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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.

Assumes Microwave background and Spectra.

18 min read 9 figures Light is the only messengerThe sky as a clock

The microwave background’s spectrum says the universe was once opaque; its acoustic peaks say it stopped being opaque abruptly enough for a sound wave to be frozen mid-oscillation. Both take the moment of transparency as given. It is worth asking when it happened and why, because the answer contains a number that turns up nowhere else in astronomy and explains a factor of fifty.

Hydrogen’s ionisation energy is 13.6 electronvolts, which is a temperature of 157,800 kelvin. A gas of hydrogen ought to be neutral once it is well below that. The universe became neutral at about 3,000 kelvin — fifty times colder — and it stayed ionised through the whole intervening range.

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.
Fig. 1 The reason, and the whole of it. 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 6.1×10−106.1\times10^{-10} baryons there are per photon. It crosses one at 5,844 K and falls by nineteen orders of magnitude across the plot, because it is the tail of an exponential. At hydrogen’s own ionisation temperature there are two billion ionising photons per atom and the gas has no chance.

The number that sets the factor of fifty

The photon-to-baryon ratio is η≈6.1×10−10\eta \approx 6.1\times10^{-10}, so there are 1.6×1091.6\times10^9 photons for every baryon. That number is fixed before nucleosynthesis and does not change afterwards — both photons and baryons are conserved and both dilute as a−3a^{-3} — which is why the same quantity is the one free parameter of big-bang nucleosynthesis, measured to a per cent from deuterium and independently from the second acoustic peak.

Here it does a different job. A Planck distribution at temperature TT has a fraction of its photons above energy BB that falls off as e−B/kTe^{-B/kT} for kT≪BkT \ll B, so the number of ionising photons per baryon is roughly η−1e−B/kT\eta^{-1}e^{-B/kT}. Setting that to one gives

BkT≈ln⁡(1η)≈21,\frac{B}{kT} \approx \ln\left(\frac{1}{\eta}\right) \approx 21,

so the temperature at which the ionising supply runs out is B/kB/k divided by something of order 20 to 40, depending on how carefully the prefactors are kept. The factor of fifty is a logarithm of the photon-to-baryon ratio. It is not an energy scale at all; it is an entropy.

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.
Fig. 2 And the same statement from the other direction, through the Saha equation, which balances ionisations against recombinations in equilibrium. Three baryon-to-photon ratios a hundredfold apart end to end, and the half-ionised point barely moves: from z=1544z = 1544 to z=1369z = 1369, which is 4,212 K against 3,735 K. Both are a factor of 37 to 42 below 157,800 K. The insensitivity is the logarithm again — a hundredfold change in η\eta shifts ln⁡(1/η)\ln(1/\eta) by 4.6 out of 21, which is a 20 per cent change in a temperature.

The abruptness follows from the same exponential. Once the tail starts running out it runs out fast: at the measured density the ionised fraction falls from nine tenths to one tenth across a redshift interval of 218, which is 16 per cent of the redshift at which it happens. A gradual transition would have destroyed the acoustic peaks, because a wave that stops ringing over many oscillation periods leaves no phase coherence. The sharpness of the peaks and the sharpness of recombination are the same fact.

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 = 1451 to z = 1296, which is 3959 K against 3535 K. Both of those are a factor of 40 to 45 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.
Fig. 3 The same three-curve construction shifted a decade lower in baryon-to-photon ratio. Fewer photons per baryon means the ionising supply runs out sooner, so every curve moves to a higher temperature — and the width of each transition, as a fraction of the redshift at which it happens, does not change. That invariance is the point. The sharpness is a property of the exponential tail rather than of where the crossing falls, so a universe with a hundredfold different entropy per baryon would still have recombined abruptly and would still have acoustic peaks.

Equilibrium is not maintained, and the gap is informative

The Saha curve above is an equilibrium calculation, and equilibrium fails near the end. The failure has a specific cause worth stating, because it is a nice piece of atomic physics doing cosmological work.

