Beyond Linear Bias Expansions for AbacusSummit Halos at z = 8

Abstract

We study the non-Gaussianity of the large-scale clustering of high-redshift halos, seeking to assess which terms of standard bias expansions are needed to understand these highly biased populations. We find that the clustering can be well modeled with only linear and quadratic bias parameters while assuming a Gaussian underlying matter field. Our analysis focuses on AbacusSummit halos at redshift $z=8$. We work with halos of mass at least $1\times10^{11}h^{-1}M_\odot$ in boxes of side length $2h^{-1}$Gpc. Measurements of bias coefficients are made by fitting bias expansions to the halo power spectrum and bispectrum. Tidal bias is not detected with only a ~$0.1σ$ deviation from $0$, but we see a $17σ$ level detection for a bias term of the form $δ^2$. A bias term of the form $δ^3$ is weakly detected at the $1.3σ$ level. Nonlinear matter is also detected at a $1.3σ$ level. To test how bias evolves, we run one test at $z=5$. We use a mass threshold for halos that gives the same variance in the halo field as our $z=8$ sample. Bias is smaller at $z=5$ and a tidal bias is detected at the $1σ$ level. Bias coefficients at $z=5$ match a linear evolution of the $z=8$ bias coefficients to within $10\%$.

Figures (6)

• Figure 1: Left: Power spectrum data with standard deviation compared to the best fit. Our data is the average of the $25$ power spectrum realizations. The model is the average of our $25$ fits, and it shows no significant deviations from the data. Right: The bias functionals that appear in Eq. \ref{['eq:power_functionals']} in the order listed there. For the LIMD terms, higher-order operators peak at smaller scales and look more similar to a constant shot noise term.
• Figure 2: Left: The bispectrum functional corresponding to $b_1b_1b_2$. The bispectrum is plotted as a function of $k_2$ and $k_3$ with the sum $k_1+k_2+k_3$ being fixed. The squeezed limit is on the bottom right, while the equilateral limit is at the top. Middle: The bispectrum functional corresponding to $b_1b_1b_{K^2}$. Right: The bispectrum functional corresponding to $b_2b_2b_2$. All three functionals have visually distinct patterns, which suggests that we will be able to use them to fit for bias coefficients.
• Figure 3: Left: constraints on the bias of halos with mass at least $10^{11}h^{-1}M_\odot$ at redshift $z=8$. The power spectrum and bispectrum are used to generate separate constraints. The constraints from both are consistent and show different degeneracies, which motivates combining them. Right: constraints on the same halos when the power spectrum is combined with the bispectrum. This is our baseline result, and as expected we see $b_1<b_2$ with a highly suppressed tidal bias. The uncertainty of $b_3$ makes it difficult to compare it directly with $b_2$.
• Figure 4: Left: bias constraints of halos with different mass thresholds at redshift $z=8$. When a lower mass is used there are more halos and they can form in less extreme environments. This leads to a reduction in the mean and variance of bias values. Right: bias constraints of halos with mass at least $10^{11}h^{-1}M_\odot$ at redshift $z=8$. An additional bias parameter $b_{F2}$ (denoted in the figure simply as $F_2$) is added to test the importance of matter nonlinearities. All bias values are consistent, and nonlinear matter is only weakly detected at a $\sim$$1\sigma$ level.
• Figure 5: Left: bias constraints of halos at different redshifts. Mass thresholds are selected so that the halo fields have the same variance. Since the matter field overdensities have grown from $z=8$ to $z=5$, bias values are reduced. Right: the same as the left but with the $z=5$ bias constraints linearly evolved to $z=8$ assuming matter domination. The high consistency suggests that we are likely measuring the same objects at the two different redshifts.