A blur that measures a depth

The surface of last scattering has a thickness: recombination is not an instant but a transition spread over a range of redshift, and the photons that arrive today left over a shell of finite depth rather than from a wall.

That essay established the thickness as a fact about the physics — hydrogen’s ionisation is held open by a tail of energetic photons long after the typical one has become useless, and the transition takes a while. This one is about where the thickness is measured, which is a different question, and the answer is the part of the power spectrum that nobody was looking at when the peaks were discovered.

Why a photon walks #

Before recombination the universe is an ionised plasma and a photon scatters off free electrons every few thousand years of conformal time — a mean free path short compared with any scale of interest, so the photons and the baryons behave as a single fluid. That fluid supports sound waves, and the peaks in the power spectrum are their standing pattern frozen at one instant.

The picture of a single fluid is an idealisation, and the idealisation fails at small scales. A photon does not stay where it is: between one scattering and the next it moves a mean free path in a random direction, and over $N$ scatterings it wanders $\sqrt{N}\lambda$ from where it started. That distance is much larger than a step and much smaller than the total path travelled.

While it wanders it carries its energy with it, and it deposits that energy wherever it ends up. So a hot region smaller than the diffusion length loses its heat to the cold region next door before the pattern can be imprinted, and a cold region smaller than the diffusion length is filled in. Structure below that scale is erased.

The scale grows through the transition, because the mean free path lengthens as the free electrons disappear, and it grows fastest at the very end. Most of the damping happens during recombination itself. It is worth putting the numbers on the walk, because they are less extreme than “random walk” suggests. At recombination the mean free path is of order a tenth of a megaparsec comoving, and the number of scatterings a photon undergoes during the transition is of order ten thousand — so the diffusion length is about a hundred times a step, or ten megaparsecs comoving. Compare that with the sound horizon, some hundred and fifty megaparsecs comoving. The damping scale is about a fifteenth of the acoustic scale, which is why the tail begins to bite around the fourth or fifth peak rather than at the first.

That ratio is not a coincidence of the parameters. It is roughly the square root of the number of scatterings’ inverse — a photon travelling at nearly $c$ for the same interval that sound travels at $c/\sqrt{3}$, but doing it as a walk rather than in a line. The difference between a walk and a line is the whole of it.

From a length to an angle #

The damping length is a comoving distance at last scattering. What is observed is an angular scale on the sky, and converting between them requires the distance to the surface — which is the same conversion the acoustic peaks require, and the same one that makes the peaks a measurement of geometry.

That is the awkwardness and the opportunity. The damping scale and the sound horizon are both lengths at the same epoch, seen through the same geometry, so their ratio is an angle-free quantity: it depends on the physics of recombination and not on the cosmology between here and there.

The ratio is therefore a much cleaner observable than either alone. The sound horizon goes as the integral of the sound speed over time, so as the square root of a time; the damping length goes as the square root of a mean free path times a time, so as the square root of a time times the square root of a length. Their ratio scales as the square root of the mean free path over the horizon — which is to say, as one over the square root of the number of scatterings, which is a pure statement about how ionised the plasma was.

There is a useful way to see the same point in reverse. The peaks and the damping are both features written on one surface by one fluid, and if the physics of recombination were different in a way that changed the number of scatterings, both would move — but not by the same factor, because one goes as a time and the other as the square root of a time times a length. So the ratio of the damping angle to the acoustic angle is the observable that isolates the recombination physics, and it is the quantity that the parameter constraints below actually use. Anything that changed the geometry between here and there would move both angles together and leave the ratio alone.

Why the cut-off is Gaussian #

Diffusion is a random walk, and a random walk’s displacement distribution is Gaussian. Convolving a temperature pattern with a Gaussian in real space multiplies its Fourier transform by a Gaussian in wavenumber, so the suppression is $\exp(-k^{2}/k_D^{2})$ — and in the angular power spectrum, $\exp(-\ell^{2}/\ell_D^{2})$.

The shape matters as much as the scale. An exponential cut-off, or a power law, would say something different about the mechanism. What is observed is Gaussian, which is a direct statement that the erasure is diffusive rather than, say, a consequence of the primordial spectrum running out of power.

