# Unification of Inflation with Dark Energy in $f(R)$ Gravity and Axion Dark Matter

## TL;DR

This paper proposes a unified model where $f(R)$ gravity with a non-minimal coupling to axion dark matter explains both inflation and late-time acceleration, predicting a primordial stiff era and aligning with current cosmological constants.

## Contribution

It introduces a novel $f(R)$ gravity model with non-minimal coupling to axions, unifying inflation, dark matter, and dark energy phenomena in a single framework.

## Key findings

- Axion behaves as cold dark matter with $
ho_a \\sim a^{-3}$.
- Late-time universe exhibits de Sitter expansion with $H \\sim \\Lambda^{1/2}$.
- Model predicts a pre-inflationary stiff era with $
ho_a \\sim a^{-6}$.

## Abstract

In this work we introduce an effective model of $f(R)$ gravity containing a non-minimal coupling to the axion scalar field. The axion field is described by the misalignment model, in which the primordial $U(1)$ Peccei-Quinn symmetry is broken during inflation and the $f(R)$ gravity is described by the $R^2$ model, and in addition, the non-minimal coupling has the form $\sim h(\phi)R^{\gamma}$, with $0<\gamma<0.75$. By appropriately constraining the non-minimal coupling at early times, the axion field remains frozen in its primordial vacuum expectation value, and the $R^2$ gravity dominates the inflationary era. As the Universe expands, when $H$ equals the axion mass $m_a$ and for cosmic times for which $m_a\gg H$, the axion field oscillates. By assuming a slowly varying evolution of the axion field, the axion energy density scales as $\rho_a\sim a^{-3}$, where $a$ is the scale factor, regardless of the background Hubble rate, thus behaving as cold dark matter. At late times, the axion still evolves as $\rho_a\sim a^{-3}$, however the Hubble rate of the expansion and thus the dynamical evolution of the Universe is controlled by terms containing the higher derivatives of $\sim R^{\gamma}$, which are related to the non-minimal coupling, and as we demonstrate, the resulting solution of the Friedman equation at late times is an approximate de Sitter evolution. The late-time de Sitter Hubble rate scales as $H\sim \Lambda^{1/2}$, where $\Lambda$ is an integration constant of the theory, which has its allowed values very close to the current value of the cosmological constant. Finally, the theory has a prediction for the existence of a pre-inflationary primordial stiff era, in which the energy density of the axion scales as $\rho_a\sim a^{-6}$.