  The baryon acoustic oscillations (BAO) provide an important bridge between
the early universe and the expansion history at late times. While the BAO has
primarily been used as a standard ruler, it also encodes recombination era
physics, as demonstrated by a recent measurement of the neutrino-induced phase
shift in the BAO feature. In principle, these measurements offer a novel window
into physics at the time of baryon decoupling. However, our analytic
understanding of the BAO feature is limited, particularly for the range of
Fourier modes measured in surveys. As a result, it is unclear what the BAO
phase teaches us about the early universe beyond what is already known from the
cosmic microwave background (CMB). In this paper, we provide a more complete
(semi-)analytic treatment of the BAO on observationally relevant scales. In
particular, we compute corrections to the frequency and phase of the BAO
feature that arise from higher order effects which occur in the tight coupling
regime and during baryon decoupling. The total phase shift we find is
comparable to a few percent shift in the BAO scale (frequency) and thus
relevant in current data. Our results include an improved analytic calculation
of the neutrino induced phase shift template that is in close agreement with
the numerically determined template used in measurements of the CMB and BAO.