When a proton and an electron recombine directly to the ground state, the photon emitted has at least 13.6 eV and immediately ionises a neighbouring atom. Direct recombination to the ground state therefore accomplishes nothing at all in a medium that is still mostly hydrogen. The only recombinations that stick are the awkward ones: the two-photon decay of the metastable 2s state, which has a rate of 8.2 per second and is a forbidden transition, and Lyman-α photons that redshift out of resonance before they can be reabsorbed.

Both are slow, both depend on the expansion rate, and neither is an equilibrium process. So recombination proceeds more slowly than Saha predicts, finishes later, and freezes out with a residual ionised fraction of about 2×10−42\times10^{-4} rather than going to zero.

The last scattering surface has a thickness. The visibility function — the probability density that a microwave background photon arriving here scattered for the last time at redshift z. It is the product of two things pulling opposite ways: the rate of scattering, which collapses as recombination proceeds, and the chance of reaching here unscattered afterwards, which rises. The product peaks at z = 1284 with a full width at half maximum of Δz = 142, and that width is what makes the surface of last scattering a surface with a depth rather than an instant — about 21 comoving megaparsecs, so any feature smaller than that is averaged away along the line of sight, which is one of the two things that damp the acoustic peaks at high multipole. The curve uses the Saha ionisation fraction with a floor at the frozen-out residual of 2e-5, and the gap between what it gives and the right answer is itself the point. A full non-equilibrium treatment puts the peak at z = 1090 and the width at about 80; equilibrium chemistry puts it at 1284. Saha assumes every recombination is balanced by an ionisation, and near the end that stops being true — a hydrogen atom formed in the ground state emits a photon that immediately ionises its neighbour, so the only recombinations that stick are the slow ones, through the two-photon decay of the 2s state and the redshifting of Lyman-α out of resonance. Those are rates rather than equilibria, they cannot keep up, and the 18 per cent by which this curve is early is the measure of how far behind they fall.
Fig. 4 The visibility function computed with the residual ionisation ten times smaller. The peak barely moves and the tail does: a lower residual means fewer free electrons after recombination, so fewer late scatterings, so the function falls away more sharply on the low-redshift side. That asymmetry is where the residual is measured from, and it matters because the same free electrons are the ones reionisation later multiplies — the frozen-out fraction is the floor the second chapter of the ionisation history starts from.
The last scattering surface has a thickness. The visibility function — the probability density that a microwave background photon arriving here scattered for the last time at redshift z. It is the product of two things pulling opposite ways: the rate of scattering, which collapses as recombination proceeds, and the chance of reaching here unscattered afterwards, which rises. The product peaks at z = 1284 with a full width at half maximum of Δz = 142, and that width is what makes the surface of last scattering a surface with a depth rather than an instant — about 21 comoving megaparsecs, so any feature smaller than that is averaged away along the line of sight, which is one of the two things that damp the acoustic peaks at high multipole. The curve uses the Saha ionisation fraction with a floor at the frozen-out residual of 2e-4, and the gap between what it gives and the right answer is itself the point. A full non-equilibrium treatment puts the peak at z = 1090 and the width at about 80; equilibrium chemistry puts it at 1284. Saha assumes every recombination is balanced by an ionisation, and near the end that stops being true — a hydrogen atom formed in the ground state emits a photon that immediately ionises its neighbour, so the only recombinations that stick are the slow ones, through the two-photon decay of the 2s state and the redshifting of Lyman-α out of resonance. Those are rates rather than equilibria, they cannot keep up, and the 18 per cent by which this curve is early is the measure of how far behind they fall.
Fig. 5 The consequence, in the quantity that matters. The visibility function is the probability density that a photon arriving here scattered for the last time at redshift zz — the product of the scattering rate, which collapses as recombination proceeds, and the chance of getting here unscattered afterwards, which rises. Computed with the Saha ionisation fraction it peaks at z=1284z = 1284 with a width of 142; a full non-equilibrium treatment puts the peak at 1090 and the width at about 80. The 18 per cent by which the equilibrium curve is early is a measure of how far behind the slow channels fall.

