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 has a solution x = ( [ ( 1 + y ) ^ { 2 } +2 L / 3 ] ^ { 1 / 2 } - ( 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 } .
The resulting EoS is w = \left ( 6 + L - 6 \sqrt { ( 1 + y ) ^ { 2 } +2 L / 3 } \right ) / \left ( L + 6 y \right ) .
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 w 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 .
The EoS 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 .
To determine the dynamics of Dark Energy we calculate the background evolution and its perturbations , since they are important to discriminate between different DE models .
Our parametrization follows closely the dynamics of a scalar field scalar fields and the function L allow us to connect it with the potential V ( \phi ) of the scalar field \phi .