A velocity that has the colour of the sky

A cluster of galaxies is a cloud of electrons at a hundred million kelvin, and the thermal distortion it leaves on the microwave background comes from how fast those electrons move at random. The same electrons are also moving together. A cluster falls towards whatever mass is near it at a few hundred kilometres a second, and a cloud of electrons carried bodily through a bath of photons scatters those photons into a second distortion, with nothing to do with the gas temperature and everything to do with the cluster’s velocity.

Sunyaev and Zel’dovich predicted this second effect in 1980, eight years after the first, and noted in the same paper that it offered something no other observation could: the velocity of a distant object along the line of sight, measured against the rest frame of the universe, with the cosmological redshift playing no part. A galaxy’s redshift is the sum of the expansion and its own motion, and the two cannot be separated from a spectrum. The scattered light separates them, because the background photons do not care how far away the cluster is — only how it is moving through them.

The prediction has been known for forty-five years. The measurement it promised, for an individual cluster, has been made convincingly perhaps once. The reason is in the shape of the spectrum, and it is not a technical difficulty that better receivers will remove.

Why a moving cloud leaves a temperature, not a shape #

The thermal distortion has a characteristic shape because the electrons doing the scattering have a spread of velocities. Each scattering boosts a photon by an amount that depends on which electron it met, and the sum over many different boosts smears the Planck spectrum into something that is no longer a Planck spectrum: a deficit below 217 GHz, a surplus above.

Bulk motion is different in kind. In the cluster’s own frame the background is not isotropic — it is slightly hotter in the direction of motion and slightly cooler behind, the same dipole the Earth sees at a hundredth of the amplitude. Scattering isotropises it. A fraction τ of the photons headed towards the observer are replaced by photons drawn from every direction in the cluster frame, whose average temperature is the background’s temperature Doppler-shifted by the cluster’s motion, and the result is a spectrum shifted by a common factor:

$$\frac{\Delta T}{T} = -\tau\,\frac{v_r}{c}$$

A Planck spectrum shifted in frequency by a common factor is a Planck spectrum at another temperature. That is the whole reason the kinematic effect has no characteristic shape of its own: it is a blackbody, and a blackbody is described by one number. A receding cluster lowers the apparent temperature behind it and an approaching one raises it, by the optical depth times the line-of-sight speed as a fraction of the speed of light.

The optical depth is small. For the cluster in the figure it is 0.0064 — about one background photon in a hundred and fifty is scattered on the way through — and the line-of-sight speed is 500 km/s, a sixth of a per cent of c. The product is a temperature shift of one part in a hundred thousand, 29 microkelvin against a sky at 2.7 kelvin.

Loudest where the gas is silent #

The kinematic signal in intensity is the temperature derivative of the Planck spectrum, $g(x) = x^4 e^x/(e^x - 1)^2$ with $x = h\nu/kT$. Its largest value falls where the logarithmic derivative of $g$ vanishes, which is where $4/x = \coth(x/2)$. The thermal distortion is $g(x)$ multiplied by $x\coth(x/2) - 4$, and it vanishes where $x\coth(x/2) = 4$.

Those are the same equation, and the coincidence is exact rather than numerical. There is a one-line reason for it that says more than the algebra does. Differentiate the kinematic shape and multiply by frequency: $-x\,g'(x) = g(x)\,[x\coth(x/2) - 4]$, which is the thermal distortion. The thermal signal in intensity is minus the frequency times the slope of the kinematic signal. Wherever the kinematic curve is flat — at its peak — the thermal curve must be zero.

That identity is physics rather than luck. Random electron motions move photons up in frequency without creating any, so the thermal distortion is the divergence of a flow of photons along the frequency axis — and the size of that flow at each frequency turns out to be exactly the kinematic shape. A flow adds or removes nothing at the point where it is largest, because as many photons arrive there from below as leave it upwards. The frequency at which the Planck spectrum responds most strongly to being warmed is therefore, necessarily, the frequency at which a thermal boost leaves the intensity unchanged.

