much smaller than the sensitivity estimated above. Neutrino-CMB scattering is even more suppressed [@Seckel:1997kk]. Near 1 PeV, the neutrino-nucleon cross section is $\sim 10^{-33} \; {\rm cm^{2}}$ [@Gandhi:1995tf; @Gandhi:1998ri], but the nucleon density is only $\sim 10^{-7} \; {\rm cm^{-3}}$, due to the small baryon asymmetry, so the interaction probability is also negligible. (For electrons as targets, both the cross section and number density are small.) Therefore, any interactions of PeV neutrinos during propagation must be due to interactions beyond the standard model.

To describe the neutrino beam, we define the co-moving number density at a time $t$ by $n(t)$ and the differential (in energy $E$) number density by $\tilde{n}(t,E) \equiv dn(t,E)/dE$. The observable neutrino number flux is $$\begin{aligned}
J(E) & \equiv & \frac{dN_{\nu}}{dA dt d\Omega dE} \\
& = & \frac{c}{4 \pi} \tilde{n}(0,E) \nonumber \,.\end{aligned}$$ The evolution of the number density is described by the propagation equation [@Lee:1996fp; @Bhattacharjee:1998qc; @Berezinsky:2002nc; @Ahlers:2009rf; @Ahlers:2010fw; @Murase:2012xs]. In the free-streaming limit, that is $$\label{eq:freestreaming}
\frac{\partial \tilde{n}(t,E)}{\partial t} = \frac{\partial}{\partial E}(b \tilde{n}(t,E) ) + {\cal L}(t,E)\, .$$ The first term on the right takes into account the continuous energy loss rate $b=H(t)E$ due to redshift, and the second is the differential number luminosity density 