A speed read from the finest spots in the map

The largest thing in a map of the microwave background is a dipole of 3.36 millikelvin, hot towards one direction and cold towards the opposite, and it is subtracted before anything else is done. It is read as the solar system’s motion through the radiation at 370 kilometres a second, and a velocity that large through a blackbody makes it hotter ahead and cooler behind by exactly the observed amount.

The reading is an assumption. Nothing in the dipole itself distinguishes a temperature difference produced by motion from one the universe might have had anyway — a very large-scale fluctuation, a gradient across the whole observable volume, or a dipole left by something before the hot big bang. Both look identical: a cosine on the sky. If part of the observed dipole were intrinsic, the inferred velocity would be wrong by that part, and nothing about the dipole would show it.

What breaks the symmetry is that motion does more than add a cosine. It also distorts everything else in the map, by an amount that depends only on the velocity, and the anisotropies are millions of independent spots each of which is distorted in the same calculable way. Measuring the distortion measures the velocity without reference to the dipole’s amplitude. If the two agree, the dipole is kinematic. If they disagree, the difference is the universe’s.

Four arcminutes, applied to a pattern #

The distortion is aberration, the same effect that moves every star in the sky by twenty arcseconds over a year as the Earth swings round the Sun. A moving observer collects light as if it arrived from slightly further forward than it did, and the displacement is the observer’s speed as a fraction of light’s, multiplied by the sine of the angle from the direction of motion. For the Earth’s orbital speed of thirty kilometres a second that is twenty arcseconds. For the solar system’s 370 kilometres a second relative to the background it is 4.2 arcminutes, about a seventh of the Moon’s diameter.

The exact relation is not a displacement but a mapping of the whole sphere onto itself, $\cos\theta' = (\cos\theta + \beta)/(1 + \beta\cos\theta)$, which crowds every source towards the apex of the motion. At this speed the first-order version is exact to a part in a thousand of itself, and the figure is computed from the exact formula anyway.

For stars the displacement is measured directly, star by star, because each star is a point with a position that can be compared with where it would be without the motion. The microwave background has no points. Its anisotropies are a statistically uniform random field — hot and cold patches with a characteristic size, but no individual patch has a known place in the absence of the observer. Moving all of them four arcminutes towards the apex would be invisible: there is nothing to compare against.

What is visible is that the displacement is not the same everywhere. It is zero at the apex and the antapex and largest at ninety degrees, so it varies across the sky, and a displacement that varies is a distortion of scale. Near the apex, neighbouring points are displaced by nearly the same tiny amount but converge on the apex, and every angle between them is reduced by a factor $1 - \beta$. Near the antapex every angle is enlarged by $1 + \beta$. A pattern of spots that is statistically the same everywhere on the true sky arrives, on the observed sky, slightly finer in front and slightly coarser behind.

The spectrum ahead is the spectrum behind, shifted #

A change of scale has a particularly simple effect on the angular power spectrum, and it is the reason the measurement works.

The power spectrum counts how much of the temperature variation sits at each angular scale, with multipole $\ell$ standing for an angle of about $180°/\ell$. Squeezing the pattern by a factor moves every scale to a smaller angle and every feature of the spectrum to a higher multipole by the same factor. For the quantity usually plotted, $D_\ell = \ell(\ell+1)C_\ell/2\pi$, which is the variance per logarithmic interval of scale, a change of scale is exactly a shift along the logarithm of the multipole, with no change of height. The local spectrum measured ahead of the motion is the true spectrum shifted right by $\ln(1+\beta)$, and the one measured behind is shifted left by the same amount.

The difference between the two is therefore the spectrum’s derivative times twice the shift. Where the spectrum is flat, at the top of a peak or the bottom of a trough, a small shift changes nothing and the two directions agree. Where the spectrum is steep, on the flanks of the peaks and throughout the damping tail, the shift moves the curve up or down measurably, and the difference ahead minus behind is a per cent or two of the local power.

A per cent is not a small effect by the standards of what is measured in the background. The acoustic peaks themselves are measured to better than a per cent, and the difference between two hemispheres’ spectra is a comparison of two measurements of similar quality. The problem is not the size of the effect but how many modes it can be measured from, and that is set by the slope.

The slope is the lever #

Every multipole carries information about the velocity in proportion to the logarithmic slope of the spectrum there, $d\ln D/d\ln\ell$, because that slope is what turns a shift into a change of power.

Through the acoustic peaks the slope swings positive and negative, reaching magnitudes of five or so on the steep flanks, and passes through zero at every extremum. In the damping tail beyond multipole 1500 it stays negative and grows steeper: the spectrum is falling exponentially as photon diffusion erases structure smaller than its random walk, and the logarithmic slope there is minus five, minus six and more.

