The first three minutes use two lifetimes

A free neutron decays, on average, after about fourteen and a half minutes, into a proton, an electron and an antineutrino. How long it lives is measured two ways. One stores ultracold neutrons — slow enough to be held by material walls, magnetic fields or gravity — for a set time and counts how many remain: 877.75 seconds, to a third of a second. The other passes a cold beam through a known volume and counts the protons produced by decays inside it: 887.7 seconds, to about two. The two numbers differ by ten seconds, which is four standard deviations, and the difference has stood since the early 2000s while each technique has been refined by several independent groups.

Big-bang nucleosynthesis depends on this number, and the dependence has been set out where the whole calculation was first written down and again where helium was used to count neutrino species: the ten seconds move the predicted helium by a couple of parts in a thousand, less than the error on any helium measurement, and move an inferred species count by about half its own error bar. That is the direction that has been asked about — what the laboratory’s disagreement does to cosmology.

The other direction is more interesting. The first three minutes are an experiment on free neutrons, run once, with a result that is still visible in every old star and metal-poor galaxy. What does that experiment say about the lifetime? And the calculation turns out to use the lifetime in two separate places, which a single number hides and one proposed resolution of the disagreement would pull apart.

Helium as a stopwatch #

The calculation behind the line is the one every account of nucleosynthesis starts from. At a temperature of a few megaelectronvolts, a second or less after the beginning, neutrons and protons are kept in equilibrium by the weak reactions — a neutron absorbing a neutrino to become a proton and an electron, and the reverse — and their ratio is the Boltzmann factor $e^{-\Delta m/kT}$, with $\Delta m = 1.293\ \mathrm{MeV}$ the mass difference. As the universe cools the reaction rates fall as the fifth power of the temperature while the expansion rate falls only as the square, and near 0.7 MeV the reactions lose the race: the ratio freezes at about one neutron to six protons. After that the only thing still acting on the neutrons is their own decay, for the two and a half minutes until the temperature has fallen far enough for deuterium to survive being broken up. Then, within minutes, essentially every surviving neutron is locked into helium-4, and the helium mass fraction is twice the neutron fraction at that moment.

The lifetime enters because the weak conversion rates are not computed from a coupling constant. They are normalised on the measured decay: the same matrix element that makes a neutron decay makes it absorb a neutrino, so the conversion rate is proportional to $1/\tau_n$. A longer-lived neutron converts more slowly, falls out of equilibrium earlier and at a higher temperature, freezes at a higher neutron fraction, and then loses fewer of its neutrons to decay before the bottleneck breaks. Every one of those pushes helium up. The integrated slope is two parts in ten thousand of helium per second of lifetime.

Read backwards, that slope makes helium a stopwatch. The helium measurement in the figure — $Y_p = 0.2449 \pm 0.0040$, from emission lines in metal-poor dwarf galaxies extrapolated to zero metallicity — says the neutron lives between 848 and 889 seconds, a laboratory constant determined from light emitted by ionised gas in galaxies, with a precision of about twenty seconds. That is twice too coarse to referee a ten-second disagreement, but it is a measurement made in a regime no laboratory can reach: neutrons at a density of about $10^{20}$ per cubic centimetre in a bath of photons at a billion kelvin, a hundred seconds after the beginning of time, measured through the most perfect blackbody ever measured for its baryon density, and the answer is the same neutron.

Two places the same number goes #

The two effects that the slope bundles together are not the same effect, and the calculation can be run with them separated.

The first effect is the normalisation of the conversion rates. What the calculation really wants there is the strength of the weak interaction between a neutron and a proton, and it takes that from the rate at which neutrons decay into protons. The second effect is free decay: after freeze-out, neutrons disappear at the rate $1/\tau_n$, whatever they disappear into. Changing the lifetime in the conversion rates alone accounts for three-quarters of the helium shift; changing it in the decay alone accounts for the other quarter. The two add because the change is small.

For a neutron that decays only one way these are the same number twice, and the distinction is bookkeeping. It stops being bookkeeping if some neutrons decay into something that is not a proton.

A resolution that makes both laboratories right #

The bottle and the beam do not measure the same thing. A bottle counts neutrons that have disappeared, by any route — decay into a proton, decay into anything else, or loss through the walls. A beam counts protons that have appeared. If the neutron has a single decay channel the two are the same measurement made two ways, and one of them has an unrecognised systematic: neutrons lost from the bottle by some route other than decay, which would make the bottle’s lifetime too short, or protons missed by the beam’s trap, which would make the beam’s too long. Both have been searched for, extensively, and neither has been found.

