On the Newtonian Limit in Gravity Models with Inverse Powers of R

TL;DR

The paper investigates the Newtonian limit in singular nonlinear gravity models with inverse powers of $R$ by expanding around a local maximally symmetric background with $R_0>0$, showing that a correct Newtonian limit on short scales requires $|f(R_0) f''(R_0)| \ll 1$ and, crucially, $f''(R_0)=0$ to suppress higher-derivative terms. It provides two explicit models with $R_0=3\mu^2$ and $R_0=5\mu^2$ that meet this condition and determines the corresponding relation between the gravitational coupling $M$ and the reduced Planck mass, along with their late-time cosmological behavior (e.g., $a(t) \propto t^2$ and $a(t) \propto t^{3.75}$). The work also discusses the possibility of exponential expansion and frames these extensions as minimal CDTT-type modifications, demonstrating that a consistent weak-field expansion can coexist with cosmic acceleration in singular NLG. Overall, the paper shows that Newtonian gravity can be recovered in singular NLG while maintaining viable late-time dynamics, by carefully choosing the background curvature and the function $f(R)$ such that $f''(R_0)=0$.

Abstract

I reconsider the problem of the Newtonian limit in nonlinear gravity models in the light of recently proposed models with inverse powers of the Ricci scalar. Expansion around a maximally symmetric local background with positive curvature scalar R_0 gives the correct Newtonian limit on length scales << R_0^{-1/2} if the gravitational Lagrangian f(R) satisfies |f(R_0)f''(R_0)|<< 1. I propose two models with f''(R_0)=0.