The Cosmic Axion Background
  Existing searches for cosmic axions relics have relied heavily on the axion
being non-relativistic and making up dark matter. However, light axions can be
copiously produced in the early Universe and remain relativistic today, thereby
constituting a Cosmic $\textit{axion}$ Background (C$a$B). As prototypical
examples of axion sources, we consider thermal production, dark-matter decay,
parametric resonance, and topological defect decay. Each of these has a
characteristic frequency spectrum that can be searched for in axion direct
detection experiments. We focus on the axion-photon coupling and study the
sensitivity of current and future versions of ADMX, HAYSTAC, DMRadio, and
ABRACADABRA to a C$a$B, finding that the data collected in search of dark
matter can be repurposed to detect axion energy densities well below limits set
by measurements of the energy budget of the Universe. In this way, direct
detection of relativistic relics offers a powerful new opportunity to learn
about the early Universe and, potentially, discover the axion.
