Cosmological evolution of general scalar fields and quintessence

We study the cosmological evolution of scalar fields with arbitrary potentials in the presence of a baryotropic fluid (matter or radiation) without making any assumption on which term dominates. We determine what kind of potentials $V\left(\phi \right)$ 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 $\lambda \equiv -V^{\prime }/V$ 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 behavior, 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 baryotropic fluid, but if $\lambda$ oscillates, then the energy density redshift depends on the specific potential.