Stochastic Biasing and Weakly Non-linear Evolution of Power Spectrum

TL;DR

The paper investigates how galaxy biasing, which can be stochastic and non-linear, evolves under gravitational dynamics using a weakly non-linear perturbation theory to one-loop order. It derives one-loop power spectra for the total mass, galaxies, and their cross-spectrum, and tracks the time evolution of the biasing parameter $b_k$ and correlation coefficient $r_k$ starting from local initial biasing with non-Gaussianity parameters $h_1$ and $h_2$. Key findings show that initial skewness can shift large-scale bias in the deterministic case, but stochastic initial conditions (low $r_0$) suppress this effect; on quasi-linear scales gravity induces scale-dependent bias with one-loop corrections that depend on $b_0 r_0$, producing either enhanced or reduced bias and affecting the correlation, in qualitative agreement with halo clustering seen in N-body simulations. The work provides a framework to interpret high-redshift galaxy clustering by linking initial stochasticity to later bias evolution and highlights the role of non-linear biasing and gravity in shaping power-spectrum measurements, while noting limitations due to neglected galaxy formation and redshift-space effects.

Abstract

Distribution of galaxies may be a biased tracer of the dark matter distribution and the relation between the galaxies and the total mass may be stochastic, non-linear and time-dependent. Since many observations of galaxy clustering will be done at high redshift, the time evolution of non-linear stochastic biasing would play a crucial role for the data analysis of the future sky surveys. In this paper, we develop the weakly non-linear analysis and attempt to clarify the non-linear feature of the stochastic biasing. We compute the one-loop correction of the power spectrum for the total mass, the galaxies and their cross correlation. Assuming the local functional form for the initial galaxy distribution, we investigate the time evolution of the biasing parameter and the correlation coefficient. On large scales, we first find that the time evolution of the biasing parameter could deviate from the linear prediction in presence of the initial skewness. However, the deviation can be reduced when the initial stochasticity exists. Next, we focus on the quasi-linear scales, where the non-linear growth of the total mass becomes important. It is recognized that the scale-dependence of the biasing dynamically appears and the initial stochasticity could affect the time evolution of the scale-dependence. The result is compared with the recent N-body simulation that the scale-dependence of the halo biasing can appear on relatively large scales and the biasing parameter takes the lower value on smaller scales. Qualitatively, our weakly non-linear results can explain this trend if the halo-mass biasing relation has the large scatter at high redshift.

Figures (7)

• Figure 1: The Evolved results of power spectra at $z=0$. The thin-dashed line shows the linear power spectrum of the total mass $P^{(11)}_m(k)$ evolved from $z=5$. The initial spectrum is given by the CDM spectrum (\ref{['BBKS-PK']}) with the appropriate model parameters (see text). The thin-solid, thick-solid and thick-dashed lines are the weakly non-linear power spectrum of the total mass, galaxies and the cross correlation between them, respectively. The initial conditions at $z=5$ are specified as $b_0=3.0$, $r_0=0.8$ and $h_1=h_2=0$.
• Figure 2: The snapshots of the evolved biasing parameter( upper panel) and the correlation coefficient( lower panel) are depicted as the function of the Fourier mode $k$. The initial condition are the same as in Figure 1. Within the interval $0.001<k<0.1~h~$ Mpc$^{-1}$, the biasing parameter and the correlation coefficient becomes almost independent of scales.
• Figure 3a: Time evolution of the biasing parameter $b_k$ ( upper panel) and the correlation coefficient $r_k$ ( lower panel) as the function of the redshift $z$, fixing the Fourier mode $k=2\pi/100~h~$ Mpc$^{-1}$. By adjusting the model parameters $b_0$ and $r_0$, the initial conditions at $z=5$ are specified as $b_k=3.0$, $r_k=0.8$, i.e, almost deterministic biasing case. The initial non-Gaussianity are respectively chosen as $h_1=h_2=0.0$( solid line), $h_1=3.0, h_2=0.0$( long-dashed line), $h_1=-3.0, h_2=0.0$( short-dashed line) and $h_1=0.0, h_2=3.0$( dotted line). The results are extrapolated to the present time $z=0$. For comparison, we also plot the linear result given by (\ref{['linear-b-r']}) with the same initial condition $b_0=3.0$, $r_0=0.8$ ( thin-solid line).
• Figure 3b: The same figure as in Figure 3a, but with the different initial conditions $b_k=3.0$, $r_k=0.2$ ( stochastic biasing case).
• Figure 4a: The one loop contribution of the biasing parameter $b_k^{loop}$ and the correlation coefficient $r_k^{loop}$ as the function of the Fourier mode $k$. The initial conditions given at $z=5$ are specified as $b_k=3.0$ and $r_k=0.8$. For simplicity, the initial non-Gaussianity are set by $h_1=h_2=0$. Because of the cutoff $k_c$, the initial parameters $b_k^{loop}$ and $r_k^{loop}$ slightly lose their power at high frequency part. However, it is obvious that the spatial dependence of the parameters $b_k^{(loop)}$ and $r_k^{(loop)}$ dynamically appears. On smaller scales, the spatial variation becomes larger. The typical behaviors are classified as $b_0r_0>1$ (Figure 4a) and $b_0r_0<1$ (Figure 4b).