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.

Assumes Reionisation and Microwave background.

15 min read 7 figures Measuring the unreachable Light is the only messenger

The microwave background’s anisotropies were imprinted at recombination and have been travelling ever since. Most of them arrive unimpeded, and some do not: once the universe reionised, free electrons were available again, and a fraction of the photons Thomson-scattered off them.

A scattered photon comes from a random direction, so scattering washes out anisotropy. The surviving fluctuations are suppressed by a factor $e^{-\tau}$ in amplitude, uniformly across every angular scale small enough to have been inside the horizon at reionisation — which is nearly all of them.

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. — Fig. 1 The consequence. The temperature power spectrum measures the primordial amplitude multiplied by $e^{-2\tau}$ , so a smaller amplitude with less scattering and a larger one with more produce the same spectrum. The locus is drawn; the constraint from the temperature alone is a curve rather than a point, and it spans a substantial range of amplitude.

The anisotropies are a standing wave frozen at one instant, and the amplitude they were frozen at is one of the handful of numbers that describe the whole universe. This essay is about the fact that the temperature map cannot measure it.

Why the damping is uniform

The suppression’s uniformity is what makes it a degeneracy rather than a measurable feature.

A photon scattered at reionisation comes from a direction uncorrelated with where it was heading, so it carries no memory of the anisotropy it had. The fraction scattered is $1 - e^{-\tau}$ , independent of angular scale, so every multipole is suppressed by the same factor in power.

The exception is the largest scales — those still outside the horizon at reionisation, which is to say multipoles below about ten. Those are not suppressed, because the anisotropy on such scales was not yet in causal contact and the scattering does not average anything away. So there is, in principle, a scale-dependent signature: a step in the spectrum between the unsuppressed largest scales and the suppressed rest.

In practice that step is unmeasurable. The largest scales are exactly where cosmic variance is largest — there are only a handful of independent modes on the sky at multipole two or three — so the uncertainty on their amplitude is tens of per cent, which is far larger than the step.

It is worth being explicit about the size of the effect, since a few per cent scattering sounds negligible. An optical depth of 0.054 means about five per cent of the photons scattered, which suppresses the power by $e^{-2\tau}\approx0.90$ — ten per cent in power, five in amplitude. That is small compared with the anisotropies and enormous compared with the precision they are measured to, which is a fraction of a per cent. So the degeneracy is severe not because the effect is large but because the measurement is good: a ten per cent suppression is a twenty-sigma effect in a spectrum measured to half a per cent, and it is completely degenerate with the amplitude.

There is a further wrinkle in the uniformity that is worth stating because it is sometimes offered as a way out. The suppression applies to the primary anisotropies — the ones imprinted at recombination. Anything generated after reionisation is not suppressed: the lensing signal, the Sunyaev–Zel’dovich effect from clusters, the integrated Sachs–Wolfe signal from evolving potentials. So the ratio of a primary to a secondary signal does carry the optical depth. Those secondaries are individually small and individually systematic-limited, and using their ratio is a real but weak route to the same parameter.

What breaks it

The route that works is polarisation, and it works for a physical reason worth following.

Thomson scattering produces polarisation when the radiation field at the scattering electron has a quadrupole. At reionisation the free electrons sat in a radiation field whose quadrupole was set by the anisotropies at the horizon scale of that epoch, so the scattering regenerated polarisation — a bump in the polarisation power spectrum at multipoles of a few tens down to a few.

The bump’s amplitude is proportional to $\tau$ itself, not to $e^{-2\tau}$ . So the temperature spectrum measures one combination and the polarisation bump measures the optical depth nearly alone, and the two together separate the parameters.

There is a second, weaker route that deserves mention because it is independent: gravitational lensing of the microwave background itself. The anisotropies are lensed by the structure they pass through, and the lensing amplitude depends on the matter fluctuations today — which depend on the primordial amplitude without the reionisation suppression, because the lensing happened after the scattering. So the lensing signal measures the amplitude directly and, combined with the temperature spectrum, gives the optical depth. It is a much weaker constraint than the polarisation bump and it has entirely different systematics, which is exactly what makes it worth having.

Why the polarisation measurement is so hard

Three difficulties stack, and all three are worst at exactly the scales the signal lives on.

The signal is tiny. The reionisation bump has an amplitude of a few hundredths of a microkelvin in polarisation, against temperature anisotropies of tens of microkelvin and a dipole of thousands.

The Galaxy is polarised. Synchrotron emission from cosmic-ray electrons in the Galactic magnetic field, and thermal emission from aligned dust grains, are both strongly polarised and both are larger than the signal over most of the sky at any single frequency. Separating them requires observing at many frequencies and exploiting their different spectra — which works, and whose residual is the limiting systematic.

