Neutron stars in Starobinsky model
  We study the structure of neutron stars in $f(R)=R+\alpha R^{2}$ theory of
gravity (Starobinsky model), in an exact and non-perturbative 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 needs to be satisfied. For an exterior Schwarzschild solution, the
curvature scalar and its derivative has to 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 that EoS, only a very
particular set of boundary conditions at the centre 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 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.
