A factor of three, and the flatness that prices every cure

Three of the four light elements agree. Deuterium inverted through a nuclear network and the height of the second acoustic peak give the same baryon density to a few per cent; helium, once its flat curve is read as a species counter rather than as a baryometer, sits where it should. Then there is lithium.

That figure disposes of the tidiest possibility before the essay begins. The light elements do not fail together, so nothing that moves the one free parameter can rescue them: whatever is wrong is wrong about mass number seven in particular. What remains are three kinds of resolution — the stars destroyed the lithium after it was made, the observation is not measuring what it is taken to measure, or the network that predicts the abundance is missing something — and the surprising thing about them is that the datum which prices all three is not the size of the discrepancy at all.

It is worth saying at the outset how unusual this failure is in its setting. Big-bang nucleosynthesis is the oldest quantitative prediction in cosmology and the one with the fewest adjustable parts: one free number, four abundances spanning nine orders of magnitude, and a reaction network whose rates were measured in accelerators for entirely unrelated reasons. Three of those four land where they should, and one of the three lands there from a measurement taken in a gas cloud twelve billion light years away. A subject with that much predictive success and one failure is in a different position from a subject with a general disagreement — the failure is localised, and a localised failure is either a mistake in one place or a discovery in one place.

The plateau, and what makes it a plateau #

The measurement is a single absorption line at 6707.8 Å in the spectra of old, warm, metal-poor dwarf stars in the Galactic halo. Monique and François Spite found in 1982 that the abundance derived from it was the same in every such star they looked at, and the constancy was the discovery rather than the value.

The flatness is doing two jobs at once and they pull in opposite directions. It is what licenses reading the plateau as primordial: an abundance that does not care how much iron a star has is not an abundance the Galaxy made, because everything the Galaxy makes tracks iron. And it is what makes the discrepancy hard to remove, because any process that removed lithium from these stars would have had to remove the same fraction from every one of them.

The reason lithium is unusual among the elements here is that the Galaxy really does make some. Cosmic-ray protons and alpha particles striking interstellar carbon, nitrogen and oxygen chip fragments off them, and a small share of those fragments are ⁶Li and ⁷Li — the same spallation process that makes beryllium and boron, neither of which is made in stars at all. That production scales with the amount of carbon and oxygen there is to hit, so it rises with metallicity, and it is visible in the disc stars: above about [Fe/H] = −1 the lithium abundance climbs away from the plateau and reaches roughly A(Li) = 3.3 in the young interstellar medium. The plateau is what is left when that rise is extrapolated down to a metallicity at which there was nothing to spall.

There is one place where the plateau is not flat, and it is the place a stellar explanation points to.

Both halves of that figure are about the same mechanism and they say opposite things. The meltdown proves that convective envelopes destroy lithium. The flat part, immediately beside it, proves that whatever destroyed 0.47 dex from the plateau stars did so without caring about the one variable the meltdown says controls the destruction.

The argument the second moment makes #

The usual way to state this is qualitative: the plateau is too flat for a stellar explanation. It is worth making quantitative, because the numbers decide how much room there is rather than whether there is any.

The gap is a factor of about seven in the induced scatter, and it does not close by adjusting the threshold — the curves are drawn with the threshold already chosen, at every mean depletion, to be the one that delivers it. What a stellar resolution therefore needs is not a mechanism that destroys lithium; the meltdown supplies one of those. It needs a mechanism that destroys lithium without depending on effective temperature, which is to say without depending on convection, in stars whose only lithium-destroying machinery is convective.

The one serious candidate is thermohaline or rotational mixing driven by the molecular-weight gradient a hydrogen-burning core leaves behind — a slow circulation that is not the convective envelope and that has its own, much weaker, dependence on the star’s surface. Models of it can reach a few tenths of a dex with acceptable scatter. Getting the full 0.47 requires the mixing efficiency to be at the upper end of what the models allow, and the efficiency is a free parameter calibrated on other things. The situation is that the stellar resolution is not excluded and is not comfortable, which is roughly where it has sat for two decades.

There is a genuine piece of evidence in its favour, and it comes from the coolest and most metal-poor stars rather than from the plateau. Below about [Fe/H] = −2.8 the plateau develops a downturn and a scatter that it does not have above that metallicity — stars that ought to be the most pristine of all show less lithium and show a spread. That is not what a primordial floor does, and it is exactly what a destruction process that becomes efficient in a particular corner of the parameter space does. Whether the mechanism responsible there can be turned up enough to account for the whole plateau, without producing scatter on the flat part, is the question the models are arguing about; the existence of the downturn at least establishes that the plateau is not perfectly inert.

What makes the whole stellar route awkward in a way no single model addresses is a coincidence it has to swallow. The mechanism must remove 0.47 dex — and 0.47 dex is what is needed only because the microwave background measured the baryon density to be what it is. Before 2003 the density was not known independently, the predicted lithium was quoted over a range, and the plateau sat comfortably inside it. A stellar process that destroys precisely the amount a measurement made twenty years later would come to require is not impossible, but it is a coincidence, and the alternative resolutions do not have to explain it because they are tied to the same number.

