A cluster weighed three ways

A cluster of galaxies is the largest object that has had time to pull itself together. It contains a thousand or so galaxies, a great deal of hot gas, and — on every measurement anyone has made — a mass some hundred times what its stars amount to.

That last statement is the most secure quantitative claim about dark matter, and its security comes from a particular structure of evidence: three ways of measuring the mass, with different assumptions, different failure modes and different instruments, giving the same answer.

The first: the galaxies #

Fritz Zwicky applied the virial theorem to the Coma cluster in 1933, and it remains the simplest of the three.

Measure the redshifts of as many member galaxies as possible. Their spread about the mean is the line-of-sight velocity dispersion, and for Coma that is about a thousand kilometres per second. Take the cluster’s radius from the extent of the galaxy distribution — a quantity that is a judgement rather than a measurement, since a cluster has no more of an edge than a galaxy does. Then, for a system in virial equilibrium,

$$M \approx \frac{2\sigma^2 R}{G},$$

which for $\sigma = 1000$ km/s and $R = 1.5$ Mpc gives about $7\times10^{14}$ solar masses.

Zwicky then divided by the light and got a mass-to-light ratio of several hundred, against the few that a population of stars can produce. He named the discrepancy dunkle Materie, and the field ignored it for four decades.

Two objections were available at the time and both were reasonable. The cluster might not be in equilibrium, in which case the theorem does not apply. And the sample of measured redshifts was small, so the dispersion was poorly determined. Neither survives modern data: dispersions are now measured from hundreds of members, and clusters that are visibly disturbed can be excluded.

The second: the gas #

A cluster’s dominant baryonic component is not its galaxies. It is a diffuse plasma at ten to a hundred million kelvin, which contains several times the mass of all the stars in all the member galaxies, and which nobody knew was there until X-ray astronomy began — a component whose existence no optical measurement of the light could have revealed.

The gas is hot because it has fallen into the cluster’s potential well and been shock-heated; its temperature is therefore a direct measure of the depth of that well. Quantitatively, if the gas is in hydrostatic equilibrium,

$$\frac{dP}{dr} = -\rho\frac{GM(<r)}{r^2},$$

and with a measured temperature and density profile the enclosed mass follows. For an isothermal gas with density falling as $r^{-2}$ — a good approximation in the outer parts — the expression simplifies to $M(<r) = 2(kT/\mu m_p)r/G$, which is the form the figure uses. The measurement requires an X-ray spectrum, because the temperature comes from the shape of the bremsstrahlung continuum and from the emission lines of highly ionised iron. Those lines are the reason the method works so well: their ratios fix the temperature nearly independently of the density, and their presence at all fixes the metallicity of the gas, which turns out to be about a third of the Sun’s — meaning this gas has been through stars.

The third: the light #

The third route was not available to Zwicky and is the one that assumes least.

A cluster with a surface density above the critical value produces multiple images and arcs of the galaxies behind it, and the Einstein radius fixes the projected mass inside it from an angle and three distances. Nothing about equilibrium enters. Weak lensing extends the same measurement outwards. Individual background galaxies are distorted far too little to notice, but averaging the shapes of thousands of them recovers the distortion statistically and maps the projected mass out to several megaparsecs — beyond where any X-ray emission is detectable and beyond where the galaxy distribution is dense enough for a dispersion.

The gas is the surprise, not the dark matter #

It is worth separating two things that are usually reported together, because one of them is a discovery about ordinary matter and gets less attention than it deserves.

When Zwicky’s measurement was made, the visible content of a cluster was its galaxies. The dark-matter conclusion was therefore stated as a ratio against starlight. X-ray astronomy then found that clusters contain five times more mass in hot gas than in stars — a component nobody had counted, made entirely of ordinary matter, and detectable only above the atmosphere.

That discovery reduced the discrepancy and did not remove it. It also changed the character of the problem: a cluster’s baryons are mostly not in galaxies at all, which means the efficiency with which gas turns into stars is low — about ten per cent — and that inefficiency is one of the central facts any theory of galaxy formation has to reproduce.

The same gas supplies the second mass measurement, so the discovery arrived with a new instrument for the problem it partly solved. That is a recurring pattern in this subject: the twenty-one centimetre line did the same thing for spirals, revealing a gas component and simultaneously providing the means to weigh the galaxy it belongs to.

