A shadow that does not get fainter with distance

Astronomy is arranged around one relentless fact: things get fainter with distance, as the inverse square, and every survey ever built is therefore a survey of the nearby. A catalogue of galaxies drawn from an image is a catalogue that thins with redshift; a catalogue of clusters drawn from an X-ray map thins faster still, because surface brightness in an expanding universe falls not as the square of the distance but as the fourth power of one plus the redshift.

There is one exception in the whole of extragalactic astronomy, and it is exact rather than approximate. It is not a clever correction or a fortunate coincidence of numbers; it follows from what kind of quantity is being measured, and it would hold in any universe with a hot uniform background in it.

What the effect is #

The intracluster medium is a gas of electrons and protons at ten to a hundred million kelvin, filling the cluster, and it is optically thin: a photon crossing it has roughly a one per cent chance of scattering off an electron and a ninety-nine per cent chance of passing through untouched.

It is also where most of a cluster’s ordinary matter is. The galaxies are the visible part and they are a minority of the baryons; the rest is between them, at ten million degrees, radiating in X-rays, and it took an X-ray telescope to find it. The effect described here is a second and entirely independent way of seeing the same gas, through a mechanism that has nothing to do with what the gas emits.

The microwave background is behind everything. So about one background photon in a hundred, on a sight line through a cluster, scatters off an electron moving at a substantial fraction of the speed of light — and comes out with slightly more energy than it went in with, because on average the electron is moving faster than the photon’s own effective temperature. This is inverse Compton scattering, and it does not create or destroy photons: it moves them up in frequency. The distinction between moving and creating is what gives the effect its signature, and it is worth pausing on, because a distortion that conserved photon number could take only one shape and a distortion that did not could take many. Moving photons up in frequency in a blackbody spectrum makes a deficit on the low-frequency side and a surplus on the high-frequency side, and between them there is a frequency at which the two exactly cancel.

The amplitude is the Compton y parameter,

$$y \;=\; \int \frac{k T_e}{m_e c^2}\, n_e \sigma_T\, \mathrm{d}\ell,$$

which is the optical depth to scattering multiplied by the fractional energy each scattering adds. It is a pure number, of order a few times ten to the minus four for a massive cluster, and it is a line integral of the electron pressure.

Two features of that integral are worth separating. The optical depth carries the density and the thermal factor carries the temperature, so the product is a pressure and not a mass — a cool dense cluster and a hot rarefied one with the same pressure produce the same signal. And the integral runs along the sight line with no weighting by distance at all, which is the formal statement of the redshift independence: nowhere in the expression does the distance to the cluster appear.

Why the shadow does not dim #

The redshift-independence follows from the form of that expression and takes one sentence to state: $y$ is a dimensionless ratio, and the observable is $y$ multiplied by the microwave background’s own brightness.

Everything else in astronomy is a flux. A flux is an energy per unit area per unit time arriving from an object, and it falls as the inverse square because the emitted energy is spread over a growing sphere. Surface brightness — flux per unit solid angle — is better behaved in a static universe, where it does not fall at all, and is worse behaved in an expanding one. The Sunyaev–Zel’dovich signal escapes even that, because it is not a surface brightness of the cluster. It is a fractional modification of a surface brightness belonging to something else, and that something else has the same surface brightness at every epoch a cluster exists in. The cluster’s angular size shrinks with distance in the ordinary way, so a distant cluster is a smaller patch of sky — but the patch is exactly as deep.

The comparison worth drawing is with the redshift itself, which is not a Doppler shift and not a measurement of a speed but a ratio of scale factors. Both quantities are dimensionless, both are properties of the light’s whole journey rather than of its source, and both are therefore immune to the distance-dependent factors that afflict everything measured in physical units.

The consequence is that a millimetre survey of the sky, at fixed sensitivity, finds clusters above a mass threshold that is very nearly independent of redshift. That is not a marginal improvement on an X-ray survey; it is a different kind of catalogue, and it is the reason cluster cosmology moved to the millimetre.

