Neutron stars in the Starobinsky model

We study the structure of neutron stars in the $f(R)=R+\ensuremath{\alpha}{R}^{2}$ theory of gravity (the Starobinsky model) in an exact and nonperturbative approach. In this model, apart from the standard general relativistic junction conditions, two extra conditions---namely, the continuity of the curvature scalar and its first derivative---need to be satisfied. For an exterior Schwarzschild solution, the curvature scalar and its derivative must be zero at the stellar surface. We show that for some equation of state (EoS) of matter, matching all conditions at the surface of the star is impossible. Hence the model brings two major fine-tuning problems: (i) only some particular classes of EoS are consistent with Schwarzschild at the surface, and (ii) given the EoS, only a very particular set of boundary conditions at the center of the star will satisfy the given boundary conditions at the surface. Hence we show that this model [and subsequently many other $f(R)$ models where the uniqueness theorem is valid] is highly unnatural for the existence of compact astrophysical objects. This is because the EoS of a compact star should be completely determined by the physics of nuclear matter at high density and not the theory of gravity.