Dynamics of interacting monomial scalar field potentials and perfect fluids

Abstract

Motivated by cosmological models of the early universe we analyse the dynamics of the Einstein equations with a minimally coupled scalar field with monomial potentials $V(\phi)=\frac{(\lambda\phi)^{2n}}{2n}$, $\lambda>0$, $n\in\mathbb{N}$, interacting with a perfect fluid with linear equation of state $p_\mathrm{pf}=(\gamma_\mathrm{pf}-1)\rho_\mathrm{pf}$, $\gamma_\mathrm{pf}\in(0,2)$, in flat Robertson-Walker spacetimes. The interaction is a friction-like term of the form $\Gamma(\phi)=\mu \phi^{2p}$, $\mu>0$, $p\in\mathbb{N}\cup\{0\}$. The analysis relies on the introduction of a new regular 3-dimensional dynamical systems' formulation of the Einstein equations on a compact state space, and the use of dynamical systems' tools such as quasi-homogeneous blow-ups and averaging methods involving a time-dependent perturbation parameter. We find a bifurcation at $p=n/2$ due to the influence of the interaction term. In general, this term has more impact on the future (past) asymptotics for $p<n/2$ ($p>n/2$). For $p<n/2$ we find a complexity of possible future attractors, which depends on whether $p=(n-1)/2$ or $p<(n-1)/2$. In the first case the future dynamics is governed by Li\'enard systems. On the other hand when $p=(n-2)/2$ the generic future attractor consists of new solutions previously unknown in the literature which can drive future acceleration whereas the case $p<(n-2)/2$ has a generic future attractor de-Sitter solution. For $p=n/2$ the future asymptotics can be either fluid dominated or have an oscillatory behaviour where neither the fluid nor the scalar field dominates. For $p>n/2$ the future asymptotics is similar to the case with no interaction. Finally, we show that irrespective of the parameters, an inflationary quasi-de-Sitter solution always exists towards the past, and therefore the cases with $p\leq(n-2)/2$ may provide new cosmological models of quintessential inflation.