Nonlinear 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 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.
