I’ve been busy. There is a lot I’d like to say here, but I’ve been writing the actual science papers. Can’t keep up with myself, let alone everything else. I am prompted to write here now because of a small rant by Maury Goodman in the neutrino newsletter he occasionally sends out. It resonated with me.

First, some context. Neutrinos are particles of the Standard Model of particle physics. They come in three families with corresponding leptons: the electron (ν_{e}), muon (ν_{μ}), and tau (ν_{τ}) neutrinos. Neutrinos only interact through the weak nuclear force, feeling neither the strong force nor electromagnetism. This makes them “ghostly” particles. Their immunity to these forces means they have such a low cross-section for interacting with other matter that they mostly don’t. Zillions are created every second by the nuclear reactions in the sun, and the vast majority of them breeze right through the Earth as if it were no more than a pane of glass. Their existence was first inferred indirectly from the apparent failure of some nuclear decays to conserve energy – the sum of the products seemed less than that initially present because the neutrinos were running off with mass-energy without telling anyone about it by interacting with detectors of the time.

Clever people did devise ways to detect neutrinos, if only at the rate of one in a zillion. Neutrinos are the template for WIMP dark matter, which is imagined to be some particle from beyond the Standard Model that is more massive than neutrinos but similarly interact only through the weak force. That’s how laboratory experiments search for them.

While a great deal of effort has been invested in searching for WIMPs, so far the most interesting new physics is in the neutrinos themselves. They move at practically the speed of light, and for a long time it was believed that like photons, they were pure energy with zero rest mass. Indeed, I’m old enough to have been taught that neutrinos *must* have zero mass; it would screw everything up if they didn’t. This attitude is summed up by an anecdote about the late, great author of the Standard Model, Steven Weinberg:

A colleague at UT once asked Weinberg if there was neutrino mass in the Standard Model. He told her “not in my Standard Model.”

Steven Weinberg, as related by Maury Goodman

As I’ve related before, In 1984 I heard a talk by Hans Bethe in which he made the case for neutrino dark matter. I was flabbergasted – I had just learned neutrinos couldn’t possibly have mass! But, as he pointed out, there were a lot of them, so it wouldn’t take much – a tiny mass each, well below the experimental limits that existed at the time – and that would suffice to make all the dark matter. So, getting over the theoretical impossibility of this hypothesis, I reckoned that if it turned out that neutrinos did indeed have mass, then surely that would be the solution to the dark matter problem.

Wrong and wrong. Neutrinos do have mass, but not enough to explain the missing mass problem. At least not that of the whole universe, as the modern estimate is that they might have a mass density that is somewhat shy of that of ordinary baryons (see below). They are too lightweight to stick to individual galaxies, which they would boil right out of: even with lots of cold dark matter, there isn’t enough mass to gravitationally bind these relativistic particles. It seems unlikely, but it is at least conceivable that initially fast-moving but heavy neutrinos might by now have slowed down enough to stick to and make up part of some massive clusters of galaxies. While interesting, that is a very far cry from being *the* dark matter.

We know neutrinos have mass because they have been observed to transition between flavors as they traverse space. This can only happen if there are different quantum states for them to transition between. They can’t all just be the same zero-mass photon-like entity, at least two of them need to have some mass to make for split quantum levels so there is something to oscillate between.

Here’s where it gets *really* weird. Neutrino mass states do not correspond uniquely to neutrino flavors. We’re used to thinking of particles as having a mass: a proton weighs 0.938272 GeV; a neutron 0.939565 GeV. (The neutron being only 0.1% heavier than the proton is itself pretty weird; this comes up again later in the context of neutrinos if I remember to bring it up.) No, there are three separate mass states, each of which are fractional probabilistic combinations of the three neutrino flavors. This sounds completely insane, so let’s turn to an illustration:

So we have three flavors of neutrino, ν_{e}, ν_{μ}, and ν_{τ}, that mix and match to make up the three mass eigenstates, ν_{1}, ν_{2}, and ν_{3}. We would like to know what the masses, m_{1}, m_{2}, and m_{3}, of the mass eignestates are. We don’t. All that we glean from the solar and atmospheric oscillation data is that there is a transition between these states with a corresponding squared mass difference (e.g., Δm^{2}_{sol} = m_{2}^{2}-m_{1}^{2}). These are now well measured by astronomical standards, with Δm^{2}_{sol} = 0.000075 eV^{2} and Δm^{2}_{atm} = 0.0025 eV^{2} depending a little bit on which hierarchy is correct.

