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A new approach to tackle Einstein equations for an isotropic and homogeneousFriedmann--Robertson--Walker Universe in the presence of a quintessence scalarfield is devised. It provides a way to get a simple exact solution to theseequations. This solution determines the quintessence potential uniquely and itdiffers from solutions which have been used to study inflation previously. Itrelays on a unification of geometry and dark matter implemented through thedefinition of a functional relation between the scale factor of the Universeand the quintessence field. For a positive curvature Universe, this solutionproduces perpetual accelerated expansion rate of the Universe, while the Hubbleparameter increases abruptly, attains a maximum value and decreases thereafter.The behavior of this cosmological solution is discussed and its main featuresare displayed. The formalism is extended to include matter and radiation.

Class of exact solutions for a cosmological model of unified gravitational and quintessence fields