Precision constraints on radiative neutrino decay with CMB spectral
  distortion
  We investigate the radiative decay of the cosmic neutrino background, and its
impact on the spectrum of the cosmic microwave background (CMB) that is known
to be a nearly perfect black body. We derive exact formulae for the decay of a
heavier neutrino into a lighter neutrino and a photon, $\nu_j \to \nu_i +
\gamma$, and of absorption as its inverse, $\nu_i + \gamma \to \nu_j$, by
accounting for the precise form of the neutrino momentum distribution. Our
calculations show that if the neutrinos are heavier than $\mathcal O(0.1)$ eV,
the exact formulae give results that differ by $\sim$50%, compared with
approximate ones where neutrinos are assumed to be at rest. We also find that
spectral distortion due to absorption is more important for heavy neutrino
masses (by a factor of $\sim$10 going from a neutrino mass of 0.01 eV to 0.1
eV). By analyzing the CMB spectral data measured with COBE-FIRAS, we obtain
lower limits on the neutrino lifetime of $\tau_{12} \gtrsim 4 \times 10^{21}$ s
(95% C.L.) for the smaller mass splitting and $\tau_{13} \sim \tau_{23} \gtrsim
10^{19}$ s for the larger mass splitting. These represent up to one order of
magnitude improvement over previous CMB constraints. With future CMB
experiments such as PIXIE, these limits will improve by roughly 4 orders of
magnitude. This translates to a projected upper limit on the neutrino magnetic
moment (for certain neutrino masses and decay modes) of $\mu_\nu < 3 \times
10^{-11}\, \mu_B$, where $\mu_B$ is the Bohr magneton. Such constraints would
make future precision CMB measurements competitive with lab-based constraints
on neutrino magnetic moments.
