Cell count moments in the halo model

Abstract

We study cell count moments up to fifth order of the distributions of haloes, of halo substructures as a proxy for galaxies, and of mass in the context of the halo model and compare theoretical predictions to the results of numerical simulations. On scales larger than the size of the largest cluster, we present a simple point cluster model in which results depend only on cluster–cluster correlations and on the distribution of the number of objects within a cluster, or cluster occupancy. The point cluster model leads to expressions for moments of galaxy counts in which the volume‐averaged moments on large scales approach those of the halo distribution and on smaller scales exhibit hierarchical clustering with amplitudes Sk determined by moments of the occupancy distribution. In this limit, the halo model predictions are purely combinatoric, and have no dependence on halo profile, concentration parameter or potential asphericity. The full halo model introduces only two additional effects: on large scales, haloes of different mass have different clustering strengths, introducing relative bias parameters; and on the smallest scales, halo structure is resolved and details of the halo profile become important, introducing shape‐dependent form factors. Because of differences between discrete and continuous statistics, the hierarchical amplitudes for galaxies and for mass behave differently on small scales even if galaxy number is exactly proportional to mass, a difference that is not necessarily well described in terms of bias.

Void statistics and hierarchical scaling in the halo model

“…We present results for statistics of voids in the distribution of dark matter, galaxies, and haloes for the numerical simulation studied in Fry et al (2011). The simulation is performed with the adaptive mesh refinement (AMR) code RAMSES (Teyssier 2002) for a ΛCDM cosmology with Ωm = 0.3, ΩΛ = 0.7, H0 = 100 h km s −1 Mpc −1 with h = 0.7, and normalization σ8 = 0.93, where σ8 is the root mean square initial density fluctuation in a sphere of radius 8 h −1 Mpc extrapolated linearly to the present time.…”

“…The simulation is performed with the adaptive mesh refinement (AMR) code RAMSES (Teyssier 2002) for a ΛCDM cosmology with Ωm = 0.3, ΩΛ = 0.7, H0 = 100 h km s −1 Mpc −1 with h = 0.7, and normalization σ8 = 0.93, where σ8 is the root mean square initial density fluctuation in a sphere of radius 8 h −1 Mpc extrapolated linearly to the present time. The simulation contains 512 3 dark matter particles on the AMR grid, initially regular of size 512 3 , in a periodic cube of size L box = 200 h −1 Mpc; The hierarchical amplitudes S k for mass, galaxies, and haloes in this simulation are presented by Fry et al (2011), and further details can be found in Colombi, Chodorowski & Teyssier (2007).…”

“…For small R the quantitiesμ k rise monotonically with scale, but on large scalesμ k and b k become constant (Fry et al 2011). In the halo model, the galaxy correlations are not simply hierarchical, but every term in equation (6), though no longer a simple power ofNξ, is a (polynomial) function of N hξh , If we assume that the halo distribution follows the hierarchical patternξ k,h = S k,hξ k−1 h , then χ is a function of the variableN hξh .…”

“…However, in data (Bouchet et al 1993;Gaztañaga 1994;Croton et al 2004b;Ross et al 2006Ross et al , 2007, and in numerical simulations (Fry et al 2011), the hierarchical clustering on which the scaling is based is not obeyed. The hierarchical normalization removes much of the variation, but the hierarchical amplitudes still depend on scale, and the premise of the scaling does not hold in detail.…”

Density split statistics: Cosmological constraints from counts and lensing in cells in DES Y1 and SDSS data

“…We note that this model can be related to the halo count and occupation distributions [54]. Our ansatz can be thought of as a model of Poisson-distributed haloes with α REDMAGIC galaxies in each one of them, similar to e.g., the relation of Poissonian DENSITY SPLIT STATISTICS: COSMOLOGICAL … PHYS.…”