The Tightest Neutrino Mass Bounds (Cosmology)
While the cosmology bound on the sum of the neutrino masses (87 meV in the most strict limitation published) is subject to all sorts of theoretical limitations, the bounds on the differences between the neutrino masses due to neutrino oscillation measurements are not. If this bound applies (ruling out the inverted hierarchy as well), then the neutrino masses are about:
mv1= 6 ± 3 meV
mv2 = 15.2 ± 3.7 meV
mv3 = 64.7 ± 4.7 meV
with mv1 neutrino mass uncertainty correlated 100% with most of the uncertainty in the other values.
Direct Measurement Bounds
The Particle Data Group lists upper bounds on the electron neutrino mass as < 1.1 eV at 90% confidence, muon neutrino mass < 0.19 MeV at 90% confidence, and tau neutrino < 18.2 MeV at 95% confidence. By itself this would limit the sum of the three neutrino masses to < 18.4 MeV.
Katrin experiment papers this summer have suggested that the electron neutrino mass is really < 0.8 eV and that this limit could fall to < 0.2 eV by the end of its data collection run.
Direct Bounds Combined With Neutrino Oscillation Bounds
But, even in an inverted hierarchy case, we know from neutrino oscillation data there is a maximum spread of 0.1 eV between the three masses. So, this places a combined limit of about 1.2 eV at 90% confidence on muon and tau neutrino masses, individually, and a sum of the three neutrino masses < 3.4 eV.
If the direct detection limit is set at 800 meV rather than 1.1 eV, as one recent Katrin paper would suggest, the sum of the three masses is limited to < 2.5 eV.
Katrin, at the end of its run, could bring the limit on the sum of the three neutrino masses to < 0.7 eV, which is still nine times the cosmological bound, but shows some convergence.
The Particle Data Group Direct Bounds For The Two Heavier Neutrino Masses Are Way Too High
As a result, the vastly higher direct detection boundaries from the Particle Data Group for the two heavier neutrino masses aren't very meaningful.
The direct detection bound on the first neutrino mass is probably at least 90 times too high.
The direct detection bound on the second neutrino mass is at least 158,000 times too high, and probably more like 8.68 million times too high.
The direct detection bound on the third neutrino mass is at least 15.2 million times too high, and probably more like 251 million times too high.