  Entropy production from scalar decay in the era of low temperatures after
neutrino decoupling will change the ratio of the relic neutrino temperature to
the CMB temperature, and, hence, the value of N_eff, the effective number of
neutrino species. Such scalar decay is relevant to reheating after thermal
inflation, proposed to dilute massive particles, like the moduli and the
gravitino, featuring in supersymmetric and string theories. The effect of such
entropy production on the relic neutrino temperature ratio is calculated in a
semi-analytic manner, and a recent lower bound on this ratio, obtained from the
WMAP satellite and 2dF galaxy data, is used to set a lower bound of ~ 1.5 x
10^-23 Gev on the scalar decay constant, corresponding to a reheating
temperature of about 3.3 Mev.
