Nonlinear Velocity-Density Coupling: Analysis by Second-Order   Perturbation Theory
  Cosmological linear perturbation theory predicts that the peculiar velocity $V(x)$ and the matter overdensity $\delta(x)$ at a same point $x$ are statistically independent quantities, as log as the initial density fluctuations are random Gaussian distributed. However nonlinear gravitational effects might change the situation. Using framework of second-order perturbation theory and the Edgeworth expansion method, we study local density dependence of bulk velocity dispersion that is coarse-grained at a weakly nonlinear scale. For a typical CDM model, the first nonlinear correction of this constrained bulk velocity dispersion amounts to $\sim 0.3\delta$ (Gaussian smoothing) at a weakly nonlinear scale with a very weak dependence on cosmological parameters. We also compare our analytical prediction with published numerical results given at nonlinear regimes. 
Naoki Seto