What their agreement is worth, and what it is not #

Three numbers within a factor of two might not sound like a triumph. It is worth being precise about why it is one.

The three methods share no assumption. The dispersion assumes the galaxies are a relaxed, isotropic tracer population; if their orbits are radially biased, the mass is overestimated. The X-ray method assumes the gas is in hydrostatic equilibrium and thermal; if a fraction of its support is turbulent or magnetic, the mass is underestimated, and simulations suggest that fraction is ten to twenty per cent. The lensing method assumes only the deflection law and the distances, and its weakness is that it measures a cylinder and is sensitive to everything else along the line of sight.

So the errors do not point the same way, and there is no common systematic that could produce all three answers if the true mass were the stellar mass alone. That is what makes the cluster case the most secure: not the precision of any one measurement, but the fact that three imprecise measurements with unrelated failure modes converge.

What the agreement is not is a measurement of the cluster’s baryon fraction to high precision. That quantity — the ratio of gas plus stars to the total — comes out near 0.15, which is a genuinely important number, and it inherits the full uncertainty of the mass rather than the agreement’s.

What lives in one #

A cluster is also an environment, and its galaxies are not a random sample of galaxies.

The population inside a rich cluster is dominated by ellipticals and lenticulars; spirals are comparatively rare, and the ones present are often deficient in hydrogen. The fraction of early-type galaxies rises smoothly with local density — the morphology–density relation, established by Alan Dressler in 1980 — and it is one of the clearest environmental effects in astronomy.

The mechanisms are known and several act at once. A galaxy falling through the intracluster gas at a thousand kilometres per second has its own gas stripped by ram pressure. Repeated fast encounters with other members heat its disc without merging with it. And the supply of fresh gas from outside is cut off, so star formation runs down as the remaining gas is consumed.

The observation behind the number #

The dispersion is a set of redshifts, and the practical difficulty is membership: a galaxy at the cluster’s redshift on the sky may be in front of or behind it, and interlopers with large velocity offsets inflate the dispersion badly. Modern analyses use iterative clipping, and the sensitivity of the answer to the clipping scheme is a real systematic.

The temperature is an X-ray spectrum, which requires a satellite. The mass depends on the temperature and on the logarithmic slopes of the density and temperature profiles, which means it depends on derivatives of noisy data — the reason X-ray masses are quoted with careful attention to the radius at which they are evaluated.

The lensing mass is a model fitted to image positions. For strong lensing that model is tightly constrained near the arcs and unconstrained elsewhere; for weak lensing it is the average shape distortion of faint galaxies, which requires the point-spread function of the instrument to be known to a part in a thousand.

Three completely different instruments, three completely different analysis traditions, and three answers within a factor of two.

A worked comparison #

The three bars in the first figure are computed rather than quoted, and it is worth walking through one of them.

The gas temperature is 8 keV. Dividing by the mean particle mass — about 0.6 proton masses for a fully ionised plasma of cosmic composition — gives a characteristic squared speed of $1.3\times10^6$ km²/s², which corresponds to about 1,130 km/s. That is comfortably above the galaxies’ 1,000 km/s dispersion, and the comparison is itself informative: the gas and the galaxies are moving at similar speeds because both are responding to the same potential, and the small difference is what the shock heating and the equilibrium assumptions do.

Multiplying by $2r/G$ at $r = 1.5$ Mpc gives $8.9\times10^{14}$ solar masses.

The stellar mass is the cluster’s luminosity — about $5\times10^{12}$ solar luminosities — times a stellar mass-to-light ratio of four, giving $2\times10^{13}$. That is 2.3 per cent of the gas-derived total.

Adding the gas itself, which is a few times $10^{13}$, brings the baryons to something near fifteen per cent, which is the number the cosmological baryon density independently predicts. That agreement is the closing of the argument: clusters are large enough to have retained a fair sample of the universe’s contents, and what they contain matches what the first three minutes are calculated to have produced. The virial estimate carries a structure constant, and the two things it depends on are worth separating.

The bias that had to be measured rather than assumed #

The X-ray route’s assumption is the weakest of the three, and the size of its failure has been the subject of a long and consequential argument, because cluster masses are not only interesting in themselves.

