It ends when the walls meet

A single quasar spectrum can show that the intergalactic medium is transparent, and the previous rung of this anchor made that argument: the absence of a saturated trough shortward of Lyman alpha requires a neutral fraction below one part in a hundred thousand, which is an extraordinarily strong statement obtained from a spectrum with structure in it rather than from a fit.

That establishes that reionisation happened. It says nothing about how, and almost nothing about when. This rung is about the accounting.

Two terms, and nothing else #

The equation the figure integrates has one line in it:

$$\frac{dQ}{dt} = \frac{\dot n_{\rm ion}}{\langle n_{\rm H}\rangle} - \frac{Q}{t_{\rm rec}}.$$

$Q$ is the fraction of the volume that is ionised. The first term is the rate at which new volume is opened, which is the number of ionising photons escaping per unit volume per unit time divided by the number of hydrogen atoms per unit volume — one photon, one atom. The second is the rate at which ionised volume closes again, which is $Q$ divided by the recombination time.

That is the whole model. It contains no radiative transfer, no geometry, and no assumption about the sizes or shapes of bubbles. It is a bookkeeping identity for volume, and its usefulness comes from the fact that both terms can be evaluated from measurements made at lower redshift.

Why the balance tips #

The two terms have opposite dependences on redshift and that is what produces the sharp ending.

The recombination time is one over the density times the recombination coefficient. Density falls as the cube of one plus the redshift — the expansion is a scaling of every length, so every density carries the same power — so the recombination time lengthens by a factor of eight for every halving of one plus the redshift. Between redshift twelve and redshift six it lengthens by a factor of nearly eight.

The source term does the opposite. The cosmic star formation rate density rises from redshift twenty to a peak near redshift two, so over the epoch of interest the supply of photons is increasing while the demand created by recombinations is falling.

Neither term is dramatic on its own. What is dramatic is the crossing, and the crossing is sharp because the two are exponentials running in opposite directions.

Bubbles, and why overlap is the event #

The volume-filling account glosses over the geometry, and the geometry is where the word “reionisation” gets its meaning.

Ionising photons have a very short mean free path in neutral hydrogen — a fraction of a proper kiloparsec at these densities — so a source does not ionise the universe around it uniformly. It carves out a bubble with a sharp edge, and inside that bubble the medium is transparent while outside it is opaque.

While the bubbles are separate, an ionising photon travels until it hits the wall of its own bubble and is absorbed. When the bubbles meet, that wall is gone, and the photon travels until it hits a dense clump — which at these redshifts is much further. So the mean free path jumps by orders of magnitude at overlap, the ionising background rises abruptly everywhere, and the residual neutral fraction inside the ionised regions falls by a further factor of a hundred.

That jump is the event. It is not the moment the last neutral atom is ionised; a substantial neutral fraction survives inside dense clumps for a long time afterwards. It is the moment the topology changes from isolated bubbles in a neutral sea to isolated neutral islands in an ionised one, and it is a percolation transition rather than a chemical one.

The two measurements that bracket it #

Reionisation is constrained from opposite ends by two observations with almost nothing in common.

The first is the quasar absorption troughs. A quasar at redshift six shows transmission spikes and a resolved forest; one at redshift seven shows a nearly complete trough. That transition marks the tail end, because the Gunn–Peterson optical depth is so large that a neutral fraction of one part in ten thousand saturates the absorption — so the troughs constrain the very end of the process and are blind to everything before it.

The second is the microwave background. Free electrons scatter its photons, so the total optical depth to Thomson scattering integrated back to the surface of last scattering measures the total column of free electrons — which is an integral over the whole history rather than a snapshot of its end. The integration in the hero figure produces both. It gives overlap at redshift five and a bit and a scattering depth of about 0.047, against a measured 0.054 — agreement at the level the simplicity of the model deserves.

The photon budget, and the parameter nobody can measure #

The source term is the product of three things: the star formation rate density, the number of ionising photons produced per unit of star formation, and the fraction of those that escape the galaxy rather than being absorbed inside it.

The first is measured, out to redshift eight or so, from deep imaging — and a star formation rate is inferred from light that almost none of the young stars emit, which puts an initial mass function into the chain before reionisation is even reached.

