Relativistic hierarchical cosmology

Abstract

An account of hierarchical cosmology is given that is based on general relativity, being split up into three pieces for ease of assimilation. Part II will treat of specific solutions relevant to the observed Universe, while Part III compares these models with available empirical data in an effort to pick a model that agrees with observation. Part I derives a metric for a system based on the assumptions of (i) spherical symmetry about any local observer, (ii) a density distribution falling off as ϕ = ϕ₀(t)r ⁻² from any local origin with the hierarchy delineated by step functions, (iii) the Universe was denser and more compact at some epoch in the past. These assumptions (Sections 1, 2) serve to derive a metric (Section 2) that allows of the definition of an effective scale factorS′(t) for Hubble's law, but which involves (Section 3) a curvature (k) and a parameter for measuring the radial coordinate (r/R ₀) that both vary with epoch:k=k(t), R ₀=R ₀ (t). The metric reduces in a special case (k=constant,R ₀=constant,S′(t)=S(t) defined by the Friedmann equations) to that of a Robertson/Walker Universe. Spacetime in the hierarchy (Section 4) can be open or closed depending on the behaviour of the dust. The equations of motion (or the field equations, equivalently) yield two equations analogous to the Friedmann equations, but including terms in the time-derivative of the curvature and a space-variable density of matter. Relativistic hierarchical cosmology provides a wider field of possible investigation with a view to describing the Universe (Section 6) than do the conventional uniform Robertson/Walker models.