Counting clusters as a function of mass and redshift measures how fast structure grew, which constrains the amplitude of the initial fluctuations. When that count is calibrated with hydrostatic masses and compared against the amplitude the microwave background predicts, the two disagree — the clusters imply less structure than the early universe seems to promise. One clean way to remove the disagreement is to suppose the hydrostatic masses are systematically low by twenty to thirty per cent, which is exactly what non-thermal support would do.

That is an uncomfortable position to argue from. A parameter introduced at the size required to remove a discrepancy, and constrained mainly by the discrepancy, is not a measurement. So the bias had to be measured some other way, and there are two.

The first is to calibrate against lensing, which assumes no equilibrium at all. Weak-lensing masses for large samples now give a hydrostatic bias of the order of twenty per cent, with an uncertainty that is itself dominated by how well galaxy shapes can be measured.

The second is to measure the turbulence directly, which means measuring the width of an X-ray emission line rather than its position — a Doppler broadening of a few hundred kilometres a second on a line at seven kilo-electronvolts. That requires a spectrometer resolving a part in a thousand at those energies, which no dispersive instrument can do for a diffuse source, and it took a microcalorimeter flown above the atmosphere. Hitomi observed the core of the Perseus cluster in 2016 and measured a velocity dispersion of about 164 kilometres a second — implying that turbulence supplies around four per cent of the pressure there, far less than the bias argument wants.

The satellite failed a month later, and that one pointing remained the whole of the direct evidence for years. It is a good illustration of how a field’s most important number can rest on a single observation: the measurement is not in dispute, it is simply of one region of one cluster, and whether the core is representative of the volume the mass is integrated over is precisely what cannot be settled from it.

Where the picture stops #

Clusters are not relaxed. Most of them are still assembling: substructure is common, and a merger in progress can inflate a virial mass by tens of per cent and disturb the gas badly. The figure’s bars are drawn for an idealised relaxed cluster and real ones are messier.

The three masses are measured within different radii. The comparison drawn here scales them to a common radius using an isothermal profile, and that scaling is an assumption — most visibly for the lensing bar, which measures a cylinder of a hundred kiloparsecs and is carried out to fifteen times that.

And a cluster’s baryons are mostly not its stars. The intracluster gas outweighs the stars by five to one, which was not known when the mass discrepancy was first reported and which reduced it substantially when it was. It did not remove it: gas and stars together are still about a seventh of the total.

One number from this essay is worth carrying away on its own, because it recurs throughout the field.

The baryon fraction of a rich cluster — gas plus stars, divided by the total — comes out near 0.15, and clusters are the only objects large enough to have kept a fair sample of what the universe is made of. Nothing smaller has: a galaxy has expelled some of its gas and a group has lost more, so their baryon fractions are lower and vary from one to the next. That makes a cluster a sampling instrument for cosmic composition, and its agreement with the value derived from the first three minutes is one of the quieter successes in the subject.

The generalisation #

The structure of this argument is the most transferable thing in the essay, and it is worth stating as a method: when a quantity cannot be measured precisely, measure it three ways whose systematic errors are unrelated.

The alternative — measuring it once, very carefully — is worth less than it appears, because the dominant uncertainty in almost every astronomical measurement is systematic rather than statistical, and a systematic error is invisible from inside the method that produces it. The only thing that reveals it is a measurement that fails differently.

This collection has now used the same structure at several scales: a planet’s density from a transit and a wobble, where the two methods measure different things about the same object; a star’s distance from a parallax and a period–luminosity relation, where the second is calibrated by the first and then checked against it; and here, at the largest scale in the collection so far.

What clusters add is the extreme case: three measurements, none of them better than twenty per cent, together making a claim that is essentially certain.

And the light profile at a smaller half-light radius, which is what the third weighing rests on.

Where the ladder goes next #

The next rung is what happens when a cluster is caught out of equilibrium on purpose — the merging systems in which the gas and the mass are separated by the collision, and what that separation can be made to test.

Later rungs on this anchor: the baryon fraction and its use as a cosmological probe; the mass–temperature and mass–richness scaling relations that let cluster counts constrain the growth of structure; the cool cores and the heating problem; the intracluster light and how many stars are outside any galaxy; the Sunyaev–Zel’dovich effect, which detects clusters through their shadow on the microwave background and is nearly independent of distance; and the outskirts, where the gas stops being in equilibrium and the assumptions of the second method visibly fail.