Distinguishing the two is not academic. A primordial spectrum with a strongly running spectral index would also suppress small scales, and the two effects are separated precisely because one is a Gaussian in $\ell$ and the other is a power law. Fitting the tail with both free is how the running is constrained, and the constraint is currently consistent with none.

The Gaussian is also a claim that could have failed, and it is worth saying what failure would have looked like. A walk gives a Gaussian only when the steps are independent, identically distributed, and numerous; had the photons decoupled abruptly, or had the mean free path jumped rather than lengthened smoothly, the displacement distribution would have kept the shape of its last few steps and the suppression would have carried a tail heavier than Gaussian at high multipole. The envelope’s shape is therefore a measurement of the smoothness of the transition, made without reference to any model of it.

What the data show is a Gaussian very slightly shallower than the pure form, and that departure is not a failure of the picture but the expected consequence of integrating over a shell of finite depth. Photons that last scattered at the near edge of the shell had longer to walk than those from the far edge, so the observed envelope is a weighted sum of Gaussians of different widths rather than any one of them, and a sum of Gaussians is broader in the wings than its broadest term. Reading a thickness off the tail means fitting that sum, which is why the single number quoted for the damping multipole depends slightly on the convention used to define it.

Why the tail is the sharpest probe of some things #

Anything that changes the free-electron density during recombination changes the diffusion length and moves $\ell_D$, while leaving the peak positions nearly alone. That makes the damping tail the sensitive observable for a specific list of questions.

The number of relativistic species. Extra light species change the expansion rate at that epoch, which changes how long recombination takes and therefore how far a photon walks. The damping tail is where the constraint on the effective number of neutrino species mostly comes from.

The primordial helium abundance. Helium recombines earlier than hydrogen and locks up its electrons, so a higher helium fraction means fewer free electrons per baryon at hydrogen recombination, a longer mean free path, and more damping. The microwave background’s own measurement of the helium abundance — independent of any stellar or nebular spectroscopy, and of the four abundances that nucleosynthesis predicts from one parameter — comes from the tail.

A varying fine-structure constant. Hydrogen’s ionisation energy depends on it, so a different value would move recombination in redshift and change the tail’s position relative to the peaks.

Early energy injection. Anything that ionises the gas early — a decaying particle, an evaporating black hole — extends the transition and damps more. The late counterpart, the reionisation of the universe by the first stars, acts differently: it rescatters a few per cent of the photons long afterwards and suppresses the whole spectrum uniformly rather than only its small scales.

Two effects that damp alike #

The list above hides a difficulty. Extra relativistic species and a higher helium fraction both push the damping to larger angular scales, and to first order they do it the same way — by lengthening the mean free path, or by lengthening the time available to walk it. The envelope alone cannot say which.

They separate on what else they touch. Extra species change the expansion rate at that epoch, so they change the sound horizon as well as the diffusion length, and the whole comb of peaks moves with the envelope. Helium changes only the free-electron density, which the sound speed barely notices, so it moves the envelope and leaves the peaks where they were. The two are told apart by whether the peak positions came along.

That is why the constraint on the effective number of species is quoted from a fit to the entire spectrum rather than from the tail alone, and why it tightened when polarisation was added rather than when the tail was pushed to higher multipole. The polarised peaks are sharper than the temperature’s — they are sourced by the fluid’s velocity rather than its density, so they are not filled in by the driving that softens the temperature peaks — and a sharper comb locates the acoustic scale better, which is the quantity the damping scale has to be measured against.

There is a third handle, and it is nearly independent of both. Neutrinos free-stream faster than the fluid’s sound speed, and their gravity pulls on the oscillations slightly ahead of where the fluid itself would put them, displacing every peak by a few multipoles in a direction no change to the ionisation history reproduces. That shift has been measured. Three routes into one number, each sensitive to a different piece of the same epoch, is what makes an answer worth quoting to a fraction of a species.

What was actually measured #

The damping tail was not measured by the experiments that found the peaks. Resolving it requires arcminute resolution and sensitivity at high multipole, which means either a large aperture or a long integration, and preferably both.

