Using Equation (157), the two-point
correlation function in the cosmological redshift space,
, is computed as
Since and
are defined in redshift space, the
proper weight should be
Note that and
, defined in
and
, are related to their comoving
counterparts at
through Equations (158
) and (154
), while those in
and
are
not specifically related to any comoving wavenumber and separation.
Rather, they correspond to the quantities averaged over the range
of
satisfying the observable conditions
and
.
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 18 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
and QSOs in
, 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]:
|
Consider first the results for the galaxy sample
(upper panels). On linear scales (),
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 (131
)). 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,
becomes larger than its real-space counterpart in linear theory. In
the SCDM model, however, this is opposite and
becomes smaller.
The power spectra at (dash-dotted
lines) are smaller than those at
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
(see Equation (158
)). If we plot the
power spectra at
taking into account the
geometrical distortion,
in the dotted lines becomes significantly larger than
. Therefore,
should take a value between those of
and
. This explains the qualitative features shown in the
upper panels of Figure 18
. 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.
|
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
%. 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.