Scale-dependent halo bias from scale-dependent growth

Abstract

We derive a general expression for the large-scale halo bias, in theories with a scale-dependent linear growth, using the excursion set formalism. Such theories include modified gravity models, and models in which the dark energy clustering is non-negligible. A scale dependence is imprinted in both the formation and evolved biases by the scale-dependent growth. Mergers are accounted for in our derivation, which thus extends earlier work which focused on passive evolution. There is a simple analytic form for the bias for those theories in which the nonlinear collapse of perturbations is approximately the same as in general relativity. As an illustration, we apply our results to a simple Yukawa modification of gravity, and use SDSS measurements of the clustering of luminous red galaxies to constrain the theory's parameters. 98.80.Es; 98.65.Dx; 95.35.+d

Thef(ℛ) halo mass function in the cosmic web

“…Currently our model is based on simply substituting the argument of the unconditional mass function from ν h to ν eff , inspired by the equivalent result found in the excursion-set formalism. However, this rescaling of the mass function is only mathematically consistent, in the sense described above, for the specific form of the Press-Schechter mass function [40]. Thus, most simple attempts at modelling the conditional mass function have only been able to provide a qualitative description of it, with a poor quantitative performance.…”

“…It is straightforward to show that this integral equation has an analytical solution for a constant barrier [40,41], including the constant case δ Λ c ≈ 1.676 which is the ΛCDM solution. Once the first crossing distribution is found after solving this equation, the procedure would be to marginalise over the possible values of the environmental density.…”

“…Compared to ΛCDM, we expect haloes to form at higher masses (earlier in the random walk) and to have more in low-density regions (where the fifth-force potential is unscreened). For a moving barrier like this, and under the assumption that trajectories in (δ, S) take uncorrelated steps, the diffuson equation admits a solution [40]:…”

Modeling halo mass functions in chameleonf(R)gravity

“…The crossing probability conditional on the Brownian motion trajectory passing δ env at S χ is φ[δ c (S, δ env ), S|δ env , S χ ]. This probability needs to be computed numerically, for which we use a code developed in [28] based on the algorithm of [45]. We refer to [28] for more details on this computation.…”

“…We shall adopt the value for the Lagrangian radius used in [28], ξ = 8h −1 Mpc such that S ξ = σ 2 8 . For comparison to Nbody simulations or observations that do not differentiate between structures formed in different environments, the conditional first-crossing distribution of the moving barrier φ[δ c (S, δ env ), S|δ env , S ξ ] computed with the algorithm of [45] needs to be integrated over all environments. In order to do so, in the following, we shall denote the distribution of δ env characterized through ξ as P ξ (δ env ), corresponding to the probability that the Brownian motion trajectory passes through δ env at S ξ never having crossed the collapse density δ Λ c at S < S ξ .…”