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We investigate the radiative decay of the cosmic neutrino background, and itsimpact on the spectrum of the cosmic microwave background (CMB) that is knownto be a nearly perfect black body. We derive exact formulae for the decay of aheavier neutrino into a lighter neutrino and a photon, $\nu_j \to \nu_i +\gamma$, and of absorption as its inverse, $\nu_i + \gamma \to \nu_j$, byaccounting for the precise form of the neutrino momentum distribution. Ourcalculations show that if the neutrinos are heavier than $\mathcal O(0.1)$ eV,the exact formulae give results that differ by $\sim$50%, compared withapproximate ones where neutrinos are assumed to be at rest. We also find thatspectral distortion due to absorption is more important for heavy neutrinomasses (by a factor of $\sim$10 going from a neutrino mass of 0.01 eV to 0.1eV). By analyzing the CMB spectral data measured with COBE-FIRAS, we obtainlower limits on the neutrino lifetime of $\tau_{12} \gtrsim 4 \times 10^{21}$ s(95% C.L.) for the smaller mass splitting and $\tau_{13} \sim \tau_{23} \gtrsim10^{19}$ s for the larger mass splitting. These represent up to one order ofmagnitude improvement over previous CMB constraints. With future CMBexperiments such as PIXIE, these limits will improve by roughly 4 orders ofmagnitude. This translates to a projected upper limit on the neutrino magneticmoment (for certain neutrino masses and decay modes) of $\mu_\nu < 3 \times10^{-11}\, \mu_B$, where $\mu_B$ is the Bohr magneton. Such constraints wouldmake future precision CMB measurements competitive with lab-based constraintson neutrino magnetic moments.

Precision constraints on radiative neutrino decay with CMB spectral distortion