Machian derivation of the Friedmann equation

TL;DR

This paper presents a Machian derivation of the Friedmann equation that remains valid even when a gravitational horizon exists, linking total energy to cosmological constant in de Sitter universe.

Contribution

It introduces a Machian approach to derive the Friedmann equation without relying on the shell theorem, applicable in universes with a gravitational horizon.

Findings

Machian total energy density is constant in de Sitter universe

Friedmann equation derivation is valid with a gravitational horizon

Total energy's identity depends on horizon evolution

Abstract

Despite all fundamental objections against Newtonian concepts in cosmology, the Friedmann equation derives from these in an astoundingly simple way through application of the shell theorem and conservation of Newtonian energy in an infinite universe. However, Friedmann universes in general posses a finite gravitational horizon, as a result of which the application of the shell theorem fails and the Newtonian derivation collapses. We show that in the presence of a gravitational horizon the Friedmann equation can be derived from a Machian definition of kinetic energy, without invoking the shell theorem. Whereas in the Newtonian case total energy translates to curvature energy density, in the Machian case total energy takes on different identities, depending on the evolution of the horizon; we show that in the de Sitter universe Machian total energy density is constant, i.e. appears as cosmological constant.