Evaluating two-point correlation functions from N-body simulation data

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 $N$ -body simulations in SCDM and $Λ$ -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 $Z$ -axis regardless of its $(X, Y )$ position. Second, they periodically duplicate the simulation box along the $Z$ -direction so that at a redshift $z$ , the position and velocity of those particles locating within an interval $χ (z) ± Δχ (z)$ are dumped, where $Δ χ(z)$ is determined by the output time-interval of the original $N$ -body simulation. Finally they extract five independent (non-overlapping) cone-shape samples with the angular radius of 1 degree (the field-of-view of $π$ degree $2$ ). In this manner, they have generated mock data samples on the light-cone continuously extending up to $z = 0.4$ (relevant for galaxy samples) and $z = 2.0$ (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 $x12$ of two objects located at $z1$ and $z2$ with an angular separation $𝜃12$ is given by

where $x ≡ D (z ) 1 c 1$ and $x ≡ D (z ) 2 c 2$ .

In redshift space, the observed redshift $zobs$ for each object differs from the “real” one $zreal$ due to the velocity distortion effect:

where $vpec$ is the line of sight relative peculiar velocity between the object and the observer in physical units. Then the comoving separation $s12$ of two objects in redshift space is computed as

where $s1 ≡ Dc(zobs,1)$ and $s2 ≡ Dc (zobs,2)$ .

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 $MB$ at $z$ (with the luminosity distance $dL (z)$ ), we applied the K-correction,

for the quasar energy spectrum $L ν ∝ ν− p$ (we use $p = 0.5$ ). In practice, we adopt the galaxy selection function $ϕ (< B ,z) gal lim$ with $B = 19 lim$ and $z = 0.01 min$ for the small box realizations, and the QSO selection function $ϕQSO (< Blim,z)$ with $Blim = 21$ and $zmin = 0.2$ 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 $Λ$ -CDM, respectively, taking account of the selection functions. It is clear that the simulation results and the predictions are in good agreement.

Figure 14: Mass two-point correlation functions on the light-cone for particles with redshift-dependent selection functions in the SCDM model, for $z < 0.4$ (upper panels) and $0.2 < z < 2.0$ (lower panels). Left panels: with selection function whose shape is the same as that of the B-band magnitude limit of 19 for galaxies (upper) and 21 for QSOs (lower); right panels: randomly selected $N ∼ 104$ particles from the particles in the results from the left panels. (Figure taken from [28

].)

Figure 15: Same as Figure 14

but for the $Λ$ -CDM model. (Figure taken from [28

].)