A Lower Limit on Ωm −ΩΛ Using the Gravitational Lensing Rate in the Hubble Deep Field

We calculate the expected strong gravitational lensing rate in the Hubble Deep Field (HDF) by using photometric redshift information for galaxies present in all four HDF passbands with a I-band magnitude limit of 27. A comparison of these predictions with the observed number of strongly lensed sources places a lower limit on the current value of Ωm − ΩΛ, where Ωm is the cosmological mass density of the universe and ΩΛ is the normalized cosmological constant. Based on current estimates of the HDF luminosity function and associated uncertainties in individual parameters, our 95% confidence lower limit on Ωm − ΩΛ is between -0.17, if there are no strongly lensed sources in the HDF, and -0.59, if there are three strongly lensed sources in the HDF. For a flat universe (Ωm + ΩΛ = 1), ΩΛ < 0.58 to 0.79 (95% C.L.). If the only lensed source in the HDF is the one presently viable candidate, then Ωm − ΩΛ > −0.39. These lower limits are compatible with estimates based on high redshift supernovae and with previous limits based on gravitational lensing. Subject headings: cosmology: observations — galaxies: luminosity function — gravitational lensing