Astronomy & Celestial Mechanics
the sky, computed
Concept

Freeze out — where it appears

The moment a reaction stops keeping up with the expansion, leaving an abundance fixed at whatever value it had reached. It is what fixes the primordial helium abundance and the relic density of any particle species, and the abundance left behind is a record of the expansion rate at that moment.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

Four abundances, one free parameter. The abundances big-bang nucleosynthesis predicts, against the one number it is free to choose: η₁₀, the ratio of baryons to photons in units of 10⁻¹⁰. Four curves spanning nine decades, from a helium mass fraction of about a quarter down to a lithium abundance of one atom in ten billion, and they are not four independent predictions — they all come out of the same reaction network run at the same density. The horizontal bands are what is measured in the sky, each at its published one sigma. The measurement that matters is deuterium, because its curve is the steep one: inverting the drawn curve at D/H = 2.527e-5 gives η₁₀ = 6.11, and the ends of the observed interval give 6.06 to 6.15. The vertical band is what the microwave background gives, 6.13 ± 0.04, from the height of the second acoustic peak relative to the first. Those two agree to 0.4 per cent, and they have nothing whatever in common: one is a nuclear-reaction network run in the first three minutes and read off a quasar absorption line, the other is a fluid oscillation at four hundred thousand years read off a sky map. Lithium is the exception and it is not a small one — the network predicts 4.70e-10 at the microwave background's density and the halo stars show 1.60e-10, a factor of 2.9 too much, which is unresolved.

Four abundances and one free parameter

In the first three minutes the universe ran a nuclear reaction network with exactly one adjustable number in it. That number predicts four abundances spanning nine orders of magnitude, three of them are observed and match, and the fourth is wrong by a factor of three and has been for twenty-five years.

cosmology · Nucleosynthesis One ionising photon per atom, and where that happens. The number of photons energetic enough to ionise hydrogen, per baryon, against temperature — with temperature falling to the right, so the universe ages to the right. The quantity is the fraction of a Planck distribution above 13.6 eV, integrated numerically, divided by the 6.13·10⁻¹⁰ baryons there are per photon. It crosses one at 5,844 K, which is z = 2143, and it falls by 19 orders of magnitude across the plot because it is the tail of an exponential. That crossing is the answer to why recombination waits until three thousand kelvin. At hydrogen's own ionisation temperature of 157,803 K there are two billion ionising photons per atom and the gas has no chance; the temperature has to fall by a factor of forty before the supply runs out, and it runs out suddenly because an exponential tail does. The number that sets the factor of forty is not an energy at all — it is the photon-to-baryon ratio, which is to say the entropy per baryon, which is to say a number fixed long before any of this and measured today from the second acoustic peak and from deuterium alike.

The surface the background actually is

Hydrogen ionises at 157,800 kelvin and the universe became transparent at 3,000. The factor of fifty between them is not an error, and it is not about energy — it is about there being two billion photons for every atom, so the far tail of the distribution can keep the gas ionised long after the typical photon has become useless.

cosmology · Microwave background The helium abundance as a count of neutrino species. The primordial helium mass fraction against the number of light neutrino species, integrated at a deuterium-bottleneck temperature of 0.0855 MeV and a neutron lifetime of 877.75 s. The curve rises at 0.01348 in Y_p per species near the standard model, and it is not quite a line: freeze-out temperature goes as the sixth root of g_*, so a species added at N_eff = 4.5 buys 25 per cent less helium than one added at 2 — a slope ratio of 0.749 against the 0.769 that scaling requires. The reading below is therefore taken off the drawn curve rather than off a slope. The horizontal band is the measurement — ⁴He (Aver 2015), Y_p = 0.2449 ± 0.004, taken from recombination lines in metal-poor dwarf galaxies and extrapolated to zero metallicity. Where the band crosses the line is the answer: N_eff = 2.84, with the ends of the observed interval giving 2.56 to 3.13. The standard model has three, and 3.046 rather than 3 because the neutrinos are not quite decoupled when the electron–positron pairs annihilate and take a sliver of the heat. The strength of this is not its precision, which is a third of a species and worse than the microwave background's; it is that the two constraints come from utterly different epochs, and that this one is a laboratory result about particle content obtained from an emission line in a galaxy.

A particle count taken from a dwarf galaxy

Nearly every neutron that survives the first three minutes ends inside a helium nucleus, so the primordial helium abundance is not chemistry — it is the reading of a race between the weak interaction and the expansion. The expansion rate carries the square root of the number of relativistic species, which is why an emission line in a metal-poor galaxy counts neutrinos.

cosmology · Nucleosynthesis The deuterium burning rate against the expansion, at three baryon densities. The rate at which a deuteron is destroyed, divided by the expansion rate, against temperature. Temperature falls to the right, so the picture reads left to right as time, and the horizontal line at one is where the burning stops mattering: above it a deuteron is destroyed many times over before the universe doubles in size, below it the reaction has effectively ceased. The rate drawn is D(p,γ)³He at the NACRE parameterisation, multiplied by the free-proton density — three quarters of the baryons by number once ⁴He has taken the rest — and the expansion rate is the same 1.66√g* T²/mPl the freeze-out calculation raced the weak interactions against. The three curves differ in one number and one only: the baryons per photon, 3, 6, 12 in units of 10⁻¹⁰. They are therefore vertical translations of each other, exactly in proportion to η, because the reaction is two-body and the expansion is not. That is the entire mechanism by which an abundance measures a density. A denser universe crosses the line later — 21.4 keV at η₁₀ = 3, 16.0 keV at η₁₀ = 6, 12.3 keV at η₁₀ = 12 — and every extra second below the crossing is deuterium that does not survive. What the figure does not do is predict the abundance itself: the residue depends on the whole reaction network and on the ⁷Be and ³He channels that feed back into it, and the curve the abundance is read off is a fit to that network rather than to this.

The residue that failed to burn

Deuterium's abundance is not a measure of what the first three minutes made. It is a measure of what escaped being used — a two-body destruction rate losing a race to a one-body expansion, which is why its curve against the baryon density is steep and helium's is flat.

cosmology · Nucleosynthesis

Named alongside it

The objects these essays reach for when they reach for this one.

Baryon-to-photon ratioBig bang nucleosynthesisBaryon densityDeuterium abundanceDeuterium bottleneckNeutron proton ratioPrimordial heliumQuasar absorption linesBinding energyColumn densityCurve of growthDetailed balance

All concepts