The Phase Of The Bao On Observable Scales

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.