The universe that was lumpy at one second

Every rung of this anchor so far has assumed one number. The baryon-to-photon ratio η is written as a single quantity, the network is run at that value, and four abundances come out; the concordance between the deuterium inversion and the acoustic peaks is a statement that two measurements of that one number agree.

The assumption is not innocuous, and for most of the 1980s it was the thing standing between cosmology and a much tidier universe. Dynamical measurements said the matter density was several times what nucleosynthesis allowed for baryons — the same gap the budget essay opens with, before there was any independent measurement of either quantity to close it. A first-order quark–hadron transition at about a hundred megaelectronvolts, which lattice calculations at the time did not exclude, would have left the baryons in clumps — quark matter is denser than hadronic matter, so the shrinking quark bubbles concentrate baryon number — and if each clump ran its own nucleosynthesis at its own density, the mean density might be raised while the abundances stayed where they are observed.

The deuterium excess is the whole appeal of the proposal, because it means a higher mean density can still match what is measured. The lithium excess is the cost. And lithium was already three times too high before anybody made the universe lumpy.

The rest of this rung is an account of why those two excesses are not independent adjustable outcomes, why the mechanism that would produce them may not survive to the moment it is needed, and what it means that a decade of careful work was closed by an inequality about curves rather than by an observation.

Why both go up, and why it is not a modelling result #

The tempting reading of that figure is that it depends on a two-zone caricature, on a volume fraction chosen out of the air, and on the fitting formulae the abundance curves are drawn from. It depends on none of them. It depends on the sign of one second derivative.

An abundance is a ratio per baryon, so what an observation adds up is not the volume average of the abundance but the baryon-weighted one: total deuterium over total hydrogen, with each region contributing in proportion to how many baryons it holds. That makes the averaged quantity η times the abundance, divided by the mean η. And for both deuterium and lithium, η times the abundance is a convex function of η.

Nothing about nucleosynthesis is used in that figure except the sign of the curvature. A chord of a convex function lies above the function, which is Jensen’s inequality, and the conclusion therefore survives every refinement that leaves the curve convex — a different transport mechanism, a different geometry of the fluctuations, a hundred zones instead of two, a continuous distribution of densities instead of any zones at all.

It is worth noticing how unusual that is as a way to argue in this subject. Almost every result on this ladder is the output of an integration: the neutron fraction comes from stepping a rate equation, the deuterium residue from a reaction network, the abundance curves from fits to that network. A conclusion that follows from a curvature rather than from a calculation is cheap in the best sense: it costs nothing to check, it cannot be wrong because of a rate, and it applies to every version of the model at once. The proposal’s proponents did the multi-zone calculations properly and arrived at the same place, which is the correct order in which to do these things — but the inequality was there the whole time.

The asymmetry has an arithmetic explanation and it is worth stating because it makes the result predictable rather than surprising. The curvature of a power law η^p in the averaged variable is set by p(p+1) with the p+1 from the baryon weighting: deuterium at p = −1.6 gives 0.96, lithium at p = 2 gives 6. Six against one. Whatever a lumpy universe does to deuterium, it does about six times as much to lithium, and it does it in the direction lithium could least afford.

The parameter that actually matters #

The volume fraction of the dense zone is the obvious free parameter, and it is not quite the right one.

A small dense zone at high contrast is the worst case, because it holds most of the baryons at a density far from the mean while occupying almost none of the volume — and it is also, unhelpfully, the configuration a first-order phase transition actually produces. Shrinking quark bubbles concentrate baryon number into what is left of them, so the dense regions are the last surviving pockets and are small by construction. The proposal’s own mechanism selects the corner of parameter space in which it damages lithium most.

Whether lumpiness survives at all #

None of the above matters unless the inhomogeneity is still there when the network fires, and there is a specific reason it might not be.

Neutrons are uncharged. Protons are not, and at these temperatures they are tightly coupled to the electron–photon plasma and cannot move relative to it. So a density fluctuation in baryon number is, for the neutrons, something they can diffuse out of, and for the protons something they are stuck in — which means the zones separate not only in density but in neutron-to-proton ratio. That is the actual mechanism, and it is what makes the proposal a distinct piece of physics rather than an averaging exercise: the dense zone becomes proton-rich and the thin zone neutron-rich, and each runs a different network for that reason as well as for its density.

