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 [28Jump To The Next Citation Point]. 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 degree2). 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]:

DD (x) − 2DR (x) + RR (x ) ξ(x) = ---------------------------. (149 ) RR (x )

The comoving separation x12 of two objects located at z1 and z2 with an angular separation 𝜃12 is given by

∘ --------∘ --------- 2 2 2 2 2 2 2 2 x12 = x1 + x2 − Kx 1x 2(1 + cos 𝜃12) − 2x1x2 1 − Kx 1 1 − Kx 2 cos𝜃12, (150 )
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:

zobs = zreal + (1 + zreal)vpec, (151 )
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
∘ --------∘ -------- s2 = s2+ s2− Ks2 s2(1 + cos2𝜃 ) − 2s s 1 − Ks2 1 − Ks2 cos𝜃 , (152 ) 12 1 2 1 2 12 1 2 1 2 12
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,

B = MB + 5log(dL(z)∕10 pc ) − 2.5(1 − p)log(1 + z), (153 )
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 14View Image and 15View Image 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.

View Image

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 [28Jump To The Next Citation Point].)
View Image

Figure 15: Same as Figure 14View Image but for the Λ-CDM model. (Figure taken from [28Jump To The Next Citation Point].)

  Go to previous page Go up Go to next page