A mass measured by what it stopped from forming

Every other entry in the cosmic budget stays in its category. Baryons and dark matter dilute as the cube of the expansion and have always been matter; photons dilute as the fourth power and have always been radiation; the cosmological constant does not dilute at all.

Neutrinos are the exception. There are about 336 of them per cubic centimetre everywhere, left over from the first second, and their energies now are set by a temperature of 1.95 kelvin — which is $1.7\times10^{-4}$ electronvolts. A neutrino of mass 0.05 electronvolts is therefore moving slowly today and was moving at essentially the speed of light for the first several million years. It was radiation, and then it became matter.

That transition is what makes cosmology sensitive to a particle mass, and the sensitivity is remarkable: the bound it produces is tighter than anything a laboratory has achieved, on a quantity no laboratory experiment has yet detected at all.

A density that follows from counting #

The relic density is the one quantity here that involves no astrophysics and no modelling, and it is worth deriving because its cleanliness is what makes the measurement possible.

Neutrinos decoupled from the rest of the plasma when the universe was about one second old and the temperature about $10^{10}$ kelvin, because the weak interaction rate fell below the expansion rate. After that they simply expanded and cooled, and their number in a comoving volume has not changed since.

Electron–positron annihilation happened shortly afterwards and dumped its entropy into the photons but not into the already-decoupled neutrinos, which is why the neutrino temperature today is lower than the photon temperature by a calculable factor of $(4/11)^{1/3}$. That gives 1.95 kelvin against 2.725, and a number density of 112 per cubic centimetre per species.

Multiply by the mass and divide by the critical density and the result is

$$\Omega_\nu h^2 = \frac{\Sigma m_\nu}{93.14\ \mathrm{eV}}.$$

Nothing in that expression is fitted. It is a number density from statistical mechanics times a mass, and the only input beyond the standard model of particle physics is the photon temperature, which is measured to six significant figures.

The consequence is that a cosmological measurement of the neutrino density is a measurement of the sum of the masses, directly, with no conversion factor that anybody has to argue about.

Why they leave a hole #

The reason neutrinos are visible at all is that they were fast when structure was starting to grow.

A gravitational overdensity grows by pulling material in. A particle moving slowly compared with the escape speed of the growing clump falls in and stays; a particle moving fast crosses the clump and leaves. Neutrinos did the second, for as long as they were relativistic, and the distance one travels before slowing down is the free-streaming scale.

Below that scale, neutrinos do not cluster. They contribute their share to the mean density of the universe, which sets the expansion rate, but they contribute nothing to the growing lumps — so a perturbation on a small scale is trying to grow against a background expansion set by all the matter while only part of the matter is participating.

The effect on the growth rate is a slight suppression, and because growth is exponential in the number of e-folds, a slight suppression maintained over a long time is a large final deficit. Working it through gives

$$\frac{\Delta P}{P} \approx -8f_\nu,\qquad f_\nu = \frac{\Omega_\nu}{\Omega_{\rm m}},$$

on scales well below the free-streaming length. The factor of eight is the amplification: a species that is one per cent of the matter removes eight per cent of the small-scale power.

That is what makes the measurement possible. A one per cent component of the budget would be undetectable if it only changed the budget; it is detectable because it changes a growth rate that has been running for thirteen billion years.

What is actually measured, and it is never a neutrino #

Four observables carry the signature and none of them is a neutrino.

The microwave background is sensitive mainly through the expansion history: neutrinos with mass contribute to the matter density at late times, which changes the distance to the last-scattering surface and therefore the angular scale of the peaks. That dependence is degenerate with the other parameters that change the same distance, so the microwave background alone gives a weak bound.

Gravitational lensing of the microwave background is much better. The temperature map is distorted by the matter between the last-scattering surface and here, and the amount of distortion measures the integrated clumpiness — precisely the quantity neutrinos suppress. This is the cleanest single probe because it is sensitive to mass rather than to light.

**Galaxy clustering measures the shape of the power spectrum directly, and the suppression appears as a downward bend below the free-streaming scale. Its weakness is that galaxies are biased tracers: the relation between how galaxies clump and how matter clumps is not known a priori, and a scale-dependent bias could mimic or mask the signal.

The Lyman-α forest reaches the smallest scales and therefore the largest suppression, and it carries the largest astrophysical uncertainties — the gas’s thermal history sets the small-scale power as much as the cosmology does.

