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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 $V(ϕ)$, while a one-dimensional subset of parameter space, for which $w_a = κ(1+w_0)$, with $κ$ constant, corresponds to a wide range of functional forms for $V(ϕ)$. 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.