In 2018 a third possibility was put forward: that about one per cent of neutrons decay into particles no detector sees — a dark-matter particle and a photon, or a dark-matter particle and an electron–positron pair, or dark particles alone. Then both laboratories are right. The bottle measures the total lifetime, 877.75 seconds; the beam measures the lifetime for decay into a proton, 887.7 seconds; and the ratio $877.75/887.7 = 0.989$ says that 1.1 per cent of neutrons decay invisibly.

The first three minutes would then use the two numbers in the two places. The conversion rates are set by the proton-producing matrix element, so they want the beam’s value; free decay removes neutrons by every channel, so it wants the bottle’s.

The history shows where the difference is made. Through the first ten seconds the two cases are indistinguishable, because both have the beam’s slower conversion rates and freeze-out is what those rates decide. By 0.3 MeV each has about 48 parts in ten thousand more neutrons than the bottle history. After that the conversions stop mattering and the two diverge on decay alone: a single-channel neutron with the beam’s lifetime decays slowly and keeps gaining on the bottle history, while a neutron with a dark channel decays at the bottle’s faster rate and gains much less. The single-channel beam case ends 78 parts in ten thousand above the bottle case; the dark-channel case ends 60 above.

So the three hypotheses about the neutron predict three different helium abundances, and the dark-channel universe is not halfway between the other two.

In the plane of the two lifetimes the contours of constant helium are steep: moving across, in the conversion lifetime, changes helium three times as fast as moving up, in the decay lifetime. That is the three-quarters-to-a-quarter split read geometrically. It means helium is mostly a measurement of the lifetime a beam measures, and the dark-channel universe, which has the beam’s conversion rates, sits much closer to the beam’s helium than to the bottle’s — 15 of the 19 parts in ten thousand that separate the two laboratories.

That is a surprising place for the argument to end up. The one resolution of the neutron-lifetime puzzle that requires new particles makes a prediction for primordial helium, and the prediction is not the bottle value that the particle-physics tables adopt by averaging. A cosmologist who used the tabulated lifetime would be using the right number for decay and the wrong one for the rates. The error is small — a part in a thousand of helium — and it has a sign.

A lifetime that can be computed instead of timed #

There is a third way to get the lifetime, and it does not time a neutron at all. Neutron decay is a weak-interaction process whose rate is fixed by three quantities measured elsewhere: the Fermi constant, from the muon’s lifetime; the element $V_{ud}$ of the quark-mixing matrix, from the superallowed decays of nuclei such as aluminium-26 and cobalt-54, in which a proton turns into a neutron without changing the nuclear spin; and the ratio $g_A$ of the axial to the vector coupling, from the asymmetry of the electrons emitted by polarised neutrons. With radiative corrections calculated to a few parts in ten thousand, the relation is

$$\tau_n \approx \frac{4908.6\ \mathrm{s}}{|V_{ud}|^2\,(1 + 3g_A^2)},$$

and with the measured values it gives a lifetime between about 879 and 881 seconds, depending on which determination of each input is used — within a few seconds of the bottle and some eight from the beam.

That does not settle the question, and why it does not is instructive. The relation assumes the neutron decays only into a proton, so it computes the partial lifetime for the ordinary channel — the quantity a beam measures. A dark channel would leave it untouched and make the bottle shorter. Agreement between the computed partial lifetime and the bottle’s total therefore argues against a dark channel of one per cent, since the partial and the total cannot both be near 879 if one per cent of neutrons go elsewhere; it argues against it only as strongly as $g_A$ is known, and the measurements of $g_A$ have themselves drifted by more than their errors over two decades, with older experiments giving smaller values. The more recent determinations are the ones that land on the bottle.

The same coupling sets a rate in a place that looks unrelated. The first step of the Sun’s energy generation, two protons fusing to deuterium, is a weak process with the same axial coupling in its matrix element, and the furnace that runs cooler than a compost heap is paced by it. A number that decides how many neutrons survive the first three minutes also decides how slowly the Sun burns, and it is measured by watching free neutrons decay in a laboratory.

How far the helium is from being able to tell #

The prediction is only useful if helium can be measured to better than the gap, and the gap is two parts in a thousand.

They are not yet close. The helium mass fraction is measured from the ratio of helium to hydrogen recombination lines in ionised gas around young stars in small galaxies, corrected for the helium the galaxy’s own stars have since made by extrapolating to zero metallicity. The correction requires the gas’s temperature, its density, the fraction of helium that is singly rather than doubly ionised, the underlying stellar absorption, collisional excitation of the helium lines, and the optical depth in them — each fitted from the same few lines, each with its own degeneracy with the others. The six emission-line determinations drawn here agree with one another at about the one-per-cent level except for the most recent, from a survey of extremely metal-poor galaxies found in wide-field imaging, which sits three per cent lower and more than two standard deviations below the others. The microwave background measures helium independently: helium recombines earlier than hydrogen, so more helium leaves fewer free electrons at last scattering, a longer photon mean free path and a stronger damping of the smallest fluctuations. Its error is five per cent.