The T⁻³ is worth unpacking, because it is the whole reason the window is where it is. The mean free path is one over a number density times a cross section, the number density of scatterers falls as the cube of the temperature, and the neutron–proton scattering cross section is roughly flat at these energies — so the path grows as T⁻³. The available time grows as T⁻², since the age in the radiation era is proportional to the inverse square of the temperature. The diffusion length is the square root of their product, so it grows as T⁻⁵ᐟ². Against that, a comoving length converts with a factor of T, leaving T⁻³ᐟ² for the comoving reach. It grows, steeply, right up to the moment the network fires.

That curve defines a window and the window is narrow. A fluctuation much smaller than a neutron’s reach is smoothed away before the network fires, and the universe is homogeneous after all. A fluctuation much larger keeps its own neutrons, so each region is simply a small universe at its own density with no separation of species — which is the pure averaging case the earlier figures draw, and which is the weaker version of the effect. Only fluctuations comparable to the drawn curve produce the mechanism as intended.

The comoving scale in question is around a tenth of a parsec, which is about a solar mass of material. Nothing measures a primordial fluctuation on that scale. It is thirteen orders of magnitude below the smallest scale the microwave background resolves and nine below anything the galaxy correlation function reaches, so the separation length is a free parameter that no observation constrains and that the model has to be told.

The zones, as two nucleosyntheses #

Two densities that far apart are not a perturbation on one calculation. They are two different regimes of the same network.

The asymmetry between the zones is larger than the density contrast alone suggests, because the burning is exponential in how long it runs and the elapsed times differ by a factor of twenty. A universe made of two such regions is not a universe with a slightly wrong density; it is two nucleosyntheses that would be described in different words if either were the only one.

That last clause is where the model earns its complexity, and it is also where it becomes fragile. The observed abundance is a ratio, so the dense zone contributes deuterium at nearly zero and hydrogen at a great deal, which drags the measured D/H down; the thin zone contributes deuterium at nearly primordial and little hydrogen, which pushes it up. Which effect wins depends on the baryon weighting, and the answer — that the mixture is above the smooth value — is exactly Jensen’s inequality applied with the correct weights. Getting the weights wrong is the standard way to get this calculation wrong, and it moves the answer by tens of per cent at high contrast.

What the proposal was for #

It is worth being explicit about the problem that made a decade of this work worthwhile, because the problem is real and the answer to it turned out to lie elsewhere entirely.

The gap was not a small anomaly. Rotation curves stay flat far outside the visible light in every disc galaxy measured; cluster galaxies move too fast for the mass their light implies. In 1985 the choices were that the dynamical mass estimates were wrong by a factor of five, that nucleosynthesis was wrong, or that most of the matter is not made of baryons. The third was a large thing to propose and the second was the cheapest thing to attack, which is why inhomogeneous nucleosynthesis received the attention it did. The proposal was not fringe: it was worked on carefully, by good people, with multi-zone codes that included the neutron transport properly rather than the mixing argument used here.

It is also worth saying what the proposal would have bought if it had worked, because the prize was larger than a tidy density. A universe in which the baryons are the whole of the matter needs no new particle, and the case for one rested at the time entirely on this arithmetic — there was no direct detection, no collider constraint that mattered, and no structure-formation argument of the modern kind. Nucleosynthesis was, and to a considerable extent still is, the single strongest reason to believe that most of the matter in the universe is something nobody has ever handled. A proposal that removed that reason would have removed the whole case.

What killed it was not one result. The lattice calculations improved and the quark–hadron transition turned out to be a smooth crossover rather than a first-order transition, which removes the mechanism that made the clumps. That is the cleanest of the three, and it is worth registering how it came about: the question was settled by numerical quantum chromodynamics on a lattice, by people working on the strong interaction, with no cosmological motive and no cosmological input. An astrophysical proposal was closed by a calculation about quarks. The multi-zone calculations, done properly, found that the parameter space giving a high baryon density with acceptable deuterium and helium always overproduced lithium — the same conclusion the convexity argument reaches, obtained the hard way and reported before the easy argument was in circulation. And the acoustic peaks eventually measured the baryon density independently, at a value agreeing with the homogeneous deuterium inversion and nowhere near the matter density.

What a failed proposal leaves behind #

Inhomogeneous nucleosynthesis is not a cautionary tale and it should not be read as one. Three things came out of it that outlasted the idea.

The first is the multi-zone machinery itself, which is now the standard tool for any question about nucleosynthesis in a region that is not uniform — including the neutron-rich ejecta of a neutron-star merger, where the same transport problem appears with the signs of everything reversed. The second is a much sharper statement of what the light elements do and do not constrain: the exercise of asking what could be changed without breaking the abundances forced the field to write down which combinations of parameters each element actually measures, and the leverage arithmetic that shows deuterium as a baryometer and helium as a species counter dates from that period.