Ground-based telescopes at very dry sites did it first — the atmosphere is opaque at these frequencies almost everywhere else — and their measurements now extend to multipoles of several thousand, well past the point where the primary anisotropy has been damped away entirely and the signal is dominated by later effects: gravitational lensing of the background by intervening structure, and the Sunyaev–Zel’dovich distortion from hot gas in clusters.

That contamination is itself the reason the tail is hard. The primary signal falls as a Gaussian and the secondary signals do not, so past $\ell \approx 3000$ the sky is no longer showing the last-scattering surface at all. Separating the components requires observing at several frequencies and using their different spectra, which is why every modern instrument is multi-band.

What was measured before it, and why the order matters #

There is a historical detail worth recording. The damping was predicted in 1968, thirty years before the first peak was resolved and forty before the tail was. It was not a prediction made in the expectation of a measurement: at the time nobody had detected any anisotropy at all, and the calculation was a piece of theory about a plasma.

When the anisotropy was finally found, the observations climbed in multipole from the largest scales downward — degrees first, then the first peak, then the second and third. The tail was the last part of the primary spectrum to be measured, and by the time it was, the parameters had already been determined well enough that the damping scale was a prediction rather than a fit.

It came out where it was supposed to. That is worth more than a new constraint would have been, because the prediction involved a chain — an ionisation history, a scattering cross-section, an expansion rate, a geometry — every link of which had been fixed elsewhere.

What the measurement is worth #

Two things, and the second is the more interesting.

The first is the constraint list above: the tail is where several parameters that are nearly degenerate elsewhere become separable, and the modern parameter constraints from the microwave background lean heavily on it.

The second is what it is a measurement of. Nearly everything else about the background is a statement about a two-dimensional surface: its temperature, its fluctuations, its geometry, the ruler written across it. The damping tail is the one feature that measures a third dimension — the depth of the shell, and therefore the duration of the event. It is the difference between a photograph and a long exposure, and reading the exposure time off the blur.

That the exposure time comes out at about eighty in redshift, and that this is what the physics of hydrogen at three thousand kelvin in a bath of two billion photons per atom predicts, is one of the quieter successes of the subject. Nobody adjusted anything to make it fit.

One more thing is worth saying about what the tail is for, because it is easy to read this essay as an account of a nuisance. The damping is not a loss of information. The unsuppressed spectrum would carry the same acoustic physics out to arbitrarily small scales and would carry it redundantly; what the tail carries instead is a second, independent quantity — the duration of recombination — written on top of the first. A measurement that ends where the damping begins has one number from the surface. A measurement that goes through it has two, and the second one is about the surface’s depth rather than its position.

What the picture leaves out #

The hero figure draws a pure Gaussian envelope and normalises it to the observed damping multipole, and the real calculation is more involved in three ways.

The damping is not a single scale. The diffusion length grows through recombination, so the suppression is an integral over the visibility function rather than a single Gaussian, and the resulting envelope is slightly shallower than $\exp(-\ell^{2})$ at large $\ell$. Fitting a pure Gaussian to real data gives a damping multipole biased by a few per cent.

Polarisation damps differently. The polarisation of the background is generated only by scattering during the transition itself, so the E-mode power spectrum’s damping envelope is not the same as the temperature’s — and the difference is another handle on the visibility function’s shape.

And the tail is lensed. Structure between here and there deflects the photons by a few arcminutes, which smooths the peaks and adds power at high multipole. That effect is a signal in its own right — it maps the intervening mass, in the same way a shear field weighs a cluster — and it has to be modelled before the damping can be read.

One more reading spans a wider range of the quantity the damping tail measures.

Where the ladder goes #

The earlier rungs of this anchor were about the background as a blackbody, as a frozen standing wave, and as a surface with a thickness. This one measures that thickness.

The next rungs go to the polarisation, where the same transition is seen through a different observable that exists only because the shell is thick — a photon scattering in a perfectly isotropic bath produces no polarisation, so every polarised photon is a record of a quadrupole in the local radiation field at the moment of last scattering, and the quadrupole exists only because the photons had begun to free-stream. Polarisation is, in that sense, an even purer measurement of the depth than the damping is: it does not merely depend on the shell’s thickness, it is caused by it.