The Irreducible Axion Background

Searches for dark matter decaying into photons constrain its lifetime to be many orders of magnitude larger than the age of the Universe. A corollary statement is that the abundance of any particle that can decay into photons over cosmological timescales is constrained to be much smaller than the cold dark-matter density. We show that an $\textit{irreducible}$ freeze-in contribution to the relic density of axions is in violation of that statement in a large portion of the parameter space. This allows us to set stringent constraints on axions in the mass range $100\rm \;eV - 100\; MeV$. At $10\rm \; keV$ our constraint on a photophilic axion is $g_{a\gamma \gamma} \lesssim 8.1 \times 10^{-14}~{\rm GeV}^{-1}$, almost three orders of magnitude stronger than the bounds established using horizontal branch stars; at $100~{\rm keV}$ our constraint on a photophobic axion coupled to electrons is $g_{aee} \lesssim 8.0 \times 10^{-15}$, almost four orders of magnitude stronger than present results. Although we focus on axions, our argument is more general and can be extended to, for instance, sterile neutrinos.