This short but systematic work demonstrates a link between Chebyshev's theorem and the explicit integration in cosmological time t and conformal time η of the Friedmann equations in all dimensions and with an arbitrary cosmological constant Δ. More precisely, it is shown that for spatially at universes an explicit integration in t may always be carried out, and that, in the non-at situation and when Δ 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 η is complementary to that in t. More precisely, it is shown in the at-universe case with Δ = 0 that an explicit integration in η 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 Δ = 0, an explicit integration can always be carried out whether the space is at, 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.
- Alternatives to inflation
- Cosmological applications of theories with extra dimensions
- String theory and cosmology
ASJC Scopus subject areas
- Astronomy and Astrophysics