We formulate and investigate a mathematical model of the S^ (0) cosmological system based on a classical scalar field with self-interaction and an ideal scalar-neutral fluid. For a nonzero fluid energy density, the system of equations for the perturbations is reduced to a Lifshitz–Khalatnikov form, which is used to study the cosmological evolution of perturbations at singular points of the S^ (0) background model. At these points, the scalar field, on the one hand, and the gravitational perturbations and the fluid, on the other, become independent subsystems. We find exact solutions for the perturbations of the scalar field, gravitational perturbations, and the perturbation of the energy-momentum of the fluid for its nonrelativistic and ultrarelativistic states. Near stable singular points of the background, the perturbations decay, while near unstable singular points, they grow exponentially rapidly. We find an asymptotic solution to the equation for the perturbation of a scalar field for sufficiently large wavenumbers. This solution is used to establish necessary and sufficient conditions for system instability and evolution. We establish laws for scaling the results of perturbation theory to the parameters of known field-theoretical interaction models.
Yu. G. Ignat’ev (Sun,) studied this question.