Direct reconstruction of the quintessence potential

We describe an algorithm which directly determines the quintessence potential from observational data, without using an equation of state parametrization. The strategy is to numerically determine observational quantities as a function of the expansion coefficients of the quintessence potential, which are then constrained using a likelihood approach. We further impose a model selection criterion, the Bayesian Information Criterion, to determine the appropriate level of the potential expansion. In addition to the potential parameters, the present day quintessence field velocity is kept as a free parameter. Our investigation contains unusual model types, including a scalar field moving on a flat potential, or in an uphill direction, and is general enough to permit oscillating quintessence field models. We apply our method to the ``gold`` Type Ia supernovae sample of Riess et al. A. G. Riess et al. (Supernova Search Team Collaboration), Astrophys. J. 607, 665 (2004) confirming the pure cosmological constant model as the best description of current supernovae luminosity-redshift data. Our method is optimal for extracting quintessence parameters from future data.