We consider invisible neutrino decay $$\nu _H \rightarrow \nu _l + \phi $$
ν
H
→
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$$\nu _l$$
ν
l
and $$\phi $$
ϕ
, we reinstate in this work the daughter neutrino mass $$m_{\nu l}$$
m
ν
l
in a manner consistent with the experimentally determined neutrino mass splittings. We find that a nonzero $$m_{\nu l}$$
m
ν
l
introduces a new phase space factor in the loss rate $$\varGamma _\mathrm{T}$$
Γ
T
proportional to $$(\varDelta m_\nu ^2/m_{\nu _H}^2)^2$$
(
Δ
m
ν
2
/
m
ν
H
2
)
2
in the limit of a small squared mass gap between the parent and daughter neutrinos, i.e., $$\varGamma _\mathrm{T} \sim (\varDelta m_\nu ^2/m_{\nu H}^2)^2 (m_{\nu H}/E_\nu )^5 (1/\tau _0)$$
Γ
T
∼
(
Δ
m
ν
2
/
m
ν
H
2
)
2
(
m
ν
H
/
E
ν
)
5
(
1
/
τ
0
)
, where $$\tau _0$$
τ
0
is the $$\nu _H$$
ν
H
rest-frame lifetime. Using a general form of this result, we update the limit on $$\tau _0$$
τ
0
using the Planck 2018 CMB data. We find that for a parent neutrino of mass $$m_{\nu H} \lesssim 0.1$$
m
ν
H
≲
0.1
eV, the new phase space factor weakens the constraint on its lifetime by up to a factor of 50 if $$\varDelta m_\nu ^2$$
Δ
m
ν
2
corresponds to the atmospheric mass gap and up to $$10^{5}$$
10
5
if the solar mass gap, in comparison with naïve estimates that assume $$m_{\nu l}=0$$
m
ν
l
=
0
. The revised constraints are (i) $$\tau ^0 > rsim (6 \rightarrow 10) \times 10^5$$
τ
0
≳
(
6
→
10
)
×
10
5
s and $$\tau ^0 > rsim (400 \rightarrow 500)$$
τ
0
≳
(
400
→
500
)
s if only one neutrino decays to a daughter neutrino separated by, respectively, the atmospheric and the solar mass gap, and (ii) $$\tau ^0 > rsim (2 \rightarrow 6) \times 10^7$$
τ
0
≳
(
2
→
6
)
×
10
7
s in the case of two decay channels with one near-common atmospheric mass gap. In contrast to previous, naïve limits which scale as $$m_{\nu H}^5$$
m
ν
H
5
, these mass spectrum-consistent $$\tau _0$$
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.