On weakly tight families

Using ideas from Shelah's recent proof that a completely separable maximal almost disjoint family exists when $\c < {\aleph}_{\omega}$, we construct a weakly tight family under the hypothesis \s \leq \b < {\aleph}_{\omega}. The case when \s < \b is handled in \ZFC and does not require \b < {\aleph}_{\omega}, while an additional PCF type hypothesis, which holds when \b < {\aleph}_{\omega} is used to treat the case \s = \b. The notion of a weakly tight family is a natural weakening of the well studied notion of a Cohen indestructible maximal almost disjoint family. It was introduced by Hru{\v{s}}{\'a}k and Garc{\'{\i}}a Ferreira \cite{Hr1}, who applied it to the Kat\'etov order on almost disjoint families

Gravitino production in an inflationary Universe and implications for leptogenesis

Models of leptogenesis are constrained by the low reheat temperature at the end of reheating associated with the gravitino bound. However a detailed view of reheating, in which the maximum temperature during reheating, \Tmax, can be orders of magnitude higher than the reheat temperature, allows for the production of heavy Majorana neutrinos needed for leptogenesis. But then one must also consider the possibility of enhanced gravitino production in such scenarios. In this article we consider gravitino production during reheating, its dependence on \Tmax, and its relevance for leptogenesis. Earlier analytical studies of the gravitino abundance have only considered gravitino production in the post-reheating radiation dominated era. We find that the gravitino abundance generated during reheating is comparable to that generated after reheating. This lowers the upper bound on the reheat temperature by a factor of 4/3.Comment: Journal version, minor change in title, 13 pages (revtex), 2 eps figure