### 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.