Neutron star models in perturbative f(R) gravity are considered with realistic equations of state. In particular, we consider the FPS, SLy and other equations of state and a case of piecewise equation of state for stars with quark cores. The mass-radius relations for f(R) = R+R(e−R/R0−1) model and for R2 models with logarithmic and cubic corrections are obtained. In the case of R2 gravity with cubic corrections, we obtain that at high central densities (ρ > 10ρns, where ρns = 2.7 × 1014 g/cm3 is the nuclear saturation density), stable star configurations exist. The minimal radius of such stars is close to 9 km with maximal mass ~ 1.9M⊙ (SLy equation). A similar situation takes place for AP4 and BSK20 EoS. Such an effect can give rise to more compact stars than in General Relativity. If observationally identified, such objects could constitute a formidable signature for modified gravity at astrophysical level. Another interesting result can be achieved in modified gravity with only a cubic correction. For some EoS, the upper limit of neutron star mass increases and therefore these EoS can describe realistic star configurations (although, in General Relativity, these EoS are excluded by observational constraints).

The so-called “hyperon puzzle” in the theory of neutron stars is considered in the framework of modified $f(R)$ gravity. We show that for simple hyperon equations of state, it is possible to obtain the maximal neutron star mass which satisfies the recent observational data for PSR J1614-2230, in higher-derivative models with power-law terms as $f(R)=R+\gamma R^2+\beta R^3$. The soft hyperon equation of state under consideration is usually treated as unrealistic in the standard general relativity. The numerical analysis of the mass-radius relation for massive neutron stars with a hyperon equation of state in modified gravity turns out to be consistent with observations. Thus, we show that the same modified gravity can solve at once three problems: a consistent description of the maximal mass of a neutron star, a realistic mass-radius relation, and a method to account for hyperons in the equation of state.

In the context of f ( R ) = R + α R 2 gravity, we study the existence of neutron and quark stars for various α with no intermediate approximation in the system of equations. Analysis shows that for positive α 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 α 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 α ’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 α and the gravitational mass weakly depends upon the chosen equation of state. Another interesting feature is the increase of the star radius in comparison with general relativity for stars with masses close to maximal, whereas for intermediate masses 1.4 – 1.6 M ⊙ the radius of star depends upon α 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 α 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.

Quark star models with realistic equation of state in nonperturbative f(R) gravity are considered. The mass-radius relation for f(R)=R+αR2 model is obtained. Considering scalar curvature R as an independent function, one can find out, for each value of central density, the unique value of central curvature for which one has solutions with the required asymptotic R→0 for r→∞. In other words, one needs a fine-tuning for R to achieve quark stars in f(R) gravity. We consider also the analogue description in corresponding scalar-tensor gravity. The fine-tuning on R is equivalent to the fine-tuning on the scalar field ϕ in this description. For distant observers, the gravitational mass of the star increases with increasing α (α>0) but the interpretation of this fact depends on frame where we work. Considering directly f(R) gravity, one can say that increasing of mass occurs by the “gravitational sphere” outside the star with some “effective mass”. On the other hand, in conformal scalar-tensor theory, we also have a dilaton sphere (or “disphere”) outside the star but its contribution to gravitational mass for distant observer is negligible. We show that it is possible to discriminate modified theories of gravity from General Relativity due to the gravitational redshift of the thermal spectrum emerging from the surface of the star

In this paper we propose and extensively study mimetic modified gravity models, with various scenarios of cosmological evolution, with or without extra matter fluids. The easiest formulation is based on the use of the Lagrange multiplier constraint. In certain versions of this theory, it is possible to realize accelerated expansion of the Universe or even unified evolution, which includes inflation with dark energy, and at the same time in the same theoretical framework, dark matter is described by the theory. This is achieved by the re-parametrization of the metric tensor, which introduces a new degree of freedom in the cosmological equations and leads to the appearance of the mimetic ‘dark matter’ component. In the context of the mimetic theory, we also provide some quite general reconstruction schemes, which enable us to find which gravity generates a specific cosmological evolution. In addition, we also provide the general reconstruction technique for the Lagrange multiplier gravity. All our results are accompanied by illustrative examples, with special emphasis on bouncing cosmologies.