Friedmann's Equations in All Dimensions and Chebyshev's Theorem

TL;DR

This paper explores the explicit integration of Friedmann equations across all dimensions using Chebyshev's theorem, revealing conditions under which solutions can be expressed explicitly for various cosmological scenarios.

Contribution

It establishes a systematic link between Chebyshev's theorem and explicit solutions of Friedmann equations in all dimensions with arbitrary cosmological constant, including conditions for flat and non-flat universes.

Findings

Explicit integration in cosmological time $t$ for flat universes is always possible.

Explicit integration in conformal time $ta$ depends on specific relations among dimension $n$ and equation of state parameter $w$.

Method can be extended to more realistic models with nonlinear equations of state.

Abstract

This short but systematic work demonstrates a link between Chebyshev's theorem and the explicit integration in cosmological time $t$ and conformal time $\eta$ of the Friedmann equations in all dimensions and with an arbitrary cosmological constant $\Lambda$. More precisely, it is shown that for spatially flat universes an explicit integration in $t$ may always be carried out, and that, in the non-flat situation and when $\Lambda$ is zero and the ratio $w$ of the pressure and energy density in the barotropic equation of state of the perfect-fluid universe is rational, an explicit integration may be carried out if and only if the dimension $n$ of space and $w$ obey some specific relations among an infinite family. The situation for explicit integration in $\eta$ is complementary to that in $t$. More precisely, it is shown in the flat-universe case with $\Lambda\neq0$ that an explicit integration in $\eta$ can be carried out if and only if $w$ and $n$ obey similar relations among a well-defined family which we specify, and that, when $\Lambda=0$, an explicit integration can always be carried out whether the space is flat, closed, or open. We also show that our method may be used to study more realistic cosmological situations when the equation of state is nonlinear.