Finite-Density Nonhomogeneous Newtonian Cosmologies

Abstract

Approximate self-consistent solutions are obtained to the coupled Vlasov and Poisson equations for a system of self-gravitating point masses subject to a general nonuniform, but spherically symmetric, initial condition. The method of solution, which is essentially the variational determination of a trial distribution function by the application of Hamilton's principle to the trial Lagrangian, is capable of interesting generalization, but, as used here, does not allow for the escape of mass. The model describes cosmologies, or evolving star clusters, in which the usual cosmological principle is modified: Isotropy and homogeneity are approximate symmetries in the interior of the system only, while, to present approximation, the Hubble law is still exact. The model agrees with the major observational results of homogeneous isotropic cosmologies, but differs from them in predicting finite (infinite) oscillations of the central density $\ensuremath{\rho}(0)$ in the gravitationally bound (unbound) case and, in all cases, a finite maximum density $\ensuremath{\rho}{(0)}_{max}$. The equation for $\ensuremath{\rho}(0)$ is that of the average density in homogeneous, but not isotropic, rotating cosmologies. The value of $\ensuremath{\rho}{(0)}_{max}$ is constrained observationally only by a requirement of compatibility with the 3\ifmmode^\circ\else\textdegree\fi{}K blackbody radiation, which yields the lower limit $\ensuremath{\rho}{(0)}_{max}\ensuremath{\gtrsim}{10}^{\ensuremath{-}21}$ g ${\mathrm{cm}}^{\ensuremath{-}3}$.