The Sun's speed counted in quasars comes out twice too large

Every argument about the expansion of the universe starts from an assumption: on large enough scales the universe is the same everywhere and looks the same in every direction. It is the reason a redshift that grows with distance can be read as a uniform expansion rather than as a centre, and on scales above about a hundred megaparsecs the distribution of galaxies bears it out. The microwave background bears it out more precisely still, uniform to a few parts in a hundred thousand — once one pattern has been removed.

That pattern is a dipole a hundred times larger than everything else in the map: the sky is warmer by about a part in a thousand in one direction and cooler in the opposite one. It is read as motion, not structure. The Sun is moving through the background radiation at 369.82 kilometres per second, and a Doppler shift of that size warms the sky ahead and cools it behind. That reading is so natural that it is rarely examined, and it has a consequence that can be examined. If the Sun is moving at that speed relative to the radiation, and the radiation and the distant matter share one rest frame, then the same motion must produce a dipole in the counts of distant sources, of an amount that can be calculated with no free parameters.

Two ways a motion crowds the sky #

A moving observer sees two things change about a population of distant sources that is, in its own frame, spread evenly over the sky.

Aberration. Light arriving from a direction at an angle to the observer’s motion appears to come from slightly further forward, just as rain falling vertically appears to slant towards a runner. Every source’s apparent position is pushed towards the direction of motion by an angle proportional to the speed, and sources that were evenly spaced crowd together ahead and spread apart behind. To first order in $\beta = v/c$ the number per unit solid angle is raised by a fraction $2\beta\cos\theta$ at an angle $\theta$ from the direction of motion.

Doppler boosting. The light from sources ahead is blueshifted, so each appears slightly brighter at the frequency being observed. A survey counts every source above a fixed flux limit, and brightening the sources ahead pushes some that were just below the limit over it. How many depends on two things about the population. One is how the counts rise towards fainter fluxes: if the number of sources brighter than $S$ goes as $S^{-x}$, then a population with a large $x$ has many sources waiting just below the limit. The other is the sources’ spectrum: if the flux density falls with frequency as $\nu^{-\alpha}$, then the blueshift both raises the intensity and moves the observed frequency to a part of the spectrum that was intrinsically fainter or brighter, and together those make the flux rise as $\delta^{1+\alpha}$. The count above the limit rises by a fraction $x(1+\alpha)\beta\cos\theta$.

Adding the two gives the result derived by George Ellis and John Baldwin in 1984:

$$\mathcal{D} = \left[\,2 + x(1+\alpha)\,\right]\beta.$$

For radio sources, with $x$ close to 1 and a spectral index near 0.75, the bracket is 3.75 and the dipole is 0.0046 — a difference of less than half a per cent between the numbers counted in opposite halves of the sky. For mid-infrared quasars, whose counts rise more steeply and whose spectra fall more steeply, it is 0.0072.

In whole sources the effect is easier to picture. For a dipole $\mathcal{D}$, the hemisphere centred on the direction of motion holds a fraction $\mathcal{D}/2$ more of the sources than an even split would give it. In a catalogue of 1.36 million quasars the expected dipole of 0.0072 puts about 4,900 more quasars in the forward half of the sky than in the backward half, out of 680,000 in each. The measured dipole puts about 10,500 more.

Four minutes of arc that nobody can see #

The aberration behind that crowding is not small in angle. A speed of 370 kilometres per second displaces the apparent position of every distant source by up to $\beta$ radians towards the direction of motion — about four minutes of arc, an eighth of the width of the full Moon. No telescope has ever measured that displacement directly, and none can. Every source on the sky is displaced in the same pattern at once, and there is nothing at rest against which to compare them; a map of the sky made by a moving observer is simply a slightly distorted map, and its distortion is invisible from inside it.

What is visible is a consequence of the distortion that does not need a fixed reference: the compression of sources towards one side of the sky, which changes how many are counted per square degree. That is the aberration term in the dipole. If the speed changes, the displacement changes with it, and that is visible too — the Sun’s acceleration round the Galaxy makes every quasar drift by a few millionths of an arcsecond a year in a pattern converging on the Galactic centre. The counts measure the speed; the drift measures its rate of change.

