Evolution of spherical perturbations in the cosmological environment of degenerate scalar-charged fermions with a scalar Higgs coupling

A mathematical model is constructed for the evolution of spherical perturbations in a cosmological one-component statistical system of completely degenerate scalar-charged fermions with a scalar Higgs coupling. A complete system of self-consistent equations for the evolution of small spherical perturbations is constructed. Singular parts in perturbation modes corresponding to a point-like mass and scalar charge are singled out. We obtain systems of ordinary differential equations for the evolution of the mass and charge of a singular source and systems of partial differential equations for the evolution of nonsingular parts of perturbations. The coefficients of partial differential equations are described by solutions of evolutionary equations for mass and charge. The problem of spatially localized perturbations for solutions that are polynomial in the radial coordinate is reduced to a recurrent system of ordinary linear differential equations for the coefficients of these polynomials. The properties of solutions are studied in the case of cubic polynomials; in particular, it is shown that the radii of localization of gravitational and scalar perturbations coincide and evolve in proportion to the scale factor. The evolution of perturbations is modeled numerically, which in particular confirms the exponential growth of the central mass of the perturbation, and also reveals the oscillatory nature of the evolution of the scalar charge.