The realistic models of relativistic stars in f(R) = R + alpha R^2
  gravity
  In the context of f(R)=R + alpha R^2 gravity, we study the existence of
neutron and quark stars with no intermediate approximations in the generalised
system of Tolman-Oppenheimer-Volkov equations. Analysis shows that for positive
alpha's the scalar curvature does not drop to zero at the star surface (as in
General Relativity) but exponentially decreases with distance. Also the stellar
mass bounded by star surface decreases when the value alpha increases.
Nonetheless distant observers would observe a gravitational mass due to
appearance of a so-called gravitational sphere around the star. The non-zero
curvature contribution to the gravitational mass eventually is shown to
compensate the stellar mass decrease for growing alpha's. We perform our
analysis for several equations of state including purely hadronic
configurations as well as hyperons and quark stars. In all cases, we assess
that the relation between the parameter $\alpha$ and the gravitational mass
weakly depend upon the chosen equation of state. Another interesting feature is
the increase of the star radius in comparison to General Relativity for stars
with masses close to maximal, whereas for intermediate masses around 1.4-1.6
solar masses, the radius of star depends upon alpha very weakly. Also the
decrease in the mass bounded by star surface may cause the surface redshift to
decrease in R^2-gravity when compared to Einsteinian predictions. This effect
is shown to hardly depend upon the observed gravitational mass. Finally, for
negative values of alpha our analysis shows that outside the star the scalar
curvature has damped oscillations but the contribution of the gravitational
sphere into the gravitational mass increases indefinitely with radial distance
putting into question the very existence of such relativistic stars.
