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 $$V(\phi )=\frac{(\lambda \phi )^{2n}}{2n}$$ V ( ϕ ) = ( λ ϕ ) 2 n 2 n , $$\lambda >0$$ λ > 0 , $$n\in {\mathbb {N}}$$ n ∈ N , interacting with a perfect fluid with linear equation of state $$p_{\textrm{pf}}=(\gamma _{\textrm{pf}}-1)\rho _{\textrm{pf}}$$ p pf = ( γ pf - 1 ) ρ pf , $$\gamma _{\textrm{pf}}\in (0,2)$$ γ pf ∈ ( 0 , 2 ) , in flat Robertson–Walker spacetimes. The interaction is a friction-like term of the form $$\Gamma (\phi )=\mu \phi ^{2p}$$ Γ ( ϕ ) = μ ϕ 2 p , $$\mu >0$$ μ > 0 , $$p\in {\mathbb {N}}\cup \{0\}$$ p ∈ N ∪ { 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.