A null that moves with the temperature

The spectrum of the thermal Sunyaev–Zel’dovich effect is usually written as a single function of frequency multiplied by a single number. The number is the Compton parameter $y$, which says how much energy the gas hands the photons; the function says how that energy is distributed across the spectrum, and it has one celebrated feature, a zero crossing at 217.5 GHz. Every cluster in the universe was supposed to share that crossing, whatever its mass, its temperature or its distance, and the kinematic effect’s whole observing strategy is built on it: observe at the null, where the gas vanishes, and whatever remains is motion.

The function was derived by taking the limit in which the scattering electrons are slow compared with light. That limit is the Kompaneets equation, and it is the right equation for the problem it was written for — the thermalisation of radiation by cool plasma in the early universe, where the electrons are at a few thousand kelvin. A cluster’s gas is at eighty to two hundred million. At 10 keV the ratio of thermal energy to rest energy is 0.0196, and the root-mean-square electron speed is almost a quarter of the speed of light.

At that speed the slow-electron expansion is not wrong so much as incomplete. The distortion depends on temperature through its shape, not only through its amplitude — and a shape that depends on temperature is two different things at once. It is a thermometer. And it is a moving target for anything that assumed the shape was fixed.

Why fast electrons change the shape #

A photon scattering off a slow electron gains, on average, an energy fraction $4kT_e/m_ec^2$, and the spread of gains is narrow. Summed over many photons and many small boosts, the result is a diffusion in frequency — a gentle drift upward that the Kompaneets equation describes with two derivatives.

A photon scattering off a fast electron is boosted by a factor that depends on the angle between them, and at a quarter of the speed of light the spread of possible boosts is wide. A single scattering can move a photon by tens of per cent in frequency, which is not a diffusion. Photons are carried past the frequencies where the slow-electron picture would have deposited them, the increment is spread over a broader range and its peak falls, and the decrement below the null is partially filled by the same broadening.

The exact result is an integral over the relativistic Maxwell–Jüttner distribution of electron momenta and over scattering angles. Its expansion in powers of $\theta = kT_e/m_ec^2$ was worked out in the late 1990s, and in intensity it reads

$$\frac{\Delta I}{I_0} = y\, g(x)\,\left[\,Y_0(x) + \theta\,Y_1(x) + \theta^2\,Y_2(x) + \dots\right]$$

where $Y_0 = x\coth(x/2) - 4$ is the non-relativistic shape and each later term is a polynomial in $x\coth(x/2)$ and $x/\sinh(x/2)$. The series is asymptotic rather than convergent — adding terms improves it only up to a point — and at second order it is good to a fraction of a per cent up to about 15 keV. Above twenty it has to be abandoned for the integral itself.

The figure’s fractional changes are not small against what they are compared with. A cluster’s integrated Compton signal is the quantity survey catalogues rank clusters by and convert into mass, and a mass bias of ten per cent is the size of effect cosmological analyses argue about for years. Read at 150 GHz with the non-relativistic shape, a 10 keV cluster’s decrement returns a Compton parameter six per cent below the true one, and a 15 keV cluster’s more than eight. Since the signal scales as mass to about the 1.8 power, that is a mass underestimate of three to five per cent — for the hottest objects, which are the most massive, which are the ones the cosmological count is most sensitive to.

The crossing, measured against a table #

The null’s position is the easiest single consequence to check, because there is something to check it against.

The shift is first order in $\theta$. That means its size at any temperature is set almost entirely by the first correction term, and a figure that drew the null at the right place for 3 keV and 14 keV is a figure whose first correction term is right. The second-order term is a small refinement of the curvature, visible only above about 10 keV, where the solid and dashed curves begin to part.

Half a gigahertz per keV is not something a receiver resolves. The bands used at these frequencies are thirty to fifty gigahertz wide, and no instrument has ever measured the position of a cluster’s null directly. What the shift does is quieter. It means that at 217.5 GHz — at the centre of the band every kinematic analysis relies on — a hot cluster’s thermal decrement has not quite finished crossing zero, and some of it is still there.

A velocity manufactured from a temperature #

That residual has a definite sign, and the sign is the sign of a cluster moving away.

At the old null the non-relativistic term is zero, so everything left is the correction: $y\,g\,\theta\,(Y_1 + \theta Y_2)$. Interpreted as a kinematic signal, it is set equal to $-\tau\,(v/c)\,g$. The optical depth is itself $y/\theta$, because the Compton parameter is the optical depth times the temperature in units of rest energy. Substituting,

$$v_{\rm false} = -c\,\theta^2\,(Y_1 + \theta\,Y_2)\Big|_{x = 3.830}$$

The Compton parameter has cancelled. A cluster’s spurious velocity does not depend on how much gas it has, how far away it is, or how deep its decrement is. It depends only on how hot the gas is, and it goes as the square of the temperature.

