Relativistic stars in f(R) gravity, and absence thereof

TL;DR

The paper investigates strong-field behavior in carefully constructed $f(R)$ gravity theories, focusing on the existence of relativistic stars. By formulating the metric and scalar-field equations in a spherically symmetric setting and employing a classical-mechanics analogy for the scalar degree of freedom, the authors show analytically that there is a maximum allowable gravitational potential for stationary stars, beyond which asymptotically de Sitter solutions cannot be sustained. Numerical results for specific Starobinsky-like models confirm that even with weak gravity, the exterior tends toward a de Sitter geometry with a suppressed PPN parameter $\gamma$, and attempts to realize relativistic stars either overshoot to curvature singularities or fail to form a stable thin-shell. The work concludes that, within this class of $f(R)$ theories, neutron stars are not viable unless one imposes special, potentially fine-tuned conditions, implying significant constraints on the viability of these models as complete theories of gravity.

Abstract

Several f(R) modified gravity models have been proposed which realize the correct cosmological evolution and satisfy solar system and laboratory tests. Although nonrelativistic stellar configurations can be constructed, we argue that relativistic stars cannot be present in such f(R) theories. This problem appears due to the dynamics of the effective scalar degree of freedom in the strong gravity regime. Our claim thus raises doubts on the viability of f(R) models.

Figures (7)

• Figure 1: The effective potential $V(\chi)$ for Starobinsky's $f(R)$ model with $n=1$ and $x_1=3.6$ ($\lambda\simeq 2$). $\chi$ is the effective scalar degree of freedom defined by $\chi :=df/dR$.
• Figure 2: The (inverted) potential $U(\chi)$ for Starobinsky's $f(R)$ model with $\lambda=2$ and $n=1$. The point A corresponds to a curvature singularity $(R=+\infty)$, and the point B is the de Sitter extremum. (See also Fig. 1 of Ref. Frolov.)
• Figure 3: Motion of a particle near the de Sitter extremum of $U(\chi)$. The particle feels the force ${\cal F}$ ($<0$) which arises from the trace of the energy-momentum tensor of the matter, ${\cal F}\propto T$.
• Figure 4: Metric for a nonrelativistic star. Parameters are given by $n=1$, $x_1=3.6$, $4\pi G\rho_0=10^6\Lambda_{{\rm eff}}$, and $p_c=5\times10^{-2}\rho_0$. The central value of the Ricci scalar is tuned to be $R_c = 3.462\times 10^{-6}\times8\pi G \rho_0$. The radial coordinate is normalized by the radius of the star ${\cal R}$.
• Figure 5: Numerical solutions of the Ricci scalar and $\chi$ for a nonrelativistic star. Parameters are the same as those in Fig. \ref{['fig:metric1.eps']}. Dashed line is a plot of the analytic approximation (\ref{['apr_sol_chi']}) and (\ref{['chiout']}). Since $R\to x_1R_0$ and $\chi\to\chi_1$ as $r\to\infty$, this is the desired solution with asymptotically de Sitter geometry.