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We consider invisible neutrino decay $\nu_H \to \nu_l + \phi$ in theultra-relativistic limit and compute the neutrino anisotropy loss rate relevantfor the cosmic microwave background (CMB) anisotropies. Improving on ourprevious work which assumed massless $\nu_l$ and $\phi$, we reinstate in thiswork the daughter neutrino mass $m_{\nu l}$ in a manner consistent with theexperimentally determined neutrino mass splittings. We find that a nonzero$m_{\nu l}$ introduces a new phase space factor in the loss rate $\Gamma_{\rmT}$ proportional to $(\Delta m_\nu^2/m_{\nu_H}^2)^2$ in the limit of a smallsquared mass gap between the parent and daughter neutrinos, i.e., $\Gamma_{\rmT} \sim (\Delta m_\nu^2/m_{\nu H}^2)^2 (m_{\nu H}/E_\nu )^5 (1/\tau_0)$, where$\tau_0$ is the $\nu_H$ rest-frame lifetime. Using a general form of thisresult, we update the limit on $\tau_0$ using the Planck 2018 CMB data. We findthat for a parent neutrino of mass $m_{\nu H} \lesssim 0.1 {\rm eV}$, the newphase space factor weakens the constraint on its lifetime by up to a factor of50 if $\Delta m_\nu^2$ corresponds to the atmospheric mass gap and up to$10^{5}$ if the solar mass gap, in comparison with naive estimates that assume$m_{\nu l}=0$. The revised constraints are (i) $\tau^0 \gtrsim (6 \to 10)\times 10^5~{\rm s}$ and $\tau^0 \gtrsim (400 \to 500)~{\rm s}$ if only oneneutrino decays to a daughter neutrino separated by, respectively, theatmospheric and the solar mass gap, and (ii) $\tau^0 \gtrsim (2 \to 3) \times10^7~{\rm s}$ in the case of two decay channels with one near-commonatmospheric mass gap. In contrast to previous, naive limits which scale as$m_{\nu H}^5$, these mass spectrum-consistent $\tau_0$ constraints areremarkably independent of the parent mass and open up a swath of parameterspace within the projected reach of IceCube and other neutrino telescopes inthe next two decades.

Weaker yet again: mass spectrum-consistent cosmological constraints on the neutrino lifetime