  A mathematical model is constructed for the evolution of spherical
perturbations in a cosmological one-component statistical system of completely
degenerate scalarly charged fermions with a scalar Higgs interaction. A
complete system of self-consistent equations for small perturbations describing
the evolution of spherical perturbations is constructed. Singular parts in
perturbation modes corresponding to point mass and scalar charge are singled
out. Systems of ordinary differential equations are obtained that describe the
evolution of the mass and charge of a singular source, and systems of partial
differential equations that describe the evolution of non-singular parts of
perturbations. In this case, 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 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 in the case of cubic polynomials are studied, in
particular, it is shown that the radii of localization of gravitational and
scalar perturbations coincide and evolve in proportion to the scale factor.
Numerical modeling of the evolution of perturbations was carried out, in
particular, it confirmed the exponential growth of the central mass of the
perturbation, and also revealed the oscillatory nature of the evolution of the
scalar charge. Keywords: scalar charged plasma, cosmological model, scalar
Higgs field, gravitational instability, spherical perturbations.
