The residue that failed to burn

The first rung of this anchor put four abundances against one free parameter and inverted the steepest of them. Deuterium did the work: its curve falls as the baryon density rises, steeply enough that measuring D/H to a per cent measures the density to less than one, and the answer agrees with a sky map made four hundred thousand years later. The second rung took the flattest curve instead and got a particle count out of it.

Neither rung asked why the steep curve is steep. It was quoted as an exponent — D/H goes as $\eta^{-1.6}$ — and an exponent lifted out of a fitting formula is a description of an answer rather than an account of it. The account is a race, and it is the same kind of race the previous rung ran between the weak interaction and the expansion, one epoch later and with the roles rearranged.

That figure is the mechanism. Everything below is an account of the three things it rests on: what deuterium is doing there at all, what a logarithmic slope is worth as a measurement, and what the two lines in a quasar spectrum that the abundance is actually read from look like.

Deuterium is the thing nothing wants to leave alone #

The chain that makes helium has to pass through mass number two, and mass number two is the weakest link in it. A deuteron is bound by 2.2246 MeV, which is a tenth of what binds an alpha particle per nucleon and is small enough that the tail of a blackbody photon distribution can break it apart long after the mean photon energy has fallen well below it. With about a billion photons for every baryon, the tail is enormous: the previous rung’s bottleneck temperature of about 85 keV is roughly a twenty-sixth of the binding energy, and the factor of twenty-six is the logarithm of the photon-to-baryon ratio.

So deuterium is built late and burned fast. Once the photodissociation stops, the reactions that consume it run at rates set by nuclear physics rather than by the radiation field, and they run downhill: two deuterons make a tritium and a proton or a helium-3 and a neutron, and those in turn take another deuteron each and make an alpha. Six deuterons in and one alpha out, with the binding energy released at every step. The chain terminates at mass four because there is no stable nucleus at mass five and none at mass eight, so the network has nowhere to go and stops.

What is left over is a residue. Not a yield, not a product — the deuterium observed today is the fraction of the deuterium ever made that was never used, and the size of that fraction is set by how long the burning had before the universe pulled the reactants apart. The distinction matters because it inverts the intuition about what a denser universe does. More baryons do not mean more deuterium. More baryons mean more collisions, more burning, and less deuterium surviving, and the whole method rests on that sign.

It is worth being precise about how small the residue is, because the number is easy to read past. About two and a half deuterons survive per hundred thousand hydrogen nuclei. Essentially every nucleon in the universe passed through mass two on its way to mass four, so what is measured is a leak of a few parts in a hundred thousand from a process that was otherwise complete. A measurement of a residue at that level is only possible because the residue is stable: nothing destroys deuterium in the intergalactic medium, and the clouds the abundance is measured in have never been inside a star. Both of those conditions fail in the Galaxy, where the local interstellar D/H varies by a factor of three from sightline to sightline, mostly because deuterium sticks to dust grains.

The other reason a residue is a good thing to measure is that it is not a competition between two large numbers. The helium mass fraction is a quarter, and getting it to a per cent means predicting a number of order one to a per cent, with every systematic in the calculation contributing at that level. Deuterium’s abundance is fixed almost entirely by one thing — the integrated destruction — and a quantity that depends steeply on one input and hardly at all on the rest is exactly the kind of quantity a measurement can profitably invert.

A two-body rate against a one-body expansion #

Every destruction channel available to a deuteron requires it to meet something. The rate per deuteron is therefore a number density times a cross section times a velocity, and the number density carries the baryon-to-photon ratio in it directly: at temperature T the photon density is fixed by the temperature alone, and the baryon density is η times it.

The expansion rate carries nothing of the kind. In the radiation era it is 1.66√g_* T²/m_Pl, and the energy density in that expression is radiation, which is photons and neutrinos and has no idea how many baryons are present. Baryons are a part in ten billion of it. So one side of the race scales with the density and the other side does not, and the ratio of the two — which is what decides whether a reaction is still doing anything — is exactly proportional to η at every temperature.

That is the mechanism in one sentence, and the drawn curves are its picture: three parallel lines separated by the logarithms of their densities. The reaction the figure draws is D(p,γ)³He, and the choice is deliberate rather than convenient. It is the channel that is first order in the deuterium abundance. The D + D channels are four orders of magnitude faster per pair, but their rate per deuteron carries a second factor of the deuterium abundance itself, and by the time the residue is being decided that abundance is ten parts in a million. A figure racing the D + D rate against the expansion would find the burning stopping before the deuterium had finished being made, which is wrong by about an hour.

The temperature dependence of the cross section is the Gamow factor and nothing else: two singly charged nuclei have to tunnel through each other’s Coulomb barrier, so the rate carries exp(−3(E_G/4kT)^(1/3)) with E_G = 0.979 MeV, and that exponential is what makes the crossing sharp. A rate falling as a power of temperature would leave the burning trailing off over decades of cooling; a rate falling as the exponential of an inverse cube root shuts off inside a factor of two.

