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

Abstract

A mathematical model is constructed for the evolution of spherical perturbations in a cosmological onecomponent 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.

Formation of Supermassive Nuclei of Black Holes in the Early Universe by the Mechanism of Scalar-Gravitational Instability. I. Local Picture$${}^{1}$$

“…To solve the first of these problems, let us consider the evolution of small localized spherical perturbations in the cosmological medium of scalarly charged degenerate fermions [4].…”

“…In [4] it is proved that the entire system of equations for disturbances can be written as a system of four linear homogeneous equations, of which the first two equations form an independent subsystem of ordinary homogeneous differential equations for the source functions m(t) and q(t) (( 8) -(9))…”

“…where ρ(r) and χ(r) are given functions. The concept of localized spherical perturbations was introduced in the works of [6] (for more details, see [4]). Localization of spherical disturbances at the initial moment of time means that at this moment of time t 0 = 0 at some radius r 0 in the metric (2) zero boundary conditions are satisfied for the perturbations ν(r, t), φ(r, t) and their first derivatives with respect to radius.…”

“…Secondly, the chain of equations obtained in [4] for the coefficients ρ k (t), χ k (t) of the expansion of nonsingular parts of perturbations ρ(r, t), χ(r, t) through boundary conditions (14) becomes inhomogeneous, -their right-hand sides are determined by solutions of the system of homogeneous equations ( 8) -( 9) with respect to m(t), q(t), defining the singular part perturbations. Note that, thus, the evolution of the singular mass and charge does not depend on the type of perturbation inside the localization radius, determined by the non-singular parts of the perturbations 4 .…”