5.1 Cosmological light-cone
effect on the two-point correlation functions
Observing a distant patch of the Universe is equivalent to
observing the past. Due to the finite light velocity, a
line-of-sight direction of a redshift survey is along the time, as
well as spatial, coordinate axis. Therefore the entire sample does
not consist of objects on a constant-time hypersurface, but rather
on a light-cone, i.e., a null hypersurface defined by observers at
. This implies that many properties of the objects
change across the depth of the survey volume, including the mean
density, the amplitude of spatial clustering of dark matter, the
bias of luminous objects with respect to mass, and the intrinsic
evolution of the absolute magnitude and spectral energy
distribution. These aspects should be properly taken into account
in order to extract cosmological information from observed samples
of redshift surveys.
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
introduce
We 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:
where
Adopting those approximations, the
direction-averaged correlation functions on the light-cone are
finally computed as
where
and
denote the redshift range of the
survey, and
Throughout 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 by
The 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 as