Halo Models of Large Scale Structure

TL;DR

This paper presents the halo model as a unifying framework for nonlinear large-scale structure, decomposing matter statistics into contributions from virialized halos via 1-halo and 2-halo terms. It combines the spherical/ellipsoidal collapse physics, halo mass functions, and halo density profiles to predict power spectra, higher-order statistics, and cross-correlations for dark matter, galaxies, velocities, and weak lensing, extending to CMB secondary anisotropies. The model is validated against simulations and applied to a wide range of observables, including galaxy clustering, velocity fields, weak lensing covariances, SZ effects, and the nonlinear ISW effect, while acknowledging limitations like halo sphericity and substructure. The halo approach enables efficient, semi-analytic exploration of nonlinear regimes and provides a framework to interpret upcoming large-scale structure surveys and CMB experiments, with quantified non-Gaussian covariances and biasing relations. Overall, it serves as a powerful bridge between linear theory and fully nonlinear clustering, guiding interpretation of observations and informing galaxy formation and cosmological parameter studies.

Abstract

We review the formalism and applications of the halo-based description of nonlinear gravitational clustering. In this approach, all mass is associated with virialized dark matter halos; models of the number and spatial distribution of the halos, and the distribution of dark matter within each halo, are used to provide estimates of how the statistical properties of large scale density and velocity fields evolve as a result of nonlinear gravitational clustering. We first describe the model, and demonstrate its accuracy by comparing its predictions with exact results from numerical simulations of nonlinear gravitational clustering. We then present several astrophysical applications of the halo model: these include models of the spatial distribution of galaxies, the nonlinear velocity, momentum and pressure fields, descriptions of weak gravitational lensing, and estimates of secondary contributions to temperature fluctuations in the cosmic microwave background.

Figures (62)

• Figure 1: The complex distribution of dark matter (a) found in numerical simulations can be easily replaced with a distribution of dark matter halos (b) with the mass function following that found in simulations and with a profile for dark matter within halos.
• Figure 2: Distribution of galaxies (in color) superposed on the dark matter distribution (grey scale) in simulations run by the GIF collaboration Kauetal99. Galaxy colors blue, yellow, green and red represent successively smaller star formation rates. Different panels show how the spatial distributions of dark matter and galaxies evolve; the relation between the two distributions changes with time, as do the typical star formation rates.
• Figure 3: The halo mass function in numerical simulations of the Virgo collaboration. The measured mass distribution is show in color; dashed line shows the Press-Schechter mass function; dotted line is a fitting formula which is similar to the Sheth-Tormen mass function. The figure is from Jenetal01.
• Figure 4: Large scale bias relation between halos and mass (from SheTor99). Symbols show the bias factors at $z_{\rm obs}$ for objects which were identified as virialized halos at $z_{\rm form}=4$,2,1 and 0 (top to bottom in each panel). Dotted and solid lines show predictions based on the Press-Schechter and Sheth-Tormen mass functions.
• Figure 5: Higher order moments of the halo distribution if the initial fluctuation spectrum is scale free and has slope $n=-1.5$. Dotted and solid curves show the result of assuming the halo mass function has the Press--Schechter and Sheth--Tormen forms.