The published bounds combine several of these, and the combination is where the strength comes from: the degeneracies each one suffers are different degeneracies.

The degeneracy that decides the bound #

A cosmological neutrino-mass bound is not a measurement of one parameter. It is the projection of a multi-dimensional fit, and the bound depends on what else was allowed to vary.

The worst offender is the dark-energy equation of state. Massive neutrinos suppress structure; dark energy that is not a cosmological constant changes how fast structure grows at late times. The two effects have similar shapes over the observable range, so allowing $w$ to float roughly doubles the neutrino bound. A bound quoted “assuming a cosmological constant” and one quoted “with $w$ free” are different numbers about the same universe.

The second is the primordial amplitude and tilt, which set how much small-scale power there was to begin with. A universe with slightly less initial small-scale power looks like one with slightly heavier neutrinos, and the two are separated only because the neutrino effect has a characteristic scale and the tilt does not.

The third, and the one that will matter most as the measurements sharpen, is the modelling of the nonlinear regime. The suppression is largest on scales where structure has gone nonlinear, and predicting the nonlinear power spectrum to the required per-cent accuracy requires simulations that include the neutrinos — and the baryonic effects that redistribute gas on the same scales.

Every one of the three is an assumption about something other than neutrinos, which is the price of measuring a particle property with a telescope.

The floor the laboratory put under it #

The reason the cosmological bound is interesting rather than merely tight is that another field has supplied a lower limit, by a route with nothing in common.

Neutrino oscillation experiments measure the differences of the squared masses, because oscillation depends on those and on nothing else. Two splittings are known: a small one of about $7.5\times10^{-5}$ eV², and a large one of about $2.5\times10^{-3}$ eV². Their square roots are 8.6 and 50 millielectronvolts.

Those splittings put a floor under the sum. If the lightest neutrino has zero mass, the other two must have at least 8.6 and 50, so the sum is at least 58 millielectronvolts. That is the normal ordering. If instead the two closely spaced states are the heavy ones — the inverted ordering — the sum is at least about 100.

So the total mass is somewhere between 58 millielectronvolts and the cosmological ceiling, and the interval is now about a factor of two wide.

Two fields that share no instruments, no techniques and no systematics have bracketed a quantity between them, and the bracket is closing from both ends. If cosmology pushes the ceiling below 100 millielectronvolts it will have excluded the inverted ordering — a statement about particle physics, made by counting galaxies.

Two numbers from the same thermal history #

There is a second cosmological neutrino observable, older than the mass bound and independent of it, and the pair are worth holding together because they constrain different things about the same particles.

The relic neutrinos contribute to the radiation density in the early universe whether or not they have mass, because at those temperatures they are relativistic regardless. That contribution is conventionally written as an effective number of species, $N_{\rm eff}$, defined so that three standard neutrinos give 3.044 — slightly above three because the decoupling is not quite instantaneous and a little of the electron–positron annihilation energy leaks into them.

The radiation density sets the expansion rate in the radiation era, which sets the epoch of matter–radiation equality, and with it the boost the early acoustic peaks carry, which sets the scale at which the matter power spectrum turns over and the damping of the small-scale acoustic peaks. So $N_{\rm eff}$ is measured from the microwave background and from the primordial abundances, and the two agree near the standard value to about five per cent.

The two measurements answer different questions. $N_{\rm eff}$ counts how many relativistic species there were at one second and would detect a fourth light particle of any kind; the mass bound measures how much those species weigh now and says nothing about how many there are. Together they are a statement that the thermal history assumed in the 93.14 eV coefficient is the right one, which is what licenses reading a density as a mass at all.

A measurement is only as good as the history it is embedded in, and here the history is checked by a second measurement rather than assumed.

What a laboratory can and cannot do #

The comparison with direct experiments makes the cosmological bound’s character clear, because the two measure genuinely different quantities.

A beta-decay endpoint experiment measures the electron mass spectrum from tritium decay near its maximum energy, where a nonzero neutrino mass removes the last few electronvolts. What it constrains is an effective electron-neutrino mass — an incoherent average over the mass states weighted by their electron-flavour content — and the best current limit is about 0.45 electronvolts, roughly four times the cosmological one.

A neutrinoless double beta decay search measures a different average again, coherent this time and carrying unknown phases, and it only exists at all if the neutrino is its own antiparticle. A null result there is compatible with any mass if the phases conspire.

So there are three different “neutrino masses” measured by three fields, and they are related by the oscillation parameters plus assumptions. The cosmological one is the sum, it is the tightest, and it is the one most dependent on a model.

