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

We consider invisible neutrino decay $ν_H \to ν_l + ϕ$ 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 $ν_l$ and $ϕ$, we reinstate in this work the daughter neutrino mass $m_{νl}$ in a manner consistent with the experimentally determined neutrino mass splittings. We find that a nonzero $m_{νl}$ introduces a new phase space factor in the loss rate $Γ_{\rm T}$ proportional to $(Δm_ν^2/m_{ν_H}^2)^2$ in the limit of a small squared mass gap between the parent and daughter neutrinos, i.e., $Γ_{\rm T} \sim (Δm_ν^2/m_{νH}^2)^2 (m_{νH}/E_ν)^5 (1/τ_0)$, where $τ_0$ is the $ν_H$ rest-frame lifetime. Using a general form of this result, we update the limit on $τ_0$ using the Planck 2018 CMB data. We find that for a parent neutrino of mass $m_{νH} \lesssim 0.1 {\rm eV}$, the new phase space factor weakens the constraint on its lifetime by up to a factor of 50 if $Δm_ν^2$ corresponds to the atmospheric mass gap and up to $10^{5}$ if the solar mass gap, in comparison with naive estimates that assume $m_{νl}=0$. The revised constraints are (i) $τ^0 \gtrsim (6 \to 10) \times 10^5~{\rm s}$ and $τ^0 \gtrsim (400 \to 500)~{\rm s}$ if only one neutrino decays to a daughter neutrino separated by, respectively, the atmospheric and the solar mass gap, and (ii) $τ^0 \gtrsim (2 \to 3) \times 10^7~{\rm s}$ in the case of two decay channels with one near-common atmospheric mass gap. In contrast to previous, naive limits which scale as $m_{νH}^5$, these mass spectrum-consistent $τ_0$ 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.