I'm going to tell you how, soon, humanity might measure the masses of neutrinos just by observing past events in the universe. I like this topic because it is one of the few situations in fundamental physics where a measurement of the greater universe might detect something about fundamental particles and/or their interactions, before we manage to measure it in a lab. Another example is the existence of dark matter; however the mass of dark matter will almost certainly be first measured in a lab. Perhaps with neutrinos it will go in the other direction?
What is a neutrino?
I guess that before telling you how to measure a neutrino's mass, it might be pertinent to tell you what a neutrino is and how we can know it has mass before we've measured that mass. Well...
When an atomic nucleus decays, the decay products we see are other nuclei, electrons and/or positrons. These visible products always carry less energy and momentum than the amount that the initial nucleus had. This suggests strongly that some unknown other particle is also being created in the decay and that we just can't see it. This hypothetical particle was dubbed the neutrino and when theories were developed for the force responsible for nuclear decays, the neutrino became an important part of them. And, eventually, neutrinos were detected directly. It took a while because neutrinos interact incredibly weakly, which means you need either a lot of neutrinos or a lot of transparent stuff for the neutrino to interact with (or both) before you will see them.
Initially, it was assumed that neutrinos are massless. They don't need to be massless, but for a long time there was no evidence that they did have mass, so the simplest assumption was that they didn't. There are three types of neutrinos: those emitted in interactions with electrons, those emitted in interactions with muons and those emitted in interactions with tau particles. If neutrinos were massless, then a neutrino emitted as an electron neutrino would always remain an electron neutrino. Similarly, a muon neutrino would always remain a muon neutrino. However, if neutrinos do have mass, then a neutrino emitted in an interaction with an electron will actually travel as a superposition of an electron neutrino, muon neutrino and tau neutrino. The net result being that this neutrino could be detected as a different type of neutrino. Therefore, a smoking gun thing to look for when determining whether neutrinos have mass is this characteristic signal whereby one type of neutrino appears to oscillate into another type of neutrino.
This effect was then seen and seen and seen again. Neutrinos appear to have mass. From the perspective of particle physics this is a bit weird. Neutrinos must have really small masses and it is unclear why these masses are so small. Unfortunately, this mechanism of neutrino oscillations doesn't directly give the masses of the neutrinos. Although, it can be used to measure the differences of the masses of the neutrinos, thus setting lower bounds on the possible masses of the neutrinos.
What has this got to do with cosmology?
Good question. Well, neutrinos exist. They interact through the weak nuclear force with electrons. Today, the temperature of the universe is nowhere near high enough for neutrinos to be emitted by the free electrons in space. However, if we go far enough backward in time, it was. And, in fact, just as the cosmic microwave background (CMB) formed when the temperature of the universe dropped to a value small enough that the thermal energy available for interactions could not strip electrons from protons, there was also a point in the universe's history when the temperature dropped by enough that electrons and neutrinos went from interacting regularly, to not interacting at all. At this point, the cosmic neutrino background (CNB) was formed.
This cosmic neutrino background is still out there in the universe doing whatever cosmic neutrino backgrounds like to do and hopefully we can spot it doing some of these things.
Unfortunately, the universe has cooled a lot since the CNB was formed and there is next to no hope of us ever detecting it directly during the lifetime of anyone writing or reading this blog. We can detect neutrinos in the lab today, but only neutrinos that were recently formed in nuclear explosions, reactors, supernovae, etc. The CNB, which has cooled to the temperature of a few Kelvin just simply lacks the energy to interact with anything on Earth. However, I suppose one should never say never and there would be all sorts of interesting cosmological information present for us in the CNB if we ever could measure it directly.
Fortunately, these neutrinos still have some energy and gravity sees energy. Therefore, just as we can detect dark matter indirectly through its gravitational effects, we can detect these neutrinos indirectly too.
And we have (and only recently has this become conclusive)!
Detecting neutrinos through the expansion history of the universe
Matter and radiation have different pressures. Cold, stationary, matter has no pressure. This makes sense. It isn't moving, it will start to move under local gravitational effects, but it won't move over cosmological distances or gain speeds that are cosmologically relevant. So it won't push against the universe's expansion. However, radiation does have pressure, a positive pressure. This also makes sense. Put radiation somewhere and it will inevitably push against whatever it is put in. Great. What this means is that the expansion of the universe is different when the universe is dominated by matter than when it is dominated by radiation. This is because the pressure that is exerted by the radiation provides an additional deceleration to the expansion of the universe, over and above the simple gravitational attraction of the energy.
At some point in the history of the universe, the dominant energy of the universe changed from radiation to matter. This is because the total energy in any radiation dilutes faster than it does in cold matter. We can see when in time this turnover point occurred by looking at things like the CMB. This is because the oscillations in the primordial hydrogen plasma were also different during the different types of expansion. Therefore, the sound waves we see in the CMB look subtly different depending on whether they were happening during radiation or matter domination.