OK, so now we guess. If the hierarchy is normal and m_{1} = 0, then m_{2} = √Δm^{2}_{sol} = 0.0087 eV and m_{3} = √(Δm^{2}_{atm}+m_{2}^{2}) = 0.0507 eV. The first eigenstate mass need not be zero, though I’ve often heard it argued that it should be that or close to it, as the “natural” scale is m ~ √Δm^{2}. So maybe we have something like m_{1} = 0.01 eV and m_{2} = 0.013 eV in sorta the same ballpark.

Maybe, but I am underwhelmed by the naturalness of this argument. If we apply this reasoning to the proton and neutron (Ha! I remembered!), then the mass of the proton should be of order 1 MeV not 1 GeV. That’d be interesting because the proton, neutron, and electron would all have a mass within a factor of two of each other (the electron mass is 0.511 MeV). That almost sounds natural. It’d also make for some very different atomic physics, as we’d now have hydrogen atoms that are quasi-binary systems rather than a lightweight electron orbiting a heavy proton. That might make for an interesting universe, but it wouldn’t be the one we live in.

One very useful result of assuming m_{1} = 0 is that it provides a hard lower limit on the sum of the neutrino masses: ∑m_{i} = m_{1} + m_{2} + m_{3} > 0.059 eV. Here the hierarchy matters, with the lower limit becoming about 0.1 eV in the inverted hierarchy. So we know neutrinos weigh at least that much, maybe more.

There are of course efforts to measure the neutrino mass directly. There is a giant experiment called Katrin dedicated to this. It is challenging to measure a mass this close to zero, so all we have so far are upper limits. The first measurement from Katrin placed the 90% confidence limit < 1.1 eV. That’s about a factor of 20 larger than the lower limit, so in there somewhere.

Katrin on the move.

There is a famous result in cosmology concerning the sum of neutrino masses. Particles have a relic abundance that follows from thermodynamics. The cosmic microwave background is the thermal relic of photons. So too there should be a thermal relic of cosmic neutrinos with slightly lower temperature than the photon field. One can work out the relic abundance, so if one knows their mass, then their cosmic mass density is

Ω_{ν}h^{2} = ∑m_{i}/(93.5 eV)

where h is the Hubble constant in units of 100 km/s/Mpc (e.g., equation 9.31 in my edition of Peacock’s text *Cosmological Physics*). For the cosmologists’ favorite (but not obviously correct) h=0.67, the lower limit on the neutrino mass translates to a mass density Ω_{ν} > 0.0014, rather less than the corresponding baryon density, Ω_{b} = 0.049. The experimental upper limit from Katrin yields Ω_{ν} < 0.026, still a factor of two less than the baryons but in the same ballpark. These are nowhere near the Ω_{CDM} ~ 0.25 needed for cosmic dark matter.

Nevertheless, the neutrino mass potentially plays an important role in structure formation. Where cold dark matter (CDM) clumps easily to facilitate the formation of structure, neutrinos retard the process. They start out relativistic in the early universe, becoming non-relativistic (slow moving) at some redshift that depends on their mass. Early on, the represent a fast-moving component of gravitating mass that counteracts the slow moving CDM. The nascent clumps formed by CDM can capture baryons (this is how galaxies are thought to form), but they are not even speed bumps to the relativistic neutrinos. If the latter have too large a mass, they pull lumps apart rather then help them grow larger. The higher the neutrino mass, the more damage they do. This in turn impacts the shape of the power spectrum by imprinting a free-streaming scale.

The power spectrum is a key measurement fit by ΛCDM. Indeed, it is arguably its crowning glory. The power spectrum is well fit by ΛCDM assuming zero neutrino mass. If Ω_{ν} gets too big, it becomes a serious problem.

Consequently, cosmological observations place an indirect limit on the neutrino mass. There are a number of important assumptions that go into this limit, not all of which I am inclined to grant – most especially, the existence of CDM. But that makes it an important test, as the experimentally measured neutrino mass (whenever that happens) better not exceed the cosmological limit. If it does, that falsifies the cosmic structure formation theory based on cold dark matter.

The cosmological limit on neutrino mass obtained assuming ΛCDM structure formation is persistently an order of magnitude tighter than the experimental upper limit. For example, the Dark Energy Survey obtains ∑m_{i} < 0.13 eV at 95% confidence. This is similar to other previous results, and only a factor of two more than the lower limit from neutrino oscillations. The window of allowed space is getting rather narrow. Indeed, it is already close to ruling out the inverted hierarchy for which ∑m_{i} > 0.1 eV – *or* the assumptions on which the cosmological limit is made.

