cm^{2}} \,.$$ This is an necessary, but not sufficient, condition for the effects of $\nu$SI on astrophysical neutrinos to be observed. The actual $\nu$SI sensitivity for an neutrino telescope depends on model details such as the resonance energy and detector details such as the analysis energy range. A larger $\sigma_{\nu\nu}$ would severely affect the incoming neutrino beam, because the unattenuated fraction scales as $e^{-\tau}$.

If the interaction is through a heavy mediator, then $\sigma_{\nu \nu} \simeq G^2 s \simeq G^2 2 E m_\nu$. For the PeV neutrinos detected by IceCube, this leads to a scale $$\label{eq:4fermi}
G \simeq 4 \; {\rm GeV^{-2}} \,.$$ This nominal sensitivity on G is a factor of $\sim 25$ times below than the CMB limit [@Archidiacono:2013dua]. Thus, the detection of astrophysical neutrinos gives an exciting opportunity to test $\nu$SI.

Neutrino propagation: free streaming case
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In the usual case, neutrinos from cosmic sources travel to the detector without interaction. The standard model neutrino self-interaction cross section grows linearly with neutrino energy when the center of mass energy is below the mass of the Z boson. It is $\sim10^{-41}\, {\rm cm^{2}}$ [@Roulet:1992pz; @Bhattacharjee:1998qc; @Barenboim:2004di; @Lunardini:2013iwa] for a $1\,{\rm PeV}$ neutrino scattering on C$\nu$B ($0.1\,{\rm eV}$ masses), 