Universality in static spherically symmetric solutions of f(R) gravity

Abstract

f(R) gravity is a well-known modification of General Relativity, that can be reduced to a scalar-tensor theory by a conformal transformation (Einstein frame). We study static spherically symmetric (SSS) asymptotically flat vacuum configurations of the f(R) gravity in the Einstein frame for three known scalaron potentials. The main attention is paid to solutions in case of astrophysically relevant configuration masses and scalaron mass $μ$ larger than several meV. Analytical and numerical analysis reveals remarkably similar properties of some elements of the SSS solutions for different M, $μ$ and sizes of the scalarization region $r_0$. In particular, the scalaron field has universal behavior regardless of the configurations mass and $r_0>100 r_g$ in case of each of the models considered. Moreover, some elements of the solutions are practically the same for the different models. Asymptotic parameters of the metric near the naked singularity at the center of the SSS configuration are obtained for all the models.

Figures (10)

• Figure 1: Potentials: SR2 (\ref{['SF-starobins']}), HT (\ref{['hilltop']}) and TT (\ref{['f(R)table-top']}) with $p=10^7$.
• Figure 2: Universal behavior of $\alpha(r)$ and $\xi(r)$ in case of potentials (\ref{['SF-starobins']}), (\ref{['hilltop']}) and (\ref{['f(R)table-top']}) regardless of $M\mu=10\div 10^{20}$ and various $r_0=(100\div 10^{10})r_g$. In case of the TT model (\ref{['f(R)table-top']}), which is intermediate between SR2 and HT, we used $p=1,10,1000$. All the curves are practically the same after rescaling to $r/r_0$; e.g., rhombs illustrate the coincidence of the SR2 model (\ref{['SF-starobins']}) and the HT case (\ref{['hilltop']}) with the same parameters. Outside the scalarization regions ($r\ge r_0$) $\alpha\approx -r_g/r$ and $\xi$ is given by (\ref{['xi_inf']}).
• Figure 3: The figure shows how $\beta(r)$ depends on the parameters in case of the SR2 model. Two upper curves show the case $\mu M=10^{15}$, for $q=100$ (black) and $q=1000$ (blue). Two lower ones: $\mu M=10^{20}$, $q=100$ (gray) and $q=1000$ (light blue). The curves are cut off at the top right corner, where for all the cases $\beta(r_0)\approx r_g/r_0$. Outside the scalarization region ($r\ge r_0$) $\beta(r)\approx r_g/r\ll 1$. The HT and TT models also show similar dependence on the parameters.
• Figure 4: Dependencies $Y(r)$ (blue) and $Z(r)$ (red) in the scalarization regions after rescaling to $r/r_0$ for the $R2$ model (\ref{['SF-starobins']}) (two upper curves) and for HT scalaron potential (\ref{['hilltop']}) (lower curves); $M\mu= 10^{20}$, the size of the scalarization region is $r_0=10^2\div10^{10}r_g$. The small circles on $Z$ curve and crosses on $Y$ curve show the same dependencies for $M\mu=10$. But the difference between models is considerable. All the curves have a smooth continuation for $r>r_0$ (not shown here).
• Figure 5: The figure shows differences between TT models (\ref{['f(R)table-top']}) with different $p=1,10, 1000$ ($M\mu=10^{15},\, q=1000$) for $Y(r)<Z(r)$ in the scalarization regions after rescaling to $r/r_0$. At the same time, the curves with fixed $p$ are practically independent of the parameters from interval $M\mu=10\div 10^{20},\, q=100\div 10^5$.