The quasi-Gaussian density-velocity relationship

The nonlinear relationship between density and divergence of the velocity field is expressed for the limit of a vanishing variance, and its astrophysical implications are considered. Attention is given to the tree shape of the correlations, approximating the Omega-dependence, and establishing the joint delta-delta(upsilon) distribution. The relation proposed by Bernardeau and Schaeffer (1992) is used to determine the divergence of the velocity field when delta is known. The relations established provide two types of nonzero variance corrections: scatter in the delta-delta(upsilon) relationship and an expected point distribution. The nonlinear density-velocity relationship is a direct consequence of the treelike correlation function in the quasi-Gaussian regime. The Omega-dependence of these expressions makes them valuable for the study of underdense regions of the universe where the number density of galaxies is weak.