  All generic, calculable models of dynamical supersymmetry breaking have a
spontaneously broken $R$ symmetry and therefore contain an $R$ axion. We show
that the axion is massive in any model in which the cosmological constant is
fine-tuned to zero through an explicit $R$-symmetry-breaking constant. In
visible-sector models, the axion mass is in the 100 MeV range and thus evades
astrophysical bounds. In nonrenormalizable hidden-sector models, the mass is of
order of the weak scale and can have dangerous cosmological consequences
similar to those already present from other fields. In renormalizable hidden-
sector models, the axion mass is generally quite large, of order $10^7$ GeV.
Typically, these axions are cosmologically safe. However, if the dominant decay
mode is to gravitinos, the potentially large gravitino abundance that arises
from axion decay after inflation might affect the successful predictions of
big-bang nucleosynthesis. We show that the upper bound on the reheat
temperature after standard inflation can be competitive with or stronger than
bounds from thermal gravitino production, depending on the model and the
gravitino mass.
