Observational manifestations of some models of modified gravity, which have been suggested to explain the accelerated cosmological expansion, are analyzed for gravitating systems with time dependent mass density. It is shown that if the mass density rises with time, the system evolves to the singular state with infinite curvature scalar. The corresponding characteristic time is typically much shorter than the cosmological time.

General features of spontaneous baryogenesis are studied. The relation between the time derivative of the (pseudo) Goldstone field and the baryonic chemical potential is revisited. It is shown that this relation essentially depends upon the representation chosen for the fermionic fields with nonzero baryonic number (quarks). The calculations of the cosmological baryon asymmetry are based on the kinetic equation generalized to the case of nonstationary background. The effects of the finite interval of the integration over time are also taken into consideration. All these effects combined lead to a noticeable deviation of the magnitude of the baryon asymmetry from the canonical results.

Gravitational instability in classical Jeans theory, General Relativity, and modified gravity is considered. The background density increase leads to a faster growth of perturbations in comparison with the standard theory. The transition to the Newtonian gauge in the case of coordinate dependent background metric functions is studied. For modified gravity a new high frequency stable solution is found

The problem of neutrino spin rotation in dense matter and in strong electromagnetic fields is solved in accordance with the basic principles of quantum mechanics. We obtain a complete system of wave functions for a massive Dirac neutrino with an anomalous magnetic moment which are the eigenfunctions of the kinetic momentum operator and have the form of nonspreading wave packets. These wave functions enable one to consider the states of neutrino with rotating spin as pure quantum states and can be used for calculating probabilities of various processes with the neutrino in the framework of the Furry picture.