Integration of D-dimensional 2-factor spaces cosmological models by reducing to the generalized Emden-Fowler equation

TL;DR

This paper reduces the Einstein equations of a D-dimensional cosmological model with two factor spaces and multicomponent perfect fluid to a generalized Emden-Fowler equation, enabling the construction of integrable cosmological solutions.

Contribution

It introduces a method to derive integrable cosmological models by reducing Einstein equations to a known differential equation, expanding the class of solvable models.

Findings

Derived explicit metrics for integrable models

Reduced complex Einstein equations to a solvable differential equation

Demonstrated the method on Ricci-flat spaces with two fluid components

Abstract

The D-dimensional cosmological model on the manifold $M = R \times M_{1} \times M_{2}$ describing the evolution of 2 Einsteinian factor spaces, $M_1$ and $M_2$, in the presence of multicomponent perfect fluid source is considered. The barotropic equation of state for mass-energy densities and the pressures of the components is assumed in each space. When the number of the non Ricci-flat factor spaces and the number of the perfect fluid components are both equal to 2, the Einstein equations for the model are reduced to the generalized Emden-Fowler (second-order ordinary differential) equation, which has been recently investigated by Zaitsev and Polyanin within discrete-group analysis. Using the integrable classes of this equation one generates the integrable cosmological models. The corresponding metrics are presented. The method is demonstrated for the special model with Ricci-flat spaces $M_1,M_2$ and the 2-component perfect fluid source.