# Universal expansion with spatially varying $G$

## TL;DR

This paper explores a cosmological model where the gravitational constant varies spatially, leading to a universe expanding linearly with time driven by inertia, with implications for dark energy and energy conservation.

## Contribution

It introduces a novel cosmological model with spatially varying G in the MOND regime, showing linear expansion and non-reducibility when including dark energy effects.

## Key findings

- Universe's scale factor increases linearly with time.
- Hubble's law emerges naturally without extra assumptions.
- Inclusion of dark energy breaks energy conservation, leading to a new Thom catastrophe.

## Abstract

We calculate the expansion of the universe under the assumptions that $G$ varies in space and the radial size $r$ of the universe is very large (we call this the MOND regime of varying-$G$ gravity). The inferred asymptotic behavior turns out to be different than that found by McCrea & Milne in 1934 and our equations bear no resemblance to those of the relativistic case. In this cosmology, the scale factor $R(t)$ increases linearly with time $t$, the radial velocity is driven by inertia, and gravity is incapable of hindering the expansion. Yet, Hubble's law is borne out without any additional assumptions. When we include a repulsive acceleration $a_{\rm de}$ due to dark energy, the resulting universal expansion is then driven totally by this new term and the solutions for $a_{\rm de}\to 0$ do not reduce to those of the $a_{\rm de}\equiv 0$ case. This is a realization of a new Thom catastrophe: the inclusion of the new term destroys the conservation of energy and the results are not reducible to the previous case in which energy is conserved.