A note on the relationship between solutions of Einstein, Ramanujan and Chazy equations

TL;DR

This paper explores the connections between the Einstein equation in cosmology and classical equations from number theory, specifically the Ramanujan and Chazy equations, to generate solutions and relate them to knot theory.

Contribution

It establishes a novel relationship between the Einstein equation and Ramanujan and Chazy equations, enabling solution construction from known number theory solutions.

Findings

Solutions of Einstein equation derived from Ramanujan and Chazy equations.

Connection established between Friedmann equation and knot theory.

Provides a new method to find Einstein solutions using number theory equations.

Abstract

The Einstein equation for the Friedmann-Robertson-Walker metric plays a fundamental role in cosmology. The direct search of the exact solutions of the Einstein equation even in this simple metric case is sometime a hard job. Therefore, it is useful to construct solutions of the Einstein equation using a known solutions of some other equations which are equivalent or related to the Einstein equation. In this work, we establish the relationship the Einstein equation with two other famous equations namely the Ramanujan equation and the Chazy equation. Both these two equations play an imporatant role in the number theory. Using the known solutions of the Ramanujan and Chazy equations, we find the corresponding solutions of the Einstein equation. The relationship between the Friedmann equation and the equations of the trefoil knot and figure-eight knot is established.