The nucleus that has to be destroyed is not lithium #

Before pricing the other two routes, one thing about the object of the exercise has to be corrected, because it changes what a mechanism has to do.

So the plateau’s lithium was, for the first several months of the universe’s existence, beryllium. A mechanism that destroys lithium during nucleosynthesis and leaves beryllium alone changes essentially nothing. And a mechanism that destroys lithium in stars is acting on the decay product, four hundred thousand years and one recombination later, which is the stellar route already discussed.

The reason the two channels split the way they do is a difference in what destroys the product rather than in what makes it. Lithium made directly has a proton-induced destruction channel, ⁷Li(p,α)⁴He, which is fast at nucleosynthesis temperatures and which the plasma has plenty of protons for; so direct lithium is made and immediately unmade, and only the fraction produced late enough to escape survives. Beryllium’s analogous channel, ⁷Be(p,γ)⁸B, runs into the fact that there is no stable nucleus at mass eight, so the product falls apart and the beryllium is effectively unmakeable-away. It survives untouched until the universe cools enough for it to hold onto an electron, which is at recombination, and then it decays. The whole of mass-7’s survival therefore turns on a nucleus that could not be destroyed because the thing it would have turned into does not exist.

That leaves the third kind of resolution, which acts on the beryllium during the interval before it decays. Everything in that class works the same way.

Neutrons are not selective #

Destroying ⁷Be requires taking it out of the plasma, and the only lever available in a bath of photons and protons is a free neutron: ⁷Be(n,p)⁷Li converts it into lithium, and a proton then removes the lithium through ⁷Li(p,α)⁴He, which is fast. Every proposal in this class — a decaying massive particle, a hadronic injection, an annihilating relic — reduces to a supply of neutrons at the right epoch.

That is the most decisive figure of the three routes, and its strength is that it does not depend on the model. The branching ratio is a ratio of two measured cross sections and an abundance, none of which a choice of particle can alter, so any mechanism that destroys mass-7 by neutron injection pays the same twenty thousand deuterons per nucleus. The deuterium measurement — a pair of absorption lines eighty-two kilometres a second apart — is what closes it, and it closes it by a wide margin rather than marginally.

There are cleverer versions. A particle that decays electromagnetically rather than hadronically photodissociates ⁷Be directly, without free neutrons, and pays in a different currency: the same photons dissociate deuterium as well, and because deuterium’s binding energy of 2.2 MeV is the lowest of any nucleus present, it goes first. The constraint reappears with the sign reversed and is if anything tighter. A particle that is negatively charged and binds to ⁷Be — a long-lived stau in some supersymmetric spectra — catalyses its destruction with no injection at all, by pulling the nuclear charge down and lowering the Coulomb barrier its destroyers have to cross. That is the one proposal in the class that is not obviously priced out, and it requires a specific new particle with a specific lifetime and a specific abundance, which is a large thing to buy for one number.

The shape of the whole class is worth extracting, because it recurs. A cure that acts on one species through a shared channel pays for the cure in every other species that uses the channel, and the price is fixed by the ratio of two rates rather than by the model. This collection meets the same structure in the neutrino count read off the helium abundance, where an exotic species cannot contribute to the expansion rate without also contributing to the helium — and in the recombination history, where anything that delays recombination to make one observable fit moves the sound horizon and breaks another. A single knob attached to two things is not a knob.

What is actually measured, and what it rests on #

The observation behind A(Li) = 2.20 is a resonance doublet of neutral lithium at 6707.8 Å, and in a halo dwarf it is a feature about five per cent deep and a tenth of an ångström wide. Converting its equivalent width to an abundance needs the star’s effective temperature, its surface gravity, its metallicity and a model atmosphere, and the sensitivity is almost entirely to the first: the abundance derived scales with temperature at roughly 0.07 dex per 100 K.

That is the quiet load-bearing quantity in the whole subject. A systematic error of 200 K in the halo-dwarf temperature scale — which is not an outrageous error for stars at [Fe/H] = −3 whose colours are calibrated against much more metal-rich analogues — moves the plateau by 0.14 dex, or a third of the discrepancy. Successive temperature scales published since 1994 differ by rather more than that from one another. The scales have not moved in a direction that would close the gap, and the discrepancy has survived every one of them, but a reader should know that the second-most-likely resolution is a mundane one and that the field has been checking it for forty years for exactly this reason.

There are two further things about the line that are worth knowing before treating its equivalent width as an abundance. It is formed out of equilibrium: neutral lithium is a trace ionisation stage — the element is more than 99 per cent ionised in these atmospheres, because its ionisation potential is 5.4 eV against hydrogen’s 13.6 — so the abundance derived depends on the ionisation balance as well as on the temperature, and on the radiation field that sets it rather than on the local gas temperature alone. Corrections for that departure are computed, are of order 0.05 dex, and go in the direction of raising the derived abundance slightly. And the feature is a doublet whose components are 0.15 Å apart, unresolved in most spectra, so what is measured is a blend whose equivalent width is well defined even when its shape is not.

