5.3 Cosmological
redshift-space distortion
Consider a spherical object at high redshift. If the wrong
cosmology is assumed in interpreting the distance-redshift relation
along the line of sight and in the transverse direction, the sphere
will appear distorted. Alcock and Paczynski [2] pointed out that this
curvature effect could be used to estimate the cosmological
constant. Matsubara and Suto [54] and Ballinger,
Peacock, and Heavens [3
] developed a
theoretical framework to describe the geometrical distortion effect
(cosmological redshift distortion) in the two-point correlation
function and the power spectrum of distant objects, respectively.
Certain studies were less optimistic than others about the
possibility of measuring this Alcock-Paczynski effect. For example,
Ballinger, Peacock, and Heavens [3] argued that the
geometrical distortion could be confused with the dynamical
redshift distortions caused by peculiar velocities and
characterized by the linear theory parameter
. Matsubara and Szalay [55, 56] showed that the
typical SDSS and 2dF samples of normal galaxies at low redshift
(
) have sufficiently low signal-to-noise, but they are
too shallow to detect the Alcock-Paczynski effect. On the other
hand, the quasar SDSS and 2dFGRS surveys are at a useful redshift,
but they are too sparse. A more promising sample is the SDSS
Luminous Red Galaxies survey (out to redshift
) which turns out to be optimal in terms of both
depth and density.
While this analysis is promising, it remains to
be tested if non-linear clustering and complicated biasing (which
is quite plausible for red galaxies) would not ‘contaminate’ the
measurement of the equation of state. Even if the Alcock-Paczynski
test turns out to be less accurate than other cosmological tests
(e.g., CMB and SN Ia), the effect itself is an interesting and
important ingredient in analyzing the clustering pattern of
galaxies at high redshifts. We shall now present the formalism for
this effect.
Due to a general-relativistic effect through the
geometry of the Universe, the observable separations perpendicular and
parallel to the line-of-sight direction,
and
, are mapped differently to the
corresponding comoving separations in real space
and
:
with
being the angular diameter distance. The difference
between
and
generates an
apparent anisotropy in the clustering statistics, which should be
isotropic in the comoving space. Then the power spectrum in
cosmological redshift space
is related to
defined in the comoving
redshift space as
where the first factor comes from the Jacobian of the volume
element
, and
and
are the wavenumber perpendicular and
parallel to the line-of-sight direction.
Using Equation (131), Equation (156) reduces to
where
Figure 16 shows anisotropic
power spectra
. As specific examples, we consider SCDM, LCDM, and
OCDM models, which have
,
,
and
, respectively. Clearly the linear
theory predictions (
; top panels) are quite
different from the results of
-body simulations (bottom
panels), indicating the importance of the nonlinear velocity
effects (
computed according to [58]; middle panels).
Next we decompose the power spectrum into harmonics,
where
are the
-th order Legendre
polynomials. Similarly, the two-point correlation function is
decomposed as
using the direction cosine
between the separation
vector and the line-of-sight. The above multipole moments satisfy
the following relations:
with
being spherical Bessel functions.
Substituting
in Equation (159) yields
, and then
can be
computed from Equation (161).
A comparison of the monopoles and quadrupoles from
simulations and model predictions exhibits how the results are
sensitive to the cosmological parameters, which in turn may put
potentially useful constraints on
.
Figure 17 indicates the
feasibility, which interestingly results in a constraint fairly
orthogonal to that from the supernovae Ia Hubble diagram.