  Of the many proposed extensions to the $\Lambda$CDM paradigm, a model in
which neutrinos self-interact until close to the epoch of matter-radiation
equality has been shown to provide a good fit to current cosmic microwave
background (CMB) data, while at the same time alleviating tensions with
late-time measurements of the expansion rate and matter fluctuation amplitude.
Interestingly, CMB fits to this model either pick out a specific large value of
the neutrino interaction strength, or are consistent with the extremely weak
neutrino interaction found in $\Lambda$CDM, resulting in a bimodal posterior
distribution for the neutrino self-interaction cross section. In this paper, we
explore why current cosmological data select this particular large neutrino
self-interaction strength, and by consequence, disfavor intermediate values of
the self-interaction cross section. We show how it is the $\ell \gtrsim 1000$
CMB temperature anisotropies, most recently measured by the Planck satellite,
that produce this bimodality. We also establish that smaller scale temperature
data, and improved polarization data measuring the temperature-polarization
cross-correlation, will best constrain the neutrino self-interaction strength.
We forecast that the upcoming Simons Observatory should be capable of
distinguishing between the models.
