Four abundances and one free parameter

A prediction with one free parameter and four outputs is worth a great deal more than four predictions with four parameters, and big-bang nucleosynthesis is the cleanest example of the first kind that astronomy has. Fix the ratio of baryons to photons and everything else follows: the temperature at each moment, the expansion rate, the reaction rates, and therefore how much of each light nucleus is left when the reactions stop.

Why there is anything to predict #

The window is narrow and its boundaries are set by two different things.

Nucleosynthesis cannot begin while the temperature is above about $10^9$ kelvin, because deuterium — the first thing that has to form, since every heavier nucleus is built through it — has a binding energy of only 2.22 MeV and is photodisintegrated as fast as it forms. And it cannot begin late, because the density and temperature are falling and the reactions freeze out. The whole of it happens between roughly one second and twenty minutes. What decides the outcome is a competition. Neutrons and protons interconvert through weak interactions, which keep their ratio at the thermal equilibrium value $e^{-\Delta m c^2/kT}$ while they are fast enough. As the universe expands the weak rates fall faster than the expansion rate, and at about $10^{10}$ K they lose — the neutron-to-proton ratio freezes out at about 1:6. Neutrons then decay with a 880-second half-life, which drops it to about 1:7 by the time deuterium can survive. Then essentially every remaining neutron gets locked into a helium-4 nucleus.

That last step is why the helium prediction is so insensitive to everything.

So the helium mass fraction is very nearly $2n/(n+p) = 2\times1/(1+7) \times \ldots$ — two nucleons of helium per neutron, over the total — which comes to about 0.25 almost regardless of the baryon density. Doubling $\eta$ changes it by less than one per cent. Helium is a superb test of the framework and a useless measurement of the density, and the flatness of its curve in the hero figure is that statement drawn.

Deuterium is the baryometer #

Deuterium’s curve is the steep one, falling as $\eta^{-1.6}$, and the steepness has a physical reason: deuterium is an intermediate. It is made and then burned through to helium, and the burning is a two-body reaction whose rate goes as the density. A denser universe burns more of its deuterium, so what survives is a sensitive measure of how dense it was.

The sensitivity runs both ways. A one per cent measurement of D/H is a 1.6 per cent measurement of the baryon density, which is why an abundance measured on a handful of objects competes with a full-sky satellite map.

The observation is reading composition out of what is missing, applied at a redshift where the whole Lyman series has been carried into the optical — which is the only reason it can be done from the ground at all, since Lyman-α at rest is in the far ultraviolet and the atmosphere is opaque to it. A cloud at $z = 3$ has its lines shifted by a factor of four, and 121.6 nanometres becomes 486, which is squarely in the visible.

That last clause about metallicity is what makes the measurement possible at all. Deuterium is fragile: it burns in any stellar interior, and no astrophysical process makes it in quantity. So every deuterium atom observed anywhere is primordial, and any measured abundance is a lower bound on the primordial one. Finding the primordial value is therefore a matter of finding the least-processed gas, which is why the measurements are made against quasars at $z \approx 3$ in systems with a thousandth of the solar metal content. The best current value is $(2.527 \pm 0.030)\times10^{-5}$, from seven such systems, and its error is dominated by the scatter between them rather than by any one spectrum.

The check that has nothing in common #

The agreement between these two is one of the results that made ΛCDM the standard model rather than a candidate. Consider what they share: nothing. One is a network of nuclear reaction cross-sections measured in accelerators, run at a temperature of $10^9$ kelvin one minute after the beginning, and read off an absorption line in a quasar spectrum. The other is a fluid oscillation at 3,000 kelvin four hundred thousand years later, read off the relative heights of two bumps in the power spectrum of a temperature map. They share the parameter and nothing else, and they agree to four parts in a thousand.

The agreement also settles something that neither measures directly. Both give the density of baryons, and both come out at about 5 per cent of the critical density — while the total matter density measured by rotation curves, by cluster dynamics and by the third acoustic peak is about 31 per cent. The dark matter cannot be ordinary matter that is merely dark, because ordinary matter at that density would have produced the wrong deuterium abundance. The nucleosynthesis constraint is the reason the exclusion is a physical one rather than a failure of imagination about faint stars.

