A dipole a hundred times the signal

The cosmic microwave background is a blackbody at 2.7255 kelvin, uniform across the sky to a part in a hundred thousand. The tiny departures from uniformity are the seeds of every structure in the universe and the basis of most of modern cosmology.

They are not the largest feature in the map. The largest feature is a dipole of 3.36 millikelvin — one part in eight hundred — and it is not cosmological at all. It is the observer’s own motion.

The most perfect blackbody ever measured is uniform to a part in a hundred thousand, and the first thing done to it is to remove a feature a hundred times larger than the uniformity.

Where it comes from #

A blackbody seen from a moving frame is still a blackbody — that is a special property of the Planck spectrum and it is not true of most radiation fields — but at a different temperature. To first order the temperature seen at an angle $\theta$ to the direction of motion is

$$T(\theta) = T_0\left(1 + \beta\cos\theta + \beta^2\cos^2\theta + \dots\right),$$

with $\beta = v/c$. The first term is a monopole, the second a dipole of amplitude $\beta T_0$, and the third contains both a monopole correction and a quadrupole.

For $v = 370$ kilometres a second, $\beta$ is $1.23\times10^{-3}$ and the dipole is 3.36 millikelvin. That is the largest single number anybody measures about the microwave sky and it is entirely local.

It is worth noticing what makes this measurement possible at all: the Planck spectrum’s special property under a boost. A general radiation field seen from a moving frame is not the same field at a different temperature — the spectrum is distorted. A blackbody is, exactly, to all orders in the boost, and the only observable effect is a direction-dependent temperature. That is why the dipole is a clean temperature map rather than a spectral mess, and it is why any spectral distortion associated with the dipole would be evidence that the field is not a perfect blackbody. Measuring the dipole’s spectrum is therefore a test of the blackbody nature at a completely different amplitude from the monopole measurement, and it passes.

It is also worth being clear about which frame is which. The dipole defines a rest frame — the one in which the radiation is isotropic — and that frame has a real meaning: it is the frame in which the universe as a whole is not moving, and it is the frame every cosmological calculation is done in. It is not a violation of relativity that such a frame exists. Relativity says the laws are the same in every frame, not that no frame is picked out by the contents of the universe, and a radiation field filling space picks one out just as an ocean picks out a frame for the fish in it.

What the harmonics do #

The second-order term is where the trouble is. It produces a kinematic quadrupole of amplitude $\beta^2 T_0$ — about four microkelvin — which is a quarter of the cosmological quadrupole and has to be subtracted separately.

Subtracting it requires knowing $\beta$, which the dipole supplies, so the correction is well determined. What is less well determined is that the boost also aberrates the map: the whole pattern is compressed slightly towards the direction of motion and expanded behind, by an angle of order $\beta$ — which is four arcminutes. At the small angular scales modern experiments reach, that is a real distortion, and it couples adjacent multipoles in a specific and predictable way.

That coupling has been detected, and it is a measurement of the same velocity by a completely different route. The two agree, which is the only available check that the dipole is kinematic rather than intrinsic.

The aberration deserves a further sentence because it is the same effect that displaces every star in the sky by twenty arcseconds, applied to a pattern rather than to point sources. For stars the displacement is measured directly because the stars are resolved; for the microwave background the pattern is statistical, so the aberration appears as a correlation between multipoles $\ell$ and $\ell+1$ that would be absent for a statistically isotropic sky. Detecting it means measuring a coupling of order $\beta$ between adjacent multipoles across thousands of them, which is possible only because there are thousands.

There is a second-order effect on the monopole as well, and it is worth mentioning because it is the one that cannot be removed. The boost raises the sky-averaged temperature by $\beta^2/3$ relative to the rest-frame value — about a microkelvin — so the measured mean temperature of 2.7255 kelvin is not quite the temperature in the frame where the radiation is isotropic. The correction is far below the measurement’s uncertainty and it is a reminder that even the monopole, which looks like the one quantity untouched by the observer, carries a term from the observer’s motion.

The contaminant is the calibrator #

The dipole’s role in a microwave experiment is stranger than “a large thing to subtract”, and it is the reason no modern instrument would want it removed at the source.

