The Power of General Relativity

TL;DR

This paper investigates extended theories of gravity based on a power-law Lagrangian, analyzing their cosmological and weak-field properties, and deriving tight observational bounds on the deviation parameter from general relativity.

Contribution

It provides a comprehensive analysis of the cosmological and weak-field implications of $f(R) o R^{1+ ext{delta}}$ theories, establishing new constraints on the deviation parameter $ ext{delta}$.

Findings

Stable matter-dominated era requires $ ext{delta} > 0$ or $ ext{delta} < -1/4$.

Helium and element synthesis constrain $ ext{delta}$ to be between -0.017 and 0.0012.

Perihelion precession yields a bound on $ ext{delta}$ of approximately $10^{-19}$.

Abstract

We study the cosmological and weak-field properties of theories of gravity derived by extending general relativity by means of a Lagrangian proportional to $R^{1+\delta}$. This scale-free extension reduces to general relativity when $\delta \to 0$. In order to constrain generalisations of general relativity of this power class we analyse the behaviour of the perfect-fluid Friedmann universes and isolate the physically relevant models of zero curvature. A stable matter-dominated period of evolution requires $\delta >0$ or $\delta <-1/4$. The stable attractors of the evolution are found. By considering the synthesis of light elements (helium-4, deuterium and lithium-7) we obtain the bound $-0.017<\delta <0.0012.$ We evaluate the effect on the power spectrum of clustering via the shift in the epoch of matter-radiation equality. The horizon size at matter--radiation equality will be shifted by $\sim 1%$ for a value of $\delta \sim 0.0005.$ We study the stable extensions of the Schwarzschild solution in these theories and calculate the timelike and null geodesics. No significant bounds arise from null geodesic effects but the perihelion precession observations lead to the strong bound $\delta =2.7\pm 4.5\times 10^{-19}$ assuming that Mercury follows a timelike geodesic. The combination of these observational constraints leads to the overall bound $0\leq \delta <7.2\times 10^{-19}$ on theories of this type.