On solving dynamical equations in general homogeneous isotropic cosmologies with scalaron

Abstract

We study general dynamical equations describing homogeneous isotropic cosmologies coupled to a scalaron $\psi$. For flat cosmologies ($k=0$), we analyze in detail the gauge-independent equation describing the differential, $\chi (\alpha )\equiv {\psi }^{\prime }(\alpha )$, of the map of the metric $\alpha$ to the scalaron field $\psi$, which is the main mathematical characteristic locally defining a `portrait' of a cosmology in `$\alpha$-version'. In the `$\psi$-version', a similar equation for the differential of the inverse map, $\bar{\chi }(\psi )\equiv {\chi }^{-1}(\alpha )$, can be solved asymptotically or for some `integrable' scalaron potentials $v(\psi )$. In the flat case, $\bar{\chi }(\psi )$ and $\chi (\alpha )$ satisfy the first-order differential equations depending only on the logarithmic derivative of the potential. Once we know a general analytic solution for one of these $\chi$-functions, we can explicitly derive all characteristics of the cosmological model. In the $\alpha$-version, the whole dynamical system is integrable for $k\ne 0$ and with any `$\alpha$-potential', $\bar{v}(\alpha )\equiv v[\psi (\alpha )]$, replacing $v(\psi )$. There is no a priori relation between the two potentials before deriving $\chi$ or $\bar{\chi }$, which implicitly depend on the potential itself, but relations between the two pictures can be found by asymptotic expansions or by inflationary perturbation theory. Explicit applications of the results to a more rigorous treatment of the chaotic inflation models and to their comparison with the ekpyrotic-bouncing ones are outlined in the frame of our `$\alpha$-formulation' of isotropic scalaron cosmologies. In particular, we establish an inflationary perturbation expansion for $\chi$. When all the conditions for inflation are satisfied and $\chi$ obeys a certain boundary (initial) condition, we get the standard inflationary parameters, with higher-order corrections.

New version: 33 pages instead 32; revised and extended Abstract, Sections 4.3, 5; edited Section 1, changed a few titles; corrected misprints

References in corpus (8)

• A Critical Review of Classical Bouncing Cosmologies
• Escher in the Sky
• Bouncing Universes in Scalar-Tensor Gravity Models admitting Negative Potentials
• Classical Inflationary and Ekpyrotic Universes in the No-Boundary Wavefunction
• Dimensional Reduction of Gravity and Relation between Static States, Cosmologies and Waves
• Some Unusual Dimensional Reductions of Gravity: Geometric Potentials, Separation of Variables, and Static - Cosmological Duality
• Conformal Universe as false vacuum decay
• Integrals of equations for cosmological and static reductions in generalized theories of gravity

Cited by in corpus (1)

• A fresh view of cosmological models describing very early Universe: general solution of the dynamical equations