An abundance with no direction to correct in

Every primordial abundance in this subject is measured in the present and used to constrain the past, and every one of them rests on the same unstated step: the material observed today has been through stars, the stars changed it, and the change has to be undone. The undoing is what makes the measurement possible, and it works for exactly the reason nobody states — for deuterium and for helium the change has a direction.

Deuterium is destroyed by every star and made by none, so the observed abundance is a floor under the primordial one and the correction is upward, whatever the star-formation history was. Helium is made by stars and destroyed by none, so the observed abundance in a metal-poor galaxy is a ceiling and the correction is downward. Neither statement requires a model of the Galaxy. Both are consequences of what a star does to a nucleus, and the extrapolation to zero metallicity that both measurements perform is legitimate because of them.

Helium-3 has no such statement attached, and the consequence is that no primordial ³He abundance appears anywhere in the literature — not as a poorly determined number with a wide error bar, but as no number at all. The network predicts one, cleanly: about one part in a hundred thousand at the density everything else agrees on. There is simply nothing to compare it with.

That absence is worth dwelling on because it is a different kind of gap from the ones this collection usually reports. A quantity can be unmeasured because the instrument is not good enough, which is a statement about the present, or because the object is too far away, which is a statement about geometry. This one is unmeasured because the arithmetic that would turn an observation into a primordial value has an unknown sign in it, and no improvement in the observation touches that.

What a star does to a helium-3 nucleus, twice #

The reason ³He has both a source and a sink is that it sits in the middle of the proton–proton chain, and a star is not one temperature.

Every deuteron entering a star is converted to ³He almost immediately — D(p,γ)³He runs at a million kelvin, which is well below the temperature at the base of any main-sequence star’s convective envelope. So a star swallows a galaxy’s deuterium and hands back helium-3 in its place. That is the source, and it is the reason the two species have to be discussed together rather than separately.

The sink is ³He(³He,2p)⁴He, which needs about seven million kelvin, and the outer parts of a low-mass star never get there. In a star of about one solar mass the ³He therefore builds up in a shell at intermediate depth — hot enough to make it out of deuterium and hydrogen, too cool to destroy it — and when the star loses its envelope on the asymptotic giant branch, that shell is returned to the interstellar medium. A star of three solar masses is hot enough throughout the region that matters and returns very little. The yield changes sign somewhere between them.

The shell exists because of a coincidence of two temperatures rather than by design, and its width is what decides everything. p(p,e⁺ν)D is the slowest step in the whole chain and it feeds ³He at a rate that rises steeply with temperature; ³He(³He,2p)⁴He removes it at a rate that rises more steeply still, because it is a reaction between two charge-two nuclei. There is therefore a band of temperature in which the first has switched on and the second has not, and in the Sun that band sits at about a third of the radius. The equilibrium ³He abundance there is a hundred times its surface value, and the material below it has essentially none. A star that keeps that band and then sheds the layers above it returns helium-3; a star that mixes the band into hotter material destroys it. Which of those a star does is a question about circulation rather than about nuclear physics, and circulation is the least well determined thing in stellar structure.

The complication nobody expected #

The account above is the one that stood until the 1990s, and it has a well-known problem: it predicts far more helium-3 in the Galaxy than is observed.

The measurements are made in H II regions and planetary nebulae, using the 8.665 GHz hyperfine transition of singly ionised ³He — the exact analogue of hydrogen’s 21 cm line, in a species with one electron and a nuclear spin of a half. The physics is identical: an unpaired electron spin can be parallel or antiparallel to the nuclear spin, the two states differ by a tiny energy, and a transition between them emits a photon whose frequency is set by nuclear magnetic moments. ³He’s moment is larger than the proton’s and the ion has a hydrogen-like orbital, which is why the line sits at 3.5 cm rather than 21.

It is a hard measurement in every respect. The line is a few parts in ten thousand of the free-free continuum it sits on, so it is detected as a small ripple on a large signal and the continuum has to be subtracted at that precision. The abundance derived is not ³He/H but ³He⁺/H⁺, so it needs the ionisation structure of the region — how much of the helium is singly ionised where the hydrogen is — and getting that wrong biases the answer in a direction that is common to every region observed. And the strongest sources are the most massive H II regions, which are the ones whose gas has been most recently processed. What comes out of it is roughly ³He/H between one and two parts in a hundred thousand, nearly everywhere, and standard chemical evolution with ³He yields from classical stellar models predicts several times that in the inner Galaxy.