Lithium, which does not work #

The fourth curve is the exception and it is not a small one. The network predicts $^7{\rm Li}/{\rm H} = 4.7\times10^{-10}$ at the microwave background’s baryon density. The oldest, most metal-poor stars in the Galactic halo show $1.6\times10^{-10}$ — a factor of three low, and the observed values sit on a remarkably flat plateau against metallicity, which is exactly the signature a primordial abundance should have.

Three kinds of explanation have been pursued for twenty-five years.

The nuclear rates could be wrong. Lithium-7 is made mostly as beryllium-7, which later captures an electron, so the prediction depends on the $^3{\rm He}(\alpha,\gamma)^7{\rm Be}$ cross-section and on the reactions that destroy it. Those have been remeasured repeatedly, including underground to suppress cosmic-ray backgrounds, and the prediction has not moved.

The stars could be destroying it. Lithium burns at 2.5 million kelvin, which is reached just below the convective envelope of a halo dwarf, so slow mixing over ten billion years could deplete the surface. This is the leading explanation, and its difficulty is the flatness of the plateau: a depletion mechanism has to remove the same factor of three from stars of different masses, temperatures and metallicities without introducing scatter, and the observed scatter is very small.

Or the physics could be incomplete — a decaying particle around the time of nucleosynthesis, for instance, which could destroy lithium selectively. Every such proposal has to avoid disturbing deuterium and helium, and that is a tight constraint.

The honest position is that it is unresolved, and it is worth saying so plainly, because a field that quietly stops mentioning its one failed prediction has stopped being checkable.

What is actually measured #

Nothing in this essay measures an abundance in the early universe. What is measured is:

• a ratio of two absorption line depths in a quasar spectrum, corrected for the velocity structure of the absorbing cloud, which is the dominant systematic;
• an emission-line ratio in a metal-poor extragalactic H II region, extrapolated to zero metallicity, which gives helium;
• an equivalent width in the spectrum of a halo dwarf, converted to an abundance through a model atmosphere, which gives lithium.

Every one of those is a line strength turned into a number of atoms by a model, and every one of them requires an extrapolation to zero metallicity, because no observed gas is genuinely primordial. The extrapolation is the reason the helium error bar is 1.6 per cent rather than the 0.1 per cent the line ratios themselves would support.

The species that is on the list and not in the test #

The network produces a fourth light nucleus in quantity — helium-3 — and it appears nowhere in the argument above. The reason is worth stating, because it is the clearest example of what makes an abundance usable.

Deuterium works because stars only destroy it. Any measured value is a lower bound on the primordial one, and the least-processed gas gives the best estimate.

Helium-4 works because stars only make it, and because the primordial value is so large that the stellar addition is a small correction. Extrapolating a measured abundance to zero metallicity removes most of that correction.

Helium-3 does neither. Stars burn it in their interiors and they also produce it, in the outer parts of the hydrogen-burning shell, and low-mass stars are thought to dredge some of that production to the surface and return it to the interstellar medium. So the observed abundance is the primordial value plus an unknown addition minus an unknown destruction, and the sign of the net is not agreed.

It is also hard to measure. Helium-3 has a hyperfine transition at 3.46 centimetres, analogous to hydrogen’s 21-centimetre line, and it is observed as emission from ionised regions in the Galaxy — a handful of measurements, each requiring a model of the region’s structure. There is no equivalent of a quasar absorption line, because the species is far too rare to show one.

A prediction is only testable against an observable whose relationship to it is monotone, and helium-3’s is not. It remains a consistency check that nothing has grossly violated rather than a measurement.

The laboratory number that will not settle #

The helium yield depends on how many neutrons survive to be captured, which depends on how fast a free neutron decays. That is a laboratory quantity, and the laboratory has not agreed with itself for two decades.

There are two ways to measure it. One traps ultracold neutrons in a bottle, waits, and counts how many are left — a measurement of disappearance. The other sends a beam of neutrons through a volume and counts the protons produced by decays inside it — a measurement of appearance. Both are careful, both are done by several groups, and the two methods give answers differing by about nine seconds out of eight hundred and eighty, which is several times the quoted uncertainties.

That is a discrepancy in a fundamental constant, and it is unresolved. Either one method has an unrecognised systematic — neutrons lost from the bottle by some route other than decay, or protons uncounted in the beam — or free neutrons occasionally decay into something the proton counter cannot see, which would be new physics and has been proposed.

