A fresh view of cosmological models describing very early Universe: general solution of the dynamical equations

TL;DR

This paper presents a novel method for solving the dynamical equations of spherical cosmological models with scalar fields by changing the time variable, allowing for complete integration in general gauges and potentials.

Contribution

It introduces a new approach to integrate the equations of spherical cosmology with scalar fields by replacing the time variable with a metric function, enabling solutions for arbitrary potentials and gauges.

Findings

Complete integration of the general spherical theory in any gauge.

Identification and explicit derivation of inflationary solutions.

Discovery of classically forbidden regions in the solution space.

Abstract

The dynamics of any spherical cosmology with a scalar field (`scalaron') coupling to gravity is described by the nonlinear second-order differential equations for two metric functions and the scalaron depending on the `time' parameter. The equations depend on the scalaron potential and on the arbitrary gauge function that describes time parameterizations. This dynamical system can be integrated for flat, isotropic models with very special potentials. But, somewhat unexpectedly, replacing the `time' variable by one of the metric functions allows us to completely integrate the general spherical theory in any gauge and with apparently arbitrary potentials. The main restrictions on the potential arise from positivity of the derived analytic expressions for the solutions, which are essentially the squared canonical momenta. An interesting consequence is emerging of classically forbidden regions for these analytic solutions. It is also shown that in this rather general model the inflationary solutions can be identified, explicitly derived, and compared to the standard approximate expressions. This approach can be applied to intrinsically anisotropic models with a massive vector field (`vecton') as well as to some non-inflationary models.