Do cosmological neutrino-antineutrino pairs annihilate at all?

Question

Do low energy cosmological/relic neutrino-antineutrino pairs annihilate to produce photons at all? Their energy is presently too low to produce electron-positron pairs but there should be an indirect, suppressed path to produce photons. Is there an estimate of the rate or time to annihilate?

Answer 1

Although cosmological neutrino-antineutrino pairs could annihilate into photons when they were hotter and denser in the very early universe, the current relic neutrino densities and energies are so low that it is unlikely to happen ever again anywhere in the universe.

In the Standard Model, neutrino annihilation into photons proceeds through loop diagrams involving charged leptons and weak bosons. The cross-section for $\sigma(\nu\bar{\nu} \rightarrow \gamma\gamma)$ annihilation is actually much smaller than for $\sigma(\nu\bar{\nu} \rightarrow \gamma\gamma\gamma)$ because although the former has one less factor of $\alpha_{QED}$, it also has additional factors of $\omega/m_W$, where $m_W$ is the W boson mass.

The cross-section for neutrino-antineutrino pairs into three photons is (if I haven't made any mistakes):

$$\begin{aligned} \sigma(\nu\bar{\nu} \rightarrow \gamma\gamma\gamma) &= \frac{136}{91125 \, \pi^4} \, \alpha_{QED}^3 \, G_F^2\,\left(\frac{1}{2}+2 \sin^2\theta_W\right)^2 \, m_e^2 \, \left(\frac{\omega}{m_e}\right)^{10} \\ &\sim \left(\frac{\omega}{m_e}\right)^{10}\,10^{-55}\,\mathrm{cm^2} \end{aligned}$$ where $\alpha_{QED}$ is the fine structure constant, $G_F$ and $\theta_W$ are the Weak Fermi constant and mixing angle, $m_e$ is the electron mass, and $\omega$ is the $\nu\bar{\nu}$ centre-of-mass energy.

The estimated density of the $1.95\,K\,(1.68\times10^{-4}\,\mathrm{eV})$ cosmic neutrino background is $\mathrm{56\,\nu/cm^3} \, + \, \mathrm{56\,\bar{\nu}/cm^3}$ for each neutrino flavour, i.e. $n_{\nu}=\mathrm{336/cm^3}$ for 3 flavours. Based on current neutrino data, we don't expect neutrinos to be heavier than about $0.1\,\mathrm{eV}$, so $\omega\sim 0.1\,\mathrm{eV}$ and the relic annihilation cross-section should be about

$$ \sigma(\nu\bar{\nu} \rightarrow \gamma\gamma\gamma) \sim 10^{-122}\,\mathrm{cm^2} $$

The cosmic relic neutrino velocities are of order $v\sim 10^3 \, \mathrm{km/s}$, so their annihilation rate should be:

$$ \frac{dn}{dt}\left(\nu\bar{\nu} \rightarrow \gamma\gamma\gamma\right) \sim n_{\nu}^2 \sigma v \sim 10^{-111}\,\mathrm{annihilations/cm^3 /s} $$

The current volume of the observable universe is $\sim 4\times 10^{80}\,\mathrm{m^3}$, so the current relic neutrino annihilation rate in the entire universe is $$\sim 10^{-18}\,\mathrm{annihilations/year/universe}$$

This means that at the current rate one would have to wait 8 orders-of-magnitude longer than the current age of the universe for even one relic $\nu\bar{\nu}$ annihilation to occur anywhere in the universe. Even that one annihilation, however, could never happen if the universe continues to expand. The expansion dilutes and cools the cosmic neutrino background, reducing the already infinitesimal rate and pushing that one annihilation farther and farther into the future.