Large-Scale Galaxy Bias

TL;DR

The paper develops a comprehensive, perturbative framework for galaxy bias on large, quasi-linear scales, showing that a finite set of local, higher-derivative, and nonlocal operators suffices to describe tracer statistics once gravitational evolution is accounted for. It weaves together local LIMD bias, peak-background split, and excursion-set/peaks formalisms, clarifying renormalization and the connection between rest-frame bias and observed statistics, including relativistic and stochastic effects. The work also provides practical measurement strategies (n-point functions, moments, scatter plots, response methods) and outlines how bias parameters evolve and can be constrained by current and future surveys, with explicit treatments of assembly bias and non-Gaussian initial conditions. Overall, it offers a unified, renormalized, multi-model approach to interpreting large-scale structure through the lens of galaxy bias, applicable to both simulations and observations. The framework has significant implications for extracting cosmological information from upcoming surveys and for understanding how complex galaxy formation processes imprint on large-scale clustering.

Abstract

This review presents a comprehensive overview of galaxy bias, that is, the statistical relation between the distribution of galaxies and matter. We focus on large scales where cosmic density fields are quasi-linear. On these scales, the clustering of galaxies can be described by a perturbative bias expansion, and the complicated physics of galaxy formation is absorbed by a finite set of coefficients of the expansion, called bias parameters. The review begins with a detailed derivation of this very important result, which forms the basis of the rigorous perturbative description of galaxy clustering, under the assumptions of General Relativity and Gaussian, adiabatic initial conditions. Key components of the bias expansion are all leading local gravitational observables, which include the matter density but also tidal fields and their time derivatives. We hence expand the definition of local bias to encompass all these contributions. This derivation is followed by a presentation of the peak-background split in its general form, which elucidates the physical meaning of the bias parameters, and a detailed description of the connection between bias parameters and galaxy statistics. We then review the excursion-set formalism and peak theory which provide predictions for the values of the bias parameters. In the remainder of the review, we consider the generalizations of galaxy bias required in the presence of various types of cosmological physics that go beyond pressureless matter with adiabatic, Gaussian initial conditions: primordial non-Gaussianity, massive neutrinos, baryon-CDM isocurvature perturbations, dark energy, and modified gravity. Finally, we discuss how the description of galaxy bias in the galaxies' rest frame is related to clustering statistics measured from the observed angular positions and redshifts in actual galaxy catalogs.

Figures (39)

• Figure 1: Two-dimensional slice projections (pie diagram) of the measured locations of galaxies in the CfA2, 2dF, and SDSS galaxy redshift surveys (top left half). The lower right half shows the location of galaxies which were assigned to dark matter halos in the Millennium gravity-only N-body simulation using a semi-analytical prescription. It is apparent that the simulation, which assumes a flat $\Lambda$CDM cosmology, qualitatively reproduces the observed large-scale structure of the Universe very well. From millennium:2006.
• Figure 2: Schematic outline of the theoretical prediction of observed galaxy statistics. Given the statistics of the initial conditions, perturbative bias expansions predict the rest-frame galaxy density as well as that of dark matter halos. This expansion can either be done using Lagrangian (left) or Eulerian frames (right). Crucially, the general bias expansion in either frame is mathematically equivalent. The bias expansion is closely connected to the perturbation theory of the matter density field (Lagrangian [LPT] and standard Eulerian [SPT] perturbation theory, respectively). The peak-background split (PBS) informs the bias expansion by relating the bias parameters to responses of the mean tracer abundance. The peak and excursion-set approaches are a special case of the Lagrangian bias expansion, and predict the proto-halo density, which is connected to the statistics of halos at low redshift through conserved evolution. The statistics of halos in turn can be related to those of galaxies through halo occupation distribution (HOD) and subhalo abundance matching (SHAM) approaches. Finally, the connection between rest-frame and observed galaxy statistics involves selection and projection effects (such as redshift-space distortions). Cosmological physics enters in the initial conditions through primordial non-Gaussianity and isocurvature perturbations between baryons and CDM. It also affects the evolution of structure, and consequently the bias expansion, through the effects of massive neutrinos, dark energy, and modified gravity.
• Figure 3: Illustration of the toy model of Sec. \ref{['sec:localbias']}. The solid blue line shows the smoothed density field $\delta_R^{(1)}(\bm{q})$, while the red line indicates a long-wavelength perturbation. The dashed horizontal line marks the threshold overdensity $\delta_\text{cr}$.
• Figure 4: Correlation function in Lagrangian space of thresholded regions in the initial density field extrapolated to $z=0$. The smoothing scale $R = 4.21 \,h^{-1}\,{\rm Mpc}$ (mass scale $2.5\cdot 10^{13}\,h^{-1}\,M_{\odot}$) is chosen to correspond to $b_1^L = 1.5$ via Eq. (\ref{['eq:bPBSthr']}). Left panel: We show the exact result Eq. (\ref{['eq:xihthr1']}), multiplied by $r$ to better show the large-scale behavior, as well as the series expansion Eq. (\ref{['eq:xihthr']}) truncated at different orders. For comparison, the cyan dotted line shows the linear, unfiltered matter correlation function. The bump at $r \approx 100 \,h^{-1}\,{\rm Mpc}$ is the BAO feature. Right panel: Relative deviation of the truncated series expansion from the exact result. The dotted line shows the relative deviation of the unfiltered linear contribution $(b_1^L)^2 \xi_{\rm L}$ from the filtered contribution $(b_1^L)^2 \xi_{{\rm L},R}$.
• Figure 5: Sketch of the setup considered in Sec. \ref{['sec:dynamics']}--\ref{['sec:evol2']}. Galaxies form instantaneously at $\tau=\tau_*$, where they are described by an initial bias relation (lower slice), and are comoving with the matter. After this, the evolution is governed by number conservation and the comoving assumption up until time $\tau$ (upper slice), where they are assumed to be observed. The grey region denotes a Lagrangian volume encompassing three galaxies which gets deformed by nonlinear gravitational evolution. Since galaxies comove with matter, their density is similarly affected.