Precision measurement of the local bias of dark matter halos

TL;DR

The paper tackles precise measurement of the local halo bias parameters $b_1$, $b_2$, and $b_3$ by employing curved separate-universe N-body simulations that realize an infinite-wavelength overdensity, effectively implementing the peak-background split exactly. It compares these measurements to predictions from the peak-background split with universal mass functions and from the excursion set peaks (ESP) framework, highlighting that a stochastic moving barrier in ESP improves high-mass bias predictions while constant-barrier PBS can fall short. The authors show that separate-universe results agree with independent measurements from halo-matter power spectrum and halo-matter bispectrum, and they provide robust fitting formulas for $b_2(b_1)$ and $b_3(b_1)$ across redshift. The work reinforces the separation between PBS and ESP as key ingredients in accurate halo bias modeling and offers a practical route for forecasts in large-scale structure analyses, including potential extensions to assembly bias and tidal biases.

Abstract

We present accurate measurements of the linear, quadratic, and cubic local bias of dark matter halos, using curved "separate universe" N-body simulations which effectively incorporate an infinite-wavelength overdensity. This can be seen as an exact implementation of the peak-background split argument. We compare the results with the linear and quadratic bias measured from the halo-matter power spectrum and bispectrum, and find good agreement. On the other hand, the standard peak-background split applied to the Sheth & Tormen (1999) and Tinker et al. (2008) halo mass functions matches the measured linear bias parameter only at the level of 10%. The prediction from the excursion set-peaks approach performs much better, which can be attributed to the stochastic moving barrier employed in the excursion set-peaks prediction. We also provide convenient fitting formulas for the nonlinear bias parameters $b_2(b_1)$ and $b_3(b_1)$, which work well over a range of redshifts.

Figures (8)

• Figure 1: Top panel: comparison between the linear halo bias from separate universe simulations (green dots), and from clustering (red crosses; displaced slightly horizontally for clarity). Error bars that are not visible are within the marker size. The solid black curve is the Tinker et al. (2010) best fit curve for $b_1$, while the dot-dashed green curve is the ESP prediction Eq. (\ref{['eq:btheo']}). We also show the result obtained by applying the PBS argument [Eq. (\ref{['eq:biasPBS']})] to the T08 and ST99 mass functions (blue dashed curves). Bottom panel: relative difference between the measurements and the Tinker et al. (2010) best fit.
• Figure 2: Top panel: same as Figure \ref{['fig:b1']}, but for the quadratic bias $b_2$. The color code is as in Figure \ref{['fig:b1']}. Bottom panel: relative difference between measurements and the theoretical prediction of the ESP. In each panel, the clustering points have been horizontally displaced as in Figure \ref{['fig:b1']}.
• Figure 3: As Figure \ref{['fig:b2']} but for $b_3$.
• Figure 4: $b_2$ and $b_3$ as a function of $b_1$ obtained from separate universe simulations and for different redshifts. The dashed curves present the third order best fit polynomial for each bias. See text for details about the fit.
• Figure 5: Comparison of $b_3$ measured directly via fitting $\delta_h$ vs the Eulerian density $\delta_\rho$ (crosses) and, the corresponding value inferred from the Lagrangian bias fits, as shown in the main text (dots). The crosses have been displaced horizontally for clarity.