A microwave detector measures a voltage. Turning a voltage into a temperature requires a source of known brightness on the sky, and there is no such source: the planets are the usual candidates and their brightness temperatures are known to a few per cent at best, which is a hundred times worse than the anisotropy measurement requires. The one signal on the sky whose amplitude is known from first principles is the dipole — because it is a boost of a blackbody whose temperature is measured, at a velocity that appears in the same map.

So the largest contaminant is the primary calibrator, and the accuracy of every anisotropy amplitude ever quoted rests on it.

The Earth’s own orbit makes the argument watertight. Superposed on the 370-kilometre solar motion is the Earth’s 30-kilometre orbital velocity, which changes direction over a year and is known from planetary ephemerides to nine significant figures. That produces an orbital dipole of 270 microkelvin whose amplitude and phase are known absolutely, with no reference to the microwave sky at all. Calibrating on it removes the last circularity: the instrument’s gain is fixed by a signal computed from Newtonian mechanics, and the solar dipole then becomes a measurement rather than an assumption.

The orbital dipole is small — a twelfth of the solar one — and it is the better calibrator for exactly that reason, because its uncertainty is the ephemeris’s rather than the map’s. Its use took the absolute calibration of the recent satellite experiments from a per cent to about a tenth of one, and that improvement propagates directly into every amplitude derived from the power spectrum, including the one that arrives multiplied by an optical depth.

There is a pleasing inversion in this. The observer’s motion is the largest thing in the map, it cannot be removed from the largest angular scale, and it is what makes every other scale measurable. A contaminant with a computable amplitude is not a contaminant; it is an instrument. The same relation holds wherever a known signal happens to sit in a measurement’s way — a flat field measured on the sky itself uses the same logic, turning an unavoidable illumination into the calibration of the thing illuminating it.

What the dipole is worth as a measurement #

The dipole is a nuisance in one respect and one of the field’s most useful measurements in another.

It gives the solar system’s velocity with respect to the frame in which the radiation is isotropic — a frame that has an absolute meaning in cosmology, being the one in which the universe looks the same in all directions. That velocity is 370 kilometres a second, known to a fraction of a per cent, and it is the reference against which every peculiar velocity in the local universe is quoted.

It also decomposes. The Sun’s motion within the Galaxy is about 220 kilometres a second, known from Galactic dynamics; the Galaxy’s motion within the Local Group is about 80; and what is left, some 620 kilometres a second, is the Local Group’s own motion with respect to the microwave background. That residual is a measurement of the gravitational pull of everything within a few hundred megaparsecs, and comparing it with the pull predicted from galaxy surveys is a test of how much matter there is.

There is one more thing the dipole supplies, and it is a constraint rather than a measurement. If the universe were expanding anisotropically — faster in one direction than another — the radiation would carry a quadrupole from that anisotropy, on top of everything else. The observed quadrupole is small, and after the kinematic contribution is removed the residual bounds any global anisotropy in the expansion to about one part in $10^{5}$ over the age of the universe. That is one of the tightest constraints on any departure from the assumed symmetry of the universe, and it comes from the same map, at the same multipole, as the term this essay is about removing.

What was actually measured #

Three results, of three different kinds.

The dipole’s amplitude and direction, to four digits. It is measured by every microwave experiment as a by-product, and the modern value has an uncertainty of a fraction of a per cent in amplitude and of arcminutes in direction. It is the best-measured quantity in the field.

The aberration and modulation signals. The boost’s effect on the small-scale anisotropies — an aberration and a modulation of the amplitude — has been detected at high significance, and gives a velocity consistent with the dipole. That is the confirmation that the dipole is a boost rather than an intrinsic feature.

And the dipoles of other populations. A distant, isotropically distributed population should show the same dipole, for the same kinematic reason. Radio-source and quasar catalogues do show a dipole in the right direction — and with an amplitude about twice what the velocity predicts. That discrepancy is unexplained, it is significant at several standard deviations, and it is one of the more interesting open anomalies in the subject.

Where the picture stops #

Three, and the third is the interesting one.

The subtraction assumes the dipole is entirely kinematic. A genuine intrinsic dipole — from a very large-scale perturbation, or from a departure from homogeneity on scales beyond the horizon — would be indistinguishable from a boost at first order. The aberration measurement breaks that degeneracy in principle, at the precision currently available, and the constraint is not tight.