The third is the escape fraction, and it is the weak link. Measuring it requires detecting radiation shortward of 912 ångströms escaping from a galaxy, which at the redshifts that matter is impossible because the intervening medium absorbs it. Every estimate is either from lower-redshift analogues, where the escape fraction may be different, or from demanding that reionisation happen on time — which makes it an output of the model rather than an input.

The clumping factor, which hides the geometry #

One symbol in the equation is doing more work than the rest and it deserves its own section.

The recombination rate depends on the square of the density, because a recombination needs an electron and a proton to meet. What appears in the equation is the mean of the square, and what is measurable is the square of the mean. The ratio between them is the clumping factor, and it is not one.

Physically it encodes the fact that recombinations happen preferentially in the densest gas, so a universe with structure in it recombines faster than a smooth one of the same mean density. The value used above is three, which is what simulations give when the gas that can actually recombine is separated from the gas that has already collapsed into galaxies. Values of ten and thirty have been used in the past, and the difference is a factor of ten in the demand for photons.

The clumping factor is also where the volume-filling model quietly stops being about volume. A bubble expanding into clumpy gas does not have a sharp edge; it has a ragged one, with dense clumps surviving inside it as neutral islands that recombine and are re-ionised repeatedly. Averaging that into a single number is the model’s one real approximation.

What was actually measured #

Three numbers, with very different provenance.

The scattering depth is measured from the microwave background’s large-scale polarisation, and its dominant uncertainty is the separation of the cosmological signal from Galactic dust and synchrotron emission. It has moved substantially over two decades as those foregrounds have been better modelled, from 0.17 in the first measurement to 0.054 now — a shift of the kind that makes two measurements of the same constant disagree at five sigma until one of them moves — a change that shifted the inferred midpoint of reionisation by several units of redshift.

The end of reionisation is measured from quasar spectra, and its dominant uncertainty is the number of quasars bright enough to observe at redshift seven, which is a few dozen. It is also complicated by the fact that a quasar ionises its own surroundings, so the region immediately in front of it is not representative.

The star formation history is measured from galaxy counts, and its dominant uncertainty at high redshift is what fraction of the star formation is in galaxies below the detection limit. The luminosity function is steep, so the integral depends on where it is truncated.

The shape of the argument #

It is worth stepping back to notice what kind of argument this is, because it is unusual in the collection.

Nothing here is a measurement of reionisation. The hero figure is an integration of a model whose two terms are calibrated on observations made at other redshifts, and its output is compared with two observations that constrain the beginning and the end. The model has one free parameter of consequence, and that parameter is adjusted until the output matches.

What makes it more than a fit is that the two constraints are independent and the model has fewer parameters than constraints. A history that ends at the right redshift and produces the wrong scattering depth would be ruled out, and histories of that kind exist — a reionisation that happened abruptly at redshift six produces a depth of about 0.04, which is too low, and one that happened gradually from redshift fifteen produces about 0.08, which is too high.

One further asymmetry between the two constraints is worth stating, because it explains why the subject has felt so unsettled for so long. The scattering depth is an integral, so it is sensitive to an early tail of ionisation and insensitive to exactly when the process finished. The quasar troughs are a threshold, so they are sensitive to the finish and blind to everything before. Neither observation constrains the middle, which is where all the interesting physics is — how fast the filling factor rose, whether it rose smoothly or in bursts, and whether the sources were the galaxies that are counted or something fainter.

A model that reproduces both constraints is therefore a model that has been fitted at its two ends and is unconstrained in between, and the differences between competing histories in the literature are almost entirely differences in that middle. That is a familiar shape in cosmology: two integrals of the same function, measured well, and the function itself measured not at all.

A closing remark about the vocabulary, because it does real damage. Reionisation is routinely described as the universe becoming transparent, and the phrase suggests a change in the gas itself — as though a fog lifted. What actually changed is the topology of a two-phase medium, and the gas at the end of the process is very largely the same gas it was at the beginning, at the same density, ionised by photons whose supply had been growing for half a billion years. The transparency is a property of the paths between the ionised regions rather than of the regions themselves. Getting that right matters for what is being measured: a survey that finds a transmission spike in a quasar spectrum at redshift seven has found a sightline that happened to run through a bubble, not evidence that reionisation was complete at that redshift. Sightline-to-sightline scatter at the end of the epoch is therefore enormous, and it is a measurement of the bubble size distribution rather than noise.

