On solving dynamical equations in general homogeneous isotropic cosmologies with scalaron
| Autor: | Alexandre T. Filippov |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2015 |
| Předmět: |
Physics
High Energy Physics - Theory Cosmology and Nongalactic Astrophysics (astro-ph.CO) 010308 nuclear & particles physics Differential equation Inverse FOS: Physical sciences Statistical and Nonlinear Physics General Relativity and Quantum Cosmology (gr-qc) Mathematical Physics (math-ph) 01 natural sciences General Relativity and Quantum Cosmology High Energy Physics - Theory (hep-th) 0103 physical sciences Boundary value problem Gauge theory Logarithmic derivative Perturbation theory Invariant (mathematics) 010306 general physics Scalar field Mathematical Physics Mathematical physics Astrophysics - Cosmology and Nongalactic Astrophysics |
| Popis: | 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 $k\neq 0$ 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. Comment: New version: 33 pages instead 32; revised and extended Abstract, Sections 4.3, 5; edited Section 1, changed a few titles; corrected misprints |
| Databáze: | OpenAIRE |
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