The consequence is a selection effect of an unusually perverse kind. A cluster’s kinematic signal is its optical depth times its velocity, so the clusters with the largest kinematic signals — the ones a velocity survey would target first — are the ones with the most gas. The clusters with the most gas are the most massive. The most massive clusters are the hottest, because temperature is set by the depth of the potential well. So the objects a velocity measurement would pick first are exactly the objects for which the correction is as large as the answer.

The first attempts to measure individual kinematic signals, in the late 1990s, targeted exactly such clusters — among them one of the hottest known, at about 12 keV — and reported velocities with error bars of several hundred to more than a thousand kilometres a second, consistent with zero. At that precision the figure’s 500 km/s was inside the noise and nothing was misread. It stopped being inside the noise once maps became deep enough to quote a single cluster’s velocity to a hundred or two, which is why the correction is now included in every fit as a matter of course. The figure’s lesson is how large it is when it is not: an analysis of the hottest clusters at the old null, with the textbook shape, reports a recession velocity for all of them.

The spectrum as a thermometer #

The other face of a temperature-dependent shape is that the shape can be read.

Take the increment at one high frequency and divide it by the decrement at one low frequency. The Compton parameter multiplies both and cancels; so does the cluster’s distance, and so does its angular size if both measurements are made with the same beam. What is left depends only on the gas temperature. That is a temperature from the microwave spectrum alone, with no X-ray photon in it — which matters because X-ray temperatures are hard to obtain for the most distant clusters and disputed for the hottest.

It is worth stating how weak that is in the terms used for every other thermometer. A thermometer’s quality is its logarithmic sensitivity — the power of the temperature that the measured quantity goes as. A star’s colour is a ratio of two fluxes on the Wien side of a Planck curve, and changes by a large fraction of itself for a modest change of temperature; the boron-8 neutrino flux goes as roughly the twenty-fourth power of the Sun’s central temperature, which is why it measures that temperature to a tenth of a per cent. At 8 keV the ratio drawn here goes as about the −0.07th power. A one per cent measurement of it is a fifteen per cent measurement of temperature, and that is the whole difficulty in one number.

Twelve per cent across the whole range is not much lever. The two frequencies were chosen because they are where millimetre surveys actually observe, and at those frequencies the relativistic changes are modest. A frequency further up the increment, around 500 to 600 GHz, would be much more sensitive — but there the ratio stops being a simple function of temperature: at 545 GHz it rises, peaks near 14 keV and turns over, so a single measured value corresponds to two temperatures. The monotone thermometer and the sensitive one are different frequency pairs.

The accuracy needed is the real difficulty. A two per cent ratio means two per cent calibration at two frequencies that are measured, in practice, by different detectors and often different instruments, against a sky whose foregrounds differ at each. At 353 GHz the dominant foreground is dust in the Galaxy and in the cluster’s own galaxies; at 150 GHz it is the primary background and radio sources. Calibration at the per-cent level across that span has been achieved by only a few experiments.

And the kinematic effect is not a small complication. A cluster receding at 300 km/s deepens the 150 GHz decrement and reduces the 353 GHz increment, pulling the ratio in the same direction that heating the gas does. Temperature and velocity are therefore partially degenerate in a two-frequency measurement, and breaking the degeneracy needs a third frequency near the null. The figure of a thermometer drawn above is the best case, with one of the unknowns set to zero.

Where a fixed shape was built into the maps #

The single-cluster measurements are the obvious casualties of a moving null. The less obvious one is every all-sky map that separates the Compton signal from everything else.

A component-separation method builds a map of $y$ by combining observations at several frequencies with weights chosen so that anything with a blackbody spectrum cancels and anything with the Sunyaev–Zel’dovich spectrum survives. The spectrum it protects is a template, and the template is the non-relativistic shape. For a cluster at 10 keV the real spectrum departs from that template by several per cent — most at the high frequencies where the weights are largest — so the map recovers a $y$ that is systematically low for hot clusters and slightly contaminated by whatever the mismatch lets through.

That matters twice over. The power spectrum of the $y$ map is itself a cosmological measurement, dominated by the most massive clusters and therefore by the hottest gas, and a few per cent low in $y$ for those objects is a bias of a few per cent on the amplitude of structure that power spectrum is used to infer. And the frequency channel near 217 GHz, which an analyst reaches for as a map free of clusters, is free of them only in the template. Neither effect is large. Both are one-sided, both grow with mass, and both sit inside error budgets that are being argued over at the few-per-cent level — which is where the same argument about the amplitude of structure from lensing is being fought too.

