General Scalar Fields as Quintessence

We study the cosmological evolution of scalar fields with arbitrary potentials in the presence of a barotropic fluid (matter or radiation) without making any assumption on which term dominates. We determine what kind of potentials V(phi) permits a quintessence interpretation of the scalar field phi and to obtain interesting cosmological results. We show that all model dependence is given in terms of lambda= - V'/V only and we study all possible asymptotic limits: lambda approaching zero, a finite constant or infinity. We determine the equation of state dynamically for each case. For the first class of potentials, the scalar field quickly dominates the universe behaviour, with an inflationary equation of state allowing for a quintessence interpretation. The second case gives the extensively studied exponential potential. While in the last case, when lambda approaches infinity, if it does not oscillate then the energy density redshifts faster than the barotropic fluid but if lambda oscillates then the energy density redshift depends on the specific potential.