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=w₀+w₀ (1-a). In the case of quintessence models, we find that over most of the w₀, w₀ parameter space the CPL parametrization maps onto a fairly narrow form of behavior for the potential V (), while a one-dimensional subset of parameter space, for which w₀= (1+w₀), with constant, corresponds to a wide range of functional forms for V (). For barotropic models, we show that the functional dependence of the pressure on the density, up to a multiplicative constant, depends only on w₈=w₀+w₀ and not on w₀ and w₀ separately. Our results suggest that the CPL parametrization may not be optimal for testing either type of model.