neutrinos of such energy may not exist; the cross section at lower energies is much smaller. Any significant neutrino self-interaction observed at lower energies must be due to $\nu$SI.

In terms of specific limits, neutrinos detected from SN 1987A were the first and, until recently, only direct detection of neutrinos from astrophysical sources beyond the Sun. Requiring that these neutrinos travel through the C$\nu$B without scattering leads to a robust upper limit on the cross section. Had the interaction strength been larger, the neutrinos would have scattered to lower energy and fallen below the detector sensitivity [@Kolb:1987qy]. The limit from KT87 corresponds to $G \lesssim 10^{8} \; {\rm GeV}^{-2}$. The average supernova neutrino energy is $\sim 10$ MeV and that of the C$\nu$B is $\sim 10^{-1}\,{\rm eV}$ (assuming small but degenerate neutrino masses), making $\sqrt{s} \sim 10^{-3}$ MeV, so the applicability of this limit would end below a vertical boundary at $M \sim 10^{-3}$ MeV (not shown). For a massless neutrino, this boundary would be at $M \sim 10^{-4}$ MeV, because the average neutrino energy is $\sim 10^{-3}\,{\rm eV}$.

Another general framework for directly testing $\nu$SI is through their effects on a gas with a high neutrino density. In the 