Interior spacetimes of stars in Palatini $f(R)$ gravity

We study the interior spacetimes of stars in the Palatini formalism of $f(R)$ gravity and derive a generalized Tolman-Oppenheimer-Volkoff and mass equation for a static, spherically symmetric star. We show that matching the interior solution with the exterior Schwarzschild-de Sitter solution in general gives a relation between the gravitational mass and the density profile of a star, which is different from the one in general relativity. These modifications become negligible in models for which $\delta F(R)\equiv \partial f/\partial R-1$ is a decreasing function of $R$ however. As a result, both Solar System constraints and stellar dynamics are perfectly consistent with $f(R)=R-{\mu }^4/R$.

Figure 1

The effective cosmological constant $\lambda (r)$ (solid) and $F(r)$ (dashed) for $f(R)=R-{\mu }^4/R$ and an assumed density profile $\rho (r)={\rho }_0/(1+e^{\xi (r-r_{\odot })})$ (dotted), where ${\rho }_0$ have been chosen so that $M=M_{\odot }$, the radius $r_{\odot }=2.3\times {10}^5$ ($3\mathrm{km}/2G{M}_{\odot }$), and $\xi =1.4\times {10}^{-5}$. Above, $\lambda$ and $\rho$ have been normalized to the observed amount of dark energy ${\Omega }_{\Lambda }\approx 0.72$ [2] and the central density $\rho (0)$, respectively.