Friedmann's equations in all dimensions and Chebyshev's theorem

Abstract

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