The right-hand panel carries something the subject’s own nickname hides. Deuterium is made in the first three minutes, and it is still being destroyed an hour later. The reason the phrase survives is that the helium mass fraction — which is what the phrase was coined about — really is decided in the first few minutes, because it is twice a neutron fraction and the neutron fraction stops changing when the network fires. The deuterium residue is a different quantity with a different clock, and the two have been quoted in the same sentence for seventy years.

What a slope is worth #

An abundance becomes a density by inversion, and inversion propagates a fractional error by the reciprocal of the logarithmic slope. If A(η) is the predicted abundance, then δη/η equals (δA/A) divided by the modulus of d ln A / d ln η. A steep curve shrinks the error; a flat one magnifies it into uselessness.

Reading that figure the wrong way is easy and worth guarding against. It does not say that deuterium is measured well. It says that whatever deuterium is measured to, the density inherits five-eighths of the error, and that whatever helium is measured to, the density inherits twenty-five times it. The two statements are about the derivative of a curve and carry no information at all about how hard the observations are.

The slopes are also not free parameters, and each has a reason attached. Helium’s near-zero comes straight from the previous rung: the yield is twice a neutron fraction fixed by a weak-interaction rate, and the baryon density enters only through when the network fires, which is logarithmic. Deuterium’s steep negative slope is the burning. Helium-3’s shallower negative slope is the same burning acting on a nucleus with a higher Coulomb barrier, so it is destroyed less completely. Lithium’s positive slope of two is a production term rather than a destruction one — most of the mass-7 that survives is built as ⁷Be by ³He(α,γ)⁷Be, and that is a two-body reaction between two species whose abundances both rise with the density, hence the square.

Four curves, four exponents, and each exponent is a count of how many factors of the baryon density the dominant reaction carries. That is the structure the fitting formulae encode and it is the reason the exponents are close to whole and half numbers rather than arbitrary.

The last sentence of that caption is the one that deserves an argument rather than an assertion. If two measurements agree and one of them is ten times more precise, the agreement is a statement about the precise one and the vague one is along for the ride. These two are within a few per cent of each other in precision, so either could have contradicted the other and neither is decorative. That is why the concordance between an absorption line at redshift three and the height of the second acoustic peak is quoted as evidence rather than as a consistency check.

Two lines, eighty-two kilometres a second apart #

The measurement itself is not an abundance. It is a pair of absorption features in the spectrum of a quasar that happens to shine through a cloud of gas at high redshift, and the whole difficulty is in separating them.

Deuterium’s spectral lines sit at slightly shorter wavelengths than hydrogen’s, and the reason is entirely a reduced mass. The Rydberg constant for a one-electron atom carries μ = m_eM/(m_e + M), so a nucleus twice as heavy pulls the levels down by a part in 3,700. The shift is 2.75 parts in ten thousand, which at Lyman-α is 0.331 Å and in velocity units is 82 km/s, and it is the same 82 km/s at every line in the series because it is a fractional shift in wavelength.

The deuterium feature is never saturated at the columns these systems have, so it reports its own column honestly; the trouble is all on the hydrogen side. A saturated line is a line whose depth has stopped responding to how much material is in front of it — the core has absorbed everything it can — and a hydrogen column measured from a saturated core is not measured at all. Ten times more hydrogen would look identical. This is the curve of growth in its least forgiving regime, and the standard escape from it is to stop looking at Lyman-α.

The two figures together explain why the usable systems are so few. A cloud with too little hydrogen has no detectable deuterium; a cloud with too much has a saturated deuterium line and the ratio becomes unreadable — above about 10^18.2 in H I column the deuterium feature stops responding to the abundance at all. The window between those bounds is roughly a decade wide, the cloud has to be metal-poor enough that its deuterium has not been processed by stars, it has to be at a redshift that puts the Lyman series into an optical spectrograph, and it has to sit in front of a quasar bright enough for a hundred-hour exposure to reach the required signal-to-noise. Rather more than a hundred thousand absorption systems are catalogued in the Lyman-α forest. Fewer than a dozen have produced a D/H measurement at the per-cent level.

The one parameter that decides whether the system is usable #

The separation between the two lines is fixed by atomic physics. The width of them is not, and the width is what the measurement lives or dies on.

This is why the field’s selection is so aggressive. A system with warm or turbulent gas is not analysed more carefully; it is discarded, because there is no amount of signal-to-noise that separates a deuterium feature from a hydrogen wing once the wing is wider than the isotope shift. Combined with the column-density window and the metallicity requirement, the selection removes better than 99.99 per cent of known absorption systems, and what survives is the sample that a per-cent measurement of the baryon density rests on.

Selecting that hard has an obvious risk attached, and the field states it rather than hides it: the systems chosen are chosen for a property — narrow lines, simple velocity structure — which is not obviously independent of the quantity being measured. The defence is that no plausible mechanism connects a cloud’s turbulence to its primordial deuterium, and that the measured values do not correlate with the selection variables. It is a defence rather than a proof, and it is the sort of argument the collection meets again wherever a sample is chosen by the thing that makes it measurable.