The steepness of the counts sets the size #

Aberration contributes the same $2\beta$ to every population. The boost is the part that depends on the sources, and it can be larger.

Doubling the steepness of the counts, from $x = 1$ to $x = 2$, adds half again to the dipole. That is why the counts themselves must be measured for each survey at its own flux limit, and why the shape of a population’s count — how the number of objects rises towards fainter ones — is not a detail but half the prediction. It also means that a measured dipole cannot be converted into a speed without that measurement, and that a mistake in $x$ or $\alpha$ is a mistake in the inferred speed.

For the quasar sample in the first figure, $x$ and $\alpha$ are measured from the catalogue itself, at the flux limit used. The prediction is then fixed. No property of the universe enters except the speed and the assumption that the quasars, on average, are at rest in the same frame as the radiation.

A few thousandths needs a million sources #

A dipole of a few parts in a thousand is very small against the random scatter of counting.

If $N$ sources are scattered randomly over the sky, a dipole fitted to them has an uncertainty of about $\sqrt{3/N}$ from the counting alone, so a three-standard-deviation detection needs $N = 27/\mathcal{D}^2$. The kinematic dipole of radio sources needs over a million, which is roughly the size of the largest radio surveys of the whole sky, and the quasar dipole half a million. That is why the test proposed in 1984 waited decades for catalogues large enough to make it.

Counting noise is the floor, not the whole error. A dipole is the largest pattern there is on the sky, and anything that affects a survey’s depth differently in different directions imprints a dipole of its own: the obscuring dust of the Milky Way, which is concentrated towards the Galactic plane and dims infrared sources as well as optical ones; the regions near the plane that have to be masked out entirely, which leave a sky that is no longer symmetric; the pattern in which a satellite scanned the sky, which gives some directions more exposures than others. Removing those is most of the work, and every one of them is a possible source of an error of the size being sought.

What the counts show #

Radio surveys of the whole sky were the first catalogues large enough. The first analysis of a survey of nearly two million radio sources, published in 2002, found a dipole consistent with the microwave background’s in both direction and size, within uncertainties that were then large. As the treatment of the survey’s systematics improved and more surveys were added, later analyses found dipoles still pointing roughly towards the microwave background’s direction but two to four times larger than expected — with uncertainties large enough, and systematics different enough between surveys, that the excess could not be treated as settled.

The quasar measurement is the one in the first figure. A catalogue of about 1.4 million quasars selected from an infrared survey of the whole sky, with its dust, masks and scanning pattern modelled and removed, gave a dipole of 0.0155 against the expected 0.0072. Its direction lay within about thirty degrees of the microwave background’s — close, given the noise, which is itself a sign the signal is related to motion. Its size was 2.16 times the prediction, and taken at face value it would mean the Sun is moving at 797 kilometres per second relative to the distant quasars, while moving at 370 relative to the radiation. The discrepancy was reported at close to five standard deviations, and a combined analysis with radio sources strengthened it. Other analyses of the same kind of data, treating the systematics differently, have found the tension weaker.

Not a matter of which local speed #

It is natural to ask whether the wrong speed has been used. The Sun is not the only candidate observer; the whole Local Group moves relative to the microwave background at about 620 kilometres per second, a velocity built up by the gravity of large-scale structure over the universe’s history.

The dipole of the counts depends on the observer’s own velocity, which is the Sun’s, so 620 kilometres per second is not the right number. But even with it the quasars’ dipole is 1.29 times the prediction: the gap is not a matter of choosing between local speeds. The speed the measurement requires, 797 kilometres per second, is larger than any velocity of the Sun, the Galaxy or the Local Group relative to the radiation.

The same comparison shows how the two catalogues differ in power. At the Sun’s real speed a radio survey needs well over a million sources to see its expected dipole at all, and the largest have about that many. The quasar catalogue, with a steeper count and a larger expected dipole, has more than twice the sources a detection needs. The quasar result is therefore the stronger test, which is why its disagreement carries more weight than the radio ones did.

