Non-linear stochastic growth rates and redshift space distortions

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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 nonlinear, stochastic relation between $\theta = \nabla \cdot v({\bf x},t)/aH$ and $\delta$. This provides a new phenomenological approach that examines the conditional mean $< \theta|\delta>$, 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 $\sim$10% at $k<0.2 h $Mpc$^{-1}$ to 25% at $k\sim0.45h$Mpc$^{-1}$ at $z = 0$. Both the stochastic relation and nonlinearity are more pronounced for halos, $M \le 5 \times 10^{12}M_\odot h^{-1}$, compared to the dark matter at $z=0$ and $1$. Nonlinear growth effects manifest themselves as a rotation of the mean $< \theta|\delta>$ away from the linear theory prediction $-f_{\tiny \rm LT}\delta$, where $f_{\tiny \rm LT}$ 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 $f_{\tiny \rm LT}$ from two point statistics in redshift space. Given that the relationship between $\delta$ and $\theta$ is stochastic and nonlinear, this will have implications for the interpretation and precision of $f_{\tiny \rm LT}$ extracted using models which assume a linear, deterministic expression.