Scaling Solutions and reconstruction of Scalar Field Potentials

TL;DR

This paper derives the general form of scalar field potentials under the hypothesis of scaling solutions, providing explicit solutions for two-fluid cases and a quadrature-based approach for three-fluid cases.

Contribution

It introduces a unified method to determine scalar field potentials for multiple fluid scenarios based on scaling solutions, including explicit forms and quadrature-based solutions.

Findings

Potential for two fluids is a negative power of hyperbolic sine.

Analytic form for three fluids is obtained via quadratures.

General exact form of scalar field potential derived from scaling hypothesis.

Abstract

Starting from the hypothesis of scaling solutions, the general exact form of the scalar field potential is found. In the case of two fluids, it turns out to be a negative power of hyperbolic sine. In the case of three fluids the analytic form is not found, but is obtained by quadratures.