The light that climbs out of a cluster of galaxies

A rich cluster of galaxies holds a thousand galaxies and a hundred times as much mass that is not in them, packed into a region a few megaparsecs across. To light, it is a hollow in time. The clock that runs slow lower down showed that a photon leaving a place of lower potential arrives reddened by the difference in potential divided by $c^2$, and the redshift that weighs a dead star followed the same shift to the surface of a white dwarf, where it is large enough to read the star’s mass from one spectral line. A cluster is a far shallower well than a white dwarf’s surface, but it has something no star has: many sources of light sitting at different depths inside the same well.

The galaxy at the centre of a cluster sits at the bottom. The others orbit through the well at every height. Light from the central galaxy climbs out of the deepest part and should arrive the most reddened; light from a galaxy out at the edge has less to climb. Relative to the central galaxy, then, every other member should look a little blue — not because it is moving, but because of where it is. That is a test of how gravity behaves across millions of light years, at distances where it has otherwise only been inferred from the motions it causes, and it is buried under those motions a hundred times over.

A well measured in kilometres a second #

The size of the effect follows from the depth of the well. The gravitational shift between two places is the difference in potential divided by $c^2$, and astronomers quote it as the velocity whose Doppler shift would be the same: a potential difference $\Delta\Phi$ appears as a velocity $\Delta\Phi/c$. The well itself is described by the profile of mass that simulations of cold dark matter produce and that the distribution of hot gas and lensing in real clusters broadly follow, the Navarro–Frenk–White profile, whose density falls as $1/r$ near the centre and as $1/r^3$ far out. Its potential is

$$\Phi(r) = -\frac{4\pi G\rho_s r_s^3}{r}\ln\!\left(1 + \frac{r}{r_s}\right),$$

finite at the centre, where it reaches $-4\pi G\rho_s r_s^2$. A cluster is labelled by $M_{200}$, the mass inside the radius $R_{200}$ within which the mean density is two hundred times the critical density of the universe, and by its concentration $c = R_{200}/r_s$, around four or five for the heaviest clusters.

The heaviest clusters are about thirty-four kilometres a second deep at the centre, and a galaxy a megaparsec from the centre sits about fifteen kilometres a second higher up. A cluster of a tenth of the mass is a quarter as deep. The numbers are tiny beside the white dwarf’s eighty kilometres a second, and they would be invisible next to the motion of any single galaxy, which in a rich cluster is typically a thousand kilometres a second along the line of sight. But they are differences in where the light started, and the motions, unlike the depths, have both signs.

What a spectrograph sees from outside #

No spectrograph sees a member galaxy at a known distance from the centre. It sees the galaxy at a projected distance on the sky, and the galaxy might be anywhere along the line of sight through the cluster — far in front, in the middle, or behind. So the prediction for a member seen at projected radius $R$ is the shift averaged along the line of sight, weighted by how many galaxies there are at each depth along it. Taking the galaxies to follow the mass out to twice $R_{200}$ gives the profile in the next figure.

Projection blunts the shift. A galaxy seen near the centre of the cluster on the sky can lie a megaparsec in front of it, higher in the well than its projected position suggests, so the curve does not begin at the full depth of the well but rises from near zero and steepens slowly. At two megaparsecs on the sky, the heaviest cluster’s members are blueshifted by about twenty-two kilometres a second relative to the centre; a cluster of $10^{14}$ solar masses manages five or six.

There is a second reason the central galaxy is the right reference. It is not a randomly chosen member. The brightest galaxy in a cluster almost always sits close to the cluster’s centre of mass and moves slowly relative to it, because it has sunk there by dynamical friction over billions of years. Its velocity is as close as the cluster offers to the velocity of the well itself, and its light starts from as close to the bottom as any light does. Using it as the zero point removes the cluster’s own motion through the universe — the expansion, and its peculiar velocity of hundreds of kilometres a second — and leaves only the differences inside.

A thousand kilometres a second of noise #

Each member galaxy’s velocity can be measured to a few kilometres a second. The difficulty is not the spectrograph. It is that the members move through the well at random, with a spread of velocities along the line of sight, $\sigma$, that in a rich cluster is near a thousand kilometres a second. The gravitational shift is a hundredth of that. It appears only as a shift of the mean, and the mean of $N$ velocities with a spread $\sigma$ is uncertain by $\sigma/\sqrt N$.

Reaching an uncertainty of three kilometres a second with a dispersion of a thousand takes more than a hundred thousand galaxies. The richest clusters have a few hundred members with measured velocities. So the measurement cannot be made on one cluster, or on ten. It can only be made by stacking: taking thousands of clusters, aligning each on its brightest galaxy, scaling or not scaling them to a common size, and adding all their members into a single distribution of velocities against projected radius.

