Bispectrum and Nonlinear Biasing of Galaxies: Perturbation Analysis, Numerical Simulation and SDSS Galaxy Clustering

TL;DR

This work demonstrates that nonlinear galaxy biasing, encapsulated by a local expansion in the mass density contrast, yields a robust correlation between the linear and quadratic bias parameters, $b_1$ and $b_2/b_1$, across analytic halo/peak models, N-body simulations, and SDSS galaxy data. By connecting the biased-field bispectrum through $Q_b = \frac{1}{b_1}[Q_m + \frac{b_2}{b_1}]$ and analyzing both real and redshift-space statistics, the authors show that $Q$ for equilateral configurations is largely insensitive to $b_1$, explaining observational hierarchies seen in SDSS clustering. The study validates this correlation with halo occupation distribution mocks and highlights the generic nature of nonlinear biasing in Gaussian initial conditions, even amid redshift-space distortions and survey geometry. The findings provide a coherent framework for interpreting higher-order galaxy clustering and for constraining biasing in cosmological analyses.

Abstract

We consider nonlinear biasing models of galaxies with particular attention to a correlation between linear and quadratic biasing coefficients, b_1 and b_2. We first derive perturbative expressions for b_1 and b_2 in halo and peak biasing models. Then we compute power spectra and bispectra of dark matter particles and halos using N-body simulation data and of volume-limited subsamples of Sloan Digital Sky Survey (SDSS) galaxies, and determine their b_1 and b_2. We find that the values of those coefficients at linear regimes (k<0.2h/Mpc) are fairly insensitive to the redshift-space distortion and the survey volume shape. The resulting normalized amplitudes of bispectra, Q, for equilateral triangles, are insensitive to the values of b_1 implying that b_2 indeed correlates with b_1. The present results explain the previous finding of Kayo et al. (2004) for the hierarchical relation of three-point correlation functions of SDSS galaxies. While the relations between b_1 and b_2 are quantitatively different for specific biasing models, their approximately similar correlations indicate a fairly generic outcome of the biasing due to the gravity in primordial Gaussian density fields.

Figures (8)

• Figure 1: biasing coefficients in the three perturbative biasing models. Thick and thin curves show $b_1$ and $b_2/b_1$ at $z=0$ (solid) and $z=2$ (dashed). We use spherical halo model ( left), ellipsoidal halo model ( center), and peak model ( right). The biasing coefficients are plotted against halo mass $M_{\rm halo}$ ( left and center), and against peak height $\nu$ ( right).
• Figure 2: Correlation of $b_1$ and $b_2/b_1$ from halo and peak biasing. Different line types are results of halo biasing of spherical collapse(solid), ellipsoidal collapse(dashed), and peak biasing (dotted), evaluated at redshifts 2 (thin) and 0 (thick). Open (filled) symbols represent the mass-averaged values, $B_1$ and $B_2/B_1$, defined in equation (\ref{['eq:b_ave']}), assuming spherical (ellipsoidal) halo model. The mass ranges correspond to L (square), and S (triangle) defined in subsection \ref{['sec:simulation_SDSS']}.
• Figure 3: $Q$ for biased fields as a function of $b_1$ in real space. We estimate $Q$ using equation (\ref{['eq:Qb']}) on the basis of perturbative predictions for peaks (dotted), ellipsoidal halos (dashed), and spherical halos (solid). We assume three different values for $Q_m$: $Q_m=1.58$, $Q_m=0.83$, and $Q_m=4/7$ from top to bottom. The first two values are computed from N-body results at $k=0.4h{\rm Mpc}^{-1}$, and $k=0.18h{\rm Mpc}^{-1}$ (see section \ref{['sec:simulation']} below), and the last value corresponds to perturbation theory [eqs.(\ref{['eq:Qtreelevel']}) and (\ref{['eq:F2']})], where equilateral triangles ($k_1=k_2=k_3=k$) are assumed. left: $z=0$, right: $z=2$.
• Figure 4: Same as figure \ref{['fig:b1_Qb']}, but for redshift space. We use equation (\ref{['eq:Qred']}) for equilateral triangles instead of equation (\ref{['eq:Qb']}). In this time, $Q_b$ is calculated directly (not in terms of $Q_m$), so we plot only one set corresponding to $Q_m=4/7$ in figure \ref{['fig:b1_Qb']}.
• Figure 5: Inverse of the linear biasing parameters of simulated halos (Table \ref{['tab:halo_catalog']}) and SDSS galaxies (Table \ref{['tab:galaxy_catalog']}). Crosses and filled circles for simulated halos (SDSS galaxies) correspond to S and L samples (blue and red), respectively. Dashed and solid lines indicate results based on the direct estimation of $P(k)$, while symbols in all panels use $P^{\rm PD,r}(k)$ or $P^{\rm PD,s}(k)$, equation (\ref{['eq:pks0']}). The quoted error bars for simulated halos are computed from three different realizations. We simply use the error bars for wedge subsamples just for reference in the case of SDSS galaxies.