The resolution that emerged is that most low-mass stars destroy their own helium-3 after all, through a mixing process that classical stellar models do not contain. Above the point on the red-giant branch where the hydrogen-burning shell erases the composition discontinuity left by the convective envelope, the shell’s own ³He-burning creates an inversion in the mean molecular weight — the material above is heavier than the material below — and that inversion is unstable to a slow, salt-fingering circulation. The circulation carries ³He down into hotter material and destroys it.

The mechanism is the same thermohaline mixing that turns up in the lithium problem’s stellar resolution, and the two are not independent problems with a shared explanation; they are one piece of missing physics appearing in two abundances. It also explains the carbon isotope ratios in evolved low-mass stars, which classical models get wrong in the same direction, and the lithium and carbon on the red-giant branch drop at the same evolutionary point rather than at points that happen to be near each other. Three discrepant abundances resolved by one absent circulation is the strongest evidence any of them has.

The efficiency of the mixing is a free parameter, and it is the same free parameter in all three cases, which is what makes the argument better than a fit. A model tuned to the carbon isotopes predicts the lithium drop and predicts a ³He survival fraction, and those predictions are checked rather than adjusted. They come out on the low side of the sign-change value — which would mean the correction to ³He is downward and the observed abundance is a floor — and they come out on the low side by less than the spread among the models. That is the honest state: the sign is probably known and is not known.

There is a second, older piece of evidence pointing the other way, and it is why nobody treats the question as settled. A handful of planetary nebulae show ³He/H around 10⁻³, a hundred times the interstellar value, which is exactly what a low-mass star that did not mix would return. If those measurements are right, some stars destroy their helium-3 and some do not, the Galactic yield is an average over two populations, and the sign question becomes a question about what fraction of stars mix. The mass a star does not keep is difficult enough to determine on its own; the composition of what it does not keep is worse.

What survives, and it is an inequality #

Nothing above is a measurement of primordial helium-3, and none of it is going to become one. What survives is a statement about the pair.

A deuteron entering a star becomes ³He. A fraction of the ³He present is returned. Nothing makes deuterium. So the sum D + ³He can only move from one member to the other and then decrease, whatever any star does, and that is enough.

Turning that into a number requires one inversion. If the observed pair is D_obs and ³He_obs, and a fraction g₃ of the ³He present survives, then the primordial deuterium satisfies D_p ≤ D_obs + ³He_obs/g₃ — the observed helium-3 could have been deuterium once, and the poorer the survival the more deuterium it could have been.

The shape of that argument is the thing worth carrying out of this rung. A quantity too poorly understood to be extrapolated at all — a quantity with no primordial value, then or now — nonetheless constrained the baryon density, because the direction of its change under processing was known even though the magnitude was not. Direction is cheaper than magnitude and it is often enough.

It is worth being clear about how much work that bound was doing at the time. Through the 1980s the baryon density had no independent determination whatever: the acoustic peaks were two decades away, the deuterium measurements in quasar absorbers were a decade away, and the local deuterium abundance was known to be contaminated by processing in an amount nobody could estimate. What existed was this inequality and the helium mass fraction. The inequality said the universe contains at least a certain density of baryons; the dynamical mass of clusters said it contains several times more matter than that; and the gap between the two numbers is where the case for non-baryonic dark matter was first made quantitative. A bound derived from an abundance with no primordial value is one of the load-bearing arguments in twentieth-century cosmology, which is not the reputation helium-3 has.

The solar-system numbers behind it are worth a sentence, because they are the one place in this ladder where the sample is a rock rather than a spectrum. Pre-solar deuterium comes from the ³He excess in meteoritic gas and in the solar wind: the Sun burned all its deuterium to ³He in its first few million years, before it reached the main sequence, so the ³He in material that has not been through the solar interior is the sum of what the nebula had and what its deuterium became. That is the same inequality being used constructively rather than as a bound, and it is why the two numbers in the caption are quoted from the same measurement.

Why the slopes differ, and why the obvious reason is wrong #

Deuterium’s predicted abundance falls as $\eta^{-1.6}$ and helium-3’s as $\eta^{-0.6}$. The tempting explanation is a Coulomb barrier: ³He carries twice the charge, its destroyer has to tunnel further, so it is destroyed less completely and depends less steeply on the density. That explanation is checkable, and it is wrong.

So helium-3 is destroyed harder and for longer than deuterium is, and it survives in comparable quantity anyway. The reason is that it is not a residue. Deuterium is a leftover: what is measured is the fraction that escaped a process, and a leftover is exponential in how long the process ran, which is what makes it steep. Helium-3 is being manufactured out of deuterium throughout, so what is measured is closer to a balance between a source and a sink — and a balance is much less sensitive to the duration than a leftover is, because lengthening the process consumes more and makes more.

