Perturbative approach to $f(R)$-gravitation in FLRW cosmology

TL;DR

This paper develops a perturbative method to derive explicit cosmological solutions in $f(R)$ gravity models within flat FLRW spacetime, linking the evolution of the Universe to an effective cosmological constant and demonstrating applicability to various models.

Contribution

It introduces a general perturbative approach to solve $f(R)$ gravity in FLRW cosmology, providing explicit solutions adaptable to any well-defined $f(R)$ function and illustrating its effectiveness with specific models.

Findings

Derived a general explicit solution for $f(R)$ cosmology.

Showed the solution's applicability to models with realistic universe evolution.

Revealed $f(R)$-gravity as an effective GR with a variable cosmological constant.

Abstract

The $f(R)$ theory of gravitation developed perturbatively around the general theory of relativity with cosmological constant (the \text{$\Lambda$}CDM model) in a flat FLWR geometry is considered. As a result, a general explicit cosmological solution that can be used for any model with an arbitrary, but well-defined, $f(R)$ function (just satisfying given perturbation conditions) is derived. This perturbative solution shows how the Hubble parameter $H (t)$ depends on time (along with the cosmological constant and the matter density) to adapt to the evolution of the Universe. To illustrate, this approach is applied to some specific test models. One of these models appears to be more realistic as it could describe three phases of the Universe's evolution. Despite the fact that the perturbation is applied for a flat FLWR geometry (according to the current cosmological observation) indicates that the obtained solution can mainly describe the evolution of the late Universe, it may also work for an early Universe. As a next step, the present method can be applied to the case with a more general FLRW geometry to increase the precision of the description of different stages in the evolution of the Universe. Finally, it is shown that in a desription of the Universe's evolution the perturbative $f(R)$-theory can be considered as an effective GR with the cosmological constant $\Lambda$ replaced by an effective parameter $ \Lambda_{eff}[\rho(t)] $. This trick leads to a simpler way of solving an $f(R)$-theory regardless its specific form.