

Nonlinear dynamics of cosmological scalar fields with singular potentials
Date/Time: 16:30 05Aug2014
Abstract:
\begin{center}
{\Large Nonlinear dynamics of cosmological scalar fields} {\Large with singular potentials} \bigskip V.A. Koutvitsky and \underline{E.M. Maslov} \textit{IZMIRAN, Russia} \end{center} We investigate the dynamics of the inflaton scalar field $\phi (t,\mathbf{r}% ) $ governed by the nonlinear KleinGordon equation in the FriedmannRobertsonWalker Universe,% \begin{equation*} \phi _{tt}+3H\phi _{t}a^{2}\Delta \phi +U^{\prime }(\phi )=0, \end{equation*}% where $a(t)$ is the scale factor, $H=a_{t}/a$ is the Hubble parameter. We consider the potentials having singularity at their minimum, $\left U^{\prime \prime }(\phi )\right \rightarrow \infty \;(\phi \rightarrow 0)$. First, the rapid oscillations of the homogeneous background $\phi (t)$ near the minimum are studied. These damped oscillations determine growth of the scale factor $a(t)$ through the Friedmann equations with the effective pressure $p$ and energy density $\rho $ [1]. To describe the oscillations we make the transformation $(\phi ,\phi _{t})\rightarrow (\rho ,\theta )$, representing the field as $\phi (t)=\varphi (\rho ,\theta )$, where the function $\varphi (\rho ,\theta )$ is given by a quadrature, $\rho $ and $% \theta $ are slow and fast variables. Equations governing the evolution of these variables are obtained in the Van der Pol approximation. As examples, we consider two potentials with logarithmic and fractional power singularities. For these potentials we calculate the equation of state parameter $w=p/\rho $ and show that in some range of $\rho $ it lies in the interval $1<w<1/3$, providing the accelerated expansion of the Universe. Then we examine the resonant amplification of the field fluctuations $\delta \phi (t,\mathbf{r})$ on the oscillating background $\phi (t)$. The Fourier $k $modes of $\delta \phi $ satisfy the singular Hill equation with slowly varying parameters $\rho $ and $k/a$. Using the stabilityinstability chart, calculated by the generalized Lindemann  Stieltjes method [2,3], we argue that the fluctuations can be significantly amplified when the representative trajectories of the parameters cross the resonance zones. At the nonlinear stage these fluctuations can transform into well localized oscillating lumps, the pulsons. \bigskip \textbf{References} \bigskip [1] M.S.~Turner, Phys. Rev.~D, \textbf{28}, 1243 (1983). [2] V.A.~Koutvitsky and E.M. Maslov, J. Math. Phys., \textbf{47}, 022302 (2006). [3] [3] В.А. Кутвицкий, Е.М. Маслов, Проблемы мат. анализа (2014) (в печати). \end{document} Attachments:
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