Weaker yet again: mass spectrum-consistent cosmological constraints on
  the neutrino lifetime
  We consider invisible neutrino decay $\nu_H \to \nu_l + \phi$ in the
ultra-relativistic limit and compute the neutrino anisotropy loss rate relevant
for the cosmic microwave background (CMB) anisotropies. Improving on our
previous work which assumed massless $\nu_l$ and $\phi$, we reinstate in this
work the daughter neutrino mass $m_{\nu l}$ in a manner consistent with the
experimentally determined neutrino mass splittings. We find that a nonzero
$m_{\nu l}$ introduces a new phase space factor in the loss rate $\Gamma_{\rm
T}$ proportional to $(\Delta m_\nu^2/m_{\nu_H}^2)^2$ in the limit of a small
squared mass gap between the parent and daughter neutrinos, i.e., $\Gamma_{\rm
T} \sim (\Delta m_\nu^2/m_{\nu H}^2)^2 (m_{\nu H}/E_\nu )^5 (1/\tau_0)$, where
$\tau_0$ is the $\nu_H$ rest-frame lifetime. Using a general form of this
result, we update the limit on $\tau_0$ using the Planck 2018 CMB data. We find
that for a parent neutrino of mass $m_{\nu H} \lesssim 0.1 {\rm eV}$, the new
phase space factor weakens the constraint on its lifetime by up to a factor of
50 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 one
neutrino decays to a daughter neutrino separated by, respectively, the
atmospheric and the solar mass gap, and (ii) $\tau^0 \gtrsim (2 \to 3) \times
10^7~{\rm s}$ in the case of two decay channels with one near-common
atmospheric mass gap. In contrast to previous, naive limits which scale as
$m_{\nu H}^5$, these mass spectrum-consistent $\tau_0$ constraints are
remarkably independent of the parent mass and open up a swath of parameter
space within the projected reach of IceCube and other neutrino telescopes in
the next two decades.