The surface has a thickness, and the thickness is visible

The word “surface” is a convenience. The visibility function has a width of Δz≈80\Delta z \approx 80, which is about 20 comoving megaparsecs, so the microwave background is not a photograph of an instant but an average over a shell of that depth.

That has an observable consequence.

The last scattering surface has a thickness. The visibility function — the probability density that a microwave background photon arriving here scattered for the last time at redshift z. It is the product of two things pulling opposite ways: the rate of scattering, which collapses as recombination proceeds, and the chance of reaching here unscattered afterwards, which rises. The product peaks at z = 1284 with a full width at half maximum of Δz = 142, and that width is what makes the surface of last scattering a surface with a depth rather than an instant — about 21 comoving megaparsecs, so any feature smaller than that is averaged away along the line of sight, which is one of the two things that damp the acoustic peaks at high multipole. The curve uses the Saha ionisation fraction with a floor at the frozen-out residual of 2e-3, and the gap between what it gives and the right answer is itself the point. A full non-equilibrium treatment puts the peak at z = 1090 and the width at about 80; equilibrium chemistry puts it at 1284. Saha assumes every recombination is balanced by an ionisation, and near the end that stops being true — a hydrogen atom formed in the ground state emits a photon that immediately ionises its neighbour, so the only recombinations that stick are the slow ones, through the two-photon decay of the 2s state and the redshifting of Lyman-α out of resonance. Those are rates rather than equilibria, they cannot keep up, and the 18 per cent by which this curve is early is the measure of how far behind they fall.
Fig. 6 And at a residual ten times larger, which is what a universe with a slower expansion at that epoch would have left. The tail is heavier, the effective surface is thicker, and the damping of the small-scale structure is correspondingly stronger. That is the chain the essay’s later section describes running in reverse: the residual ionisation sets the tail of the visibility function, the tail sets the thickness, and the thickness sets how much of the small-scale power survives — so a measurement of the damping tail is a measurement of a freeze-out fraction from an atomic physics calculation.

The two effects are usually quoted together as “damping”, and separating them matters because they respond to different things: the diffusion length depends on the mean free path and therefore on the electron density, while the shell thickness depends on how fast the visibility function rises and falls. Fitting the tail constrains both.

What is being photographed

The material on that surface is not distant in any permanent sense. It is now some 46 billion light years away, having been about 42 million light years away when the light left, and it has spent the intervening time doing whatever it did — forming stars and galaxies, presumably, much as everything else has. An observer there sees a microwave background of the same temperature, centred on themselves, whose surface passes through here.

What was actually measured

The temperature of last scattering is not measured directly and cannot be. What is measured is the angular power spectrum, and z∗z_* comes out of the fit as a derived parameter — 1089.92±0.251089.92 \pm 0.25, which sounds like a measurement and is a consequence of the baryon and matter densities that the peaks constrain.

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.
Fig. 7 The same crossing computed for a gas seven tenths hydrogen by mass rather than three quarters. Fewer hydrogen atoms per baryon means slightly fewer targets for the ionising photons, and the crossing moves by tens of kelvin out of nearly six thousand. That insensitivity is worth drawing once because it bounds what the derived redshift of last scattering depends on: the helium fraction is known from nucleosynthesis to a per cent, and it enters this calculation weakly, so the 1089.92 quoted above inherits almost nothing from it.

There is one thing about the ionisation history that is measured directly, and it is at the other end. The universe did not stay neutral: the first stars and quasars re-ionised it, and free electrons after that scatter a small fraction of microwave background photons. That scattering polarises the largest angular scales, and measuring the polarisation gives the optical depth to reionisation, τ=0.054±0.007\tau = 0.054 \pm 0.007 — which says about 5 per cent of the photons were rescattered, and places reionisation around z≈8z \approx 8.

