Understanding higher-order nonlocal halo bias at large scales by combining the power spectrum with the bispectrum

TL;DR

This work addresses the halo-bias problem at large scales by extending perturbation theory to include nonlocal bias terms up to third order, encapsulated by the four parameters $b_1$, $b_2$, $b_{s^2}$, and $b_{3\rm nl}$. It demonstrates that the halo-matter power spectrum and the halo–matter bispectrum can be jointly described within this four-parameter framework, with $b_{3\rm nl}$ playing a crucial role for $k \lesssim 0.1\,h\,\mathrm{Mpc}^{-1}$ and across multiple halo masses and redshifts, in agreement with a simple coevolution picture and Galileon-invariant arguments. The results show that nonlocal bias cannot be neglected for precision cosmology and that the inclusion of nonlocal terms improves consistency between different statistics, informing the modeling of upcoming galaxy surveys. Overall, the paper provides a physically motivated, renormalized bias formulation that captures the leading nonlocal effects of gravity on halo statistics and offers a pathway to more accurate cosmological parameter inference from large-scale structure data.

Abstract

Understanding the relation between underlying matter distribution and biased tracers such as galaxy or dark matter halo is essential to extract cosmological information from ongoing or future galaxy redshift surveys. At sufficiently large scales such as the BAO scale, a standard approach for the bias problem on the basis of the perturbation theory (PT) is to assume the `local bias' model in which the density field of biased tracers is deterministically expanded in terms of matter density field at the same position. The higher-order bias parameters are then determined by combining the power spectrum with higher-order statistics such as the bispectrum. As is pointed out by recent studies, however, nonlinear gravitational evolution naturally induces nonlocal bias terms even if initially starting only with purely local bias. As a matter of fact, previous works showed that the second-order nonlocal bias term, which corresponds to the gravitational tidal field, is important to explain the characteristic scale-dependence of the bispectrum. In this paper we extend the nonlocal bias term up to third order, and investigate whether the PT-based model including nonlocal bias terms can simultaneously explain the power spectrum and the bispectrum of simulated halos in $N$-body simulations. We show that the power spectrum, including density and momentum, and the bispectrum between halo and matter in $N$-body simulations can be simultaneously well explained by the model including up to third-order nonlocal bias terms up to k~0.1h/Mpc. Also, the results seem in a good agreement with theoretical predictions of a simple coevolution picture, although the agreement is not perfect. These demonstration clearly shows a failure of the local bias model even at such large scales, and we conclude that nonlocal bias terms should be consistently included in order to model statistics of halos. [abridged]

Figures (12)

• Figure 1: A comparison of the PT correction terms at $z=0$ for $P^{\rm hm}_{\,00}$ ( left) and $P^{\rm hm}_{\,01}$ ( right). The data points are the nonlinear matter power spectrum directly measured from our simulations described in Sec. \ref{['sec: simulation']}.
• Figure 2: The best-fitting values of $b_{3{\rm nl}}$ as a function of $k_{\rm max}$. We present results at $z=1$ ( top three), at $z=0.5$ ( middle four), and at $z=0$ ( bottom four), for light to heavy (from I to IV) halo mass bins. In each panel, we show results in the case of $P^{\rm hm}_{\,00}$ only (red), $P^{\rm hm}_{\,01}$ only (blue), and both of two (black). The goodness of fit, $\chi^{2}_{P(k)}/{\rm d.o.f}$, is also plotted in the lower part of each panel. Note that we jointly fit the bispectrum together with the power spectrum. For comparison, the prediction from the coevolution picture (local Lagrangian bias model), $32/315$, is indicated by the horizontal line (cyan solid).
• Figure 3: The power spectra with best-fitting bias parameters at $z=1$. We here plot $P^{\rm hm}_{X}(k)/(b_{1}P^{\rm mm}_{X}(k))-1$ where $X$ is '00' ( left) or '01' ( right) with the best-fitting values of $b_{1}$ and $b_{\rm 3nl}$ at $k_{\rm max}=0.125\,h$/Mpc (specified as an arrow). Namely, zero values ( black dotted) mean it matches to the linear bias term, any deviation from zero represents deviation from the linear bias model. The red solid line corresponds to the case including all contributions. The blue dashed line includes only local bias terms up to second order, while the green dashed line includes local and nonlocal bias terms up to second order.
• Figure 4: Same as Fig. \ref{['fig: estimate b3nl vs Pk z=1']}, but at $z=0.5$. The best-fitting values are derived at $k_{\rm max}=0.1\,h$/Mpc (specified as an arrow).
• Figure 5: Same as Fig. \ref{['fig: estimate b3nl vs Pk z=1']}, but at $z=0$. The best-fitting values are derived at $k_{\rm max}=0.08\,h$/Mpc (specified as an arrow).