Coincidence Problem and Holographic $f(R)$ Gravity in Spatially flat and Curved Universes

TL;DR

This paper explores how holographic $f(R)$ gravity models in flat and curved universes can address the coincidence problem by producing nearly constant ratios of dark matter to dark energy, depending on the universe's curvature.

Contribution

It demonstrates that holographic $f(R)$ gravity models can naturally explain the near constancy of dark matter and dark energy densities in cosmology.

Findings

In flat universes, the ratio of dark matter to dark energy remains constant.

In curved universes, the ratio varies but can be nearly constant in viable models.

The approach offers a potential solution to the coincidence problem.

Abstract

The $f(R)$ gravity models formulated in Einstein conformal frame are equivalent to Einstein gravity together with a minimally coupled scalar field. The scalar field couples with the matter sector and the coupling term is given by the conformal factor. We apply the holographic principle to such an interacting model in spatially flat and curved universes. We show that the model leads to a constant ratio of energy densities of dark matter to dark energy in a spatially flat universe. In a spatially curved universe, the ratio is not a constant and the evolution seems to be model-dependent. However, we argue that any cosmologically viable $f(R)$ model can lead to a nearly constant ratio of energy densities and therefore alleviate the coincidence problem.