Self-similar cosmological solutions with a non-minimally coupled scalar field

TL;DR

This work analyzes self-similar cosmologies in a Brans-Dicke-type gravity with a power-law scalar potential $V(\phi)=V_0(\kappa\phi)^{2n}$, showing that the dynamics reduce to a two-dimensional autonomous phase space in the Einstein frame. Fixed points corresponding to matter–kinetic scaling, kinetic-dominated, scalar-field–dominated, and a novel three-way scaling with matter, kinetic, and potential energy are identified, with explicit expressions for their exponents and stability. A conformal relation links these Brans-Dicke frame solutions to Einstein-frame scenarios with exponential potentials and fluid interaction; two limiting procedures recover standard GR and reveal how a decoupled radiation era and a strongly coupled dark-matter sector can drive late-time acceleration without fine-tuning. The analysis suggests a viable quintessence-like mechanism where acceleration emerges from non-minimal coupling to dark matter and scale-invariant dynamics, though it requires careful treatment of the dark matter coupling and nucleosynthesis constraints. Overall, the paper provides a comprehensive phase-plane framework for scalar-tensor quintessence with power-law potentials and highlights conditions under which late-time acceleration can arise in a scale-invariant setting.

Abstract

We present self-similar cosmological solutions for a barotropic fluid plus scalar field with Brans-Dicke-type coupling to the spacetime curvature and an arbitrary power-law potential energy. We identify all the fixed points in the autonomous phase-plane, including a scaling solution where the fluid density scales with the scalar field's kinetic and potential energy. This is related by a conformal transformation to a scaling solution for a scalar field with exponential potential minimally coupled to the spacetime curvature, but non-minimally coupled to the barotropic fluid. Radiation is automatically decoupled from the scalar field, but energy transfer between the field and non-relativistic dark matter can lead to a change to an accelerated expansion at late times in the Einstein frame. The scalar field density can mimic a cosmological constant even for steep potentials in the strong coupling limit.

Figures (1)

• Figure 1: $\Omega_{M,0}$-$w_q$ parameter space for 3-way matter--kinetic--potential scaling solution with $\gamma=1$ and $\lambda<0$. Limits from nucleosynthesis constraints requiring $\tilde{\Omega}_{\psi,{\rm nuc}}^{\rm Max}=0.2$ are shown by a dot--dash line. The solid line shows the constraint from requiring acceleration at present.