region. Because both the astrophysical neutrino beam and the C$\nu$B targets are expected to contain all flavors of neutrinos, the IceCube sensitivity is complementary to the flavor-dependent laboratory limits.

Astrophysical Neutrino Interactions with the C$\nu$B {#sec:nucnub}
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Sensitivity estimate for $\nu$SI
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We first make an order-of-magnitude estimate of the sensitivity of IceCube to $\nu$SI with cross section $\sigma_{\nu\nu}$; we present a more detailed calculation in later sections. If the neutrino mass scale is $\simeq 0.1$ eV, target C$\nu$B neutrinos have degenerate masses and are nonrelativistic today. Because of neutrino mixing, all flavors of neutrinos and antineutrinos should be present with comparable fractions in both the beam and target. To be conservative, we assume that only one species (flavor or mass eigenstate) of neutrinos and antineutrinos in the beam interacts, each with only half of the targets of a given species, so the target density is $n_t \simeq 56$ cm$^{-3}$ [@Hannestad:2006zg; @Wong:2011ip]. A typical distance for astrophysical sources is the Hubble length, $c/H_{0} \simeq 4$ Gpc. The optical depth for $\nu$SI interactions is $\tau \simeq n_t \sigma_{\nu\nu} c/H_{0}$. For $\nu$SI to affect neutrino propagation appreciably, one would require $\tau \simeq 1$, and therefore $$\label{eq:sigma_0}
\sigma_{\nu\nu} \simeq
1\times 10^{-30} \, {\rm 