Large scale bias and stochasticity of halos and dark matter

TL;DR

This work tests the standard assumptions that halos and galaxies trace dark matter with a constant bias and that dark matter follows linear growth. Using high dynamic range N-body simulations, the authors quantify significant stochasticity between the initial and final density fields and between halos and matter, even on scales where the nonlinear power spectrum is close to linear. They derive a new, accurate halo-bias–mass relation expressed in units of the nonlinear mass M_nl, with bias remaining roughly constant for M < M_nl (around 0.68) and increasing for larger masses, and provide a cosmology-dependent correction for broader models. The findings highlight substantial limitations for bias determination from cross-correlations and weak lensing and offer a practical bias model to improve amplitude measurements of the matter power spectrum in upcoming surveys.

Abstract

On large scales galaxies and their halos are usually assumed to trace the dark matter with a constant bias and dark matter is assumed to trace the linear density field. We test these assumption using several large N-body simulations with 384^3-1024^3 particles and box sizes between 100-1000h/Mpc, which can both resolve the small galactic size halos and sample the large scale fluctuations. We explore the average halo bias relation as a function of halo mass and show that existing fitting formulae overestimate the halo bias by up to 20% in the regime just below the nonlinear mass. We propose a new expression that fits our simulations well. We find that the halo bias is nearly constant, b~0.65-0.7, for masses below one tenth of the nonlinear mass. We explore next the relation between the initial and final dark matter in individual Fourier modes and show that there are significant fluctuations in their ratio, ranging from 10% rms at k~0.03h/Mpc to 50% rms at k~0.1h/Mpc. We argue that these large fluctuations are caused by perturbative effects beyond the linear theory, which are dominated by long wavelength modes with large random fluctuations. Similar or larger fluctuations exist between halos and dark matter and between halos of different mass. While these fluctuations are small compared to the sampling variance, they are significant for attempts to determine the bias by relating directly the maps of galaxies and dark matter or the maps of different galaxy populations, which would otherwise be immune to sampling variance.

Figures (8)

• Figure 1: Ratio of final to initial density perturbations as a function of wavemode amplitude $k$. There is a large scatter between the two quantities even on large scales, where linear theory is usually assumed to be valid.
• Figure 2: Relative rms fluctuation $\sigma_b/b=[2(1-r)]^{1/2}$ between the final and the initial density field for HOT3 (solid). Also shown is the power spectrum ratio between the two fields (dashed). Results for HOT2 are similar, but show a somewhat larger supression of nonlinear versus linear power spectrum for $k<0.1h/Mpc$.
• Figure 3: Relative rms fluctuation $\sigma_b/b=\sqrt{2(1-r)}$ between the halo density field and the initial (solid) or final (dashed) matter density field. Lower curves have been obtained by applying the shot noise subtraction from the halo power spectrum. Average masses are $4.5\times 10^{11}h^{-1}M_{\sun}$ (a), $10^{12}h^{-1}M_{\sun}$ (b), $2\times 10^{12}h^{-1}M_{\sun}$ (c) and $10^{13}h^{-1}M_{\sun}$ (d). The corresponding halo densities are $7\times 10^{-3}h^3/Mpc^3$, $2.7\times 10^{-3}h^3/Mpc^3$, $1.5\times 10^{-3}h^3/Mpc^3$ and $3.5\times 10^{-4}h^3/Mpc^3$.
• Figure 4: Ratio of 2-d projected halo density perturbations ($M=10^{11}h^{-1}M_{\sun}$) to the initial density field as a function of wavemode amplitude $k$. The projections are along each of the three axes (288$h^{-1}$Mpc for HOT2 simulation used here). The scatter is larger than the corresponding 3-d case in figure \ref{['fig3']}.
• Figure 5: Ratio of halo density perturbations $\delta_{h1}/\delta_{h2}$ as a function of wavemode amplitude $k$. The halos are of mass $10^{11}h^{-1}M_{\sun}$ (h1) and $10^{12}h^{-1}M_{\sun}$ (h2).