Non-local halo bias with and without massive neutrinos

TL;DR

This paper addresses how halos trace the underlying matter distribution by extending bias modeling beyond locality to non-local terms generated by gravity and peak constraints. It develops a perturbative, third-order framework that expresses the halo density in terms of non-local operators, with explicit mappings from Lagrangian to Eulerian biases, including a linear $k^2$ term from peak constraints. The main finding is that gravity-induced non-local bias, together with a peak-derived $k^2$ contribution, is essential to reproduce the scale dependence observed in N-body data, especially in cosmologies with massive neutrinos, achieving about 3% accuracy up to $k \approx 0.3\, h\,\mathrm{Mpc}^{-1}$ for $\sum m_\nu \le 0.6$ eV without free parameters. This work provides a parameter-free, physically motivated description of halo bias across neutrino masses and lays groundwork for refined analyses in upcoming galaxy surveys. The results underscore the importance of non-local bias terms for precise cosmological inference from halo clustering.

Abstract

Understanding the biasing between the clustering properties of halos and the underlying dark matter distribution is important for extracting cosmological information from ongoing and upcoming galaxy surveys. While on sufficiently larges scales the halo overdensity is a local function of the mass density fluctuations, on smaller scales the gravitational evolution generates non-local terms in the halo density field. We characterize the magnitude of these contributions at third-order in perturbation theory by identifying the coefficients of the non-local invariant operators, and extend our calculation to include non-local (Lagrangian) terms induced by a peak constraint. We apply our results to describe the scale-dependence of halo bias in cosmologies with massive neutrinos. The inclusion of gravity-induced non-local terms and, especially, a Lagrangian $k^2$-contribution is essential to reproduce the numerical data accurately. We use the peak-background split to derive the numerical values of the various bias coefficients from the excursion set peak mass function. For neutrino masses in the range $0\leq \sum_i m_{ν_i} \leq 0.6$ eV, we are able to fit the data with a precision of a few percents up to $k=0.3\, h {\rm \,Mpc^{-1}}$ without any free parameter.

Figures (4)

• Figure 1: Halo bias at redshift $z=0$ as a function of wavenumber in the case $\sum m_\nu =0$. The different terms that contribute to the scale-dependence bias in Eq.(\ref{['eq:bhm']}) are labelled according to the bias parameter they are proportional to (see text). The solid black curve represents the sum of all the contributions, Eq.(\ref{['eq:bhm']}). All the bias factors have been computed consistently from the ESP halo mass function and the relations Eq.(\ref{['eq:biask']}).
• Figure 2: Halo bias (left) and fractional scale-dependence (right) at $z=0$ as a function of wavenumber for values of $\sum_i m_{\nu_i}=0$, 0.1, 0.2, 0.3 and 0.6 eV. In the left panel, only the local bias terms are included in the predictions. In the right panel, the models with non-zero neutrino masses still assume local bias, whereas the solid (black) curve represents the case in which massive neutrinos are absent but all non-local terms are accounted for. The non-local bias contributions induced by gravity and by the peak constraint generate a sharp rise beyond $k\sim 0.1\, h {\rm \,Mpc^{-1}}$ substantially steeper than the effect of non-zero neutrino mass.
• Figure 3: Halo bias at $z=0$ as a function of wavenumber. Data points are from Villaescusa-Navarro:2013pva. In the upper left panel, we show for $\sum_i m_{\nu_i}=0.3$ eV the local bias prediction as the dashed (blue) curve, and our full non-local model as the solid (red) curve. The difference between the solid (red) and the dotted (magenta) curve represents the effect of turning off the contribution $b_{01} k^2$ arising from the peak constraint. In the upper right panel, we compare our non-local prediction with the numerical data for $\sum_i m_{\nu_i}=0$, 0.3 and $0.6$ eV. The lower panels show the fractional deviation between theory and simulations. Note that these predictions have no free parameter.
• Figure 4: Same as Fig.\ref{['fig:viel']} but at redshift $z=0.5$.