That measurement matters here because it is a partial erasure. A fraction of what the microwave background shows is not from z=1090z = 1090 at all, and the amplitude of the whole spectrum is suppressed by e−2τe^{-2\tau}, which is a 10 per cent effect that has to be separated from the primordial amplitude. It is the largest degeneracy in the parameter fit: a larger τ\tau and a larger initial amplitude produce almost the same spectrum, so the two are measured together or not at all, and the only thing that breaks them apart is the polarisation at the very largest angular scales — where there is only one sky and the sample is smallest.

Two things are worth taking from that. The first is that the transparency of the universe is not a single event but a history with two disconnected chapters, opaque then transparent then partly ionised again, and the second chapter is measured by an entirely different observable from the first. The second is that the quantity most often quoted as “the amplitude of primordial fluctuations” is, strictly, the combination Ase−2τA_{\rm s}e^{-2\tau}, and separating the two costs the largest error bar in the Planck parameter set.

The surface that came first

The microwave background is the oldest light, and it is not the oldest signal in principle. There is an earlier decoupling, of a different particle, and its surface sits at a redshift ten million times higher.

Neutrinos interact with matter only through the weak force, whose cross-section falls steeply as the temperature drops. In the first second of the universe, at temperatures around a few MeV, the interaction rate fell below the expansion rate and the neutrinos stopped scattering — precisely the same argument as recombination, applied to a different interaction at a much earlier time.

Those neutrinos are still here. Their temperature has fallen with the expansion to about 1.95 kelvin — slightly below the photons’ 2.725, because the photons were later reheated when electrons and positrons annihilated and the neutrinos were not — and their number density is about 336 per cubic centimetre, comparable to the photons’.

The cosmic neutrino background has never been detected directly, and detecting it is among the hardest problems anybody has proposed. The particles carry a fraction of a milli-electronvolt of energy apiece, far below the threshold of every neutrino detector ever built, and the only serious proposal is capture on a radioactive nucleus already at its decay threshold, where a captured neutrino produces an electron slightly above the endpoint of the ordinary decay spectrum.

Its existence is nevertheless established indirectly and rather well. The neutrinos contribute to the energy density during the radiation era, which changes the expansion rate, which changes both the primordial abundances and the shape of the acoustic peaks. Both measurements return an effective number of neutrino species close to three — which is a detection of the background’s gravitational effect, if not of a single particle from it.

The thickness, and what it does to the small scales

The surface’s finite thickness is not merely a caveat on the word “surface”; it is a measurable feature of the map.

While the photons are still scattering, they random-walk through the plasma, and a random walk carries them a distance far greater than the mean free path. Any temperature or density variation on a scale smaller than that diffusion length is smoothed out, because photons from the hot and cold regions mix before decoupling.

The effect is a progressive damping of the small-scale structure, and it appears in the power spectrum as an envelope suppressing the higher acoustic peaks — visible from about the third peak onward, and unmistakable by the seventh.

That damping tail is not a nuisance. Its shape depends on how thick the surface is and on how fast the universe was expanding while the photons were diffusing, so measuring it constrains the expansion rate at that epoch and therefore the energy density in relativistic species — which is one of the two routes to the neutrino count mentioned above. The blurring of the picture is itself one of the picture’s most informative features, which is an unusual thing to be able to say about a limit on resolution.

What a photon actually did on its way here

It helps to state the journey plainly, because the phrase “last scattering” invites a picture that is wrong in one respect.

A photon in the plasma before recombination scattered off free electrons every few years of cosmic time — a mean free path far shorter than the horizon, so it moved diffusively rather than in a straight line. At recombination the electrons were captured into atoms, the free-electron density collapsed by orders of magnitude, and the mean free path exceeded the size of the observable universe. From that moment the photon travelled in a straight line.

The wrong picture is of a wall. There is no physical surface at that redshift and there never was: the plasma filled space, and the “surface” is simply the set of points from which a photon could reach here in the time available. It is defined by the observer, not by the universe — an observer somewhere else sees a different sphere of the same radius around themselves, and the two overlap.

