### Abstract

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 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.

Original language | English (US) |
---|---|

Article number | 020 |

Journal | Journal of Cosmology and Astroparticle Physics |

Volume | 2015 |

Issue number | 5 |

DOIs | |

State | Published - May 13 2015 |

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### Keywords

- cosmological applications of theories with extra dimensions
- dark energy theory
- physics of the early universe

### ASJC Scopus subject areas

- Astronomy and Astrophysics

### Cite this

*Journal of Cosmology and Astroparticle Physics*,

*2015*(5), [020]. https://doi.org/10.1088/1475-7516/2015/05/020

**Explicit integration of Friedmann's equation with nonlinear equations of state.** / Chen, Shouxin; Gibbons, Gary W.; Yang, Yisong.

Research output: Contribution to journal › Article

*Journal of Cosmology and Astroparticle Physics*, vol. 2015, no. 5, 020. https://doi.org/10.1088/1475-7516/2015/05/020

}

TY - JOUR

T1 - Explicit integration of Friedmann's equation with nonlinear equations of state

AU - Chen, Shouxin

AU - Gibbons, Gary W.

AU - Yang, Yisong

PY - 2015/5/13

Y1 - 2015/5/13

N2 - 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 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.

AB - 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 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.

KW - cosmological applications of theories with extra dimensions

KW - dark energy theory

KW - physics of the early universe

UR - http://www.scopus.com/inward/record.url?scp=84934873791&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84934873791&partnerID=8YFLogxK

U2 - 10.1088/1475-7516/2015/05/020

DO - 10.1088/1475-7516/2015/05/020

M3 - Article

AN - SCOPUS:84934873791

VL - 2015

JO - Journal of Cosmology and Astroparticle Physics

JF - Journal of Cosmology and Astroparticle Physics

SN - 1475-7516

IS - 5

M1 - 020

ER -