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The next generation of axion direct detection experiments may rule out orconfirm axions as the dominant source of dark matter. We develop a generallikelihood-based framework for studying the time-series data at suchexperiments, with a focus on the role of dark-matter astrophysics, to searchfor signatures of the QCD axion or axion like particles. We illustrate how inthe event of a detection the likelihood framework may be used to extractmeasures of the local dark matter phase-space distribution, accounting foreffects such as annual modulation and gravitational focusing, which is theperturbation to the dark matter phase-space distribution by the gravitationalfield 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 at5$\sigma$ global significance, the unique time-dependent features imprinted bythe dark matter component of the Sagittarius stream, even if only a few percentof 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 colderbut as-of-yet unknown substructure may be even more important. Our likelihoodformalism, and the results derived with it, are generally applicable to anytime-series based approach to axion direct detection.

Revealing the Dark Matter Halo with Axion Direct Detection