Mapping the Chevallier-Polarski-Linder parametrization onto Physical
  Dark Energy Models
  We examine the Chevallier-Polarski-Linder (CPL) parametrization, in the
context of quintessence and barotropic dark energy models, to determine the
subset of such models to which it can provide a good fit. The CPL
parametrization gives the equation of state parameter $w$ for the dark energy
as a linear function of the scale factor $a$, namely $w = w_0 + w_a(1-a)$. In
the case of quintessence models, we find that over most of the $w_0$, $w_a$
parameter space the CPL parametrization maps onto a fairly narrow form of
behavior for the potential $V(\phi)$, while a one-dimensional subset of
parameter space, for which $w_a = \kappa (1+w_0)$, with $\kappa$ constant,
corresponds to a wide range of functional forms for $V(\phi)$. For barotropic
models, we show that the functional dependence of the pressure on the density,
up to a multiplicative constant, depends only on $w_i = w_a + w_0$ and not on
$w_0$ and $w_a$ separately. Our results suggest that the CPL parametrization
may not be optimal for testing either type of model.