The whole calculation is a race between two rates, and each of the two parameters that set them is worth reading at a second value, because neither of them is measured.

The sources that might not be galaxies #

The photon budget above credits every ionising photon to star formation, and that is an assumption rather than a measurement. The alternative has been argued about for fifty years.

Quasars produce enormous quantities of ionising radiation, and unlike a galaxy’s they escape freely — the surrounding gas has already been ionised by the source itself, so the escape fraction is essentially one. A population of accreting black holes at high redshift would supply photons far more efficiently per unit mass than stars do.

What decides the question is how many faint quasars there were. The bright ones are counted and are far too few: their integrated output falls short of the requirement by an order of magnitude at redshift six. Whether there is a steep faint end that surveys have missed has been claimed and disputed repeatedly, and the difficulty is that a faint quasar at high redshift is photometrically hard to distinguish from a compact star-forming galaxy.

Two constraints argue against a large quasar contribution. Accreting black holes emit X-rays as well as ultraviolet, and X-rays penetrate far further than ionising photons do, so a quasar-dominated reionisation would have heated the intergalactic medium substantially — and the temperature measured later, from absorption-line widths, is lower than that would produce. And the same population would have over-produced the observed X-ray background.

There is one thing quasars demonstrably did do, and it is worth stating because it is the clean case. Helium requires photons of four times the energy hydrogen does, which stars produce in negligible quantity and quasars produce readily. The second ionisation of helium happened around redshift three, long after hydrogen’s, and it is observed by exactly the same technique — a helium absorption trough in the ultraviolet spectra of distant quasars, patchy at redshift three and absent below.

The same argument applied to the same medium at a different ionisation threshold gives a different epoch and a different source population, which is the strongest available evidence that the machinery is being used correctly.

The epoch before this one #

Everything above concerns the period in which the hydrogen was being ionised. There is an earlier interval that is entirely unobserved and that the same instrument would reach first.

Before the first stars, the gas is neutral and cold — colder than the microwave background, because it cooled adiabatically with the expansion while the radiation cooled more slowly. Neutral hydrogen at a temperature below the background’s absorbs at 21 centimetres rather than emitting, so the sky should show a broad absorption feature at the corresponding redshifted frequency.

Whether that absorption appears depends on a coupling. The relative populations of hydrogen’s two hyperfine levels are set by the radiation field unless something else drives them, and the something else is scattering of Lyman alpha photons from the first stars. So the absorption switches on when the first stars form, deepens as they multiply, and switches off again as X-rays from the first accreting objects heat the gas above the background temperature.

The signal is therefore a trough in the sky-averaged radio spectrum, with its onset marking the first stars, its depth set by how cold the gas got, and its recovery marking the onset of heating.

A detection was reported in 2018 — an absorption profile centred near seventy-eight megahertz, corresponding to redshift seventeen — and it was immediately contested, on two grounds. Its depth was about twice what any standard model allows, which would require either a colder gas than adiabatic cooling gives or a radio background brighter than the microwave background. And the measurement requires separating a signal of a fraction of a kelvin from a Galactic foreground four orders of magnitude brighter, using a fitted model of that foreground; an independent experiment reported no detection with the sensitivity to have seen it.

The measurement remains open, and it is the only observation anyone has proposed that would date the first stars directly.

And the observable itself over a redshift range extending past the transition, since it is the measurement everything else is checked against.

Where the ladder goes #

The next rung is the 21-centimetre signal, which is the only observation that would see the neutral gas rather than infer it: an absorption or emission signal against the microwave background, redshifted into the metre-wave band, whose sky-averaged evolution traces the neutral fraction and whose fluctuations trace the bubbles.

The other direction is what else the photons did. The same ultraviolet radiation that ionised hydrogen heated the intergalactic medium to about twenty thousand kelvin, and that temperature is measured in the widths of absorption lines in the forest at lower redshift — a thermal record of an epoch that is otherwise almost unobservable.