The distance-redshift equation in quintessence cosmology and the estimation of H-0 through time delays

We calculate analytically and numerically the distance-redshift equation in perfect fluid quintessence models and give an accurate fit to the numerica l solutions for all the values of the density parameter and the quintessenc e equation of state. Then we apply our solutions to the estimation of H-0 f rom multiple image time delays and find that the inclusion of quintessence modifies significantly the likelihood distribution of H-0, generally reduci ng the best estimate with respect to a pure cosmological constant. Marginal izing over the other parameters (Omega (m) and the quintessence equation of state) we obtain H-0 = 71 +/- 6 km s(-1) Mpc(-1) for an empty beam and H-0 = 64 +/- 4 km s(-1) Mpc(-1) for a filled beam. These errors, however, do n ot take into account the uncertainty on the modelling of the lens. We also discuss the future prospects for distinguishing quintessence from a cosmolo gical constant with time delays.