That is what Radosław Wojtak, Steen Hansen and Jens Hjorth did in 2011 with about eight thousand clusters from the Sloan Digital Sky Survey and about a hundred and twenty-five thousand member galaxies. The distribution of member velocities, centred on the brightest galaxies, is very nearly symmetric, as it must be if the motions are random. Its mean is not at zero. The members were blueshifted relative to the central galaxies by about eight kilometres a second on average, with an uncertainty of about three, and the shift grew with projected distance from the centre in roughly the way the curves above grow. Later stacks from larger surveys found shifts of the same size.

The stack carries more than its mean, and the rest of it is what makes the mean believable. The width of the velocity distribution at each projected radius is the members’ dispersion, and it falls outwards in the way the Jeans balance of the well predicts; that width, measured from the same galaxies, fixes the typical mass of the stacked clusters without any appeal to the shift. The shape of the distribution — how much it departs from a Gaussian in its wings — says something about whether the orbits are mostly radial or mostly round. And the distribution has to be symmetric about its own centre if the motions are random, so an asymmetry would signal contamination or an error in the reference velocities rather than gravity. The gravitational shift is the one property of the stack that motion alone cannot produce: a displacement of the whole distribution, growing with radius, in the direction of the blue.

The stacked number is an average over clusters of many masses, mostly lighter than $10^{15}$ solar masses, and over the full range of projected radii, which is why it sits below the heaviest cluster’s curve. The prediction it is compared with is the same average taken over a modelled population of clusters, and the agreement depends on the masses assigned to that population. A survey that has measured the clusters’ masses independently — from the X-ray gas, from lensing, or from the velocity dispersion itself — can turn the comparison into a test.

A second shift of the same size #

The figures so far treat the member galaxies as if they were at rest in the well. They are not, and a moving source is a moving clock. The shift that survives at right angles found the part of the relativistic Doppler shift that does not depend on direction: a source moving at speed $v$ in any direction is redshifted by $v^2/2c^2$, because its time runs slow. The first-order Doppler shifts of the members cancel in the average, half towards and half away. The second-order shift does not cancel, because it has only one sign. The clock that runs slow because it is warm found exactly this in a gas of atoms: the first-order shifts only broaden the line, and the time dilation moves its centre.

A cluster’s galaxies are a gas of that kind. If their motions are isotropic with a one-dimensional dispersion $\sigma$, the mean square of the full speed is $3\sigma^2$ and the average redshift from their motion is $3\sigma^2/2c$, which for $\sigma$ near a thousand kilometres a second is about five kilometres a second. The central galaxy, nearly at rest, contributes almost nothing. So the members’ own motion redshifts them relative to the centre, against the gravitational blueshift.

The dispersion in the figure is not a guess. A cluster in equilibrium supports itself against its own weight by the motions of its members, as a gas supports itself by its pressure, and the Jeans equation — the hydrostatic balance of a gas of galaxies — gives the dispersion at each radius from the profile of mass alone. For isotropic orbits it peaks a few hundred kiloparsecs from the centre and falls slowly outwards, which is why the transverse Doppler curve is so flat. At one megaparsec from a cluster of $10^{15}$ solar masses it adds four kilometres a second to the members’ redshift and takes away nearly a quarter of the gravitational signal.

Zhao, Peacock and Li pointed out in 2013 that this term had been left out of the first comparison. Nick Kaiser added two more of the same order in the same year, both from the way galaxies are selected rather than how they move. A galaxy moving towards the observer is seen at a slightly different moment of its history from one moving away, so a snapshot along the past light cone samples approaching and receding members unequally; and the motion of a galaxy brightens or dims it through relativistic beaming, which the brightness that is not the same for everyone worked out for a moving source, so a survey limited by brightness picks out more of one direction than the other. Each is of order $\sigma^2/c$, each has a sign that depends on the details of how the survey selected its galaxies, and together they are as large as the gravitational signal they surround.

Why heavier clusters do not help #

The obvious way to pull the two effects apart would be to compare clusters of different masses, in the hope that gravity’s shift and the motion’s grow differently. They do not.

