Compact star in general F(R) gravity: Inevitable degeneracy problem and non-integer power correction

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 Rs 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 dF(R)dR|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(R). This consequence follows that the equation of motion in F(R) gravity includes the second-order derivative of Ricci scalar R, and thus, it is applicable to general F(R) models.