So a telescope observing at 217 GHz sees no hot gas, and it sees the cluster’s motion at the largest intensity that motion can produce. That is the observing strategy the whole programme was built on, and it would be a clean one if the gas were the only thing in the way.

It is not. The kinematic spectrum is a temperature change, and so is every other feature of the microwave background. The primary anisotropies — the fluctuations frozen in at recombination — are temperature changes with exactly this spectrum, across every frequency, at an amplitude of tens of microkelvin on the angular scales of a cluster. A velocity of 500 km/s in a cluster with an optical depth of 0.006 produces 29 µK. A cold spot in the background behind the cluster, laid down 13.8 billion years earlier, produces a signal of the same shape and the same size, and there is no frequency at which the two can be told apart.

The spectral trick that removes the gas cannot remove the sky. The only handle left is angular scale, and the primary background helps a little there: its fluctuations are damped below several arcminutes by photon diffusion at recombination, while a cluster at intermediate redshift is an arcminute or two across. A filter matched to the cluster’s profile suppresses the background by the ratio of their power at those scales. It does not suppress it to zero, and what is left is a noise of a few to ten microkelvin per cluster that no amount of integration time reduces — because it is not instrument noise. It is the sky.

When the motion is large enough to see #

There is one regime in which an individual velocity stands clear of that floor, and it is the one in which the velocity stops being a property of the cluster as a whole.

Clusters grow by merging, and during a merger the infalling subcluster and the main body can differ in line-of-sight velocity by thousands of kilometres a second. The optical depth of a subclump is smaller than that of a whole cluster, but the velocity is ten times larger, and a map at a resolution of half an arcminute can separate the clumps spatially.

The measurement that established the effect on an individual object was of exactly this kind. The merging system MACS J0717.5+3745, at redshift 0.55, was mapped at two millimetre-wave frequencies in 2013, one either side of the thermal null, and one of its four identified subclusters showed a ratio between the two that its thermal emission could not account for and a line-of-sight velocity of about +3,500 km/s could. The same subcluster’s galaxies had independently measured redshifts that put it moving away from the main body at a comparable speed. That agreement — a velocity from scattered microwave light matching one from optical spectra of the galaxies embedded in the gas — is the closest thing the method has to a demonstration on a single object.

It also illustrates what the measurement actually delivers. The galaxies already gave the velocity. The microwave measurement confirms that the gas is moving with them, which in a merger is not guaranteed: gas collides and is slowed by ram pressure while galaxies pass through one another almost unimpeded, so the two components separate, and the Bullet Cluster’s separation of gas from mass is the famous case. A kinematic map of the gas is therefore not a redundant velocity. It is the velocity of the component whose redshift cannot otherwise be measured, because X-ray spectrographs have only recently acquired the resolution to see a Doppler shift of a few hundred kilometres a second in a line from hot iron.

What else multiplies the velocity #

Even with the background filtered, a kinematic signal is $\tau v_r$, and turning it into a velocity needs the optical depth. Nothing measures the optical depth directly. It is inferred from the thermal signal, which gives $y = \tau\, kT_e/m_e c^2$, so

$$v_r = -c\,\frac{\Delta T}{T}\,\frac{kT_e}{m_e c^2}\,\frac{1}{y}$$

and the temperature enters linearly. A cluster’s temperature measured by X-ray spectroscopy is an emission-weighted average dominated by the dense core, while the optical depth relevant here is weighted by density along the whole line of sight. A twenty per cent error in the temperature is a twenty per cent error in the velocity, with no averaging to reduce it, because the error is in the conversion and not in the noise.