Both shifts are set by $GM/R$. The depth of the well is $GM/R$ by definition, and the speed at which galaxies move through a well in equilibrium is set by its depth: the virial theorem, which the third law that falls out of a change of scale derived from the scaling of the action, says that for an inverse-square force twice the kinetic energy balances the potential energy, $2\langle K\rangle = -\langle V\rangle$. So $\sigma^2$ is a fixed fraction of the depth, and the transverse Doppler shift, $3\sigma^2/2c$, is a fixed fraction of the gravitational one. The radius $R_{200}$ grows as the cube root of the mass, so both rise as $M/M^{1/3} = M^{2/3}$, two parallel lines on the logarithmic plot. At a fixed concentration their ratio is the same for a group of a few dozen galaxies as for the richest cluster known.

What the ratio does depend on is everything the mass does not fix: the concentration, which sets how much of the depth is near the centre; the shape of the orbits, since radial and circular orbits of the same energy give different dispersions along the line of sight; and the way members are selected. Those are exactly the things a stacked measurement has to assume. The gravitational redshift of clusters is therefore less a measurement of gravity alone than a measurement of gravity together with the dynamics of the galaxies it holds, and the comparison with theory has to model both at once.

Gravity at a distance it has not been tested at #

What the measurement buys, once the kinematic terms are modelled, is a test of how light responds to gravity a megaparsec from the source of the field, which no other test reaches in the same way. Almost everything known about gravity on those scales comes from motion: the velocities of galaxies, the temperature of the gas, the time it takes things to orbit. Motion measures the gradient of the potential. Lensing measures how light is bent, which in general relativity is twice the Newtonian deflection because space is curved as well as time, as the radius a mass adds that no circumference shows found around the Sun. The gravitational redshift measures the potential itself — the time part of the geometry — at the places where the light starts.

Theories that modify gravity on cosmological scales, invented to explain the accelerating expansion of the universe without a cosmological constant, often change how potentials relate to the masses that make them, and some change the time part and the space part differently. A cluster’s redshift profile, set against the mass measured by lensing in the same clusters, compares the two parts directly, in the way the clocks that must all slow together compared two clocks to test whether the redshift is a property of time at all. With the current stacks, the profiles agree with general relativity at the level of their uncertainty, about thirty per cent, and the kinematic corrections are one of the largest parts of that uncertainty.

What the model leaves out #

The figures describe a smooth, spherical well in equilibrium, with galaxies following the mass on isotropic orbits and a central galaxy at rest at the bottom. Real clusters are none of these things exactly. Many are still assembling, with groups falling in along filaments, and a cluster that has recently merged has two or more bottoms and a brightest galaxy that may be moving at a few hundred kilometres a second relative to the well. Stacking averages over that, and the average is what is compared, but an average over disturbed clusters is not the same as the curve for an undisturbed one.

The members are not guaranteed to be members either. Galaxies seen in the direction of a cluster at a similar redshift include interlopers in front and behind, moving with the cosmic expansion rather than in the well, and they spread the distribution of velocities into broad wings. The stacked analyses fit the distribution as a narrow cluster component plus a broad background, and the shift is read off the narrow part; how the two are separated affects the mean at the level of a few kilometres a second.

And the figures cannot show the thing the measurement is ultimately about: whether light leaving a galaxy in a cluster responds to the potential exactly as light leaving the surface of the Earth does. They assume it, and draw what follows. The stacked spectra test it, through a chain of modelling in which the gravitational shift, the motions, the selection and the membership all enter at the same size. The domain of the drawings is clusters of $10^{13}$ to $3\times10^{15}$ solar masses with the stated profile, within twice $R_{200}$, and general relativity’s weak-field redshift.

Still open: whether the stack can separate gravity from the galaxies #

The next generation of spectroscopic surveys will measure tens of millions of galaxies, and the statistical uncertainty on the stacked shift will fall below a kilometre a second. At that point every one of the kinematic and selection effects will matter more than the noise. Whether they can be modelled well enough — from simulations that include the real distribution of orbits, mergers, the motions of central galaxies and the precise selection of each survey — to leave a clean measurement of the potential, or whether the cluster redshift will remain a consistency check rather than a sharp test, is not yet known. A complementary route compares the shift between two populations of galaxies with different mean depths in the same structures, where some of the kinematic terms cancel, and its systematics are being worked out as well.

The physics underneath is the same as in a lift shaft. A galaxy cluster is a well a few tens of kilometres a second deep, so its members’ light, leaving shallower parts of the well than the central galaxy’s, arrives blueshifted relative to it by 5 to 22 km/s — a hundredth of their thousand-kilometre-a-second motions, readable only in stacks of a hundred thousand galaxies, and accompanied by a redshift from the slowing of the members’ moving clocks that is a fixed fraction of it, 4.45 times smaller at every mass, because both are set by GM/R. The cluster weighs its own light, and the galaxies’ motion pays part of the bill.