The thermal view of $f(R)$ cosmology

TL;DR

This paper develops and applies a thermal view of gravity to $f(R)$ cosmology, treating the scalar degree of freedom as a dissipative fluid with a gravity temperature product ${\cal K}{\cal T}$ that cools under expansion and heats in strong gravity. By recasting the vacuum $f(R)$ dynamics in FLRW spacetimes, it derives the evolution equation for ${\cal K}{\cal T}$ and analyzes how ${\cal K}{\cal T}$ behaves across polynomial, power-law, and Starobinsky models, identifying regimes where convergence to Einstein gravity (${\cal K}{\cal T}\to0$) occurs and where higher-curvature effects drive departures. The study recovers and unifies a range of known results (e.g., Barrow–Ottewill, STAROBINSKY dynamics) within the thermal framework, clarifying the roles of $R\to0$, $R\to R_0$, and $R\to\infty$ asymptotics and the influence of matter content on the fate of GR. The findings imply that Starobinsky gravity lacks stable de Sitter vacua and that late-time GR behavior is favored in expanding solutions, while strong-gravity phases can sustain heating and departures from GR, offering a cohesive lens on $f(R)$ cosmology and a platform for extending the approach to broader scalar-tensor settings.

Abstract

A new thermal view of scalar-tensor gravity, in which general relativity is the zero-temperature state of gravity, is applied to the specific subclass of $f(R)$ gravity theories and, specifically, to spatially homogeneous and isotropic universes. Within the limits of application of the new thermal formalism, results on the convergence to Einstein cosmology (or lack thereof) are first obtained for general $f(R)$ theories, and then illustrated with power-law and Starobinsky $f(R)$ gravity.

Figures (3)

• Figure 1: In the region of the $\left(\Theta, {\cal K}{\cal T} \right)$ plane between the half-line ${\cal K}{\cal T} = \Theta/(8\pi)$ (black, solid upper line) and the positive $\Theta$-axis, gravity "cools" and approaches GR. Outside of this region, gravity "heats up" and departs from GR. The red, dashed line corresponds to the solution (\ref{['eq:5.5']})-(\ref{['eq:5.6']}) of vacuum $R^n$ gravity; the blue, dash-dotted line corresponds to the solution (\ref{['eq:5.15']})-(\ref{['eq:5.17']}) of $R^n$ gravity with radiation, In both cases, $n=1.3$ and gravity moves towards the origin and GR.
• Figure 2: The orbits of the solutions of Starobinsky gravity are forced to lie on the surface ${\cal K}{\cal T} \left( \Theta,R \right)$ illustrated here in the 3D phase space $\left( \Theta, R, {\cal K}{\cal T} \right)$ (for illustration, we use $\alpha=10^{-3}$ and arbitrary units).
• Figure 3: In the region of the $\left(\Theta, {\cal K}{\cal T} \right)$ plane between the graphs of the functions ${\cal K}{\cal T}_{+}(\Theta)$ (solid) and ${\cal K}{\cal T}_{-}(\Theta)$ (dashed), gravity "cools" and approaches GR. Outside of this region, gravity "heats up" and departs from GR (here we set $\alpha=1$ for illustration).