A GAUGE-INVARIANT FLUID DESCRIPTION OF PRIMORDIAL DENSITY FLUCTUATIONS OF COLLISIONLESS PARTICLES

Abstract

Formulas are derived for describing the evolution of fluctuations in the density of collisionless particles in the expanding universe using the gauge-invariant fluid description. The formulas use the “gauge-invariant” variables proposed by Bardeen to describe cosmological perturbations. These variables are hydrodynamic in nature and we show the behavior of the equations when the particles have streaming lengths large compared to the scales of interest. We also show how these equations couple gravitationally when other species of matter are present in significant densities. Using the “fourteen moment” approximation for the particle distribution function, we get a simple set of ordinary differential equations which are much easier to use than a direct integration of the Boltzmann equation. This formulation is especially useful when we are considering universes with more than one cosmologically significant component of matter density. An example of a numerical integration of the evolution equations is presented for comparison of this method to other work. A formula for calculating fluctuations in the cosmic microwave background radiation is also given.