Compact Star in General $F(R)$ Gravity: Inevitable Degeneracy Problem and Non-Integer Power Correction

TL;DR

This paper explores how the mass-radius relation of compact stars in $F(R)$ gravity can be degenerate with the equation of state, and shows that boundary conditions imply non-integer power corrections to Einstein gravity, constraining $F(R)$ models.

Contribution

It introduces a novel formulation for compact stars in $F(R)$ gravity, revealing degeneracy issues and deriving boundary conditions that restrict the functional form of $F(R)$ with non-integer power corrections.

Findings

Mass-radius relation is insufficient to constrain $F(R)$ gravity.

Boundary conditions imply non-integer power corrections to Einstein gravity.

The equation of motion involves fourth-order derivatives of the metric.

Abstract

We investigate a compact star in the general $F(R)$ gravity. Developing a novel formulation in the spherically symmetric and static space-time with the matter, we confirm that an arbitrary relation between the mass $M$ and the radius $R_s$ of the compact star can be realized by adjusting the functional form of $F(R)$. Such a degeneracy with a choice of the equation of state (EOS) suggests that only mass-radius relation is insufficient to constrain the $F(R)$ gravity. Furthermore, by solving the differential equation for $\left. \frac{dF(R)}{dR}\right|_{R=R(r)}$ near and inside the surface of the compact star with the polytropic EOS, the boundary condition demands a weak curvature correction to the Einstein gravity could be non-integer power of the scalar curvature, which gives a stringent constraint on the functional form of $F\left(R\right)$. This consequence follows that the equation of motion in $F(R)$ gravity includes the fourth-order derivative of metric, and thus, it is applicable to general $F(R)$ models.