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

Abstract

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 Λ. 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 Λ 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 flat-universe case with Λ = 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 Λ = 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.Keywords: Astrophysical fluid dynamics, cosmology with extra dimensions, alternatives to inflation, initial conditions and eternal universe, cosmological applications of theories with extra dimensions, string theory and cosmology.

Friedmann-Lemaitre cosmologies via roulettes and other analytic methods

“…In the recent systematic works [1,2], we explored a link between the Chebyshev theorem [3,4] and the integrability of the Friedmann equations and obtained a wealth of new explicit solutions when the barotropic equation of state is either linear or nonlinear, in both cosmic and conformal times, and for flat and non-flat spatial sections with and without a cosmological constant. The purpose of the present paper is to present several analytic methods which may be used to obtain either exact expressions or insightful knowledge of the solutions of the Friedmann equations beyond the reach of the Chebyshev theorem.…”

“…Recall also that the Chebyshev theorem applies only to integrals of binomial differentials [1,2]. In cosmology, one frequently encounters models which cannot be converted into such integrals.…”

Traveling wave solutions of degenerate coupled multi-KdV equations

“…Here 11) and the graph of this solution for −2 ≤ ξ < π/2 − 2 is given in Figure 7 Figure 7: Graph of a solution of (y ) 2 = y(1 + y) 4 3) LetP 5 (y) = y 2 (α + βy 3 ). This form corresponds to the case of one simple and one double zeros.…”

Explicit integration of Friedmann's equation with nonlinear equations of state

“…Furthermore, we illustrated with an example that our method is also useful in the study of more realistic cosmological situations when the equation of state is nonlinear. The purpose of our current paper to explore the method further to get exact integrations of some other important cases which are not covered in [1] and give a few examples whose integrability is beyond the reach of the Chebyshev theorem but can be obtained by other means. We shall also see how the integration helps understand some general situations when integration is not possible.…”

“…When the equation of state of the perfect-fluid universe is nonlinear, the situations include the generalized Chaplygin gas, two-term energy density, the trinomial Friedmann, 1 Born-Infeld, and two-fluid models. Specifically, for the generalized Chaplygin gas model, we work on the flat-universe situation with zero cosmological constant and identify all integrable cases.…”