Analytic halo approach to the bispectrum of galaxies in redshift space

We present an analytic formula for the galaxy bispectrum in redshift space on the basis of the halo approach description with the halo occupation distribution of central galaxies and satellite galaxies. This work is an extension of a previous work on the galaxy power spectrum, which illuminated the significant contribution of satellite galaxies to the higher multipole spectrum through the nonlinear redshift space distortions of their random motions. Behaviors of the multipoles of the bispectrum are compared with results of numerical simulations assuming a halo occupation distribution of the low-redshift (LOWZ) sample of the Sloan Digital Sky Survey (SDSS) III baryon oscillation spectroscopic survey (BOSS) survey. Also presented are analytic approximate formulas for the multipoles of the bispectrum, which is useful to understanding their characteristic properties. We demonstrate that the Fingers of God effect is quite important for the higher multipoles of the bispectrum in redshift space, depending on the halo occupation distribution parameters.

Figure 1

Case analysis of galaxy distribution in halos. (A) Case where one galaxy in a halo is a central galaxy in one halo. (B) Case where two galaxies in a halo are a pair of a central galaxy and a satellite galaxy in one halo. (C) Case where three galaxies in a halo are a combination of a central galaxy and two satellite galaxies in one halo.

Figure 2

Contributions of a (1) one-halo term, (2) two-halo term, and (3) three-halo term to the bispectrum. (1) A one-halo term represents the contribution from three galaxies in one halo. (2) A two-halo term represents the contribution from the combination of two halos with one galaxy and two galaxies in each halo. (3) A three-halo term represents the contribution from three halos with each galaxy.

Figure 3

Definition of variables for the bispectrum (see also Table 2).

Figure 4

Left figure shows $Q^{0,0}(\theta )$ with the LRG sample by fixing (a) $k_1=0.05$, (b) $k_1=0.1$, (c) $k_1=0.2$, and (d) $k_1=0.5$ in the unit $h/\mathrm{Mpc}$, and $k_2/k_1=2$. In each panel the (green) dotted curve is the one-halo term contribution, the (blue) short-dashed curve is the two-halo term contribution, the (red) long-dashed curve is the three-halo term contribution, and the (black) solid curve is the total combination. The right figure shows the same as the left figure but for the LOWZ sample.

Figure 5

Same as Fig. 4, but for $Q^{2,0}(\theta )$.

Figure 6

Same as Fig. 4, but for $Q^{4,0}(\theta )$.

Figure 7

Comparison of $Q^{\ell ,0}(\theta )$ between our theoretical model and numerical simulations adopting the HOD of the LOWZ sample. The top panels, the middle panels, and the bottom panels show $Q^{0,0}$, $Q^{2,0}$ and $Q^{4,0}$ as functions of $\theta$. The wave number $k_1/[h{\mathrm{Mpc}}^{-1}]$ is fixed as 0.05, 0.1, 0.2 and 0.3, from left to right, while the ratio of $k_1$ to $k_2$ is fixed equally as $k_2/k_1=2$. In each panel, the solid curve is our theoretical model, while the dash–dotted curve is the same except for setting ${\sigma }_{v,\mathrm{off}}=0$. The meaning of ${\sigma }_{v,\mathrm{off}}=0$ is that the random velocity of the satellite galaxy is neglected, thus the FoG effect is neglected. The data with error bars are the results from mock catalogs from numerical simulations. The thick (green) data points and thin (red) data points respectively correspond to the cases in the presence and absence of ${\sigma }_{v,\mathrm{off}}$, corresponding to the solid curve and the dash–dotted curve. All values in the bottom left two panels are multiplied by a factor of 5.