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

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.