Cosmological and astrophysical signatures of dark matter annihilations into pseudo-Goldstone bosons

We investigate a model where the dark matter particle is a chiral fermion field charged under a global $U(1)$ symmetry which is assumed to be spontaneously broken, leading to a pseudo-Goldstone boson (PGB). We argue that the dark matter annihilation into PGBs determine the dark matter relic abundance. Besides, we also note that experimental searches for PGBs allow either for a very long lived PGB, with a lifetime much longer than the age of the Universe, or a relatively short lived PGB, with a lifetime shorter than one minute. Hence, two different scenarios arise, producing very different signatures. In the long lived PGB scenario, the PGB might contribute significantly to the radiation energy density of the Universe. On the other hand, in the short lived PGB scenario, and since the decay length is shorter than one parsec, the s-wave annihilation into a PGB and a $CP$ even dark scalar in the Galactic center might lead to an intense box feature in the gamma-ray energy spectrum, provided the PGB decay branching ratio into two photons is sizable. We also analyze the constraints on these two scenarios from thermal production, the Higgs invisible decay width and direct dark matter searches.