The third is the shape of the result. A model was proposed to fix one number, it was pursued honestly, and it was closed because it made a second number worse — not because it was inelegant, and not because the community lost interest. That is what a falsifiable astrophysical proposal looks like when it is falsified, and there are fewer clean examples of it than the subject would like.

What the picture cannot show #

The mixing model behind six of these figures is not inhomogeneous nucleosynthesis. It is the averaging half of it, with the neutron transport that constitutes the actual mechanism left out, and everything it computes is therefore a statement about what mixing alone does at a stated pair of densities.

Leaving out the transport matters in a specific direction, and honesty requires saying which. Neutron diffusion makes the thin zone neutron-rich, and a neutron-rich region makes more helium and can make substantially more of the heavier products — some of the multi-zone calculations found measurable production of elements up to carbon, which homogeneous nucleosynthesis does not reach at all. Those are effects the model here cannot produce, and one of them was for a time the proposal’s most distinctive prediction. What the mixing argument does establish is the direction and the rough size of the two effects that the proposal’s fate turned on, and that establishment does not depend on any of the machinery it omits.

There is a second thing left out that cuts the other way, and it is the reason the multi-zone calculations were worth doing rather than being replaced by the inequality. Diffusion does not stop at the end of nucleosynthesis. Neutrons continue to move between zones while the network is running, so the zones are not sealed and the abundances in each are not the abundances a sealed region at that density would have; and after the network finishes, ordinary diffusion and eventually the growth of structure mix the products themselves. The mixing figures here assume the products are averaged and the reactants were not, which is the limiting case of fast late mixing and slow early mixing. The opposite limit — everything mixed before the network fires — gives the homogeneous answer exactly, and the truth is somewhere between, at a place decided by the fluctuation scale that nothing measures.

The other limitation is the abundance curves themselves. They are fits to a homogeneous network over a decade of density, evaluated here at densities up to fifty times the standard value, which is at or beyond the edge of the range they were fitted over. The exponents are physical and hold well outside the fitted range for deuterium and helium; lithium’s η² is a good description near η₁₀ = 6 and becomes an underestimate at high density, where the ⁷Be channel saturates. The lithium excursions drawn are therefore not quantitatively reliable at contrasts of a hundred, and they are conservative in the direction that matters.

The shape of the argument, and where else it applies #

The pattern this rung is really about is not lumpy nucleosynthesis. It is that an average of a nonlinear function is not the function of the average, and that the difference has a sign whenever the curvature has one.

It turns up wherever a measurement of an unresolved region is treated as a measurement of its mean. The Sunyaev–Zel’dovich decrement is linear in the pressure, and X-ray emission goes as the square of the density, so a clumpy cluster gives a larger X-ray signal than a smooth one of the same mass — and every cluster mass derived from X-rays carries a clumping correction for exactly that reason. The emission measure of an H II region has the same square in it and the same bias. In each case the correction has a known direction before anything is computed, and knowing the direction is often enough to say whether an anomaly could be explained that way.

Here it settled a decade of work. Lithium was low, so the resolution had to be one that lowers lithium; convexity says that mixing raises it; so no mixing model of any complexity could be the resolution. That argument could have been made on the first day, and the point of writing it out is not to disparage the calculations that made it the hard way — it is that a curvature is a thing worth checking before a code is written.

Where this ladder goes next #

Six rungs have now taken the standard picture and its one free parameter, asked what each abundance measures, found the one place it fails, and dropped the assumption of uniformity to see whether that failure was an artefact of it. It was not.

What remains is the input every one of those rungs quietly shares. The conversion rate between neutrons and protons is normalised on the free-neutron lifetime, which is measured in a laboratory rather than in the sky — and it is measured two ways that disagree by ten seconds and better than four standard deviations, a discrepancy that has stood for two decades while both techniques improved. The next rung is that disagreement: what it does to the helium prediction, what it does to the deuterium prediction, and why a laboratory result about a bottle of ultracold neutrons is the largest single uncertainty in the oldest quantitative prediction in cosmology.

Beyond it: the constraint the same helium band places on a light sterile neutrino, which is the particle-physics reading of a measurement already made; and whether the concordance between the light elements and the microwave background is tight enough to constrain a time-varying gravitational constant, which is the one modification of the expansion rate that the abundances see and the acoustic peaks do not.