Friedman vs Abel equations: A connection unraveled

TL;DR

This paper uncovers a bidirectional link between Einstein-Friedmann equations for scalar field cosmology and a special form of Abel equations, enabling new solution methods and transformations for these models.

Contribution

It establishes a novel connection allowing solutions of Abel equations to inform Friedmann equations and vice versa, enhancing analytical tools in cosmological modeling.

Findings

Derived general solutions for Friedmann equations from Abel equations.

Constructed Backlund auto-transformations for Abel equations.

Facilitated analysis of inflation scenarios with exit.

Abstract

We present an interesting connection between Einstein-Friedmann equations for the models of universe filled with scalar field and the special form of Abel equation of the first kind. This connection works in both ways: first, we show how, knowing the general solution of the Abel equation (corresponding to the given scalar field potential) one can obtain the general solution of the Friedman Equation (and use the former for studying such problems as existence of inflation with exit for particular models). On the other hand, one can invert the procedure and construct the B\"{a}cklund auto-transformations for the Abel equation.