Abstract. In the present paper we investigate non-perturbatively and self-consistently the structure of neutron stars in R-squared gravity by simultaneously solving the interior and exterior problem. The mass-radius relations are obtained for several equations of state and for wide range of the R-squared gravity parameter a. Even though the deviation from general relativity for nonzero values of a can be large, they are still comparable with the variations due to different modern realistic equations of state. That is why the current observations of the neutron star masses and radii alone can not put constraints on the value of the parameter a. We also compare our results with those obtained within the perturbative method and we discuss the differences between them.

In the present paper we investigate self-consistently slowly rotating neutron and strange stars in Rsquared gravity. For this purpose we first derive the equations describing the structure of the slowly rotating compact stars in f (R)-gravity and then simultaneously solve the exterior and the interior problem. The structure of the slowly rotating neutron stars is studied for two different hadronic equations of state and a strange matter equation of state. The moment of inertia and its dependence on the stellar mass and the R-squared gravity parameter a is also examined in details. We find that the neutron star moment of inertia for large values of the parameter a can be up to 30% larger compared to the corresponding general relativistic models. This is much higher than the change in the maximum mass induced by R-squared gravity and is beyond the EOS uncertainty. In this way the future observations of the moment of inertia of compact stars could allow us to distinguish between general relativity and f (R) gravity, and more generally to test the strong field regime of gravity.

We study the oscillations of neutron and strange stars in R 2 gravity. More precisely the nonradial f -modes are examined and the differences with pure general relativity are investigated. Using these results we build several gravitational wave asteroseismology relations. Our goal is to determine up to what extend these relations are equation of state independent and whether they deviate enough from general relativity in order to produce an observable effect. The results show that the differences coming from R 2 gravity are up to 10% and that will be difficult to be observed in the near future. On the other hand the small deviations in some of the asteroseismology relations show that they are not only equation of state independent, but they are also quite insensitive to the gravitational theory. That is why solving the inverse problem can give us quite robust estimates of the neutron star parameters.

We study how rapid rotation influences the relation between the normalized moment of inertiaĪ and quadrupole momentQ for scalarized neutron stars. The questions one has to answer are whether the EOS universality is preserved in this regime and what are the deviations from general relativity. Our results show that theĪ −Q relation is nearly EOS independent for scalarized rapidly rotating stars, but the differences with pure Einstein's theory increase compared to the slowly rotating case. In general, smaller negative values of the scalar field coupling parameters β lead to larger deviations, but these deviations are below the expected accuracy of the future astrophysical observations if one considers values of β in agreement with the current observational constraint. An important remark is that although the normalizedĪ −Q relation is quite similar for scalar-tensor theories and general relativity, the unnormalized moment of inertia and quadrupole moment can be very different in the two theories. This demonstrates that although theĪ −Q relations are potentially very useful for some purposes, they might not serve us well when trying to distinguish between different theories of gravity.