This brings us finally to Dr. Goodman’s rant, which I quote directly:

In the normal (inverted) mass order, s=m1+m2+m3 > 59 (100) meV. If as DES says, s < 130 meV, degenerate solutions are impossible. But DES “…model(s) massive neutrinos as three degenerate species of equal mass.” It’s been 34 years since we suspected neutrino masses were different and 23 years since that was accepted. Why don’t cosmology “measurements” of neutrino parameters do it right?

Maury Goodman

Here, s = ∑m_{i} and of course 1 eV = 1000 meV. Degenerate solutions are those in which m_{1}=m_{2}=m_{3}. When the absolute mass scale is large – say the neutrino mass were a huge (for it) 100 eV, then the sub-eV splittings between the mass levels illustrated above would be negligible and it would be fair to treat “massive neutrinos as three degenerate species of equal mass.” This is no longer the case when the implied upper limit on the mass is small; there is a clear difference between m_{1} and m_{2} and m_{3}.

So why don’t cosmologists do this right? Why do they persist in pretending that m_{1}=m_{2}=m_{3}?

Far be it from me to cut those guys slack, but I suspect there are two answers. One, it probably doesn’t matter (much), and two, habit. By habit, I mean that the tools used to compute the power spectrum were written at a time when degenerate species of equal mass was a perfectly safe assumption. Indeed, in those days, neutrinos were thought not to matter much at all to cosmological structure formation, so their inclusion was admirably forward looking – or, I suspect, a nerdy indulgence: “neutrinos probably don’t matter but I know how to code for them so I’ll do it by making the simplifying assumption that m_{1}=m_{2}=m_{3}.”

So how much does it matter? I don’t know without editing & running the code (e.g, CAMB or CMBEASY), which would be a great project for a grad student if it hasn’t already been done. Nevertheless, the difference between neutrino mass states and the degenerate assumption is presumably small for small differences in mass. To get an idea that is human-friendly, let’s think about the redshift at which neutrinos become non-relativistic. OK, maybe that doesn’t sound too friendly, but it is less likely to make your eyes cross than a discussion of power spectra Fourier transforms and free-streaming wave numbers.

Neutrinos are very lightweight, so start out as relativistic particles in the early universe (high redshift z). As the universe expands it cools, and the neutrinos slow down. At some point, they transition from behaving like a photon field to a non-relativistic gas of particles. This happens at

1+z_{nr} ≈ 1987 m_{ν}/(1 eV)

(eq. 4 of Agarwal & Feldman 2012; they also discuss the free-streaming scale and power spectra for those of you who want to get into it). For a 0.5 eV neutrino that is comfortably acceptable to the current experimental upper limit, z_{nr} = 992. This is right around recombination, and would mess everything up bigly – hence the cosmological limit being much stricter. For a degenerate neutrino of 0.13 eV, z_{nr} = 257. So one way to think about the cosmological limit is that we need to delay the impact of neutrinos on the power spectrum for at least this long in order to maintain the good fit to the data.

How late can the impact of neutrinos be delayed? For the minimum masses m_{1} = 0, m_{2} = 0.0087, m_{3} = 0.0507 eV, zero mass neutrinos always remain relativistic, but z_{2} = 16 and z_{3} = 100. These redshifts are readily distinguishable, so maybe Dr. Goodman has a valid point. Well, he definitely has a valid point, but these redshifts aren’t probed by the currently available data, so cosmologists probably figure it is OK to stick to degenerate neutrino masses for now.

The redshifts z_{2} = 16 and z_{3} = 100 are coincident with other important events in cosmic history, cosmic dawn and the dark ages, so it is worth considering the potential impact of neutrinos on the power spectra predicted for 21 cm absorption at those redshifts. There are experiments working to detect this, but measurement of the power spectrum is still a ways off. I am not aware of any theoretical consideration of this topic, so let’s consult an expert. Thanks to Avi Loeb for pointing out these (and a lot more!) references on short notice: Pritchard & Pierpaoli (2008), Villaescusa-Navarro et al. (2015), Obuljen et al. (2018). That’s a lot to process, and more than I’m willing to digest on the fly. But it looks like at least some cosmologists are grappling with the issue Dr. Goodman raises.

Any way we slice it, it looks like there are things still to learn. The direct laboratory measurement of the neutrino mass is not guaranteed to be less than the upper limit from cosmology. It would be surprising, but that would make matters a lot more interesting.