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 $V(\varphi )=s\varphi$. 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 data set 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 $\sqrt{3}{M}_{\mathrm{P}}{H}_0^2\simeq {10}^{-38}\mathrm{e}{\mathrm{V}}^3$ where $M_{\mathrm{P}}$ 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.

Figure 1

The potential energy evolution for quintessence and phantom models with linear potential of slope $s=1$. In this plot $t_0=0.96$.

Figure 2

The scale factor evolution for representative ($s=1$) quintessence and phantom models with linear potential. The present time corresponds to $t_0=0.96$ as in Fig. 1.

Figure 3

The redshift evolution of the equation of state parameter $w(z)$ for phantom and quintessence models and for several values of the slope $s$.

Figure 4

The differences $\Delta {\chi }^2(s)\equiv {\chi }^2(s)-{\chi }^2(s\simeq 0)$ for quintessence models. The curve has been sampled at $s=0,0.1,0.2,0.5,0.75,1.0,1.2,1.5,2.0,2.5,3.0$ and the corresponding points have been joined.

Figure 5

The differences $\Delta {\chi }^2={\chi }^2(s)-{\chi }^2(s\simeq 0)$ for phantom models. The curve has been sampled at $s=0,0.2,0.5,0.7,1.0,1.5,2.0,2.35$ and the corresponding points have been joined.

Figure 6

The evolution of $w$ for two representatives of the field theory models studied here superposed with two better fits obtained by arbitrary parametrization: a linear ansatz $w(z)=w_0+w_{1}z$ and the ansatz $w(z)=w_0+w_1\frac{z}{1+z}$.

Figure 7

The quintom model of the Lagrangian (4.1) with natural initial conditions and $s=0.2$ can mimic quintessence and phantom models by changing the sign of the coupling $q$.