Reading it as a mass #

A cluster is useful cosmologically because its mass can be compared with a prediction. The abundance of clusters above a given mass is exquisitely sensitive to how much structure has grown, so a catalogue with masses is a measurement — provided the masses are right. The sensitivity comes from the same place as the knee in a luminosity function: a distribution falling steeply above a characteristic scale turns a small change in the scale into a large change in the count, which is what makes counting worth doing and what makes calibration the whole difficulty. What Sunyaev–Zel’dovich contributes is a mass proxy with a specific virtue. The integral of $y$ over the cluster’s solid angle, written $Y$, is proportional to the total thermal energy of the gas — pressure integrated over volume — and therefore to mass times temperature. It does not care how the gas is arranged, which is unusual.

The comparison to make is with the third method above.

A distance out of two measurements of one gas #

There is a use of the effect that predates the survey applications and that is worth setting out, because it delivers a quantity nothing else in this collection gets without a ladder underneath it.

The Compton parameter is a line integral of the electron pressure, which is linear in density. The X-ray surface brightness of the same gas is a line integral of the emission, which goes as density squared. Two integrals of different powers of the same quantity, over the same path.

Divide the square of the first by the second and the density cancels, leaving the path length — a physical length, in metres, for the depth of the cluster along the line of sight. Compare that with the cluster’s measured angular size across the sky, and if the cluster is assumed spherical, the ratio is a distance.

That is a genuinely absolute distance to an object at cosmological redshift, obtained with no standard candle, no standard ruler and no calibration from anything nearer. Combined with the redshift it is a Hubble constant, and the method was pursued through the 1990s and 2000s for exactly that reason.

Its systematics are severe and worth naming, because they explain why it is not the field’s leading determination. The sphericity assumption is the largest: a cluster elongated along the line of sight looks deeper than it is round, and returns a distance that is too large. Averaging over a sample removes the bias if the orientations are random, which they are not for a sample selected on X-ray brightness — an elongated cluster pointed at the observer is brighter and more likely to be in the catalogue.

Clumping enters with the opposite sign, since it raises the X-ray integral more than the pressure integral. And the electron temperature has to be measured from the X-ray spectrum, so the method inherits the calibration of an X-ray instrument.

The values obtained cluster around sixty to seventy, with uncertainties of ten per cent or so, which was competitive when the measurement was made and is not now. What it retains is its independence: it shares no rung, no calibrator and no zero point with any other route, so it remains one of the few checks on the distance scale that could fail without implicating anything else.

It is worth being clear about which part of the chain the independence buys. The redshift is measured the ordinary way, so the method is not independent of spectroscopy; the temperature comes from an X-ray spectrum, so it is not independent of atomic physics. What it is independent of is the distance ladder — no parallax, no period–luminosity relation, no standard candle, and no assumption that a class of object is uniform. The measurement is a geometry, and geometries do not need calibrating.

That is the same claim a time-delay lens makes and the same claim a gravitational-wave amplitude makes, and the three fail in entirely different ways. Where a subject has one disputed number and three independent routes to it, the value of each route is not its precision but its unlikeness to the others.

What the effect cannot do #

It cannot supply a distance.

That sounds like a small complaint and it is a structural one. The signal that does not know how far away the cluster is also cannot say. Every cluster in a millimetre catalogue still needs a redshift measured the ordinary way, from a spectrum of a galaxy inside it, which means an optical follow-up campaign whose depth is exactly the limitation the millimetre survey was supposed to escape.

How large the signal is, in the units an instrument works in #

It is worth converting the abstraction into something an antenna would report. A Compton parameter of three parts in ten thousand, at the low-frequency end where the temperature distortion saturates at minus twice $y$, is a decrement of six ten-thousandths of the background temperature — about 1.6 millikelvin against 2.7255 kelvin.

That is not a small signal by the standards of microwave background work. The primary anisotropies themselves are tens of microkelvin, so a massive cluster is a fifty-sigma object against the very fluctuations that cosmology was built on measuring. The difficulty is never sensitivity; it is that the sky already contains structure at the same angular scale, and separating a cluster from a chance superposition of primordial fluctuations is done by the spectrum rather than by the amplitude. A patch of the primary background is a temperature fluctuation, so it scales with frequency exactly as the background does. A cluster does not, and it changes sign at 217 gigahertz. Observing in several bands turns a detection problem into an algebra problem.

