Exact scaling solutions in normal and Brans-Dicke models of dark energy

A linear relationship between the Hubble expansion parameter and the time derivative of the scalar field is explored in order to derive exact cosmological, attractor-like solutions, both in Einstein's theory and in Brans-Dicke gravity with two fluids: a background fluid of ordinary matter and a self-interacting scalar-field fluid accounting for the dark energy in the universe. A priori assumptions about the functional form of the self-interaction potential or about the scale factor behaviour are not necessary. These are obtained as outputs of the assumed relationship between the Hubble parameter and the time derivative of the scalar field. A parametric class of scaling quintessence models given by a self-interaction potential of a peculiar form, a combination of exponentials with dependence on the barotropic index of the background fluid, arises. Both normal quintessence described by a self-interacting scalar field minimally coupled to gravity and Brans-Dicke quintessence given by a non-minimally coupled scalar field are then analysed and the relevance of these models for the description of the cosmic evolution is discussed in some detail. The stability of these solutions is also briefly commented on.