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 theta = del center dot v(x, t)/a H and delta. This provides a new phenomenological approach that examines the conditional mean <theta vertical bar delta >, together with the fluctuations of theta around this mean. We measure these stochastic components using N-body simulations and find they are non-negative and increase with decreasing scale from similar to 10 per cent at k < 0.2 h Mpc(-1) to 25 per cent at k similar to 0.45 h Mpc(-1) at z = 0. Both the stochastic relation and non-linearity are more pronounced for haloes, M <= 5 x 10(12) M-circle dot h(-1), compared to the dark matter at z = 0 and 1. Non-linear growth effects manifest themselves as a rotation of the mean <theta vertical bar delta > away from the linear theory prediction -f(LT)delta, where f(LT) 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 theta-delta relation is not so simply described by 2LPT, and we discuss its impact on measurements of f(LT) from two-point statistics in redshift space. Given that the relationship between delta and theta is stochastic and non-linear, this will have implications for the interpretation and precision of f(LT) extracted using models which assume a linear, deterministic expression.