To separate the bottle from the beam at three standard deviations, helium needs an error of about six parts in ten thousand. To separate the dark channel from the beam, about a part in ten thousand. The first is a factor of four beyond the best current determinations; the second is a factor of twenty, and would need the systematic scatter between groups — the part the error bars do not include — to fall by more than that. Neither is impossible in principle, and the microwave-background route, whose systematics share nothing with those of emission lines, is the one that improves automatically as the damping tail is mapped more finely. For now the honest statement is that the sky measures the neutron’s lifetime to twenty seconds and cannot yet say which laboratory is right, let alone whether both are.

Deuterium does not help much. Its primordial abundance is measured to about one per cent, from absorption in the spectra of distant quasars, but it depends on the neutron fraction only weakly, because the deuterium that survives is set by how fast it is burned rather than by how many neutrons made it. A ten-second change in the lifetime moves it by about half a per cent — below its own error bar — and the uncertainty in the baryon density moves it by more.

Two other places the neutron has been timed #

The first three minutes are not the only astronomical measurement of the neutron lifetime, and the others are both stranger and more direct.

A neutron star is the other place in nature where free neutrons are abundant, and it constrains the dark channel rather than the lifetime. If a neutron could decay into a dark particle lighter than itself, the dense neutron matter in a neutron star would convert partly into dark matter until the two were in equilibrium, and the dark particles, if they interact only weakly, would contribute mass without contributing pressure. That softens the equation of state enough that the maximum mass of a neutron star falls to about 0.7 solar masses — below even the limit on a cold star held up by electrons. Neutron stars of two solar masses are observed, weighed by the delay their companions’ gravity imposes on the pulses. So the simplest dark decay is excluded, and what survives needs the dark particles to repel one another strongly enough to hold the star up — an additional force invented to rescue the first invention.

The laboratories have also looked for the products. A dark decay with a photon would give a monoenergetic line between 0.78 and 1.66 MeV, and a search in a bottle apparatus excluded it at the branching ratio required. A dark decay with an electron–positron pair was excluded the same way. What remains open is a decay into dark particles alone, which leaves nothing to detect.

And the lifetime has been measured from orbit. Cosmic rays striking the surface of a planet or moon knock neutrons out of the ground, and a spacecraft carrying a neutron spectrometer counts them at altitude. The slowest neutrons take long enough to climb against gravity to the spacecraft that a measurable fraction decay on the way, so the count falls with altitude at a rate set by the lifetime. Flybys of Venus and Mercury gave 780 seconds with an uncertainty of about ninety; archival data from a lunar orbiter gave 887 with an uncertainty of about fifteen. Neither is precise enough to choose, but it is a measurement made with no bottle and no beam, and its systematics — the planet’s composition, the spacecraft’s altitude — share nothing with either.

What the calculation leaves out #

The figures here integrate only the neutron-to-proton ratio, not the whole reaction network. The helium fraction is taken as twice the neutron fraction at a fixed bottleneck temperature, and the absolute level of the line is set to a full network code’s value at the bottle lifetime; what is computed is the slope and the separation of its two parts. The conversion rates are the standard approximation with massless electrons and no radiative, finite-mass or thermal corrections, which together shift the absolute helium by about one per cent and its sensitivity to the lifetime by much less. The dark channel is treated as removing neutrons and nothing else: the dark particles’ own energy density, which would add to the expansion rate if they were light, is not included. And the helium measurements are drawn with symmetric error bars where several are asymmetric, and with only their quoted errors, which the scatter among them shows to be incomplete.

Still open: whether the neutron has a second way to die #

The bottle and the beam still disagree, and the newest measurements have not closed the gap: a method in Japan counting decay electrons from a pulsed beam has reported values closer to the bottle, with errors still several seconds wide, and a beam experiment rebuilt to address its predecessor’s largest systematics is being built. If the beam converges on the bottle, the neutron has one lifetime, the first three minutes use it twice, and helium becomes a check on nothing more than the standard calculation. If the gap survives every beam systematic, the dark channel is what is left, and then the universe’s primordial helium was made with a conversion rate set by one lifetime and a decay set by another. Helium measured to a part in ten thousand — a precision neither the galaxies nor the sky maps yet approach — would be the one measurement that could see the difference — and it would be reading, in the light of the first atoms, whether a neutron can decay into something no detector has ever caught.