On Euclidean Limit Cycles in Cosmological Models Based on Scalar Fields

Abstract

A detailed analysis of the phase trajectories of cosmological models based on classical and scalar fields near surfaces of zero effective energy has been carried out. A study of the differential parameters of the convergence of phase trajectories to a zero-energy surface boundary shows that the phase trajectories merge within a finite time with the phase trajectories of free oscillations corresponding to zero effective energy. This confirms the assumption formulated in a number of previous works by one of the authors about the existence of Euclidean limit cycles in cosmological models based on scalar fields with a Higgs interaction potential.

Complete cosmological model based on a asymmetric scalar Higgs doublet

“…Obviously, these conditions coincide with the equations( 47), (48). Calculating the second derivatives, we find…”

“…Calculating the matrix of the dynamic system (36) -( 38) a ik = ∂ k Pi, where Pi are the right-hand sides of the dynamic system equations, we find with on account of ( 47) - (48):…”

“…On the other hand, in a number of later but independent works [45] - [48], the methods of the qualitative theory of dynamical systems and numerical-analytical methods were used to study cosmological models based on scalar fields, both with a quadratic potential and with Higgs potential. In particular, in the papers [46,47], two-field cosmological models, called in these papers the asymmetric scalar doublet, were investigated.…”

“…In this case, space-time becomes pseudo-Euclidean, although the scalar fields are nonzero and perform free nonlinear oscillations. Such vacuum oscillations were studied in detail in the work [48].…”

“…In a private conversation with one of the Author on the sidelines of the conference on gravity, cosmology and astrophysics of the BRICS countries (Kazan, September 2019), Sergei Vernov expressed the opinion that the system of dynamic equations considered in the works [45] - [48] describing cosmological evolution may not be equivalent to the original Einstein system of equations. More precisely, the restriction of the Hubble constant to non-negative values (1) may contradict the original system of Einstein's equations.…”