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.