That has a consequence for what the map can show. Two points on opposite sides of the sky are separated by twice the distance to the surface, which is far more than either could have communicated across before recombination — so the fact that their temperatures agree to a part in a hundred thousand is a statement about initial conditions rather than about equilibration. The horizon problem is a statement about this geometry, and inflation is a proposal about how the initial conditions came to be what they are.

The surface is a horizon rather than an object, which is why it recedes as time passes: it is at a fixed redshift, and a later observer sees a larger sphere and a slightly different pattern on it, changing by a fraction of a per cent over millions of years.

What the pictures cannot show

The visibility figure uses an approximation and says so. The Saha ionisation fraction with a floor at the frozen-out residual is the one place in this collection’s cosmology figures where a full treatment is replaced by a stated shortcut, and the resulting peak is 18 per cent early. The alternative would have been to integrate the Peebles three-level atom, which is a real calculation and could not have been checked against anything here — and a well-formed picture of an unverifiable calculation is worse than an honest approximation with its error stated.

None of these figures shows recombination happening in helium, which occurs earlier — at z≈6000z \approx 6000 for the second electron and z≈2000z \approx 2000 for the first — and contributes about 8 per cent of the electrons. It is included in the parameter fits and omitted here.

And the hero figure’s crossing at 5,844 K is one of three defensible criteria, which is worth flagging because the essay’s argument does not depend on which is chosen. One ionising photon per baryon gives 5,844 K; Saha half-ionisation gives 3,735; the actual visibility peak gives 2,970. All three are a factor of 27 to 53 below 157,800, and the spread between them is itself informative — it is the difference between “the supply of ionising photons has run out”, “equilibrium has shifted” and “the reactions have stopped keeping up”, which are three different statements about the same transition.

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 = 2337, 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.
Fig. 8 The same count in a universe whose background is at 2.5 kelvin today rather than 2.7255. Every temperature on the curve scales with the present one, because the whole calculation is a Planck distribution at T0(1+z)T_0(1+z) against a fixed ratio — so the crossing moves and the redshift at which it happens does not. That is the invariance the essay’s argument actually rests on: the factor of fifty is a logarithm of the photon-to-baryon ratio, and the present temperature enters only as a conversion between redshift and kelvin.

How it was worked out

The Saha equation is from 1920 and was written for stellar atmospheres, where it works: the disappearance of hydrogen lines in hot stars is Saha’s result, and it is what let Cecilia Payne conclude in 1925 that stars are mostly hydrogen. Applying it to the early universe is a change of setting and not of physics — and the change of setting is precisely the photon-to-baryon ratio, which in a stellar atmosphere is of order unity and here is two billion, which is why the same equation displaces the transition by a factor of fifty in one case and not in the other.

The non-equilibrium calculation is Peebles’s, and Zel’dovich, Kurt and Sunyaev’s, both in 1968 — three years after the background was discovered and long before the anisotropies were. Both papers identified the two-photon decay and the Lyman-α escape as the rate-limiting channels, and both got the freeze-out ionisation fraction right to within a factor of two.

Their result mattered thirty years later for a reason neither paper anticipated. The width and position of the visibility function set the damping scale, and the damping scale had to be predicted to sub-per-cent accuracy before the microwave background could be used for precision cosmology. That drove a substantial effort through the 2000s to compute recombination including hundreds of atomic levels, and the current codes agree to a few parts in ten thousand. A 1968 calculation of a chemical equilibrium became, by 2010, a limiting systematic in a satellite mission.

The generalisation

The lesson is one this collection meets whenever a transition is set by the tail of a distribution rather than by its typical member: when the number of particles is enormous, a process the average particle cannot drive can still be driven by the exponentially rare ones, and the threshold is displaced by a logarithm of the number.

Fusion in a stellar core is the same argument run in the other direction: the Sun’s centre is at 15 million kelvin and the Coulomb barrier is at billions, and fusion proceeds anyway because there are 105610^{56} protons and the tail of the Maxwell distribution is not empty. The main sequence lifetime depends on this exponentially, which is why a small change in mass makes an enormous change in lifetime.

