Constraints on linear-negative potentials in quintessence and phantom   models from recent supernova data
  We study quintessence and phantom field theory models based on linear-negative potentials of the form $V(\phi)=s \phi$. We investigate the predicted redshift dependence of the equation of state parameter $w(z$ for a wide range of slopes $s$ in both quintessence and phantom models. We use the gold 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 phantom models (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 both cases the best fit is very close to $s\simeq 0$ corresponding to a cosmological constant. 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 are not allowed in any phantom or quintessence single field model minimally coupled to gravity. Mixed models (coupled phantom-quintessence fields) can in principle lead to a $w(z)$ crossing the PDL but a preliminary investigation indicates that this does not happen for natural initial conditions. 
L. Perivolaropoulos