There are three further contaminants, and they compound. Radio galaxies within the cluster emit at the low-frequency end of the observing bands. Dusty star-forming galaxies behind the cluster emit at the high-frequency end, and gravitational lensing by the cluster magnifies them. And internal motions within the gas — rotation after an off-axis merger, turbulence — produce kinematic signals of both signs in different parts of the cluster, which cancel in the total but add scatter to any fit that assumes the gas moves as one body.

Stacked together, those are why the individual measurement has been reported as upper limits for most of its history. When the Planck satellite’s maps were used to average the kinematic signal over many known clusters, what came out was a limit on the root-mean-square velocity — below about 800 km/s — and on any coherent bulk flow of the clusters as a whole, both consistent with the few hundred km/s predicted by the growth of structure and neither a detection of any particular cluster moving.

The average that does not cancel #

An average of velocities with random signs is zero, and the noise on it falls as one over the square root of the number of clusters. That is a measurement of nothing, made more precisely. What rescued the kinematic effect as a probe of cosmology was finding a velocity statistic whose expectation is not zero.

Gravity supplies it. Two massive haloes a few tens of megaparsecs apart are, on average, falling towards each other, because the region between them is overdense — that is what it means for them to be correlated. Their line-of-sight velocities are individually random, but the difference between them, projected along the line joining them, has a mean that is negative and a size set by how strongly matter clusters on that scale and how fast that clustering is growing.

In linear theory the pairwise velocity is

$$v_{12}(r) = -\frac{2}{3}\,\frac{H(z)}{1+z}\; r\, f(z)\; \frac{b\,\bar\xi(r)}{1 + b^2 \xi(r)}$$

where $\bar\xi$ is the matter correlation function averaged inside the separation, $b$ is how much more strongly the tracers cluster than matter does, and $f$ is the logarithmic rate at which structure is growing. The expression is the continuity equation applied to pairs: if the number of pairs at separation $r$ is increasing — and it is, because structure grows — then pairs must on average be moving inward across that separation, and the flux needed is set by how fast the excess count is building.

The measurement follows directly. Take a spectroscopic survey of luminous galaxies, which gives positions and redshifts but no velocities. For every pair at separation $r$, take the difference in the microwave temperature at the two positions, weighted by how much of the pair’s separation lies along the line of sight. Galaxies behind which the background happens to be cold are as common as those behind which it is warm, so the primary anisotropies average away. The kinematic signal does not, because a receding member of an approaching pair is always the nearer one.

The first detection came in 2012, from pairs of luminous red galaxies in a large spectroscopic survey against maps from the Atacama Cosmology Telescope, at a significance of about three standard deviations and an amplitude of a few tenths of a microkelvin — the size the figure predicts. Later analyses, with more pairs and deeper maps, have reached well above five.

The velocity measurement that became a census of gas #

The pairwise signal is proportional to $\bar\tau\, f\, \sigma_8^2$, roughly: an optical depth that nobody knows, a growth rate that cosmology wants, and an amplitude of structure measured elsewhere. The hope was that the second could be extracted, since a growth rate measured by velocities would test gravity on scales where redshift-space distortions are the only competing probe.

It turned out the other way round. The growth rate and $\sigma_8$ are known well enough from other data that the pairwise measurement is more useful read backwards — as a measurement of the mean optical depth of electrons around the galaxies in the sample. And that optical depth is, directly, how much ionised gas surrounds a massive galaxy, integrated out to whatever aperture was used.

That quantity matters because a large fraction of the universe’s ordinary matter has never been seen. The baryons counted in stars, cold gas and the hot gas of clusters add up to well under half of what the light-element abundances require. The rest is thought to be in diffuse gas around and between galaxies, too thin and too cool to emit X-rays that anything can detect. Fast radio bursts found it along sight lines by its dispersion. The kinematic effect finds it by its motion: the scattering does not care whether the gas is hot, only that the electrons are free and moving with the halo.

