Generator

The recombination generator

Recombination, and the photons that delay it

The fraction of hydrogen still ionised, from the Saha equation, with redshift running to the right so that time runs to the right too. Three baryon-to-photon ratios are drawn, a hundredfold apart end to end, and the curve barely moves: the half-ionised point shifts only from z = 1544 to z = 1369, which is 4212 K against 3735 K. Both of those are a factor of 37 to 42 below hydrogen's ionisation temperature of 157,803 K, and that gap is the whole point of the figure. The gas stays ionised long after the typical photon is far too feeble to ionise anything, because there are 1.6·10⁹ photons for every baryon and the far tail of the Planck distribution still holds more than enough of them. The equilibrium breaks only when that tail runs out, and because it is an exponential tail it runs out abruptly: at the measured baryon density the curve falls from nine tenths ionised to one tenth across Δz = 218, about 16 per cent of the redshift at which it happens. Saha gets the end of the story wrong and the essay says how: it drives the ionised fraction to nothing, and the real one freezes out near 2e-4 because the recombination rate cannot keep up with the expansion.

2 essays call recombination. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Every figure listed here is the same construction drawn at different numbers, so a correction to one is a correction to all of them.

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

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.

One curve the temperature fixes, and one line the polarisation adds. The plane of the optical depth to reionisation against the amplitude of the primordial fluctuations. The temperature power spectrum of the microwave background measures the product of the amplitude and the exponential of minus twice the optical depth, so it constrains a curve rather than a point: more electrons scattering the photons out is indistinguishable from fewer fluctuations to begin with, and the two trade along the drawn locus across a factor of 1.22 in amplitude. What breaks it is the polarisation at the largest angular scales, where rescattered photons regenerate a signal whose amplitude is proportional to the optical depth itself rather than to its exponential. That constraint is nearly vertical here, it comes from a handful of multipoles at the very largest scales, and it is the single hardest measurement the microwave background has demanded — because at those scales the Galaxy's own polarised emission is larger than the signal.

Cosmology

An amplitude and a depth that arrive multiplied

The microwave background's temperature fluctuations are the primordial ones damped by everything that scattered them since. The damping is uniform, so a smaller starting amplitude and more scattering produce identical maps — and separating them requires a signal from a handful of the largest angular scales, where the Galaxy's own emission is larger than what is being measured.

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