Can higher curvature corrections cure the singularity problem in f(R) gravity?

TL;DR

The paper analyzes whether adding higher-curvature corrections $R^m/\mu^{2(m-1)}$ to viable $f(R)$ gravity models can cure the curvature-singularity problem that prevents neutron-star-like strong-field solutions. By recasting the theory as a scalar-tensor model with a scalar $\chi=f_R$ and studying the effective potential, it shows that regular relativistic stars depend sensitively on the UV scale $\mu$; achieving such stars requires an unnaturally small $\mu$, around $\mu\lesssim10^{-19}$ GeV. It also uncovers an intermediate curvature scale $R_*$ where the UV term starts to dominate, which can spoil laboratory tests via a light scalar, undermining the chameleon mechanism in realistic settings. Overall, the results indicate that constructing viable $f(R)$ models that both pass local gravity tests and permit strong-field stars demands very careful and unnatural fine-tuning, challenging the feasibility of this approach.

Abstract

Although $f(R)$ modified gravity models can be made to satisfy solar system and cosmological constraints, it has been shown that they have the serious drawback of the nonexistence of stars with strong gravitational fields. In this paper, we discuss whether or not higher curvature corrections can remedy the nonexistence consistently. The following problems are shown to arise as the costs one must pay for the $f(R)$ models that allow for neutrons stars: (i) the leading correction must be fine-tuned to have the typical energy scale $μ\lesssim 10^{-19}$ GeV, which essentially comes from the free fall time of a relativistic star; (ii) the leading correction must be further fine-tuned so that it is not given by the quadratic curvature term. The second problem is caused because there appears an intermediate curvature scale and laboratory experiments of gravity will be under the influence of higher curvature corrections. Our analysis thus implies that it is a challenge to construct viable $f(R)$ models without very careful and unnatural fine-tuning.

Figures (9)

• Figure 1: The potential $V$. The inset shows the structure around the de-Sitter minimum. The potential of the original model (without $R^m$ term) is shown by a blue line for purpose of comparison. Parameters are given by $\lambda=2$, $n=1$, $m=2$, and $\varepsilon=5\times10^{-4}$. The point $\chi=1$ corresponds to a curvature singularity in the original model, but the $R^m$ term pushes the curvature singularity toward infinity, $\chi=\infty$.
• Figure 2: The effective potential $U$. The inset shows the structure around the de-Sitter extremum. The effective potential of the original model (without $R^m$ term) is shown by a blue line for purpose of comparison. Parameters are given by $\lambda=2$, $n=1$, $m=2$, and $\varepsilon=5\times10^{-4}$. The dangerous curvature singularity is pushed toward $\chi=\infty$ by the $R^m$ term.
• Figure 3: Plots of the Ricci scalar $R(r)$ for different $\varepsilon$. Parameters are given by $\lambda=2$, $n=1$, $m=2$. The energy density is $4\pi G\rho_0=10^6\Lambda_{{\rm eff}}$ and the central pressure is $p_c = 0.3\rho_0$. Solid (dashed) lines correspond to the region inside (outside) the star. These examples typically give $\hat{G} M/{\cal R} \simeq 0.25$ -- $0.26$.
• Figure 4: Plots of $\chi(r)$ for different $\varepsilon$. Parameters are the same as those in Fig. \ref{['fig:curvature.eps']}. Solid (dashed) lines correspond to the region inside (outside) the star.
• Figure 5: Plots of the metric functions for $\varepsilon=5\times10^{-9}$. Solid (dashed) lines correspond to the region inside (outside) the star.