On the Quintessence Scalar Field Potential

In this work we propose a new analytical method for determining the scalar field potential $V(\phi)$ in FRW type cosmologies containing a mixture of perfect fluid plus a quintessence scalar field. By assuming that the equation of state parameters of the perfect fluid, $\gamma -1 \equiv p_{\gamma}/\rho_{\gamma}$ and the quintessence, $\omega \equiv p/\rho$ are constants, it is shown that the potential for the flat case is $V(\phi) = A\rho_{\phi_{0}}\sinh^{B}(\lambda \phi)$, where $A$, $B$ and $\lambda$ are functions of $\gamma$ and $\omega$. This general result is a pure consequence of the Einstein field equations and the constancy of the parameters. Applying the same method for closed and open universes, the corresponding scalar field potentials are also explicitly obtained for a large set of values of the free parameters $\gamma$ and $\omega$. A formula yielding the transition redshift from a decelerating to an accelerating regime is also determined and compared to the $\Lambda$CDM case.