And there are few modes. At multipole ten there are twenty-one independent modes on the whole sky. Even a perfect measurement of them carries a cosmic-variance uncertainty of about twenty per cent, so the optical depth cannot be measured to better than about that from this signal alone, ever.

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. — Fig. 3 The spectrum whose overall amplitude carries the degenerate combination: the acoustic peaks, measured to a fraction of a per cent over two decades in angular scale. Every feature of it — the peak positions, their relative heights, the damping tail — is measured superbly and constrains the cosmology tightly. What none of it constrains is the overall normalisation independently of the scattering, because the scattering multiplies the whole curve.

Put the difficulty in one comparison. The temperature spectrum’s peaks are measured over multipoles from two to three thousand, which is of order ten million independent modes, and the resulting parameters are known to fractions of a per cent. The reionisation bump lives at multipoles from two to thirty, which is about a thousand modes in polarisation. Six orders of magnitude fewer modes, at an amplitude four orders of magnitude smaller, against a foreground that is brighter. That ratio is why one parameter of the standard model is known ten to a hundred times less well than the others.

What the optical depth is worth

Two things depend on it, and neither is about reionisation.

The neutrino mass. Massive neutrinos suppress the growth of structure on small scales, so measuring the amplitude of structure today and comparing it against the amplitude at recombination constrains the mass. The amplitude at recombination is the primordial amplitude — which is degenerate with $\tau$ — so the neutrino mass bound is directly limited by how well the optical depth is known. Halving the uncertainty on $\tau$ tightens the neutrino bound substantially.

The tensor-to-scalar ratio. Any primordial gravitational-wave signal is quoted relative to the scalar amplitude, so the same degeneracy enters that measurement too.

And one thing depends on it that is about reionisation: the optical depth is an integral of the free-electron density over time, so it constrains when reionisation happened. It does so weakly — it is one number and reionisation is a history — but it is the only constraint that integrates over the whole epoch rather than sampling its end.

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. 4 The other epoch of scattering, for comparison: the visibility function at recombination, which is the surface the anisotropies were imprinted on. Reionisation adds a second, much thinner scattering surface at low redshift, and the optical depth in this essay is its thickness. The two surfaces are separated by thirteen billion years and by an enormous factor in optical depth — one is where the photons last scattered and the other is where a few per cent of them scattered again.

What was actually measured

There are three, and the history is instructive.

The first measurement was too high. An early determination gave an optical depth of about 0.17, which implied reionisation at redshift seventeen — much earlier than any other evidence supported. It was later shown to be contaminated by imperfectly removed Galactic foregrounds.

The revisions all went downward. Successive analyses with better foreground data brought it to 0.09, then 0.07, then 0.054. Each revision was a better treatment of the Galaxy rather than a new measurement of the sky, and each moved the inferred reionisation redshift later.

And the current value agrees with the forest. An optical depth of 0.054 corresponds to reionisation ending around redshift eight, which is consistent with what quasar absorption spectra indicate independently. That agreement between two completely different observations is the main reason the current value is believed, and it was not available when the first measurement was made.

The damping tail measures a thickness: ℓ_D = 1400 for a shell 80 deep in redshift. The suppression of small-scale structure in the microwave background, against multipole, on logarithmic axes. Every other feature of the power spectrum measures the last-scattering surface as a surface — its distance, the sound horizon written on it, the ruler it provides. This one measures how thick it is. Recombination takes time: the ionised fraction falls over a range of redshift rather than all at once, and while it is falling the photons are still scattering, so each one random-walks. A photon taking N steps of a mean free path λ diffuses √N λ, which is much further than a single step and much less than the whole interval, and any temperature fluctuation smaller than that distance is mixed away before it can be frozen in. What is left is a Gaussian cut-off, drawn here for three shell thicknesses. A thicker shell means more steps and a longer walk, so it damps at a smaller multipole: Δz = 40 gives ℓ_D = 1980, Δz = 80 gives ℓ_D = 1400, Δz = 160 gives ℓ_D = 990. The observed cut-off is near ℓ = 1400, corresponding to a diffusion length of about 0.03 proper megaparsecs at the time — a scale the reader should compare with the sound horizon, some hundred and fifty comoving megaparsecs, which is what the peaks measure. The tail is therefore a genuine probe of the inside of the transition rather than of its position, and because the damping depends on the free-electron density it is also one of the cleanest constraints on anything that changes it: extra relativistic species, a varying fine-structure constant, or an early energy injection all move ℓ_D while leaving the peak positions nearly alone. The normalisation here is set once by the observed value, so what the figure asserts is the scaling with thickness and the shape of the cut-off, not the absolute number. — Fig. 6 The part of the spectrum where a scale-dependent suppression would show, for contrast: the damping tail, where photon diffusion at recombination erases small-scale structure. That suppression is scale-dependent and therefore measurable and separable; the reionisation suppression is not. The distinction between a scale-dependent damping, which is information, and a scale-independent one, which is a degeneracy, is the whole of what this essay is about.