The other Sunyaev–Zel’dovich effect #

There is a second, smaller signal from the same gas, and it measures something no other technique in this collection can reach.

If the cluster as a whole is moving with respect to the microwave background, the scattering imprints a shift with a different spectral shape — a pure change in the background’s temperature rather than a redistribution across frequency, so it does not vanish at 217 gigahertz where the thermal effect does. Measuring it gives the cluster’s peculiar velocity: its motion relative to the smooth expansion, directly, at any redshift.

That is a remarkable quantity to have. A radial velocity measured from galaxy spectra gives the recession plus the peculiar motion mixed together, and separating them requires assuming a distance. The kinematic effect gives the peculiar part alone. It is roughly a tenth of the thermal signal, it has been detected statistically rather than cluster by cluster, and it is the main reason to build instruments that observe at the null.

What it would give, if it could be measured object by object, is a velocity field: not a map of where things are but a map of how they are moving, which is a direct measurement of the gravitational field that has been accelerating them. That is a strictly stronger constraint than a census of positions, and it is the same argument that makes a survey plotted in velocity rather than in position a measurement of growth rather than of geometry.

The same signal as a nuisance #

Everything above treats the distortion as the measurement. For the experiments the microwave background was mapped by, it is a contaminant, and the way it is handled is instructive about what a foreground is.

A map of the microwave sky contains the primordial anisotropies, the emission of the Galaxy, the emission of distant galaxies, and this. The clusters are a small fraction of the sky by area and their signal is large where they sit, so a temperature map contains a population of cold spots at the arcminute scale that have nothing to do with the early universe.

Separating them is done on the spectrum, exactly as the detection is. The primordial fluctuations have the spectrum of a temperature change to a blackbody; the Sunyaev–Zel’dovich distortion does not; Galactic dust and synchrotron have their own. An experiment observing in six or seven bands solves, pixel by pixel, for the amplitude of each component, and the outputs are a cleaned temperature map and — as a by-product — a full-sky map of the Compton parameter.

That by-product turned out to be worth more than the cleaning. A full-sky $y$ map contains every massive cluster, and it also contains a diffuse background from the pressure of gas that is not in any identified cluster: the filaments, the group-scale haloes, and the warm gas between them. The total thermal energy of the universe’s ionised baryons is measurable from it, integrated over all epochs, which is a quantity no pointed observation could assemble.

A component-separation step performed to remove something produced a measurement of it, and the separation is possible only because the two components differ in a way that has nothing to do with their amplitude or their angular scale. That is the same structural argument the whole essay rests on: a signal with its own spectral signature is a signal that can be extracted from anything, however much larger the thing it sits inside.

Where the picture stops #

The gas is not isothermal and it is not smooth. Real clusters have temperature gradients, cold fronts, cavities blown by a central radio source, and clumping — dense knots that raise the X-ray emission more than the pressure, since one goes as density squared and the other does not. Clumping biases the two mass proxies in opposite directions, which is useful and is not a solution.

Point sources contaminate it. A radio source in the cluster fills in the decrement and a dusty star-forming galaxy behind it fills in the increment, and both are common enough to matter. The radio contaminant is synchrotron emission, whose spectrum carries no temperature and which therefore rises towards low frequency exactly where the decrement is deepest. The frequency dependence separates them in principle; in practice it costs observing bands.

And the hydrostatic bias is unresolved. Masses from the gas assume the gas is supported by thermal pressure alone, and simulations say a fifth of the support is bulk and turbulent motion left over from assembly. That number is a correction applied to every hydrostatic mass, it is taken from simulations rather than measured, and it moves the cosmological answer.

Where this ladder goes next #

Later rungs on this anchor: the kinematic effect and what a direct peculiar velocity is worth; the relativistic corrections to the spectral shape, which are a thermometer for the hottest clusters; the all-sky $y$ map, which contains not only clusters but the diffuse pressure of the filaments between them; the cross-correlation with lensing maps, which is how the mass scale gets fixed without resolving individual clusters; and the use of a redshift-independent catalogue to measure the growth of structure, which is the reason for building any of it.