  We consider a model with a small explicit breaking of a global symmetry, as
suggested by gravitational arguments. Our model has one scalar field
transforming under a non-anomalous U(1) symmetry, and coupled lo matter and to
gauge bosons. The spontaneous breaking of the explicitly broken symmetry gives
rise to a massive pseudo-Goldstone boson. We analyze thermal and non-thermal
production of this particle in the early universe, and perform a systematic
study of astrophysical and cosmological constraints on its properties. We find
that for very suppressed explicit breaking the pseudo-Goldstone boson is a cold
dark matter candidate.
