Clustering of dark matter tracers: generalizing bias for the coming era of precision LSS

TL;DR

The paper addresses the need for a generalized galaxy bias model beyond local density by incorporating the local tidal field and velocity divergence within an Eulerian framework. It develops a renormalized bias theory in which two new bias parameters capture non-local, velocity-tidal effects, enabling accurate predictions for the galaxy-mass cross-spectrum, galaxy-galaxy power spectrum, and the bispectrum up to 4th order. The authors provide explicit, renormalized expressions for P_mg, P_gg, and B_g, discuss short-range non-locality, redshift-space distortions, and primordial non-Gaussianity within the same framework, and emphasize the substantial information content available in beyond-linear scales for future surveys. This approach offers a practical, scalable way to extract cosmological information from high-precision LSS data while controlling small-scale physics through renormalization and a minimal set of bias parameters.

Abstract

On very large scales, density fluctuations in the Universe are small, suggesting a perturbative model for large-scale clustering of galaxies (or other dark matter tracers), in which the galaxy density is written as a Taylor series in the local mass density, delta, with the unknown coefficients in the series treated as free "bias" parameters. We extend this model to include dependence of the galaxy density on the local values of nabla_i nabla_j phi and nabla_i v_j, where phi is the potential and v is the peculiar velocity. We show that only two new free parameters are needed to model the power spectrum and bispectrum up to 4th order in the initial density perturbations, once symmetry considerations and equivalences between possible terms are accounted for. One of the new parameters is a bias multiplying s_ij s_ji, where s_ij=[nabla_i nabla_j \nabla^-2 - 1/3 delta^K_ij] delta. The other multiplies s_ij t_ji, where t_ij=[nabla_i nabla_j nabla^-2 - 1/3 delta^K_ij](theta-delta), with theta=-(a H dlnD/dlna)^-1 nabla_i v_i. (There are other, observationally equivalent, ways to write the two terms, e.g., using theta-delta instead of s_ij s_ji.) We show how short-range (non-gravitational) non-locality can be included through a controlled series of higher derivative terms, starting with R^2 nabla^2 delta, where R is the scale of non-locality (this term will be a small correction as long as k^2 R^2 is small, where k is the observed wavenumber). We suggest that there will be much more information in future huge redshift surveys in the range of scales where beyond-linear perturbation theory is both necessary and sufficient than in the fully linear regime.

Figures (5)

• Figure 1: Cumulative number of modes with $k<0.1/\left[D\left(z\right)/D\left(0\right) \right]\, h\, {\rm Mpc}^{-1}$ up to a given redshift. The largest reasonably well-sampled LSS survey, the SDSS LRGs, probe only an tiny fraction of the available modes.
• Figure 2: Weighting kernel over which $\Delta^2\left(q=r~k\right)$ is integrated to obtain the contribution of several terms to $P_{mg}\left(k\right)$. The dotted line shows $I\left(r\right)$, defined by Eq. (\ref{['eqIdef']}), which is sensitive to high-$k$ power. The solid line shows the kernel after renormalization of the linear bias, $I_R\left(r\right) =I\left(r\right)+5/6$, which now acts as a filter to produce the variance of the density field smoothed on scale $~k$.
• Figure 3: Bias terms in Eq. (\ref{['eqfinalPmg']}), for the galaxy-mass cross-power spectrum, at $z=1$. The black (solid) line shows the term proportional to $\tilde{b}_{\delta^2}$, red (dashed) shows $\tilde{b}_{s^2}$, and green (dotted) shows $\tilde{b}_3$. The coefficient values are chosen to match those in the more important galaxy-galaxy power spectrum shown in Fig. \ref{['figbasicgg']}.
• Figure 4: Effect of various kinds of bias on the auto-power spectrum of a single type of galaxy (Eq. \ref{['eqfinalPab']}), at $z=1$. The black (solid) line shows the term proportional to $\tilde{b}_{\delta^2}$, red (dashed) shows $\tilde{b}_{s^2}$, and green (dotted) shows $\tilde{b}_3$, with values of the coefficients labeling the curves (all of the other coefficients are zero in each case). The blue (long-dashed) line shows the effect of $N$ (white noise), when similarly normalized by the mass power spectrum. The coefficient values are largely arbitrary, i.e., the lines are only intended to show the shape of the effect, not to imply anything about the magnitude. The error bars show approximate fractional errors on band power measurements from a 100 cubic Gpc/h survey (e.g., $\sim 3/4$ of the sky at $1<z<2$).
• Figure 5: Quantities contributing to the reduced bispectrum, $Q_g$ (Eq. \ref{['eqQg']}), as a function of $\mu_{12}=\mathbf{k}_1\cdot\mathbf{k}_2/k_1 k_2$. The left panel shows $k_1=0.1 \, h\, {\rm Mpc}^{-1}$, $k_2=0.2 \, h\, {\rm Mpc}^{-1}$, while the right shows $k_1=k_2=0.1\, h\, {\rm Mpc}^{-1}$ (in this case, recall that $\mathbf{k}_3=-\left(\mathbf{k}_1+\mathbf{k}_2\right)$, so $k_3=0$ when $\mu_{12}=-1$). The blue, dot-dashed, curve shows the mass bispectrum $Q_m$, the black, solid, lines represent $\tilde{b}_{\delta^2}$, and the red, dashed, curves are the new $\tilde{b}_{s^2}$ term (new to Eulerian bias models, although already present in Lagrangian bias models 2000MNRAS.318L..39C2002PhR...367....1B1998MNRAS.297..692C). The parameter values were chosen to match the power spectrum figures.