Within the framework of an explicit dynamical model, in which we calculate the radiatively-corrected, tree-level potential that sets up inflation, we show that the inflaton can be a significant part of dark matter today. We exhibit potentials with both a maximum and a minimum. Using the calculated position of the potential minimum, and an estimate for fluctuations of the inflaton field in the early universe, we calculate a contribution to the matter energy density of $(1-2)\times 10^{-47}\GeV^4$ in the present universe, from cold inflatons with mass of about $6\times 10^9\GeV$. We show that the inflaton might decay in a specific way, and we calculate a possible lifetime that is several orders of magnitude greater than the present age of the universe. Inflaton decay is related to an interaction which, together with a spontaneous breakdown of CP invariance at a cosmological energy scale, can give rise to a neutrino-antineutrino asymmetry just prior to the time of electroweak symmetry breaking.

17 pages, 4 eps-figures. To be published in Zeitschr. fuer Physik C. Some comments concerning vacuum energy added