The consequence for the argument here is modest and real. A nine-second change in the lifetime moves the predicted helium mass fraction by a few parts in a thousand, which is inside the observational uncertainty on helium and therefore does not currently matter. It would matter if the helium measurement improved, and improving the helium measurement is a stated goal.

A cosmological prediction with one free parameter still has a dozen fixed inputs, and the accuracy of the prediction is bounded by the worst of them. Here the worst is a number measured in a laboratory by two methods that disagree.

What the picture cannot show #

The hero figure’s curves are fitting formulae, not a network integration. Each is a power law fitted to a full reaction-network calculation and accurate to a per cent or two over the plotted decade; the exponents are the physics and the coefficients are somebody else’s computation. A generator that hand-rolled a caricature of the network would draw a well-formed picture of nothing, which this collection tries hard not to do.

The observed bands are drawn as horizontal strips and the measurements are not. Each is an extrapolation from a handful of objects with correlated systematics, and the helium band in particular has had its central value move by more than its error bar twice in the last twenty years.

And the figure has one free parameter drawn and four held fixed. The expansion rate during nucleosynthesis depends on the number of relativistic species, the neutron lifetime enters the helium yield directly, and both are held at their laboratory values. Varying them is how this measurement becomes a constraint on particle physics — the helium abundance bounds the effective number of neutrino species at $N_{\rm eff} = 2.9 \pm 0.3$ — and none of that is visible on a plot against $\eta$ alone.

Both of those are laboratory problems rather than astronomical ones, and both are cases where a cosmological measurement is waiting on an experiment.

How it was worked out #

The idea is Gamow’s, from 1946, and the first calculation is the Alpher–Bethe–Gamow paper of 1948 — whose middle author was added for the pun and had not worked on it. The ambition was to make every element in the big bang, and it failed, on the mass-5 and mass-8 gaps. That failure was decisive twice over: it sent Fred Hoyle to stellar interiors and to the triple-alpha resonance, and it left big-bang nucleosynthesis responsible for exactly the four species in the hero figure.

The prediction of a residual radiation field came out of the same 1948 work, as a corollary: a hot early universe leaves photons behind, and Alpher and Herman estimated them at about 5 kelvin. Nobody looked.

The measurement side took much longer, and deuterium was last. Its abundance in the local interstellar medium was measured in 1973 and was known to be a lower bound; a genuinely primordial value needed high-redshift quasar absorbers and eight-metre telescopes, and the first credible measurements are from the mid-1990s. The one-parameter prediction was seventy years old before the parameter could be measured well enough to test it properly, and when it was, it agreed with a satellite that had not been conceived when the calculation was done.

Neither of the two difficulties above is a difficulty about the early universe, which is worth saying plainly: the framework is limited by what a laboratory can pin down and by what a star has not ruined.

The generalisation #

The structure worth extracting is about what makes a prediction strong, and it is not precision.

A prediction is strong in proportion to the number of independent things it constrains per adjustable parameter. Four abundances from one number is a ratio of four; those four span nine decades, so no single scaling can accommodate them; and the parameter is separately measurable by an unrelated route. Each of those three properties does work, and the third is the one that converts a consistency check into a test.

The same structure appears wherever this collection finds a one-parameter family fitting many observables. Mass decides everything about a star — luminosity, radius, temperature and lifetime all follow from it, so a cluster’s colour–magnitude diagram is many constraints on one number per star. A transit and a radial velocity together give a density, which over-determines the planet and catches the cases where one of them is wrong.

The corollary is the uncomfortable one. A one-parameter model that fits three observables and misses the fourth cannot be rescued by adjusting the parameter, because there is only one, and every observable moves together. That is precisely the lithium situation, and it is why a factor of three there is taken more seriously than a factor of three would be in a model with room to breathe.

Where the ladder goes next #

Three densities are now in hand from three unrelated measurements — baryons from deuterium and from the acoustic peaks, total matter from rotation curves and clusters, dark energy from the supernovae. The next essay puts them in one budget and asks what the budget is a statement about.

Later rungs on this anchor: the neutron lifetime, and why a disagreement between two laboratory measurements of it propagates into cosmology; $N_{\rm eff}$ as a constraint on light particles beyond the standard model; the lithium problem’s proposed resolutions in detail; helium-3 and why its abundance is not usable; and inhomogeneous nucleosynthesis, which was pursued seriously in the 1980s as a way to make the baryon density equal the matter density and does not work.