General Relativistic Relation Between Density Contrast and Peculiar Velocity

Abstract

Concepts like peculiar velocity, gravitational force, and power spectrum and their interrelationships are of utmost importance in the theories of structure formation. The observational implementation of these concepts is usually based on the Newtonian hydrodynamic equations, but used up to scales where general relativistic effects come in. Using a perturbation of FRW metric in harmonic gauge, we show that the relativistic effects reduce to light cone effects including the expansion of the universe. Within the Newtonian gravitation, the linear perturbation theory of large scale structure formation predicts the peculiar velocity field to be directly proportional to gravitational force due to the matter distribution. The corresponding relation between peculiar velocity field and density contrast has been given by Peebles. Using the general relativistic perturbation we have developed, this familiar relation is modified by doing the calculation on the light cone in contrast to the usual procedure of taking a space-like slice defined at a definite time. The velocity and density--spectrum are compared to the familiar Newtonian expressions. In particular, the relativistic $β$--value obtained is reduced and leads to an increased bias factor or a decreased expected amount of the dark matter in a cluster.