Nonlinear velocity-density coupling: Analysis by second-order perturbation theory

Cosmological linear perturbation theory predicts that the peculiar velocity, V(x), and matter overdensity, delta(x), at the same point x will be statistically independent quantities, as long as the initial density fluctuations are random Gaussian distributed. However, nonlinear gravitational effects might change the situation. Using a framework of second-order perturbation theory and the Edgeworth expansion method, we study the local density dependence of bulk velocity dispersion that is coarse-grained at a weakly nonlinear scale. For a typical cold dark matter (CDM) model, the first nonlinear correction of this constrained bulk velocity dispersion amounts to similar to 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.