Parameterizing and Measuring Dark Energy Trajectories from Late-Inflatons

Bulk dark energy properties are determined by the redshift evolution of its pressure-to-density ratio, $w_{de}(z)$. An experimental goal is to decide if the dark energy is dynamical, as in the quintessence (and phantom) models treated here. We show that a three-parameter approximation $w_{de}(z; \epsilon_s, \epsilon_{\phi\infty}, \zeta_s)$ fits well the ensemble of trajectories for a wide class of late-inflaton potentials $V(\phi)$. Markov Chain Monte Carlo probability calculations are used to confront our $w_{de}(z)$ trajectories with current observational information on Type Ia supernova, Cosmic Microwave Background, galaxy power spectra, weak lensing and the Lyman-${\alpha}$ forest. We find the best constrained parameter is a low redshift slope parameter, $\epsilon_s \propto (\partial \ln V / \partial \phi)^2$ when the dark energy and matter have equal energy densities. A tracking parameter $\epsilon_{\phi\infty}$ defining the high-redshift attractor of $1+w_{de}$ is marginally constrained. Poorly determined is $\zeta_s$, characterizing the evolution of $\epsilon_s$, and a measure of $\partial^2 \ln V / \partial \phi^2$ . The constraints we find already rule out some popular quintessence and phantom models, or restrict their potential parameters. We also forecast how the next generation of cosmological observations improve the constraints: by a factor of about five on $\epsilon_s$ and $\epsilon_{\phi\infty}$, but with $\zeta_s$ remaining unconstrained (unless the true model significantly deviates from $\Lambda$CDM). Thus potential reconstruction beyond an overall height and a gradient is not feasible for the large space of late-inflaton models considered here.