Viability of f(R) Theories with Additional Powers of Curvature

TL;DR

This work assesses the viability of modified gravity models of the form $f(R)=R-rac{a}{R^{n}}+bR^{m}$ by deriving both Jordan- and Einstein-frame formulations and examining the scalar degree of freedom's potential, mass, and matter coupling. It analyzes local (Earth/Solar System) and cosmological regimes, including fifth-force constraints, Big Bang Nucleosynthesis, early and late-time acceleration, and the existence of stable minima across media. The key finding is a pronounced tension: no single set of parameters $a,b,m,n$ simultaneously satisfies the requirements of early and late acceleration, stable minima in diverse densities, and BBN limits; BBN constraints are particularly stringent and often incompatible with local minima. Overall, the study suggests that this class of $f(R)$ models struggles to satisfy all observational and experimental constraints, motivating alternative constructions or stronger screening mechanisms and more comprehensive numerical analyses of full field equations for extended bodies.

Abstract

We consider a modified gravity theory, f(R)=R-a/R^n+bR^m, in the metric formulation, which has been suggested to produce late time acceleration in the Universe, whilst satisfying local fifth-force constraints. We investigate the parameter range for this theory, considering the regimes of early and late-time acceleration, Big Bang Nucleosynthesis and fifth-force constraints. We conclude that it is difficult to find a unique range of parameters for consistency of this theory.

Figures (4)

• Figure 1: Numerical solutions showing the evolution of $R(r)$ for two choices of $r_c$. In the left-hand plot, $r_c$ is $10^4$ bigger than in the right-hand plot, so that $\delta_R\rightarrow1$ and $\delta_R\rightarrow0$ respectively. All other parameters are kept constant. The analytic solutions are shown with dotted lines and exactly follow the numerics.
• Figure 2: The ratio of forces due to the scalar field, $\sigma$ and Newtonian potential, shown for different $m$.
• Figure 3: Numerical calculation of the effective field strength, $\alpha_{\rm eff}$ as a function of radius. A constant $\alpha(r)$ denotes a pure Yukawa regime. $R_E=10^{-53}{\rm GeV}^2$, $R_\infty=10^{-55}{\rm GeV}^2$. For the numerics shown: $\lambda/r_E=24-7600$($m=\frac{3}{2}$), $\lambda/r_E=10^{-2}-700$($m=2$), $\lambda/r_E=10^{-2}-70$($m=\frac{5}{2}$), $\lambda/r_E=10^{-1}-3800$($m=3$).
• Figure 4: Schematic diagram showing the relations between regimes for $b$,$\lambda$ and $\alpha$.