scales, the nonlinear gravitational evolution increases the power spectrum in real space, while the finger-of-God effect suppresses that in redshift space. Thus, the net result is sensitive to the shape and the amplitude of the fluctuation spectrum, itself; in the LCDM model that we adopted, the nonlinear gravitational growth in real space is stronger than the suppression due to the finger-of-God effect. Thus, P (S)(k;z = 0) 0 becomes larger than its real-space counterpart in linear theory. In the SCDM model, however, this is opposite and (S) P0 (k;z = 0) becomes smaller.

The power spectra at z = 0.2 (dash-dotted lines) are smaller than those at z = 0 by the corresponding growth factor of the fluctuations, and one might expect that the amplitude of the power spectra on the light-cone (solid lines) would be in-between the two. While this is correct, if we use the comoving wavenumber, the actual observation on the light-cone in the cosmological redshift space should be expressed in terms of ks (see Equation (158View Equation)). If we plot the power spectra at z = 0.2 taking into account the geometrical distortion, P(CRD )(ks;z = 0.2) 0 in the dotted lines becomes significantly larger than (S) P 0 (k;z = 0.2). Therefore, (LC,CRD) P0 (ks) should take a value between those of (CRD) (S) P 0 (ks;z = 0) = P0 (k;z = 0) and (CRD ) P0 (ks;z = 0.2). This explains the qualitative features shown in the upper panels of Figure 18View Image. As a result, both the cosmological redshift-space distortion and the light-cone effect substantially change the predicted shape and amplitude of the power spectra, even for the galaxy sample [60]. The results for the QSO sample can be basically understood in a similar manner, except that the evolution of the bias makes a significant difference, since the sample extends to much higher redshifts.

View Image

Figure 19: Same as Figure 18View Image on angle-averaged two-point correlation functions. (Figure taken from [84].)

Figure 19View Image shows the results for the angle-averaged (monopole) two-point correlation functions, exactly corresponding to those in Figure 18View Image. The results in this figure can also be understood by an analogy of those presented in Figure 18View Image at k ∼ 2π ∕x. Unlike the power spectra, however, two-point correlation functions are not positive definite. The funny features in Figure 19View Image on scales larger than 30h−1 Mpc (−1 100h Mpc) in SCDM (LCDM) originate from the fact that (R,lin) ξ (x,z = 0) becomes negative there.

In fact, since the resulting predictions are sensitive to the bias, which is unlikely to quantitatively be specified by theory, the present methodology will find two completely different applications. For relatively shallower catalogues, like galaxy samples, the evolution of bias is not supposed to be so strong. Thus, one may estimate the cosmological parameters from the observed degree of the redshift distortion, as has been conducted conventionally. Most importantly, we can correct for the systematics due to the light-cone and geometrical distortion effects, which affect the estimate of the parameters by ∼ 10%. Alternatively, for deeper catalogues like high-redshift quasar samples, one can extract information on the object-dependent bias only by correcting the observed data on the basis of our formulae.

In a sense, the former approach uses the light-cone and geometrical distortion effects as real cosmological signals, while the latter regards them as inevitable, but physically removable, noise. In both cases, the present methodology is important in properly interpreting the observations of the Universe at high redshifts.

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