Now, because neutrinos are so light, they actually contribute to both the radiation and matter contents of the universe at different times. They have mass, which contributes to the overall mass in the universe, but if they are light enough they will travel fast enough that they add pressure to the universe too, thus increasing the total radiation component. Therefore, the simple existence of neutrinos changes where this turnover between radiation and matter domination occurs. And, if one assumes that there was no CNB, the fit to the CMB is much worse than if one assumes there are three types of neutrinos in the CNB. That is, the turnover point happened measurably later on than it would have if electromagnetic radiation was the only radiation present during the earlier periods of the universe's history. It actually took Planck's measurements to see this conclusively, although it was always expected to turn out to be true.
But that detection assumed neutrinos were massless and thus contributed only to the radiation at this early time. If they had mass, their contribution would be a little more subtle.
The effect of neutrino masses on the expansion history of the universe
(Note: I'm only 90% sure that I got all the bigger/smaller, earlier/later, higher/lower bits of this section absolutely correct, it can get confusing...)
Things now get a little ambiguous. If neutrinos have mass, then some of the dark matter in the universe is actually neutrinos. Therefore, when we take the standard cosmological model and decide to add neutrino masses to it, something needs to change. We could keep the overall density of dark matter the same today, in which case, when the CMB formed, it would be smaller (because neutrinos would appear like radiation back then due to their very large kinetic energies). Or, we could keep the total amount of dark matter the same at early times, in which case it is actually slightly higher today because the neutrinos have cooled down. Either change will have observational consequences.
If we insist that the dark matter density today must remain fixed, then we see that the turnover point between matter and radiation will occur later than it would for massless neutrinos. In fact, the more massive we make the neutrinos, the later the turnover will happen. Conversely, if we fix the matter to be the same at early times as it is in the standard cosmological model, then this means that after this point, the density of dark matter would increase. This would mean that the deceleration of the expansion of the universe from then until now would be more pronounced. This would mean that the universe is younger than we currently measure it to be (though not by much), which would itself have consequences for relating distance scales between the current universe and the early universe.
In reality, if neutrinos do have a large enough mass to be having an effect here, it is likely that the density of dark matter will change a little both at early times and at late times. This is one of the reasons why measuring neutrino masses is very difficult using just the expansion history of the universe. That is, the effects of neutrino mass are degenerate with other things (such as the current expansion rate and the total density of dark matter).
Is there another probe?
The effect of neutrino masses on the growth of cosmological structures
There is another probe, and it is a powerful probe. It's the one that might just win the race against laboratory experiments.
Dark matter seeds the growth of structure. Where there were peaks in the early dark matter field we now find galaxies and clusters of galaxies and superclusters of galaxies. Where there were troughs, we now find vast empty regions of space (i.e. "voids").
How would massive neutrinos affect this story? Well, as I wrote above, if neutrinos have mass, then some of the dark matter that is seeding this growth of structure is actually made up of neutrinos. But, neutrinos are quite light and although they have cooled to very low temperatures, they still have relatively large velocities and earlier in time had even larger velocities. These velocities aren't large enough to be providing a measurable pressure in the universe today, but they are large enough to produce one other characteristic effect.
To understand this effect, consider something like the Earth. The Earth is at the centre of a large gravitational well. If I throw a ball up, it must come down, right? This is because of the gravitational pull of the Earth. Great. But, if ESA makes a rocket with enough fuel, such that they give a satellite enough energy, then they can send out a device with enough energy that it is no longer bound to Earth. It need not come down. Give something enough energy and it will no longer be bound to a gravitational well.
The same can happen with the growth of structure. If neutrinos have enough energy, they can evade being bound to regions of the universe that might otherwise collapse into structures. Cold dark matter can't do this because it has no kinetic energy (that's why it is called "cold"), but neutrino dark matter can. What does this mean? Well, it means that sufficiently shallow gravitational wells won't trap neutrinos and sufficiently deep ones will. If we translate this into distance scales, it means that very small structures won't trap neutrinos and very large ones will.
Hopefully the final picture is starting to form. Neutrinos can't make up much of dark matter today, but if they have mass they inevitably make up some of it. Therefore, the smoking gun signal that neutrinos with mass would give us is a characteristic deficit in the quantity/amplitude of structures on the smallest distance scales. How big this deficit is depends on the mass of the neutrinos. The bigger the mass, the bigger the deficit (simply because a bigger fraction of dark matter is neutrinos). Where this characteristic drop-off occurs also depends on the masses of the neutrinos but there is the same ambiguity as there was in the previous section. If we add neutrino mass, something else must change in the standard cosmological model and these changes also effect the location of this characteristic scale.
However, once the values for the other parameters are chosen, this characteristic scale can be predicted and thus the neutrino masses measured.
Given current lower limits on the mass of the neutrinos from measurements in laboratories, cosmology will be able to measure the sum of the neutrino masses within two decades, and much earlier if the masses are large enough.