Limits of self-interacting neutrinos from the BAO and CMB phase shift

Neutrinos with Standard Model interactions free-stream in the early Universe, leaving a distinct phase shift in the pattern of baryon acoustic oscillations (BAO). When isolated, this phase shift allows one to robustly infer the presence of the cosmic neutrino background in BAO and cosmic microwave background (CMB) data independently of other cosmological parameters. While in the context of the Standard Model, this phase shift follows a known scale-dependent relation, new physics in the cosmic neutrino background could alter the overall shape of this feature. In this paper, we discuss how changes in the neutrino phase shift could be used to constrain self-interactions among neutrinos. We produce simple models for this phase-shift assuming universal self-interactions, and use these in order to understand what constraining power is available for the strength of such interactions in BAO and CMB data. We find that, although challenging, it may be possible to use a detection of the phase to put a more robust limit on the strength of the self-interaction, $G_{\mathrm{eff}}$, which at present suffers from bimodality in cosmological constraints. Our forecast analysis reveals that BAO data alone will not provide the precision needed to tightly constrain self-interactions; however, the combined analysis of the phase shift signature in both CMB and BAO can potentially provide a way to detect the impact of new neutrino interactions. Our results could be extended upon for models with non-universal interactions.