In order to predict quantitatively the two-point statistics of objects on the light-cone, one must take account of

- nonlinear gravitational evolution,
- linear redshift-space distortion,
- nonlinear redshift-space distortion,
- weighted averaging over the light-cone,
- cosmological redshift-space distortion due to the geometry of the Universe, and
- object-dependent clustering bias.

The Effect 5 comes from our ignorance of the correct cosmological parameters, and Effect 6 is rather sensitive to the objects which one has in mind. Thus the latter two effects will be discussed in the next Section 5.2.

Nonlinear gravitational evolution of mass density fluctuations is now well understood, at least for two-point statistics. In practice, we adopt an accurate fitting formula [67] for the nonlinear power spectrum in terms of its linear counterpart. If one assumes a scale-independent deterministic linear bias, furthermore, the power spectrum distorted by the peculiar velocity field is known to be well approximated by the following expression:

where and are the comoving wavenumber perpendicular and parallel to the line-of-sight of an observer, and is the mass power spectrum in real space. The second factor on the r.h.s. comes from the linear redshift-space distortion [38], and the last factor is a phenomenological correction for the non-linear velocity effect [67]. In the above, we introduceWe assume that the pair-wise velocity distribution in real space is approximated by

with being the 1-dimensional pair-wise peculiar velocity dispersion. Then the finger-of-God effect is modeled by the damping function : where is the direction cosine in -space, and the dimensionless wavenumber is related to the peculiar velocity dispersion in the physical velocity units:Since we are mainly interested in the scales around , we adopt the following fitting formula throughout the analysis below which better approximates the small-scale dispersions in physical units:

Integrating Equation (131) over , one obtains the direction-averaged power spectrum in redshift space:

whereAdopting those approximations, the direction-averaged correlation functions on the light-cone are finally computed as

where and denote the redshift range of the survey, andThroughout the present analysis, we assume a standard Robertson–Walker metric of the form

where is determined by the sign of the curvature as where the present scale factor is normalized as unity, and the spatial curvature is given as (see Equation (13)). The radial comoving distance is computed byThe comoving angular diameter distance at redshift is equivalent to , and, in the case of , is explicitly given by Mattig’s formula:

Then , the comoving volume element per unit solid angle, is explicitly given ashttp://www.livingreviews.org/lrr-2004-8 |
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