Closed-form or parametric solutions for the coupled higher-dimensional cosmological equations

Derive closed-form or parametric solutions for the coupled modified Friedmann equations governing the four-dimensional scale factor a(t) and the time-dependent radius R(t) of the n-dimensional hyperspherical extra dimensions in the 4+n-dimensional cosmological model.

Background

The paper formulates a system of three coupled dynamical equations for a four-dimensional FLRW scale factor a(t), the radius R(t) of n compact hyperspherical dimensions, and the matter density. The extra-dimensional radius and its derivatives enter the modified Friedmann equations nontrivially, coupling the evolution of the observable four-dimensional universe to the dynamics of the compact dimensions.

Although the authors obtain a series-expansion solution for the spatially flat case κ=0, they state that it is too complicated for practical use. A general closed-form or parametric solution of the coupled equations therefore remains unresolved and would be needed for a more complete analytic characterization of the model’s cosmological dynamics.