The distance measured from a cluster’s two line integrals is exposed in a different way. That method squares the Compton parameter, so a six per cent underestimate of $y$ becomes a twelve per cent underestimate of the distance and a twelve per cent overestimate of the Hubble constant — larger than the entire disagreement between the two leading determinations of that constant, from a correction to a spectral shape.

What has been measured #

Detecting the relativistic shape on an individual cluster requires either an extremely hot object or a measurement of the increment at submillimetre wavelengths, where the shape is most sensitive. Both have been tried. Observations of the hottest merging clusters from space-based submillimetre photometers, combined with ground-based decrement measurements, have produced temperatures consistent with the X-ray values at low significance — the approach works in principle and delivers errors of several keV. Stacking hundreds of clusters from the Planck all-sky maps has shown the relativistic distortion statistically, with an average temperature consistent with X-ray estimates for the same objects.

A consistent detection is less trivial than it sounds, because the two temperatures are not the same quantity. The X-ray temperature is weighted by emission, which goes as the square of density, and so describes the dense core — one of several reasons a cluster weighed three ways does not come out the same three times. The spectral temperature is weighted by pressure along the line of sight, linear in density, and describes the gas out to large radii where the density is lower and, in most clusters, the temperature is too. Where the two measurements disagree for an individual cluster, the disagreement is information about the temperature structure, not necessarily an error in either.

There is a sense in which this is a very old problem arriving at a new wavelength. The microwave background’s own spectrum matches a blackbody to fifty parts in a million precisely because the early universe’s electrons were plentiful and, by the standards of a cluster, slow; the tightest limits on any departure are limits on Compton $y$-type distortions from an era when the electrons were thousands of times cooler than a cluster’s. At cluster temperatures and the same scattering physics that once erased distortions now writes one, with a temperature carried in its shape.

What the figures cannot show #

Every curve here is a single-temperature distortion, and no cluster has a single temperature. The observed spectrum is a sum over the line of sight weighted by pressure, and a sum of relativistic shapes at different temperatures is not the relativistic shape at the average temperature, because the correction is nonlinear in θ. The false-velocity figure is the most affected: it goes as $\theta^2$, so a cluster with a hot core and cooler outskirts has a larger pressure-weighted $\langle\theta^2\rangle$ than the square of its mean, and the spurious velocity is larger than the curve drawn at the mean.

The expansion itself fails above about 20 keV, which is not a hypothetical limit — the hottest merging clusters have shock-heated regions above it. And the figures ignore non-thermal electrons entirely. A population of relativistic electrons from past shocks or active nuclei — the population whose power-law radio spectrum carries no temperature at all — produces its own Sunyaev–Zel’dovich distortion, with a different spectrum that crosses zero somewhere else or not at all; even a small pressure fraction in such a population moves every quantity drawn above.

Finally, none of this touches the frequency bands. Every measurement is an integral of the spectrum over a passband thirty to fifty gigahertz wide, and the relativistic shift moves the curve by less than the band. Converting any of these figures into what an instrument records needs the passband, and a passband whose edges are known to a gigahertz is not a given.

A template is a limit, and its zero is where the limit shows #

A fixed-shape template is a statement about a limit. The non-relativistic Sunyaev–Zel’dovich spectrum is the limit of small θ, and every analysis that treats its null as a property of the Planck spectrum is quietly assuming that the gas is cool enough for the limit to hold. The assumption is invisible until a measurement is made precisely at the point where the template’s feature was supposed to make everything else vanish — and then the correction is not second order in the thing being measured. It is first order.

The same shape of problem appears whenever a method depends on a zero. A radial-velocity measurement that assumes a star’s lines shift only when the star moves is undone by a spot that shifts their centroid with no motion at all; an exact identity in orbital mechanics is fine until the relativistic term it neglected is the effect being sought. A null is the most sensitive place to look for anything, and for exactly that reason it is the most sensitive place for a neglected term to be found.

Still open: whether a count of clusters is a count of mass #

The integrated Compton signal is how millimetre surveys build their catalogues, and a catalogue of clusters above a signal threshold, counted by redshift, is one of the most direct measurements of how fast structure has grown. The relativistic correction is one small term in how that signal becomes a mass. The dominant term is larger and has been argued about for more than a decade: the masses the signal is calibrated against are hydrostatic, and hydrostatic masses are low by an amount nobody has pinned down. What a survey’s threshold means as a mass — and whether a shortfall in the count is a universe with less structure or a scale with the wrong zero — is the question the counts themselves cannot settle.