Friedmann-Lemaitre cosmologies via roulettes and other analytic methods

Abstract

In this work a series of methods are developed for understanding the Friedmann equation when it is beyond the reach of the Chebyshev theorem. First it will be demonstrated that every solution of the Friedmann equation admits a representation as a roulette such that information on the latter may be used to obtain that for the former. Next the Friedmann equation is integrated for a quadratic equation of state and for the Randall-Sundrum II universe, leading to a harvest of a rich collection * Email address: chensx@henu.edu.cn † Email address: gwg1@damtp.cam.ac.uk ‡ Email address: yisongyang@nyu.edu 1 of new interesting phenomena. Finally an analytic method is used to isolate the asymptotic behavior of the solutions of the Friedmann equation, when the equation of state is of an extended form which renders the integration impossible, and to establish a universal exponential growth law.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.

Analogy between equilibrium beach profiles and closed universes

“…The qualitative study and the search for analytic solutions of the Einstein-Friedmann equations (4.2)-(4.4) are reviewed in [30,36,37]), while [14][15][16] report new efforts in this direction. A mathematical property of the Friedmann equation (4.2) demonstrated in [15] is that the graphs of all solutions of this equation are roulettes. A roulette is the locus of a point that lies on, or inside, a curve that rolls without slipping on a straight line.…”

“…(5.2)). Chen et al [15] study explicitly the Friedmann equation for a closed (K = +1) universe to derive the equation of the solution in polar coordinates (r, ϑ). We do not repeat their analysis, reporting only the results.…”

“…where α, β, and δ are arbitrary constants [15] (although it must be α ≥ 0 and β ≥ 0 to avoid negative densities). Our case is reproduced for α = 0, β = ρ 0 , and δ = −( 3n+ 7 Particularly simple solutions correspond to w = 0 (n = −1/3, discussed separately in [15]) and w = 1/9 (n = −1/9) which are excluded in the model of Ref. [4].…”

Analogy between freezing lakes and the cosmic radiation era

“…By introducing This equation is analogous to the Friedmann equation of relativistic cosmology for a spatially flat Friedmann-Lemaître-Robertson-Walker (FLRW) universe filled with blackbody radiation, as discussed in the next section. Indeed, the Friedmann equation, which resembles an energy conservation equation for a conservative mechanical system, lends itself to analogy with equations arising in the study of many different and completely unrelated physical systems, ranging from particles in one-dimensional motion [15]- [19] to optical systems [20,21], condensed matter systems [22][23][24][25], the transverse profiles of glacial valleys [20,21,26], and equilibrium beach profiles [27].…”

Wheeler--DeWitt Universe Wave Function in the Presence of Stiff Matter

“…Refs. [23,24]), but not in their quantum version. Moreover, an equivalent term in the classical FRW equation appears also in the presence of so-called kination domain [25,26], in which a massless (and homogenous) scalar field φ(t) is considered.…”