astrophysics - Neutrinos vs. Photons: Who wins the race across the galaxy?

Inspired by the wording of this answer, a thought occurred to me. If a photon and a neutrino were to race along a significant stretch of our actual galaxy, which would win the race?

Now, neutrinos had better not be going faster than the speed of light in vacuum. However, an energetic enough neutrino can have a velocity arbitrarily close to $c$. Say we took a neutrino from a typical core-collapse supernova. It would have a speed $$ v_\nu = (1 - \epsilon_\nu) c $$ for some small $\epsilon_\nu > 0$. What is the order of magnitude for $\epsilon_\nu$?

At the same time, photons can also travel slower than $c$. The interstellar medium is not completely devoid of matter, and in fact much of this matter is ionized plasma. As such, it should have a plasma frequency $\omega_\mathrm{p}$, and so it should effectively have an index of refraction depending on the ratio $\omega/\omega_\mathrm{p}$. Then the speed of a photon will be $$ v_\gamma = (1 - \epsilon_\gamma) c, $$ where $\epsilon_\gamma$ is in general frequency-dependent. What is the order of magnitude for this deviation? I know it comes into play at radio frequencies, where in fact even the variation of $v_\gamma$ with frequency is detected: Pulses from pulsars suffer dispersion as they travel over hundreds to thousands of parsecs to reach us.

For simplicity, let's assume there are no obstructions like giant molecular clouds or rogue planets to get in the way of the photon. Is it possible that some photons will be outpaced by typical neutrinos? How big is this effect, and how does it depend on photon frequency and neutrino energy?

Answer

Cute question!

For a neutrino with mass $m$ and energy $E\gg m$, we have $v=1-\epsilon$, where $\epsilon\approx (1/2)(m/E)^2$ (in units with $c=1$). IceCube has detected neutrinos with energies on the order of 1 PeV, but that's exceptional. For neutrinos with mass 0.1 eV and an energy of 1 PeV, we have $\epsilon\sim10^{-32}$.

The time of flight for high-energy photons has been proposed as a test of theories of quantum gravity. A decade ago, Lee Smolin was pushing the idea that loop quantum gravity predicted measurable vacuum dispersion for high-energy photons from supernovae. The actual results of measurements were negative: http://arxiv.org/abs/0908.1832 . Photons with energies as high as 30 GeV were found to be dispersed by no more than $\sim 10^{-17}$ relative to other photons. What this tells us is that interactions with the interstellar medium must cause $\epsilon \ll 10^{-17}$, or else those interactions would have prohibited such an experiment as a test of LQG.

According to WP, the density of the interstellar medium varies by many order of magnitude, but assuming that it's $\sim 10^{-22}$ times the density of ordinary matter, we could guess that it causes $\epsilon\sim 10^{-22}$. This would be consistent with the fact that it wasn't considered important in the tests of vacuum dispersion.

For a neutrino with a mass of 0.1 eV to have $\epsilon\sim 10^{-22}$, it would have to have an energy of 10 GeV. This seems to be within but on the high end of the energy scale for radiation emitted by supernovae. So I think the answer is that it really depends on the energy of the photon, the energy of the neutrino, and the density of the (highly nonuniform) interstellar medium that the particles pass through.