Astronomy & Celestial Mechanics
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Reionisation — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. A neutral fraction of 2.4·10⁻⁶ is already opaque. The Gunn–Peterson optical depth against the neutral fraction of the intergalactic medium, at z = 3, 5, 6.3, for Ω_b = 0.0493 and h = 0.674. Note the range of the vertical axis. A fully neutral medium at z = 6.3 gives τ = 4.1·10⁵, which is not absorption but extinction of everything; the medium reaches τ = 1 — the point at which it stops transmitting most of the light — at neutral fractions of 6.1·10⁻⁶ at z = 3, 3.3·10⁻⁶ at z = 5, 2.4·10⁻⁶ at z = 6.3. That is why the argument runs from the flux that survives rather than from the flux that does not. A spectrum showing any transmission at all between Lyman α and Lyman β is a measurement that the medium is ionised to better than one part in 164,727, and no fit to any absorption line is needed to establish it.

    A trough that proves the forest survived

    A uniform neutral medium at redshift six would absorb Lyman alpha with an optical depth of four hundred thousand. So the existence of any transmitted light in a quasar's spectrum is a measurement — of a neutral fraction below one part in ten thousand — made from a detection rather than from an absorption.

    part 1 · cosmology
  2. Bubbles that meet at z = 5.3, and a scattering depth of 0.047. The fraction of the volume of the universe filled by ionised bubbles, integrated from redshift 20 down to 4.5. The equation has two terms and no others: photons escaping from young galaxies open new volume, at a rate taken from the measured cosmic star formation history with an escape fraction of 0.2; recombinations inside the bubbles close it again, on a timescale that is one over the density times the recombination coefficient times a clumping factor of 3. Early on the density is high and recombination wins almost everything; the curve is nearly flat. As the universe expands the recombination time lengthens as the cube of one plus the redshift while the star formation rate is still rising, the balance tips, and the filling factor runs to one in under half a billion years. It reaches unity at redshift 5.31, which is overlap — the moment the bubbles meet and the last neutral walls between them disappear. The same integration gives an electron-scattering optical depth of 0.0465 for the microwave background, against the 0.054 that is measured, and that agreement is the check: the two observations constrain the same history from opposite ends, one fixing when it finished and the other how long it took.

    It ends when the walls meet

    Reionisation is usually described as the universe becoming transparent, which makes it sound like a change of state. It is not. Each young galaxy opens a bubble of ionised gas around itself and recombination closes it again, and what ends the epoch is geometric — the bubbles meet, and the last neutral walls between them disappear.

    part 2 · cosmology
  3. A 5 per cent continuum error, and an optical depth wrong by 1.1. The mean transmitted flux of the Lyman-alpha forest against redshift, with a 5 per cent uncertainty in the quasar continuum drawn as a band. The continuum is not observed: at these redshifts every part of the spectrum blueward of the emission line is absorbed, so the level has to be extrapolated from the red side across a region where the quasar's own spectrum has structure. A fractional error in that level is a fractional error in the flux, and since the optical depth is minus the logarithm of the flux, the resulting error in the optical depth is the fractional error divided by the flux — which grows without bound as the forest goes black. At z = 2 it is 0.06; at z = 6.2 it is 1.1. That is why measurements of when reionisation ended are quoted as limits rather than values above about redshift six.

    A forest with no continuum left

    Measuring how much neutral hydrogen sits between here and a distant quasar means measuring the fraction of its light that survives, which means knowing how much light there was. At high redshift nothing survives at the wavelengths that would show it, so the level is extrapolated across the region being measured — and the optical depth is the logarithm of a number divided by a guess.

    part 3 · cosmology
  4. 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.

    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.

    part 4 · cosmology

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