The two cases are worth holding together because they run opposite ways. In a star the tail enables a process the typical particle cannot manage; here the tail delays a transition the typical photon has long since stopped being able to prevent. In both, the quantity that decides the displacement is a logarithm of the particle number, and in both, the naive estimate from the typical energy is wrong by a large factor in a predictable direction.

One more reading covers baryon-to-photon ratios well above the measured one.

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 = 1451, which is 4212 K against 3959 K. Both of those are a factor of 37 to 40 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.
Fig. 9 Recombination for baryon densities ten times above and at the measured value. A denser universe recombines earlier, because recombination is a balance between the density of protons and the density of ionising photons — and the measured ratio is what puts the surface at redshift 1090 rather than anywhere else.

The surface is a probability distribution rather than a surface, and its width is what limits how sharply anything on it can be seen — which is the physical origin of the damping tail and of the smallest scale the microwave background contains.

Where the ladder goes next

This is the last rung of the microwave background anchor written in this phase and the last essay of the field. What it leaves is a surface that has been treated as a boundary throughout and is not one: there is a universe behind it, opaque to photons and not to everything.

Later rungs on this anchor: the Peebles equation in full, and the atomic physics the precision codes had to add; helium recombination; reionisation as an epoch rather than an optical depth, and the twenty-one centimetre observations that will map it; the cosmic neutrino background, which decoupled at one second and would be a surface of last scattering at z=6×109z = 6\times10^9 if anyone could detect it; and what can in principle be learned about the epoch before recombination from gravitational waves, which decoupled earlier still.

Where this goes next

A blackbody at 2.7255 kelvin, filling the sky. The Planck function at 2.7255 K in the units the measurement is reported in, with the peak marked where Wien's law in frequency puts it: x = hν/kT = 2.8214, so ν = 160.2 GHz and the intensity there is 384 MJy per steradian. The points are drawn at the twenty-one frequencies across the FIRAS band, displaced from the curve by a Gaussian of 50 parts per million of the peak, which is the root-mean-square deviation the instrument actually reported. At the scale of this plot that displacement is a fifth of a pixel and the points sit on the line — which is the entire finding. Nothing else in astronomy is a blackbody to a part in twenty thousand: a stellar spectrum is a blackbody with absorption lines cut into it and a continuum that is the wrong shape at both ends. A thermal spectrum this exact requires that the radiation was once in equilibrium with matter, which requires that the universe was once opaque, which requires that it was once hot and dense. The most perfect blackbody ever measured The whole sky glows at 2.7255 kelvin, and its spectrum matches the Planck curve to fifty parts in a million. Nothing else in astronomy is thermal to a part in twenty thousand, and a spectrum that exact is not a description of the radiation — it is a constraint on the history of everything that could have disturbed it. The acoustic peaks, and where the geometry says they should be. The temperature angular power spectrum of the microwave background. The drawn curve is a monotone interpolation through the published positions and heights of the six peaks and five troughs of the Planck 2018 TT measurement — it is a representation of data, and nothing between two extrema is claimed. The marks along the top are not: they are computed from this collection's own cosmology as ℓₐ(m − 0.267), where ℓₐ = π × 13866 / 144.43 = 301.6 — π times the comoving distance to last scattering divided by the sound horizon there — is the angle the sound horizon subtends at last scattering turned into a multipole. The two agree to 2.8 per cent at worst across six peaks, which is the whole of what makes this a measurement of geometry: a wave of known physical wavelength, seen at a known distance, is a protractor. The first peak at ℓ = 220 corresponds to about 0.82 degrees on the sky — roughly twice the width of the full Moon, which is the largest hot and cold patch the sky has. A standing wave frozen at one instant The temperature of the microwave background varies across the sky by one part in a hundred thousand, and the sizes of the patches are not random. There is a preferred angular scale near one degree, and it is a sound wave that stopped ringing four hundred thousand years after the beginning. 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.

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Freeze outLast scattering surfaceOptical depthPhoton to baryon ratioRecombinationReionisationSaha equationSilk dampingThomson scatteringVisibility function