The baryons that are not in the galaxies

The rung below weighed a cluster three ways — from the speeds of its galaxies, from the temperature of its gas, and from the bending of light behind it — and found the three agreeing within a factor of two and all three exceeding the mass of the stars by about a hundred.

This rung is about the second of those measurements, and about the substance doing the measuring. The gas is not a diagnostic that happens to be lying around. It is most of the ordinary matter in the cluster, it outweighs everything in every galaxy by a factor of six or seven, and the only reason it took until the 1970s to notice is that it emits nothing an optical telescope can see.

What the gas is and how it is seen #

A cluster’s potential well is deep. A galaxy falling into it arrives at two thousand kilometres a second — the speeds that weighed the first cluster, and gas falling in at that speed shocks and thermalises — so its temperature is whatever corresponds to that kinetic energy, which is ten to a hundred million kelvin.

Gas at that temperature is fully ionised, and a fully ionised plasma radiates by bremsstrahlung: free electrons decelerating in the fields of ions, emitting a continuum whose cut-off is at the thermal energy. For ten million kelvin that is a few kilo-electron-volts, which is X-rays, and nothing at all in the optical.

Two properties of that emission make it a good instrument.

The emissivity goes as the square of the density, because it takes two particles to make a photon. So an X-ray image is weighted towards the densest gas and the cluster core stands out sharply — which is why clusters were found in X-ray surveys before anyone was looking for them.

And the spectrum gives the temperature directly. The exponential cut-off of the bremsstrahlung continuum sits at $kT$, and the ratio of a few line strengths refines it. No model is needed; the temperature is read off the shape.

Hydrostatic equilibrium, at a megaparsec #

The gas is not falling in and not flowing out, so it is holding itself up, and the equation for that is the one that holds up a star:

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

For an ideal gas, $P = \rho kT/\mu m_p$, and rearranging gives the mass in terms of two logarithmic derivatives:

$$M(<r) = -\frac{kTr}{G\mu m_p}\left[\frac{d\ln n}{d\ln r} + \frac{d\ln T}{d\ln r}\right].$$

That is a remarkable expression to be able to write for an object a megaparsec across. Nothing about the composition of the mass appears in it, nothing about the galaxies, and nothing about the dark matter. Two measured profiles and a constant give the total.

The standard description of the density is the beta model,

$$n(r) = n_0\left[1 + (r/r_c)^2\right]^{-3\beta/2},$$

which is what an isothermal gas in an isothermal-sphere potential produces and which fits nearly every relaxed cluster with $\beta$ near two thirds. It flattens inside a core radius of a few hundred kiloparsecs and falls as $r^{-2}$ outside it.

The assumption that makes it possible is the one that biases it #

Hydrostatic equilibrium requires that all of the support be thermal pressure. Some of it is not.

A cluster is assembled by mergers, and the last one leaves the gas sloshing. Bulk motions and turbulence carry a pressure of their own, in the way random motions hold up an elliptical galaxy — non-thermal, invisible in the spectrum except as a small line broadening — and gas partly supported by motion needs less thermal pressure to hold it up. So a mass computed from the thermal pressure alone comes out low.

Simulations put the deficit at ten to twenty per cent, and it is called the hydrostatic bias. Direct measurement of it requires resolving the gas motions spectroscopically, which needs a resolving power that only a calorimeter delivers, and the handful of clusters measured that way find turbulent pressures consistent with the low end.

The bias matters far beyond the individual mass, because the cluster mass function — the number of clusters above a given mass — is a steep function of mass and a sensitive cosmological probe. A systematic fifteen per cent in the masses is a large systematic in the inferred amplitude of density fluctuations, and it was for years the leading candidate explanation for a disagreement between cluster counts and the microwave background.

The third measurement, which assumes nothing about equilibrium #

Gravitational lensing weighs whatever is in the way with no assumption about whether it is settled. That is its whole value, and it is the check that makes the bias measurable rather than merely modelled.

Comparing lensing masses with hydrostatic masses for the same clusters gives the bias directly, and the answer has hovered near fifteen per cent with a large scatter — large because lensing masses have their own difficulties, chiefly the projection of everything along the line of sight into a single measured surface density.

What a cluster is worth as a cosmological probe #

Before the SZ effect, the two things a cluster survey could measure were both compromised by the same problem: the observable is not the mass.

An X-ray survey measures a luminosity, which goes as the square of the gas density integrated over the volume, so it depends on how the gas is distributed as well as on how much there is. Converting it to a mass uses a scaling relation calibrated on a subsample, and the relation has a scatter of tens of per cent and an evolution with redshift that has to be assumed.

An optical survey measures a richness, the number of galaxies of a given brightness within some radius, which is even further from a mass and whose scatter is larger still.

