A Preface to Cosmogony. I. The Energy Equation and the Virial Theorem for Cosmic Distributions.

This paper deals with the dynamics of an unbounded, statistically homogeneous, and isotropic distribution of gravitating particles. As is well known, the mean density and velocity fields depend on a single function, the cosmic scale factor R(t), given by relativistic cosmology. in an approximation that is adequate for all astronomical applications, the peculiar velocity i' of a test particle is shown to satisfy the equations of motion d(Rv) = dt Ox' where the potential is related to the peculiar density field by Poisson's equation. in a uniform medium ((T 0) the velocity of a test particle decays as R-1. More generally, the form of the equations of motion implies that "initial" conditions cannot have an enduring influence on particle motions, which must accordingly be caused mainly by the action of the fluctuating gravitational force field 3 /ax. This idea finds its mathematical expression in two coupled equations for Tm = 21 (v2 )rn, the mean peculiar kinetic energy per unit mass, and Urn = - ((p )m, the mean peculiar potential energy per unit mass: d(T + 1? dt rn Um)+ (2Tm+Um)=O 2Trn(t)+Urn( ) =0, where E « t. The first equation (the cosmologic energy equation) was first obtained in this form by irvine. it reduces to a more familiar form if one sets = p (Trn+ Urn), 3P=p(2Tm+Um), where e is the mean internal energy density and p the cosmic pressure. The second equation (the cosmologic virial theorem) is here derived by a fundamentally different method from the one used to derive the virial theorem for a bounded system. The present method applies also to bounded systems, but the conventional method breaks down when applied to an unbounded system. The cosmologic virial theorem is expected to be valid as long as non-gravitational forces and the effects of radiation are negligible. The specific energies Tm and Urn are then nearly constant in time The constancy of Urn, which may be written in the form Urn = - Ga2pX2, where x is a clustering scale and a2 = (cr2 )/p ' is a measure of the density contrast, implies that a2 increases at least as fast as R-a conclusion reached previously (Layzer 1954a) by a less rigorous argument.