The one that would have been most interesting #

The remaining possibility is that the network is wrong: a nuclear reaction rate, measured in a laboratory at energies above the ones that matter and extrapolated down, is off by enough to change the mass-7 yield.

It has been looked for hard, because it would be the cheapest resolution available and because the relevant cross sections are genuinely difficult. ³He(α,γ)⁷Be is the production channel and would have to be roughly a factor of three smaller than measured; it is the same reaction that sets the solar neutrino flux from ⁷Be and ⁸B, and it is known to about five per cent from underground measurements at solar energies. A destruction channel large enough to matter — ⁷Be(d,p)2α was the best candidate — was measured directly and came out too small by an order of magnitude.

So the resolution that would have been most satisfying is the one most thoroughly closed. The remaining rate uncertainties in the whole network propagate to about a five per cent uncertainty in the predicted ⁷Li, against the factor of three that has to be explained. That closure is worth a moment, because it is the sort of result that reads as a negative and is not one: the reason a nuclear-rate resolution can be ruled out at all is that the rates were remeasured at the energies that matter, in underground accelerators built for solar-neutrino physics, by people who were not thinking about lithium. A prediction is only falsifiable to the extent that its inputs are independently known, and this one now is.

The isotope that turned out to be a line shape #

There is a fourth episode, now closed, that is worth telling because of how it closed rather than because of what it claimed.

Between 2006 and 2010 several groups reported a detection of ⁶Li in halo dwarfs at about five per cent of the ⁷Li — a second plateau, at roughly a thousand times what big-bang nucleosynthesis predicts for that isotope, which the network makes only in traces because the ⁶Li channels are weak and its destruction by protons is fast. Nothing in standard physics makes that much ⁶Li at that metallicity, and the claim was taken seriously enough to generate a substantial theoretical literature about late-decaying particles producing it.

The detection was not a line. The two isotopes’ 6708 Å features are separated by 0.16 Å, which no spectrograph resolves in a star this faint, so the measurement was of a slight asymmetry in the blended profile: a ⁶Li component sitting a fifth of an ångström to the red would make the line’s red wing marginally deeper. The asymmetry was measured, at the level of a per cent of the line depth, and interpreted as isotopes.

What it was instead was convection. A stellar photosphere is a boiling surface: rising granules are hot, bright and blueshifted while sinking lanes are cool, dark and redshifted, and the resulting line is intrinsically asymmetric with a redward-leaning wing — the convective blueshift that shifts every line in a stellar spectrum and shifts each one differently. One-dimensional model atmospheres, which is what the ⁶Li analyses used, have no granulation in them at all and therefore predict a symmetric profile. Three-dimensional hydrodynamic models produce an asymmetry of very nearly the observed size with no ⁶Li in them whatever, and when the analyses were redone against those models the detections mostly evaporated.

The instructive part is what the false signal was made of. A model atmosphere with a missing physical effect does not fail loudly; it produces a residual, and a residual has to be attributed to something. The something available was an isotope with a plausible story attached. That is the same failure this collection describes in a planet that was the star’s own rotation — the same convective asymmetry, on the same kind of line, producing a different spurious object because a different object was being looked for.

Where the model stops #

Nothing above rules the lithium problem solved or unsolvable, and it would be dishonest to imply a verdict the field has not reached. What the figures do is price the three routes against each other, and the prices are unequal in an instructive way: the particle-physics route is closed by a number that no model choice can move, the nuclear-rate route is closed by direct measurement, and the stellar route is not closed at all — it is merely uncomfortable, and it is uncomfortable about a second moment that carries an error bar of its own.

That last point deserves saying plainly, because it is where the argument here is weakest. The plateau’s scatter of 0.06 dex is an observed dispersion in a sample selected for being well-behaved, measured through a temperature scale that is itself uncertain. If the true intrinsic scatter were 0.10 dex rather than 0.06, the pricing above would loosen by a factor of nearly two and the steep destruction laws would come within sight of the required depletion. Nobody has measured the intrinsic scatter of the plateau to a precision that settles this, and doing so — rather than measuring more stars, or measuring them better — is what would actually move the subject.

Where this ladder goes next #

The three light elements that agree do so because each carries the baryon density in a different way and each can be extrapolated back to its primordial value along a trend with a known direction. Deuterium only falls under processing; helium only rises. That one-sidedness is what makes the extrapolation legitimate, and it is not a property every light element has.

The next rung is the one that does not have it. Helium-3 is produced by low-mass stars out of the deuterium they swallow and destroyed by more massive ones, so the sign of the correction from what is observed to what was primordial changes with the mass of the star doing the processing — and there is consequently no quoted primordial ³He abundance anywhere in the literature. What survives is an inequality on the sum of deuterium and helium-3, which for most of the 1980s was the strongest statement anybody had about the density of ordinary matter in the universe.

Beyond that: inhomogeneous nucleosynthesis, and the convexity argument that says why a lumpy universe makes more lithium rather than less; and the neutron lifetime, whose two laboratory measurements disagree by ten seconds and whose disagreement propagates into every number on this ladder.