Dynamics of a spherical object of uniform density in an expanding universe

Abstract

We present Newtonian and fully general-relativistic solutions for the evolution of a spherical region of uniform interior density ρ_i(t), embedded in a background of uniform exterior density ρ_e(t). In both regions, the fluid is assumed to support pressure. In general, the expansion rates of the two regions, expressed in terms of interior and exterior Hubble parameters H_i(t) and H_e(t), respectively, are independent. We consider in detail two special cases: an object with a static boundary, H_i(t)=0; and an object whose internal Hubble parameter matches that of the background, H_i(t)=H_e(t). In the latter case, we also obtain fully general-relativistic expressions for the force required to keep a test particle at rest inside the object, and that required to keep a test particle on the moving boundary. We also derive a generalised form of the Oppenheimer-Volkov equation, valid for general time-dependent spherically-symmetric systems, which may be of interest in its own right.