This paper is a continuation of our earlier study on the integrability of the Friedmann equations in the light of the Chebyshev theorem. Our main focus will be on a series of important, yet not previously touched, problems when the equation of state
for the perfect-fluid universe is nonlinear. These include the generalized Chaplygin gas, two-term energy density, trinomial Friedmann, Born--Infeld, and two-fluid models. We show that some of these may be integrated using Chebyshevs result while other are out of reach by the theorem but may be integrated explicitly by other methods. With the explicit integration, we are able to understand exactly the roles of the physical parameters in various models play in the cosmological evolution. For example, in the Chaplygin gas universe, it is seen that, as far as there is a tiny presence of nonlinear matter, linear matter makes contribution to the dark matter, which becomes significant near the phantom divide line. The Friedmann equations also arise in areas of physics not directly related to cosmology. We provide some examples ranging from geometric optics and central orbits to soap films and the shape of glaciated valleys to which our results may be applied.
This short but systematic work demonstrates a link between Chebyshevs 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 $Lam
bda$. 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 $Lambda eq0$ 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.
