The Large Scale Bias of Dark Matter Halos: Numerical Calibration and Model Tests

TL;DR

This work delivers a numerically calibrated, $Δ$-dependent large-scale halo bias function $b(ν,Δ)$ for spherical-overdensity halos, anchored to the Tinker et al. mass function. The authors introduce a flexible form for $b(ν)$ expressed in terms of peak height $ν ≡ δ_c/σ(M)$ and enforce mass-function normalization, fitting it to extensive N-body simulations and finding ~6% cross-simulation scatter with minimal redshift evolution up to $z ≈ 2.5$. Their results place high-ν halos between spherical-collapse and ellipsoidal-collapse predictions—approaching SMT around moderate ν and SC at the highest ν—while simple $Δ$-scaling via NFW profiles fails to reproduce the measured bias. The peak-background-split provides a useful guide but leaves ~20% residuals, indicating missing physics or higher-order effects; overall, the calibrated bias model strengthens the link between halo abundance, clustering, and cosmology, while motivating larger-volume simulations and refined theory.

Abstract

We measure the clustering of dark matter halos in a large set of collisionless cosmological simulations of the flat LCDM cosmology. Halos are identified using the spherical overdensity algorithm, which finds the mass around isolated peaks in the density field such that the mean density is Delta times the background. We calibrate fitting functions for the large scale bias that are adaptable to any value of Delta we examine. We find a ~6% scatter about our best fit bias relation. Our fitting functions couple to the halo mass functions of Tinker et. al. (2008) such that bias of all dark matter is normalized to unity. We demonstrate that the bias of massive, rare halos is higher than that predicted in the modified ellipsoidal collapse model of Sheth, Mo, & Tormen (2001), and approaches the predictions of the spherical collapse model for the rarest halos. Halo bias results based on friends-of-friends halos identified with linking length 0.2 are systematically lower than for halos with the canonical Delta=200 overdensity by ~10%. In contrast to our previous results on the mass function, we find that the universal bias function evolves very weakly with redshift, if at all. We use our numerical results, both for the mass function and the bias relation, to test the peak-background split model for halo bias. We find that the peak-background split achieves a reasonable agreement with the numerical results, but ~20% residuals remain, both at high and low masses.

Figures (5)

• Figure 1: Upper Panel: Large-scale bias as determined by the ratio $(P_h/P_{\rm lin})^{1/2}$ for $\Delta=200$. Results from the smaller boxes are represented by the gray circles. For these simulations, only measurements with less than 10% error are shown to avoid crowding. The larger-volume simulations are represented by the colored symbols. Each point type indicates a different simulation. The different colors, from left to right, go in order of increasing redshift from $z=0$ to $z=2.5$ (see Table 1 for the redshift outputs of each simulation). Like colors between simulations imply the same redshift. For these large-volume simulations, measurements with less than 25% errors are shown. Lower Panel: Fractional differences of the N-body results with the the fitting function shown in the upper panel.
• Figure 2: Large-scale bias as determined by the ratio $(P_h/P_{\rm lin})^{1/2}$ for four values of $\Delta$. The solid line in each panel represents equation (\ref{['e.bias']}) with the $\Delta$-dependent parameters listed in Table 2. The dotted curve in panel (a) is the bias formula of SMT. The dashed curve in panels (c)-(d) is the $\Delta=200$ results (i.e., the solid curve in panel a).
• Figure 3: Panel (a): The $\Delta=200$ bias function in the high-$$ regime. The points with error bars represent our large-volume simulations at the redshifts listed in Table 1. Only points with fractional errors less than 25% are shown. The different colors, from left to right, go in order of increasing redshift: ( red, green, yellow, blue, cyan)$=$(0.0, 0.5, 1.0, 1.25, 2.5). Like colors between simulations imply the same redshift. The dotted line is the spherical collapse prediction. The dashed line is the SMT function. The lower panel shows the fractional difference with respect to equation (\ref{['e.bias']}), $_b=b_{\rm Nbody}-b_{\rm fit}$. Panel (b): Same as (a), but now using bias defined by the ratio of the $P_{hm}/P_{\rm lin}(k)$. Results are shown for the L1000W simulation. Colors represent the same redshifts as in panel a. Panel (c): Bias of halos identified using the FOF algorithm with linking length 0.2. Bias is calculated from equation equation (\ref{['e.pkbias']}). Results are shown for the L1000W simulation. Different colors match to different redshifts as before. The dotted curve in this Figure is the fitting function of pillepich_etal:08, which is calibrated on FOF(0.2) halos.
• Figure 4: The fractional difference between the bias from fitting functions and the bias obtained from rescaling the $\Delta=200$ fitting function to higher overdensities assuming NFW profiles and the concentration-mass relation of zhao_etal:09.
• Figure 5: Comparison of halo bias calibrated from our numerical simulations, equation (\ref{['e.bias']}), with results from the peak-background split, equation (\ref{['e.pb_split']}). At $\Delta=200$, the peak-background split calculation is $\sim 20\%$ high/low and low/high $$. As $\Delta$ increases, the residuals at $>1$ become smaller while the residuals at $<1$ become larger.