The decomposition into local motions requires those motions. Extracting the Local Group’s velocity requires the Sun’s motion within the Galaxy and the Galaxy’s within the group, both of which are measured with their own uncertainties. The residual is therefore less precisely known than the dipole itself.

The calibration and the science share a channel. Because the dipole sets the gain, an error in the assumed dipole amplitude rescales every anisotropy amplitude in proportion. That is a fully correlated error across the entire power spectrum — it moves no peak relative to another and moves all of them together — so it is invisible in every internal consistency check and shows up only against an external calibrator. This is the reason the orbital dipole matters so much: it is the only calibration in the chain that does not come from the map being calibrated.

And the radio dipole anomaly is unresolved. If the excess is real, it means the frame in which distant sources are isotropic is not the frame in which the microwave background is — which would be a violation of the assumption that the universe is the same everywhere, at a level nothing else has detected. The alternatives are a selection effect in the catalogues or an unrecognised local structure, and neither has been demonstrated.

There is a fourth, and it concerns what happens at very small scales. The aberration compresses the pattern by four arcminutes towards the direction of motion, which is a large fraction of a modern experiment’s beam. That means the effective resolution of a map is slightly direction-dependent, and the power spectrum measured over the whole sky is a slightly smeared version of the true one. The correction is small, it is calculable from the known dipole, and it is applied — and it is worth noticing that the largest local contaminant reaches into the smallest-scale measurement rather than staying at the largest scale where it lives.

Why the largest signal is the least interesting one #

The general shape is worth naming, because it recurs whenever an observer is inside what is being observed.

A measurement made from inside a system carries the observer’s own state as its largest term. Thirty kilometres a second of the Earth’s motion sits under every radial velocity; twenty arcseconds of aberration sits under every position; three millikelvin of dipole sits under every microwave map. In each case the term is calculable, is removed as a matter of routine, and is far larger than the signal.

What is distinctive here is that the term is not known independently. The Earth’s orbital velocity comes from an ephemeris; the microwave dipole’s velocity comes from the dipole. So the subtraction is a fit rather than a correction, and anything intrinsic at the dipole scale is absorbed into it by construction.

That is a structural blind spot rather than a limitation of the data, and it is worth stating plainly: cosmology has no measurement of the dipole component of the universe’s largest-scale structure, because the observer’s motion occupies that channel entirely. Every result about large-scale isotropy is a result about the quadrupole and above.

A parting observation about how the dipole was received when it was found. It was measured in the 1970s, before the anisotropies were detected at all, and for over a decade it was the only departure from uniformity anybody had seen. That put the field in an awkward position: the theory required fluctuations of some amplitude, none had been found, and the one thing visible was known to be local. The upper limits on the intrinsic anisotropy tightened for fifteen years and the theoretical predictions were revised downward to match, until a detection at eighteen microkelvin finally arrived. Reading that history now, the dipole looks like a distraction; at the time it was the only evidence that the instrument worked.

It is worth ending with the one number that summarises the whole essay. The dipole is 3.36 millikelvin, the kinematic quadrupole is 4 microkelvin, the cosmological anisotropies are 18 microkelvin, and the Galaxy’s own emission at the frequencies used is comparable to the anisotropies. Four quantities spanning three orders of magnitude, of which the largest is the observer, the second-largest is the observer squared, and only the third is what the experiment was built for. That ordering is not unusual — it is the ordinary condition of a measurement made from inside the thing being measured — and the discipline it requires is that each term be known well enough to be removed rather than merely small enough to ignore. The whole sky drifting towards one point is the same statement about positions rather than temperatures, and it took two centuries to separate from the parallax it was hiding.

One number is worth keeping from all of this: the dipole is measured to four significant figures, and every one of those figures is a statement about where the solar system is going rather than about the universe.

Where the ladder goes next #

The natural next rung is the aberration and modulation measurement in detail: how a boost couples adjacent multipoles, what the coupling’s signature is, and how precisely it can separate a kinematic dipole from an intrinsic one. The rung beyond it is the foreground problem — the Galaxy’s own emission, which is the next-largest thing to be removed and which, unlike the dipole, does not have a known spectrum.