Friedmann's equations in all dimensions and Chebyshev's theorem
- Institute of Contemporary Mathematics, School of Mathematics and Statistics, Henan University, Kaifeng, Henan 475004 (China)
- D.A.M.T.P., University of Cambridge, Cambridge CB3 0WA (United Kingdom)
- School of Mathematics and Statistics, Henan University, Kaifeng, Henan 475004 (China)
- Department of Mathematics, Polytechnic School, New York University, Brooklyn, New York 11201, U.S.A (United States)
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.
- OSTI ID:
- 22382070
- Journal Information:
- Journal of Cosmology and Astroparticle Physics, Vol. 2014, Issue 12; Other Information: Country of input: International Atomic Energy Agency (IAEA); ISSN 1475-7516
- Country of Publication:
- United States
- Language:
- English
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