5.2 Evaluating two-point
correlation functions from N-body
simulation data
The theoretical modeling described above was tested against
simulation results by Hamana, Colombi, and Suto [28
]. Using cosmological
-body simulations in SCDM and
-CDM models, they generated light-cone samples as
follows: First, they adopt a distance observer approximation and
assume that the line-of-sight direction is parallel to the
-axis regardless of its
position.
Second, they periodically duplicate the simulation box along the
-direction so that at a redshift
,
the position and velocity of those particles locating within an
interval
are dumped, where
is determined by the output time-interval of the
original
-body simulation. Finally they extract five
independent (non-overlapping) cone-shape samples with the angular
radius of 1 degree (the field-of-view of
degree
). In this manner, they have generated mock data
samples on the light-cone continuously extending up to
(relevant for galaxy samples) and
(relevant for QSO samples) from the small and large
boxes, respectively.
The two-point correlation function is estimated
by the conventional pair-count adopting the following
estimator [43]:
The comoving separation
of two objects located at
and
with an angular separation
is given by
where
and
.
In redshift space, the observed redshift
for each object differs from the “real” one
due to the velocity distortion effect:
where
is the line of sight relative peculiar velocity
between the object and the observer in physical units. Then the comoving separation
of two objects in redshift space is computed as
where
and
.
In properly predicting the power spectra on the
light-cone, the selection function should be specified. For
galaxies, we adopt a B-band luminosity function of the APM galaxies
fitted to the Schechter function [44]. For
quasars, we adopt the B-band luminosity function from the 2dF QSO
survey data [7]. To compute the
B-band apparent magnitude from a quasar of absolute magnitude
at
(with the luminosity distance
), we applied the K-correction,
for the quasar energy spectrum
(we use
). In practice, we adopt the galaxy selection
function
with
and
for the small box
realizations, and the QSO selection function
with
and
for the large box realizations. We do not introduce
the spatial biasing between selected particles and the underlying
dark matter.
Figures 14 and 15 show the two-point
correlation functions in SCDM and
-CDM, respectively,
taking account of the selection functions. It is clear that the
simulation results and the predictions are in good agreement.