Where a systematic could hide #

Dust. Infrared light is dimmed far less by interstellar dust than visible light is, but not by nothing, and the quasar selection uses colours that dust changes. A residual pattern of dust across the sky, imperfectly removed, adds a dipole whose direction depends on where the Galaxy’s dust lies.

Nearby structure. Sources in a catalogue are not all at great distances. The ones that are nearby trace the clustering of the local universe, whose own uneven distribution adds a dipole unrelated to motion. For quasars at redshifts of one or two that contribution is expected to be small, and for radio catalogues, which contain many nearby galaxies, it is larger.

The survey’s own geometry. A satellite that scans the sky in great circles through the ecliptic poles observes those poles most often and reaches fainter fluxes there, and a catalogue with a flux limit that varies over the sky has a count that varies too. That pattern has to be modelled from the survey’s exposure map, and any error in the model is a dipole.

And the reference frame. The quasars are the most distant fixed points available, and they define the celestial reference frame against which spacecraft are steered and the Sun’s acceleration round the Galaxy is measured. Their positions are exquisitely known; their numbers, near a survey’s faint limit, are much harder to make uniform.

If the matter and the radiation disagree #

If the excess survives, there are two broad readings. One is that the microwave background’s dipole is not entirely kinematic: part of it is a real difference in the radiation across the sky, left from the early universe, and the Sun’s true speed relative to the distant matter is larger than 370 kilometres per second. That is testable within the background itself, because a genuine motion also aberrates and boosts the small-scale pattern of the background’s fluctuations, and the Planck satellite detected that effect, at a speed consistent with the dipole but with an uncertainty of about a third — not yet enough to exclude a large intrinsic part.

The other is that the matter at the distance of the quasars is not at rest in the frame of the radiation: a flow on scales of billions of light years, or an anisotropy in the universe itself. The quasars in the catalogue lie mostly at redshifts above one, so their light left them more than eight billion years ago, and the frame they define is that of matter spread across a large fraction of the observable universe. Closer in, galaxies within a few hundred megaparsecs do move together in coherent flows, velocities that are not the expansion, and some surveys of those flows have reported them faster and larger than the standard model expects. None of them reaches the scale of the quasars, and a flow that did would have to be of a kind no model of structure formation produces. Either reading means that the universe does not look the same in every direction in the way the whole standard account assumes, on the largest scales anyone can measure, and it would be the first time that assumption had failed a direct test. That is why the measurement is worth the difficulty, and why the systematics are argued over with such care.

A test of the frame, not of the speed #

The Ellis–Baldwin test does not measure the Sun’s speed; that is known far better from the microwave background. It asks whether two very different components of the universe — the relic radiation and the distant matter — agree about which frame is at rest. The expected answer is that they do, to within the precision of the counts. The measured answer, so far, is that they agree about the direction and disagree about the size by a factor of two. The ways in which that could be wrong are specific and checkable, and the next generation of surveys of the whole sky, with more sources and independent systematics, will count the quasars again.

Those surveys change the arithmetic of the third figure by orders of magnitude. Radio telescopes now being built are expected to catalogue hundreds of millions of sources, enough that counting noise falls far below the kinematic dipole and the measurement becomes limited entirely by systematics. Optical and infrared surveys of the whole sky will provide galaxies and quasars selected in different ways, with dust, depth and scanning patterns that are different from the catalogues used so far. A dipole that comes out at the same excess in catalogues with unrelated systematics would be very hard to attribute to any of them. One that shrinks towards the prediction as the catalogues improve would say that the matter and the radiation agree after all, and that the principle survived its sharpest test.

Still open: whether the light from distant galaxies is dimmed as expansion requires #

The cosmological principle is one assumption behind reading redshifts as expansion; another is that the redshift is caused by expansion at all. An expanding universe dims the surface brightness of distant galaxies by a specific factor, the fourth power of $(1+z)$, far more steeply than any static universe with light that merely tires on its journey — and that dimming can be measured in galaxies whose sizes and luminosities are understood well enough to compare.