  The next generation of axion direct detection experiments may rule out or
confirm axions as the dominant source of dark matter. We develop a general
likelihood-based framework for studying the time-series data at such
experiments, with a focus on the role of dark-matter astrophysics, to search
for signatures of the QCD axion or axion like particles. We illustrate how in
the event of a detection the likelihood framework may be used to extract
measures of the local dark matter phase-space distribution, accounting for
effects such as annual modulation and gravitational focusing, which is the
perturbation to the dark matter phase-space distribution by the gravitational
field of the Sun. Moreover, we show how potential dark matter substructure,
such as cold dark matter streams or a thick dark disk, could impact the signal.
For example, we find that when the bulk dark matter halo is detected at
5$\sigma$ global significance, the unique time-dependent features imprinted by
the dark matter component of the Sagittarius stream, even if only a few percent
of the local dark matter density, may be detectable at $\sim$2$\sigma$
significance. A co-rotating dark disk, with lag speed $\sim$50 km$/$s, that is
$\sim$20$\%$ of the local DM density could dominate the signal, while colder
but as-of-yet unknown substructure may be even more important. Our likelihood
formalism, and the results derived with it, are generally applicable to any
time-series based approach to axion direct detection.
