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 / 3 H ^ { 2 } ) and with a solution x = ( [ 1 + 2 L / 3 ( 1 + y ) ] ^ { 1 / 2 } -1 ) ( 1 + y ) / 2 where y \equiv \rho _ { m } / V and L \equiv ( V ^ { \prime } / V ) ^ { 2 } ( 1 + q ) ^ { 2 } , q \equiv \ddot { \phi } / V ^ { \prime } .
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 ^ { \prime } / 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 ) .