We develop the general scheme for modified $f(R)$ gravity reconstruction from any realistic Friedmann-Robertson-Walker (FRW) cosmology. We formulate several versions of modified gravity compatible with solar system tests where the following sequence of cosmological epochs occurs: (a) matter dominated phase (with or without usual matter), transition from deceleration to acceleration, accelerating epoch consistent with recent WMAP data, (b) $\Lambda \mathrm{CDM}$ cosmology without cosmological constant. As a rule, such modified gravities are expressed implicitly (in terms of special functions) with late-time asymptotics of known type (for instance, the model with negative and positive powers of curvature). In the alternative approach, it is demonstrated that even simple versions of modified gravity may lead to the unification of matter dominated and accelerated phases at the price of the introduction of compensating dark energy.

Dark energy cosmology is considered in a modified Gauss-Bonnet (GB) model of gravity where an arbitrary function of the GB invariant, $f(G)$, is added to the general relativity action. We show that a theory of this kind is endowed with a quite rich cosmological structure: it may naturally lead to an effective cosmological constant, quintessence, or phantom cosmic acceleration, with a possibility for the transition from deceleration to acceleration. It is demonstrated in the paper that this theory is perfectly viable, since it is compliant with the solar system constraints. Specific properties of $f(G)$ gravity in a de Sitter (dS) universe, such as dS and SdS solutions, their entropy, and its explicit one-loop quantization are studied. The issue of a possible solution of the hierarchy problem in modified gravities is also addressed.

We consider modified $f(R)$ gravity which may unify $R^m$ early-time inflation with late-time $\Lambda \mathrm{CDM}$ epoch. It is shown that such a model passes the local tests (Newton law, stability of Earth-like gravitational solution, very heavy mass for additional scalar degree of freedom) and suggests the realistic alternative for general relativity. Various scenarios for the future evolution of $f(R)$ $\Lambda \mathrm{CDM}$ era are discussed.

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).