The clean statement is that the three are consistent and that only one of them is close to a detection. If the cosmological bound falls below the inverted-ordering floor before either laboratory experiment reaches its own, the ordering will have been decided by a galaxy survey — which is not how anybody expected the question to be settled.

Where the picture stops #

A sum is not three masses. Cosmology is sensitive only to the total, because what matters is the total energy density and the total free-streaming suppression. Distinguishing the individual masses requires the splittings from oscillations, and even then the assignment depends on the ordering. No cosmological measurement will ever return a mass for a single species.

The relic density assumes a standard thermal history. The 93.14 eV coefficient follows from neutrinos decoupling at the standard time with the standard number of species. Extra light species, a non-thermal production mechanism, or a decay into something lighter all change it, and a cosmological “mass” measured under a wrong thermal history is a measurement of the wrong thing. The effective number of relativistic species is constrained separately for exactly this reason.

And a null result is not a zero. The floor from oscillations means the sum is not zero, so a cosmological fit that prefers zero is preferring something the laboratory has excluded. Several analyses now find exactly that — a best fit at or below the floor — which is either a statistical fluctuation, a systematic in one of the probes, or a sign that something in the model is wrong. It is the most interesting current tension in the subject and it is usually reported as a bound.

The neutrinos are there and nobody has touched one #

It is worth saying plainly that the population this essay is about has never been detected by any direct means, and probably never will be.

The relic neutrinos are everywhere — a few hundred per cubic centimetre, passing through everything — and their energies are of order $10^{-4}$ electronvolts, which is far below the threshold of any detector that works by watching a neutrino hit something. A weak interaction cross-section falls steeply with energy, and at these energies it is smaller than anything a target can compensate for.

One proposal exists. A nucleus that is already unstable to beta decay can capture a relic neutrino instead, producing an electron with an energy slightly above the decay endpoint rather than below it, so a relic capture event is separated from the enormous beta-decay background by an energy gap equal to twice the neutrino mass. Tritium on a graphene substrate, a hundred grams of it, and an energy resolution better than the mass being sought: that is the design. It has not been built, and the required resolution is a factor of several beyond what exists.

So the strongest evidence for a population of $10^{89}$ particles is that structure did not form quite as fast as it would have without them, and that is what makes the cosmological bound worth stating carefully rather than quoting. Every other statement about the cosmic neutrino background is a prediction of the same thermal history that gives the coefficient.

The one thing that is directly observed is their effect on the early expansion rate through $N_{\rm eff}$, and that is a statement about a density rather than about a particle.

Why the bound is a measurement of an era #

There is a way of reading all this that makes the sensitivity less surprising.

The neutrinos’ effect is set by how long they spent relativistic, and that is set by the ratio of their mass to the temperature at matter–radiation equality. A heavier neutrino becomes non-relativistic earlier, spends less time free-streaming, and suppresses less — but it also contributes more density. The two effects partly cancel, and what survives is the linear dependence in the first figure.

The scale at which the suppression sets in is the horizon size when the neutrinos slowed down, and that scale is imprinted on the matter power spectrum as a bend. So the measurement is, at bottom, a measurement of when during the first few million years a particular species stopped moving at the speed of light — read off the sizes of the structures that exist now.

A particle mass has been converted into a date, and the date into a length, and the length into a deficit in a correlation function. That chain is four steps long and every step is a piece of standard physics, which is why the result is trusted despite no neutrino ever being detected in the process.

Still open: whether the data want a negative mass #

The most uncomfortable current result is easy to state and hard to interpret.

Several recent combinations of galaxy-survey and microwave-background data give a best-fit neutrino mass sum below the 58 millielectronvolt floor, and some prefer a formally negative value when the parameter is allowed to go there. A negative mass is not physical; what the fit is reporting is that the data want more small-scale structure than the model with minimal neutrinos predicts.

There are three readings and no way yet to choose. It may be a fluctuation, and the significance is currently modest. It may be a systematic in one of the probes — the calibration of a lensing amplitude, or the modelling of galaxy bias. Or it may be a real preference for a growth history that is not the one a cosmological constant plus cold dark matter gives, in which case the neutrino mass is an accidental reporter of a different problem.

What would settle it is a probe of the same quantity with different systematics, which is the same answer the S8 discrepancy has been waiting on for a decade. The two may well be the same anomaly seen twice.