Non-linear stochastic growth rates and redshift space distortions

The linear growth rate is commonly defined through a simple deterministic relation between the velocity divergence and the matter overdensity in the linear regime. We introduce a formalism that extends this to a non-linear, stochastic relation between θ=∇⋅v(x,t)/aH and δ. This provides a new phenomenological approach that examines the conditional mean 〈θ|δ〉, together with the fluctuations of θ around this mean. We measure these stochastic components using N-body simulations and find they are non-negative and increase with decreasing scale from ∼10 per cent at k < 0.2 h Mpc−1 to 25 per cent at k ∼ 0.45 h Mpc−1 at z = 0. Both the stochastic relation and non-linearity are more pronounced for haloes, M ≤ 5 × 1012 M⊙ h−1, compared to the dark matter at z = 0 and 1. Non-linear growth effects manifest themselves as a rotation of the mean 〈θ|δ〉 away from the linear theory prediction −fLTδ, where fLT is the linear growth rate. This rotation increases with wavenumber, k, and we show that it can be well-described by second-order Lagrangian perturbation theory (2LPT) for k < 0.1  h Mpc−1. The stochasticity in the θ–δ relation is not so simply described by 2LPT, and we discuss its impact on measurements of fLT from two-point statistics in redshift space. Given that the relationship between δ and θ is stochastic and non-linear, this will have implications for the interpretation and precision of fLT extracted using models which assume a linear, deterministic expression.