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We study quintessence and phantom field theory models based onlinear-negative potentials of the form $V(\phi)=s \phi$. We investigate thepredicted redshift dependence of the equation of state parameter $w(z$ for awide range of slopes $s$ in both quintessence and phantom models. We use thegold dataset of 157 SnIa and place constraints on the allowed range of slopes$s$. We find $s=0\pm 1.6$ for quintessence and $s=\pm 0.7\pm 1$ for phantommodels (the range is at the $2\sigma$ level and the units of $s$ are in$\sqrt{3}M_p H_0^2\simeq 10^{-38}eV^3$ where $M_p$ is the Planck mass). In bothcases the best fit is very close to $s\simeq 0$ corresponding to a cosmologicalconstant. We also show that specific model independent parametrizations of$w(z)$ which allow crossing of the phantom divide line $w=-1$ (hereafter PDL)provide significantly better fits to the data. Unfortunately such crossings arenot allowed in any phantom or quintessence single field model minimally coupledto gravity. Mixed models (coupled phantom-quintessence fields) can in principlelead to a $w(z)$ crossing the PDL but a preliminary investigation indicatesthat this does not happen for natural initial conditions.

Constraints on linear negative potentials in quintessence and phantom models from recent supernova data