That distinction is the general lesson of this rung and it is worth stating separately from the astronomy. A quantity produced and destroyed by the same process is a poor probe of how long the process ran and a decent probe of the ratio of its rates; a quantity only destroyed is a sharp probe of the duration and says nothing about the ratio. Deuterium and helium-3 come out of the same reaction network and are informative about different things for exactly this reason.

The same division sorts a good deal of the rest of this collection. The neutron fraction at freeze-out is a leftover and measures a duration, which is why it counts particle species. The ionisation fraction in a stellar atmosphere is a balance and measures a ratio of rates, which is why it is a thermometer and not a clock. Anything that has reached a steady state has forgotten how long it took to get there, and anything still running down remembers nothing else.

What the picture cannot show #

The one-pass model behind three of these figures is a caricature, and its limits should be stated rather than left for a reader to discover. It processes gas once, with a single survival fraction, at a single moment; a real galaxy processes some gas many times and some not at all, with a survival fraction that depends on the mass of the star doing the processing and therefore on the initial mass function and on the epoch. The sign-change value of 0.282 is exact within the model and approximate outside it.

What the model does get right is the topology, and that is what the argument uses. Deuterium’s response is negative for every parameter choice; helium’s is positive for every one; helium-3’s changes sign inside the range that stellar models allow. No refinement of the model moves a sign change out of a range it sits in the middle of, and refining it further would produce a more accurate crossing point rather than a reason to stop worrying about the crossing.

There is one thing the figures cannot address at all. Everything here treats the Galaxy’s ³He as a chemical-evolution problem, and the comparison is between a model of the Galaxy and a radio line. If the radio measurements are systematically low — if the ionisation corrections for ³He⁺ in H II regions are wrong in a common direction — then the flat, low distribution is an artefact and the whole discussion changes. The measurements are few, they are difficult, and there is no independent route to the same number. A subject in which one hard measurement carries the entire empirical weight is a subject with a single point of failure, and this one has been in that position since the 1970s.

The gradient figure also hides a choice that a reader should be able to see. The gas fraction it runs on is a straight line from 0.04 to 0.50 across the disc, which is a plausible shape and not a measurement. Real determinations of the gas fraction come from the atomic and molecular surveys and disagree with one another about the inner few kiloparsecs by more than the whole range drawn, largely because the molecular gas there is inferred from carbon monoxide through a conversion factor that itself depends on metallicity. If the inner disc has processed less of its gas than the figure assumes, the predicted gradients flatten and the observations stop constraining the survival fraction even in principle. The figure’s negative conclusion — that the gradient does not settle the sign — survives that, because it survives making the models more similar to each other, but the size of the effect drawn should be read as illustrative.

What is worth measuring next, and it is not helium-3 #

The natural instinct on reading all of the above is that the ³He observations should be improved. They should, and it would not help with the question this rung is about.

An abundance measured perfectly still has to be corrected by a factor whose sign is unknown, so its primordial value stays a range that spans the prediction and says nothing. What would help is a measurement of the survival fraction — the astrophysical quantity, not the cosmological one. Two things could deliver it. Helium-3 in a planetary nebula whose central star’s initial mass is independently known would measure the yield of one star rather than an average over a population, and the initial mass is knowable from cluster membership. And asteroseismology of red giants is now sensitive to the internal rotation and to the composition gradients that drive thermohaline circulation, which is the mechanism itself rather than its consequence.

Both of those are stellar-physics measurements, and that is the point of the rung rather than an aside. The cosmology of helium-3 is finished; what remains is a question about what happens inside a one-solar-mass star between the main sequence and the white dwarf it becomes, and until that is answered the abundance is a bound rather than a number.

Where this ladder goes next #

Three rungs of this anchor have now taken the four abundances one at a time and asked what each is a measurement of. Deuterium measures the density, helium counts relativistic species, lithium disagrees with everything, and helium-3 supplies a one-sided bound and no value. Every one of those readings assumed the universe was the same everywhere at one second.

The next rung drops that assumption. If the baryons were unevenly distributed when the network fired — as a first-order quark–hadron transition would have left them — then each region ran its own nucleosynthesis at its own density, and what is observed is a volume average. The proposal existed for a decade as the one way to make ordinary matter account for all the matter dynamics demanded, and the reason it fails is a theorem about convex functions rather than a detail of any calculation.

Beyond that: the neutron lifetime, whose two laboratory measurements disagree by ten seconds and whose disagreement propagates into every number on this ladder; and the constraint the same machinery places on a light sterile neutrino, which is the particle-physics reading of the helium band.