Adventures in Friedmann cosmology: A detailed expansion of the cosmological Friedmann equations

In this paper we study the integrability of the Friedmann equations, when the equation of state for the perfect-fluid universe is nonlinear, in the light of the Chebyshev theorem. A series of important, yet not previously touched, problems will be worked out which include the generalized Chaplygin gas, two-term energy density, trinomial Friedmann, Born-Infeld, two-fluid models, and Chern-Simons modified * Email address: chensx@henu.edu.cn † Email address: gwg1@damtp.cam.ac.uk ‡ Email address: yisongyang@nyu.edu 1 gravity theory models. With the explicit integration, we are able to understand exactly the roles of the physical parameters in various models play in the cosmological evolution which may also offer clues to a profound understanding of the problems in general settings. For example, in the Chaplygin gas universe, a few integrable cases lead us to derive a universal formula for the asymptotic exponential growth rate of the scale factor, of an explicit form, whether the Friedmann equation is integrable or not, which reveals the coupled roles played by various physical sectors and it is seen that, as far as there is a tiny presence of nonlinear matter, conventional 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.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.

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“…One has 20) and therefore if B < ρ 2 , the dominant energy condition, i.e. ρ ≥ |P |, holds, which is consistent with the sound speed v s = ∂P ∂ρ…”

Section: Examplessupporting

confidence: 58%

“…To proceed further, we set w = v 2 in (4.19) and use the partial fraction decomposition 20) to arrive at…”

Section: The Chaplygin Gas Modelmentioning

confidence: 99%

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

Chen

Gibbons

Yang

2015

J. Cosmol. Astropart. Phys.

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“…The evolution of the universe as predicted by the Einstein-Friedman equations [7] when dominated by a single, isotropic, stable, static, perfect-fluid energy form is considered for different values of the gravitational pressure to density ratio . The main result, which should be outlined, is that the dependence of density of interstellar matter in expanding Universe is proved to be given by the appropriate quasielliptical integral.…”

Section: Discussionmentioning

confidence: 99%

“…While the value of  may in principle change with redshift, it is often assumed that  is independent of time just for simplicity [5][6][7].…”

Section: Reduction Of the Initial System Of Equationsmentioning

confidence: 99%

Quasi-Periodic Solutions of Einstein-Friedman Equations

Ershkov¹

2014

Int. J. of Pure and Appl. Math.

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A new derivation for the ultra-elliptic solutions of Einstein-Friedman equations is presented here. The equation for evolution of the density of inter-stellar matter is reduced to linear ODE in the case of arbitrary equation of state.Moreover, the dependence of density of inter-stellar matter in expanding Universe is proved to be given by the appropriate elliptical integral in case of the linear equation of state. Also we obtain that in case of the adiabatic expansion of inter-stellar matter the evolution of the universe as predicted by the Einstein-Friedman equations is proved to be given by the appropriate ultra-elliptical integral.Thus, by a proper obtaining of re-inverse dependence of a solution from time-parameter we could present the entire evolution of Universe as a set of periodic cycles (it means a periodic character for the evolution of radius of space curvature).

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“…Roughly speaking, there are two ways. One is to introduce some small but nonzero components besides the cosmological constant , such as imperfect fluid cosmology [38][39][40][41] and cosmic strings [35,42]; whereas the other is to assume the cosmological constant exactly zero and the dark energy characterized by scalar fields evolving with time. In this paper, we pick the second way.…”

Section: P Dmentioning

confidence: 99%

Exploring the low redshift universe: two parametric models for effective pressure

et al. 2015

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Astrophysical observations have put unprecedentedly tight constraints on cosmological theories. The CDM model, mathematically simple and fits observational data sets well, is preferred for explaining the behavior of universe. But many basic features of the dark sectors are still unknown, which leaves room for various nonstandard cosmological hypotheses. As the pressure of the cosmological constant dark energy is unvarying, ignoring contributions from radiation and curvature terms at low redshift, the effective pressure keeps constant. In this paper, we propose two parametric models for a non-constant effective pressure in order to study the tiny deviation from CDM at low redshift. We recover our phenomenological models in the scenarios of quintessence and phantom fields, and we explore the behavior of the scalar field and potential. We constrain our model parameters with SNe Ia and BAO observations, and we detect subtle hints of ω de < −1 from the data-fitting results of both models, which indicates possibly a phantom dark energy scenario at present.

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Adventures in Friedmann cosmology: A detailed expansion of the cosmological Friedmann equations

Cited by 31 publications

References 42 publications

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

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

Quasi-Periodic Solutions of Einstein-Friedman Equations

Exploring the low redshift universe: two parametric models for effective pressure

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