The failure mode that does not look like one #

The systematic that has sunk more than one published deuterium abundance is not noise. It is a second, weaker cloud of ordinary hydrogen sitting by coincidence at −82 km/s relative to the main one, which produces an absorption feature indistinguishable from deuterium in a single line.

Nothing about the strength of that feature betrays it. Its width might, if the resolution is high enough: a hydrogen cloud has thermal motions of a given Doppler parameter and a deuterium feature in the same gas is narrower by √2, because the thermal width goes as the inverse square root of the mass. But a cold hydrogen interloper can be that narrow, and the discriminant is weak.

What actually settles it is the series. A real deuterium feature has to appear at −82 km/s in every Lyman line, with its own strengths scaling as the same oscillator strengths that scale the hydrogen lines, and with a column consistent across all of them. An interloping hydrogen cloud does the same thing — which is why the check works on something else: an interloper has its own higher-order Lyman lines whose ratios describe a hydrogen cloud rather than a deuterium one, and the two hypotheses part company once the column is high enough for the curve of growth to bend. The published measurements that were later withdrawn were withdrawn on exactly this, and it is why a modern paper reports the whole series rather than the ratio.

The other measurement of the same number #

The reason any of this is worth doing at per-cent precision is that there is a completely independent route to the same quantity, and the two have no physics in common.

Those two determinations share no equipment, no epoch, no physical process and no systematic. One is a nuclear cross section applied to a plasma at ten million kelvin, read off a spectrograph as the ratio of two absorption lines in a cloud twelve billion light years away. The other is the relative height of two bumps in the angular correlation function of a temperature map. They agree to a few per cent, and if the light-element abundances had come out wrong the disagreement would have been the most interesting result in the subject rather than a footnote.

What the picture cannot show #

The burn-race figure draws the competition that decides when the deuterium stops being destroyed. It does not, and cannot, predict how much survives. That number is the output of a full reaction network — a dozen coupled species, forty-odd rates, and the ⁷Be channel feeding back through electron capture months later — and every abundance curve in this collection is a fit to such a network rather than to anything drawn here.

The gap between the two is instructive rather than embarrassing, because it is a specific gap with a specific size. A single two-body destruction channel racing a one-body expansion gives a residue exactly proportional to 1/η, an exponent of −1.00. The network gives −1.60. The missing 0.6 is the chain: deuterium is not only destroyed by protons, it is destroyed by the ³He and ³H that the D + D reactions have just produced, and those species have their own abundances that are themselves proportional to the baryon density. An effective destruction rate that carries the density more than once is steeper than one that carries it once, and that is the whole of the difference between the drawn slope and the real one.

The honest summary is that the figure explains the sign and the shape and hands the magnitude to a calculation it does not perform. That is a weaker claim than the picture might seem to make, and stating it is the only way the picture stays useful: a reader who took −1.6 to be derivable from three parallel lines would be wrong about the subject in a way nothing in the drawing would correct.

The shape of the argument, elsewhere #

A quantity that survives a process is often a better measurement than a quantity the process produced, and the reason is always the same: what survives is exponential in how long the process ran, and an exponential is a lever.

The pattern recurs across this collection with no relationship between the physical settings. The neutron fraction left behind at freeze-out is a residue in exactly this sense, and it counts particle species because the expansion rate that ended the race carries the square root of their number. The white dwarf that has not yet cooled is a residue of a thermal reservoir, and its temperature is an age. In each case the measurable is small, is what escaped, and is steep in the parameter of interest — and in each case the quantity that was produced rather than left over turns out to be the flat one.

The corollary is the warning attached to all three. A residue is steep in the duration and therefore steep in everything that sets the duration, so an error in an input that is barely visible in a production yield is magnified in a leftover. Deuterium’s exponent of −1.6 is what makes it the baryometer, and it is also why the published network calculations disagree with each other by a per cent or two on the same input: the same steepness that turns a good abundance into a good density turns a small rate uncertainty into a visible one.

Where this ladder goes next #

The concordance holds for three of the four species and fails for the fourth. Predicted ⁷Li at the density everything else agrees on is about three times what the oldest stars in the Galaxy show, the discrepancy has stood since the abundance was first measured in 1982, and it is the only place in this subject where a number that ought to be right is not.

The next rung is that failure, taken seriously rather than noted. It has exactly three kinds of resolution — the stars destroyed the lithium, the observation is measuring something else, or the network is missing something — and the datum that prices all three is not the plateau’s height but its flatness.

Beyond it: helium-3, and why an abundance that low-mass stars produce and massive ones destroy has no direction of correction and therefore no quoted primordial value at all; inhomogeneous nucleosynthesis, pursued for a decade as the one way to make the baryons account for all the matter, and the convexity argument that says why it aggravates the lithium problem instead of fixing anything; and the neutron lifetime, whose two laboratory measurements disagree by ten seconds and whose disagreement propagates into every number on this ladder.