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

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 in finite 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 not equal 0 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.