Exact solutions of Friedmann equation

TL;DR

This paper derives exact solutions to the Friedmann equation with scalar fields for various potentials, providing closed-form, series, and asymptotic solutions, and proves their existence.

Contribution

It presents new exact solutions for the Friedmann equation with scalar fields for multiple potentials, including constant, exponential, quadratic, and arbitrary forms.

Findings

Exact solutions for constant and exponential potentials in closed form.

Series solutions for quadratic potentials near spiral and attractor regions.

Existence proofs for all derived classical solutions.

Abstract

The cosmological Friedmann equation for the universe filled with a scalar field is reduced to a system of two equations of the first order, one of which is an equation with separable variables. For the second equation the exact solutions are given in closed form for potentials as constants and exponents. For the same equation exact solutions for quadratic potential are written in the form of a series in the spiral and attractor areas. Also exact solutions for very arbitrary potentials are given in the neighborhood of endpoint and infinity. The existence of all these classical solutions is proven.