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TMF, 2016, Volume 188, Number 1, Pages 121–157 (Mi tmf9038)  

This article is cited in 1 scientific paper (total in 1 paper)

Solving dynamical equations in general homogeneous isotropic cosmologies with a scalaron

A. T. Filippov

Joint Institute for Nuclear Research, Dubna, Moscow Oblast, Russia

Abstract: We consider gauge-dependent dynamical equations describing homogeneous isotropic cosmologies coupled to a scalar field $\psi$ (scalaron). For flat cosmologies $(k=0)$, we analyze the gauge-independent equation describing the differential $\chi(\alpha)\equiv\psi'(\alpha)$ of the map of the metric $\alpha$ to the scalaron field $\psi$, which is the main mathematical characteristic of a cosmology and locally defines its portrait in the so-called $\alpha$ version. In the more customary $\psi$ version, the similar equation for the differential of the inverse map $\bar\chi(\psi) \equiv\chi^{-1}(\alpha)$ is solved in an asymptotic approximation for arbitrary potentials $v(\psi)$. In the flat case, $\bar\chi(\psi)$ and $\chi^{-1}(\alpha)$ satisfy first-order differential equations depending only on the logarithmic derivative of the potential, $v'(\psi)/v(\psi)$. If an analytic solution for one of the $\chi$ functions is known, then we can find all characteristics of the cosmological model. In the $\alpha$ version, the full dynamical system is explicitly integrable for $k\ne0$ with any potential $\bar v(\alpha)\equiv v[\psi(\alpha)]$ replacing $v(\psi)$. Until one of the maps, which themselves depend on the potentials, is calculated, no sort of analytic relation between these potentials can be found. Nevertheless, such relations can be found in asymptotic regions or by perturbation theory. If instead of a potential we specify a cosmological portrait, then we can reconstruct the corresponding potential. The main subject here is the mathematical structure of isotropic cosmologies. We also briefly present basic applications to a more rigorous treatment of inflation models in the framework of the $\alpha$ version of the isotropic scalaron cosmology. In particular, we construct an inflationary perturbation expansion for $\chi$. If the conditions for inflation to arise are satisfied, i.e., if $v>0$, $k=0$, $\chi^2<6$, and $\chi(\alpha)$ satisfies a certain boundary condition as $\alpha\to-\infty$, then the expansion is invariant under scaling the potential, and its first terms give the standard inflationary parameters with higher-order corrections.

Keywords: isotropic cosmology, scalar field, dynamical system, integrable model, gauge invariance, inflationary model

DOI: https://doi.org/10.4213/tmf9038

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English version:
Theoretical and Mathematical Physics, 2016, 188:1, 1069–1098

Bibliographic databases:

Received: 09.09.2015
Revised: 21.10.2015

Citation: A. T. Filippov, “Solving dynamical equations in general homogeneous isotropic cosmologies with a scalaron”, TMF, 188:1 (2016), 121–157; Theoret. and Math. Phys., 188:1 (2016), 1069–1098

Citation in format AMSBIB
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\by A.~T.~Filippov
\paper Solving dynamical equations in general homogeneous isotropic cosmologies with a~scalaron
\jour TMF
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\vol 188
\issue 1
\pages 121--157
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\vol 188
\issue 1
\pages 1069--1098
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. T. Filippov, “A fresh view of cosmological models describing very early universe: general solution of the dynamical equations”, Phys. Part. Nuclei Lett., 14:2 (2017), 298–303  crossref  isi  scopus
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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