Baryons make the odd peaks taller. Three baryon densities, and what each does to the alternation between odd and even acoustic peaks. Baryons add inertia to the photon–baryon fluid without adding pressure, so it falls further into a gravitational well than it rebounds out of one: compressions are deeper than rarefactions, the odd-numbered peaks are compressions, and the ratio of the first peak to the second scales as (1 + 6R) with R = 3ρ_b/4ρ_γ at last scattering. At the measured Ω_b h² = 0.02237 that gives R = 0.622, and the drawn first-to-second ratios run 1.59, 2.26, 3.03 across Ω_b h² = 0.014, 0.02237, 0.032. Only that ratio is derived: the absolute heights come from the Planck measurement and the geometric mean of each adjacent pair is held fixed, because the envelope is set by radiation driving and photon diffusion, which no figure of this kind computes. What the alternation buys is a weighing of the ordinary matter in the universe from the shape of a sky map — a number that agrees, to better than a per cent, with the one deuterium gives. — Fig. 7 For contrast, a parameter the temperature spectrum measures superbly: the baryon density, which changes the *relative* heights of the odd and even acoustic peaks and is therefore constrained by a shape rather than by an amplitude. That is the general rule — the microwave background measures ratios and positions extraordinarily well and overall normalisations poorly, because a normalisation is degenerate with anything that scales the whole spectrum and a shape is not.

And there is a fourth result that is really a caution about the first three. Every value quoted above was obtained by fitting the whole six-parameter model to the whole data set, so the optical depth’s value depends on the other parameters and on the model being right. Fitting an extended model — one with a varying dark-energy equation of state, or extra relativistic species — shifts the inferred optical depth, sometimes by more than its own uncertainty. So the number is a model-dependent quantity in a way the temperature spectrum’s peak positions are not, and comparisons between analyses have to be made within the same model. That is a general feature of a parameter constrained by a small amount of data inside a large fit: it is the one that moves when anything else changes. The same caution applies to any parameter recovered from a global solution rather than measured directly.

Where the picture stops

Three, and the second is what will improve.

The cosmic-variance floor is absolute. No experiment can measure the reionisation bump better than about ten per cent, because there are not enough modes. That translates into a floor on the optical depth’s precision and therefore on the neutrino-mass bound from this route.

The foregrounds are the current limit and are not at the floor. Present measurements are limited by foreground residuals rather than by cosmic variance, so better frequency coverage and better Galactic models still buy precision. That is the argument for the next generation of large-scale polarisation experiments.

And the optical depth is one number describing a history. Two reionisation histories with the same integrated electron column give the same $\tau$ and different bump shapes, in principle — and the shape is unmeasurable for the same cosmic-variance reason. So the constraint on the history is one number, and everything else about when and how reionisation happened has to come from elsewhere.

A fourth belongs with them, and it is about what “the amplitude” means. The primordial amplitude is quoted at a particular scale — a pivot — and the spectrum has a tilt, so the amplitude and the tilt trade against each other depending on where the pivot is placed. Choosing the pivot near the middle of the measured range makes the two nearly independent, which is why the convention exists, and it means a quoted amplitude is only comparable between analyses that used the same pivot. That is a smaller version of the same lesson: when a measurement constrains a combination, the combination has to be stated, and a number quoted without its convention is not comparable with anything.

There is a fifth limit that belongs with those and it is about what a single number can describe. The optical depth is $\int n_e \sigma_T c\, dt$ — an integral of the free-electron density over time — so a universe that reionised early and slowly and one that reionised late and quickly can have the same value. Distinguishing them needs the shape of the polarisation bump, whose peak multipole depends on the horizon size at the epoch of scattering, and that shape is buried under cosmic variance at exactly the multipoles where it lives. So the microwave background constrains one moment of the history and nothing about its form, which is a much weaker statement than “the universe reionised at redshift eight” makes it sound.

Why an exponential degeneracy is worse than a linear one

The general point deserves separating from the instance.

A degeneracy in which one parameter enters exponentially is unusually stubborn, because a small change in it corresponds to a large change in the other, and the constraint’s shape in the plane is a curve rather than a line. The optical depth ranges over a few hundredths and the amplitude it can absorb ranges over tens of per cent, so a poorly measured $\tau$ is not a small correction to the amplitude but a substantial one.