The SZ signal is closer to the thing wanted, because the integrated distortion is proportional to the integrated electron pressure, which is the gas mass times the temperature — and that product turns out to be a remarkably tight proxy for the total mass, with a scatter around ten per cent, because it is exactly the quantity hydrostatic equilibrium relates to the potential.

The measurement that does not fade #

The most useful property of the gas is one it has by accident.

The Sunyaev–Zel’dovich effect is what happens when microwave background photons pass through the hot gas: the electrons are so energetic that a photon scattering off one gains energy, and the effect on the background’s spectrum is a distortion — a deficit below about 218 GHz and an excess above it — proportional to the integrated electron pressure along the line of sight.

That distortion is a fraction of the background’s brightness, not an absolute flux. And the background’s own surface brightness rises as $(1+z)^4$, exactly cancelling the $(1+z)^{-4}$ dimming that every extended source suffers.

So the SZ signal’s surface brightness does not fall with distance at all. A cluster at redshift one is as easy to find as one at redshift a tenth, and a millimetre survey of the sky is a mass-limited cluster survey to arbitrary redshift — which no optical or X-ray survey can be.

What the accounting comes to #

Add up what a rich cluster contains.

Stars, in all its galaxies: about two per cent of the total mass.

Hot gas: about twelve per cent, six times the stars.

Everything else: about eighty-six per cent, and not made of anything that emits or absorbs.

So the baryon fraction is around fourteen or fifteen per cent — and the universal ratio $\Omega_b/\Omega_m$, from the microwave background’s acoustic peaks and from primordial nucleosynthesis independently, is 15.7 per cent.

A cluster’s own accounting reproduces a number read off the sky at redshift 1090. That agreement is the argument that a cluster is large enough to have kept everything it started with — its escape velocity exceeds any wind a galaxy can produce — so its composition is a fair sample of the universe’s, and measuring it is cosmology done with an object a person can point at.

How much of it is actually seen #

There is a distinction worth insisting on between the gas that is measured and the gas that is inferred, because the accounting above blurs them.

Inside about half the virial radius the X-ray surface brightness is well above the instrumental background, the density profile is measured directly, and the temperature is measured spectroscopically in several annuli. That is genuinely observed.

Between there and the virial radius the surface brightness has fallen by two orders of magnitude and the measurement becomes a fit of a model to a signal comparable with the background. Beyond the virial radius nothing is measured at all in emission.

And roughly half the gas mass is outside half the virial radius, because the density falls as $r^{-2}$ and the volume grows as $r^3$. So about half of the substance this essay is about has never been imaged, and the baryon fraction quoted at the virial radius is an extrapolation of a fitted profile.

The SZ effect helps here for a reason directly related to its distance-independence: its signal goes as the density to the first power rather than the square, so it falls off more slowly and is measurable further out. The two techniques therefore probe different parts of the same object, and their disagreement in the overlap region is one of the current measurements of gas clumping.

Two of the difficulties below are about the parts of a cluster that emission cannot reach, and the third is about the assumption the whole measurement runs on.

What is left unsettled #

The outskirts are barely measured. Everything above is quoted inside a radius where the X-ray surface brightness is still above the background, which is well inside where the accretion shock is. Beyond that the gas is clumpy, the emission goes as density squared and therefore overweights the clumps, and the inferred density is biased high by an amount nobody has measured well.

The centres do something that was not predicted. The cooling time of the densest gas in a cluster core is shorter than the age of the universe, so it should be cooling and flowing inward at hundreds of solar masses a year. Spectra show almost no gas below about a third of the cluster temperature: something reheats it, almost certainly the central galaxy’s black hole, and the energy balance between the two is a feedback loop nobody can compute from first principles.

And the temperature is not one number. The expression above assumes an isothermal gas; real clusters have temperature profiles that fall by a factor of two from a third of the virial radius outward, and the second logarithmic derivative in the mass expression is therefore not zero. Including it changes the masses by tens of per cent, and measuring it requires spatially resolved spectroscopy, which is exactly what limits how far out the profiles reach.

None of the three is a reason to distrust the accounting, and all three are reasons to attach a radius to it.

One reason the accounting can be closed at all deserves stating, because every step above quietly assumes it. A cluster is a closed box. Its escape velocity is two thousand kilometres a second or more, and nothing a galaxy can do accelerates gas to that: a supernova-driven wind leaves a dwarf galaxy at a few hundred kilometres a second and an active nucleus drives outflows at a thousand or two. So gas expelled from a member galaxy does not leave the cluster; it joins the intracluster medium. That is why the baryon fraction of a rich cluster can be compared with the universal value at all — the comparison is only meaningful for a system that has neither lost baryons nor gained them preferentially, and clusters are the largest objects for which that is true. It fails for groups, whose shallower wells do let enriched gas escape, and their baryon fractions are correspondingly lower and more scattered.

