The astronomy observation based constraints on the number of neutrino types (which also bounds possible number of light sterile neutrino species), and on the sum of the neutrino masses, are robust to a wide range of assumptions about the dark sector of cosmology made in the calculations made based upon observations.
Both support Standard Model assumption that there are exactly three kinds of neutrinos and strongly favors the proposition that the sum of their masses arise from a "normal" neutrino hierarchy (i.e. in which the most frequent mass eigenstate of electron neutrinos is smaller than the most frequent mass eigenstate of muon neutrinos, which in turn is less massive than the most frequent mass eigenstate of tau neutrinos).
Thus, unless there is something profoundly wrong at a theoretical level which that way that we infer the number of neutrino types form astronomy observations, the equivocal hints of "sterile neutrino" species that oscillate with the three active neutrino types from more direct nuclear reactor produced neutrino experiments are extremely strongly disfavored. But, astronomy observations sensitive to neutrinos don't "see" neutrino types of more than about 10 electron volts in mass (which is roughly a factor of two hundred more massive the the most massive of the three Standard Model neutrino mass eigenstates). A sterile neutrino type heavier than that would escape the astronomy observation based constraints, but would also be pretty much outside the mass range suggested by nuclear reactor produced neutrino experiments. The reactor anomalies, where they have been observed (not consistently with each other), favor a fourth "sterile neutrino" that oscillates with the active neutrinos with a mass on the order of 1 electron volt.
"Active neutrinos" (i.e. those that interact via the weak force at full strength) are ruled out experimentally up to about 62,500,000,000,000 milli-electron volts (because W and Z boson decays rule them out up to about 45,000,000,000,000 milli-electron volts, and an active neutrino with a mass of more than that would radically disturb the decays of the Higgs boson to a far greater extent than is consistent with observations of Higgs boson decays to date).
We know to a fair precision the differences in mass between the least massive and second least massive, and the second least massive and most massive neutrino mass eigenstate from neutrino oscillation experiments. This sets a floor to neutrino masses and also insures that the three neutrino masses are highly correlated.
This, together with the cap of the sum of the three neutrino masses that is derived from astronomy observation constraints, leaves a quite narrow range of masses for the possible lightest neutrino mass eigenstate of about 17 milli-electron volts (with 95% confidence), favoring the middle to low end of that range. By comparison, the mass of an electron is approximately 511,000,000 milli-electron volts. The is a very small absolute margin of error, although the relative error is very high for the first neutrino mass eigenstate (accurate to roughly a factor of 100), and on the order of 150%+ for the second neutrino mass eigenstate, and on the order of 35%+ for the third neutrino mass eigenstate.
Upper bounds on neutrino mass (1) from direct measurements and (2) from the lowest frequencies for which neutrinoless beta decay has been ruled out, are much less constraining than those derived from astronomy observations.
As it is currently understood, any of the three kinds of neutrino types can have one of three neutrino masses called eigenstates, but the probabilities of each type of neutrino type having a particular mass varies by type.
A new article and its abstract on the topic are as follows:
Dynamical Dark sectors and Neutrino masses and abundances(Submitted on 27 Mar 2020)We investigate generalized interacting dark matter-dark energy scenarios with a time-dependent coupling parameter, allowing also for freedom in the neutrino sector. The models are tested in the phantom and quintessence regimes, characterized by an equation of state and , respectively. Our analyses show that for some of the scenarios the existing tensions on the Hubble constant and on the clustering parameter can be significantly alleviated. The relief is either due to (a) a dark energy component which lies within the phantom region; or (b) the presence of a dynamical coupling in quintessence scenarios.
The inclusion of massive neutrinos into the interaction schemes does not affect neither the constraints on the cosmological parameters nor the bounds on the total number or relativistic degrees of freedom , which are found to be extremely robust and, in general, strongly consistent with the canonical prediction . The most stringent bound on the total neutrino mass is eV and it is obtained within a quintessence scenario in which the matter mass-energy density is only mildly affected by the presence of a dynamical dark sector coupling.