Scalar Field Dark Energy Parametrization

We propose a new Dark Energy parametrization based on the dynamics of a scalar field. We use an equation of state $w=(x-1)/(x+1)$, with $x=E_k/V$, the ratio of kinetic energy $E_k=\dot\phi^2/2$ and potential $V$. The eq. of motion gives $x=(L/6)(V/3H^2)$ and with a solution $x=([1+2 L/3(1+y)]^{1/2}-1)(1+y)/2$ where $y\equiv \rm/V$ and $L\equiv (V'/V)^2 (1+q)^2,\, q\equiv\ddot\p/V'$. Since the universe is accelerating at present time we use the slow roll approximation in which case we have $|q|\ll 1$ and $L\simeq (V'/V)^2$. However, the derivation of $L$ is exact and has no approximation. By choosing an appropriate ansatz for $L$ we obtain a wide class of behavior for the evolution of Dark Energy without the need to specify the potential $V$. In fact $w$ can either grow and later decrease, or other way around, as a function of redshift and it is constraint between $-1\leq w\leq 1$ as for any canonical scalar field with only gravitational interaction. Furthermore, we also calculate the perturbations of DE and since the evolution of DE is motivated by the dynamics of a scalar field the homogenous and its perturbations can be used to determine the form of the potential and the nature of Dark Energy. Since our parametrization is on $L$ we can easily connect it with the scalar potential $V(\phi)$.