The thawing quintessence model with a nearly flat potential provides a natural mechanism to produce an equation of state parameter , w , close to -1 today .
We examine the behavior of such models for the case in which the potential satisfies the slow roll conditions : [ ( 1 / V ) ( dV / d \phi ) ] ^ { 2 } \ll 1 and ( 1 / V ) ( d ^ { 2 } V / d \phi ^ { 2 } ) \ll 1 , and we derive the analog of the slow-roll approximation for the case in which both matter and a scalar field contribute to the density .
We show that in this limit , all such models converge to a unique relation between 1 + w , \Omega _ { \phi } , and the initial value of ( 1 / V ) ( dV / d \phi ) .
We derive this relation , and use it to determine the corresponding expression for w ( a ) , which depends only on the present-day values for w and \Omega _ { \phi } .
For a variety of potentials , our limiting expression for w ( a ) is typically accurate to within \delta w { \raise - 2.15 pt \hbox { $ \buildrel < \over { \sim } $ } } 0.005 for w < -0.9 .
For redshift z { \raise - 2.15 pt \hbox { $ \buildrel < \over { \sim } $ } } 1 , w ( a ) is well-fit by the Chevallier-Polarski-Linder parametrization , in which w ( a ) is a linear function of a .