That structure recurs wherever an absorption or a suppression is being fitted alongside a source amplitude. A continuum error in the Lyman-alpha forest has the same form — an optical depth is the logarithm of a ratio, so an error in the denominator is an error in the exponent. A transmission spectrum’s reference radius is another. In each case the observable is a product of an amplitude and an exponential, and the two cannot be separated without an observable that depends on the exponent differently.

The escape is always the same and it is worth stating as a rule: find a quantity proportional to the optical depth rather than to its exponential. Here it is the polarisation bump; in the forest it is the damping wing; in the transmission spectrum it is the resolved line core. Each is a much harder measurement than the one it rescues, and each is the only thing that works.

End on how much rests on one poorly measured number. The optical depth is the least well determined of the six parameters of the standard cosmological model — known to about ten per cent, where the others are known to fractions of a per cent — and it limits the neutrino-mass bound, the tensor-to-scalar constraint, and every comparison between the amplitude of structure at recombination and today. The disagreement between lensing surveys and the microwave background about how much structure there is is a comparison of exactly that kind, and part of its uncertainty is this parameter. One number, measured from a few hundred modes on the sky, at an amplitude below the Galaxy’s own emission, propagates into several of the field’s most-discussed results.

One remaining observation about what would settle it. The cosmic-variance floor on the optical depth from the reionisation bump is about ten per cent, and current measurements are close to it — so this route is nearly exhausted. What is not exhausted is the alternative: a direct measurement of the reionisation history from the twenty-one centimetre line of the neutral gas itself, which would give the electron column as a function of redshift rather than as one integral. That measurement is being attempted and is extraordinarily hard for its own reasons — the Galactic foreground at those frequencies is four orders of magnitude brighter than the signal — and if it succeeds it supplies the optical depth as a by-product, with no cosmic-variance limit at all. Two independent routes to one parameter, each limited by a completely different Galactic foreground, is the ordinary shape of progress on a hard number.

One thing worth adding is what the degeneracy does to a comparison rather than to a number, since that is where it does its real damage. The lensing surveys and the microwave background are compared through the amplitude of structure today, and the microwave background’s contribution to that comparison is the primordial amplitude propagated forward. Since the primordial amplitude is what the optical depth is degenerate with, an error in the optical depth moves the predicted present-day amplitude by twice as much in the exponent — and the difference between the two sides of that comparison is a few per cent. So the optical depth, measured from a handful of multipoles through a foreground larger than the signal, is one of the terms in a tension whose discussion is usually conducted in terms of dark matter and gravity. That is a long way for a single poorly measured nuisance parameter to reach, and it is the reason a dedicated large-angle polarisation measurement is worth a mission of its own.

The foreground problem this measurement lives inside is worth putting beside the other large local contaminant in the same map. A dipole a hundred times the signal is removed exactly, because its amplitude is computable from a velocity that the map itself supplies; the Galaxy’s polarised emission is not, because nothing computes it and it has to be separated by its spectrum. The two are the same shape of difficulty and they have opposite outcomes, and the difference is entirely whether the contaminant’s amplitude is predictable from something else.

Where the ladder goes next

The immediate next rung is the foreground separation itself: how synchrotron and dust are distinguished from the signal by their spectra, why the separation is harder in polarisation than in temperature, and what the residual costs. Above it again sits the reionisation history — what the optical depth constrains about when it happened, and why one number cannot describe a process that took several hundred million years.

About the same objects

Not linked from either essay — found by the objects both name.

A mass measured by what it stopped from forming degeneracy · neutrino mass
A velocity that has the colour of the sky optical depth · thomson scattering
Two skies where the paradox comes out right optical depth · thomson scattering

What links here

Essays that link to this one from their own argument.

A constant that is an angle divided by a length cosmology
A forest with no continuum left cosmology
A residual that is somebody else's velocity cosmology
A dipole a hundred times the signal cosmology
A ratio that is an energy and a distance cosmology
The tilt knows the slope and not the height cosmology

The objects this essay names

Each one links to every other essay that touches it.

Cosmic variance DegeneracyE mode polarisationForeground emissionLow multipoleNeutrino mass Optical depthPrimordial amplitudeReionisation bumpThomson scattering

Why the damping is uniform #

What breaks it #

Why the polarisation measurement is so hard #

What the optical depth is worth #

What was actually measured #

Where the picture stops #

Why an exponential degeneracy is worse than a linear one #

Where the ladder goes next #

Where this goes next