Stacked kinematic profiles around hundreds of thousands of galaxies have found the gas extending well beyond the radius at which X-ray emission falls below detection, with an optical depth profile that is flatter than the dark matter’s — as though feedback from supernovae and active nuclei had pushed a substantial part of each galaxy’s gas outward. The measurement designed to find velocities has become one of the better constraints on where the missing baryons went and how far galaxies have blown them.

Why the signal survives what defeats the individual case #

The structure of that last move is worth separating out, because it is the central idea here and it is not specific to microwaves.

Start from the case that fails, drawn for an ordinary cluster rather than the extreme one the opening figures used.

Eleven microkelvin is the entire signal at the one frequency built to isolate it. The primary background on the angular scale of a cluster core, an arcminute or two, is heavily damped, but it still fluctuates by several microkelvin, and the lensing of that background by the cluster itself, the unresolved glow of dusty galaxies and whatever thermal residual a hot cluster leaves at its moved null all add more of the same order. Every one of those contaminants except the dust has exactly the spectral shape of a moving cluster. Adding frequencies scales them together; observing for longer measures them better. For a cluster of this kind the limit on its velocity is set by the sky rather than by the instrument, and it is about as large as the velocity.

A tenfold larger velocity would clear it, which is why the individual detections came from mergers. A tenfold larger number of clusters does not, taken one at a time, because each estimate carries its own share of the background. What changes the arithmetic is a statistic in which that share averages to zero.

An individual kinematic measurement fails because its contaminant has the same form as its signal — a temperature change — and a random sign. No filter in frequency helps, and integration only measures the contaminant better. The pairwise measurement succeeds because it constructs a statistic in which the contaminant still has a random sign and the signal no longer does. The primary anisotropies are uncorrelated with which member of a galaxy pair is nearer; the infall is perfectly correlated with it.

That is the same step that recovers a phase round a triangle of telescopes — build a combination the corruption cannot enter — applied to a sign rather than an additive error. And it has the same cost. The closure phase gives up absolute position; the pairwise velocity gives up every individual velocity, and with them the peculiar velocity field that was the original goal.

What the picture leaves out #

The figures above treat the cluster as one body moving at one speed and the correlation function as linear. Both fail in instructive places.

A real cluster’s kinematic signal is weighted by the density of its gas along the line of sight and across the beam, so it measures the velocity of the densest gas, which in a relaxed cluster is the core and in a merger is whichever clump dominates the column. Rotation produces a dipole pattern across the cluster whose total cancels, and the dipole’s amplitude is itself a measurement — of angular momentum in the intracluster gas, which simulations predict and nothing has yet measured directly.

The pairwise formula is linear theory, and below about ten megaparsecs the pairs are inside the same collapsed structures, where virial motions dominate and the mean infall is replaced by a turnover the linear expression cannot describe. The figures stop at eight megaparsecs for that reason. At the other end, the power spectrum used carries no baryon acoustic feature, so the pairwise velocity near a hundred megaparsecs misses the small bump that feature would add.

And the relativistic correction to the thermal spectrum — neglected throughout — is not small for hot gas. It moves the thermal null upward by about half a gigahertz for every keV of temperature, which leaves a residual of the thermal decrement at exactly the frequency this entire strategy is built around. For the clusters hot enough to be seen, that residual is the size of the velocity being sought.

Still open: whether the null is where it was drawn #

The next question is that residual. The thermal distortion’s shape was written down for slow electrons, and electrons at 10 keV move at a fifth of the speed of light; the exact spectrum depends on temperature, which means it carries a thermometer in it, and it means the frequency at which the gas was supposed to be silent is not quite that frequency. Computing where the null actually sits, and what velocity a hot cluster appears to have if the shift is ignored, is the step that decides whether any single-cluster velocity measured at 217 GHz can be trusted.

Beyond it lie the uses of the effect as a catalogue: the integrated signal as a proxy for mass, and the count of clusters above a signal threshold as a function of redshift, which is a measurement of how fast structure has grown — and one whose answer depends entirely on whether the mass behind the threshold is the mass it is assumed to be.