The evidence that the box has been filled from inside as well as from outside is written in the gas itself. The X-ray spectrum carries emission lines of iron, silicon and sulphur, and the iron line at 6.7 keV is strong enough to be measured in an ordinary observation: the gas is about a third of the solar iron abundance, roughly uniformly, out to large radius. Primordial gas has no iron at all. So a third of a solar abundance across a megaparsec is a direct measurement that a substantial part of the intracluster medium has been inside a star, and the uniformity says the enrichment happened early and was mixed, rather than late and local. The metal content is the only part of the accounting that distinguishes gas that fell in from gas that was thrown out, and it is measured from the same spectrum that gives the temperature.

Why the gas is where it is, and the stars are not #

One number in the accounting is worth an explanation rather than a statement: why is the gas six times the stars, when in a galaxy like the Milky Way the ratio is closer to one to six the other way?

The answer is that star formation is inefficient and a cluster remembers it. Gas that falls into a galaxy can form stars, and most of it does not: it is heated by supernovae and by an active nucleus, driven out as a wind, and lost from the galaxy’s potential well — which is one reason a disc runs out of gas long before it runs out of time. A cluster’s well is a thousand times deeper, so the same wind that escapes a galaxy is retained by the cluster — it simply ends up as hot gas between the galaxies instead of cold gas inside them.

So the intracluster medium is partly primordial infall and partly processed material that a galaxy expelled, and the two can be told apart because the second carries metals. The gas is enriched to about a third of the solar iron abundance, roughly uniformly out to large radii — which is far more metal than the current galaxies could have supplied recently and points to enrichment early, before the cluster assembled.

A cluster is therefore a record of star formation’s failure rate, integrated over everything that fell into it, and the ratio of hot gas to stars is the measurement of that rate.

The instrument the bias is waiting on #

The hydrostatic bias is a statement about non-thermal pressure, and non-thermal pressure means gas moving. Measuring how fast it moves requires resolving the Doppler width of an X-ray emission line, and that has been the limiting instrument problem in the subject for thirty years.

An X-ray CCD does two jobs at once: it records where each photon landed and, from the charge it deposited, roughly what its energy was. The energy resolution is a few per cent, which is enough to identify the lines and to fit a continuum temperature, and nowhere near enough to see a line shifted or broadened by a few hundred kilometres a second out of the speed of light.

What is needed is a microcalorimeter: a detector cooled to a few hundredths of a kelvin, which measures the tiny temperature rise produced by a single absorbed X-ray photon and therefore its energy directly. The resolution is better by a factor of thirty, which turns a blend into a resolved line and a line width into a velocity.

The history of getting one into orbit is discouraging. One mission’s calorimeter lost its cryogen shortly after launch. Its successor broke up a few weeks into the mission from an attitude-control fault, after returning a single deep observation of one cluster — which measured a turbulent velocity of about 160 kilometres a second, far quieter than most simulations had predicted, implying a non-thermal pressure of only a few per cent.

That single result is the entire direct measurement of the quantity the bias is about, and it came from one cluster, chosen because it is bright and relaxed — which is to say chosen from the population least likely to be turbulent.

A correction of fifteen per cent applied to every cluster mass in the literature rests on simulations plus one observation, and that is the honest state of it. A later mission carrying the same kind of detector is now returning data, and the number will either firm up or move.

One more thing the metal lines settle is worth adding, because it is the sharpest constraint on when the enrichment happened. The iron in a cluster’s gas is not uniformly distributed in time: the abundance measured out to large radius is nearly flat, but the centres of relaxed clusters carry an additional iron peak that traces the brightest galaxy. The flat part must have been produced and mixed before the cluster assembled, since nothing since would have distributed it so evenly across a megaparsec; the central excess is being produced now, by the stars of the central galaxy. So a single spectrum separates two enrichment epochs by geometry alone, and it puts most of the metal production before the cluster existed as a cluster at all.

Where this ladder goes next #

This rung has taken the substance that dominates a cluster’s ordinary matter, weighed it and the total with one equation, and found that a cluster’s own composition reproduces the universe’s.

The rung above is the population. The number of clusters above a given mass, as a function of redshift, is exponentially sensitive to how fast structure grows out of a nearly smooth beginning — so counting clusters measures the amplitude of density fluctuations and the growth rate, and the whole enterprise stands or falls on whether the masses are right by ten per cent.

Beside it lies the gas that is not in clusters: most of the universe’s baryons at low redshift are in a warm-hot intergalactic medium too diffuse to see in emission, found instead in absorption against background quasars, and accounting for them was an open problem for two decades.

And below it, the habit this rung is the clearest case of: the assumption that makes a measurement possible is the one to look at first for its bias. Hydrostatic equilibrium is what turns two profiles into a mass, and departures from it are what makes that mass fifteen per cent too small — not by coincidence, but because the same physics is doing both jobs.