Two things therefore push the information towards the smallest scales together. The lever is largest there, and the number of independent modes the sky contains at multipole $\ell$ is $2\ell + 1$, so a band at $\ell = 2000$ contains ten times as many as one at $\ell = 200$. A measurement of the velocity from the anisotropies is a measurement from the damping tail, and the finest spots in the map, those of a few arcminutes, carry most of it.

That is an unusual situation for the background. Almost every other parameter it measures — the curvature, the baryon density, the spacing of the acoustic peaks — is read mostly from the first few peaks, at scales of a degree or less where the signal-to-noise per mode is highest. The velocity is read from scales where each mode is weak and the measurement is a statistical average over millions of them.

The same thing, as a coupling between neighbours #

The same effect has a second description, in the harmonic coefficients rather than the local spectrum, and the second description is how it is actually measured.

A statistically isotropic sky expanded in spherical harmonics has coefficients $a_{\ell m}$ that are independent of one another: knowing the coefficient at one multipole and one orientation says nothing about any other. That independence is what isotropy means in harmonic space. A boost breaks it in a specific way. Aberration mixes each coefficient with its neighbours at $\ell \pm 1$ and the same $m$, measured about the direction of motion, by an amount proportional to $\beta$ — so the boosted sky has small correlations between adjacent multipoles that the true sky does not.

The two descriptions are one statement. A dipolar modulation of local power across the sky is, in harmonic space, a coupling of each multipole to the ones next to it, because a dipole is the $\ell = 1$ harmonic and multiplying by it moves every other harmonic by one. Estimators built to measure the coupling sum the products $a_{\ell m}a^*_{\ell+1,m}$ over every multipole and orientation with weights set by the spectrum’s slope and the noise, and they find the velocity’s three components at once.

It is the same kind of estimator, adapted, that measures the lensing of the background by the matter between here and the last-scattering surface — the effect that also limits how cleanly the damping tail can be read — which likewise couples modes that isotropy would leave independent. Lensing couples them over a range of separations set by the lensing structures. A boost couples only neighbours, with one direction and one amplitude for the whole sky, and that is how the two are told apart.

A second effect of the same size #

Aberration is not the only thing a boost does to the anisotropies. It also changes their amplitude.

A moving observer sees radiation from the forward direction blueshifted and from behind redshifted. For a perfect blackbody the shift preserves the blackbody shape and changes only its temperature, which is the dipole. But the anisotropies are departures from a blackbody’s temperature, and a fluctuation measured at a fixed observing frequency is boosted along with everything else: a hot spot ahead of the motion appears very slightly hotter than it would at rest, and one behind slightly less so. That modulates the amplitude of the anisotropies by a factor $1 + b\beta\cos\theta$ across the sky.

The coefficient $b$ depends on the observing frequency and on the units the map is made in, because what is being modulated is the intensity at a fixed frequency and the conversion between intensity and temperature fluctuation depends on where on the blackbody curve the measurement sits. It is of order one at the frequencies where the background is brightest and rises towards higher frequencies. The modulation has the same dipolar pattern on the sky as the aberration and is of the same size, but it does not depend on the slope of the spectrum — it is a uniform change of amplitude, present at every multipole including the peaks.

A real measurement fits both together, and the frequency dependence of the modulation is a check: aberration is the same at every frequency, because it is geometry, and the modulation is not. The figures here compute the aberration alone.

How many standard deviations the sky holds #

An estimate of how well the anisotropies can measure the velocity comes from counting. At each multipole there are $2\ell + 1$ independent modes, each giving an estimate of local power with an uncertainty equal to the power itself. A dipolar modulation of fractional size $\beta$ times the slope, over a whole sky, is measured from those modes with an information content proportional to their number times the slope squared, divided by six — a factor of three from averaging the square of the cosine over the sphere and two from the variance of a power estimate.

Summed to the smallest scales a map can resolve, that counting gives the signal-to-noise on the speed. For a map with Planck’s noise and its five-arcminute resolution, the aberration alone is worth about two and a half standard deviations, and almost nothing is gained beyond multipole 2000 because the beam has smeared those scales below the noise. A noiseless whole sky to multipole 2500 is worth about six. A deep, sharp ground-based map over a smaller area falls between the two at multipole 2500 and keeps gaining beyond it, where the measured spectrum used here stops but the damping tail does not.

These numbers are a scaling rather than a forecast to quote. The estimate leaves out the sky that has to be masked because the Galaxy is in front of it, the amplitude modulation, which adds information, and the polarisation, which adds more, and it treats every multipole as independent when a real mask couples them.

What was actually measured #

The measurement was made in 2013, from Planck’s first-year maps at 143 and 217 gigahertz, and it was the first detection of the effect.

The estimator was the harmonic one: the coupling between each multipole and its neighbours, weighted by the spectrum’s slope, summed from multipole 500 to 2000 over the parts of the sky not masked for the Galaxy and for point sources. Aberration and modulation were fitted together, as a single boost with frequency-dependent amplitude.

The result was a speed of 384 kilometres a second, with a statistical uncertainty of 78 and a systematic one of 115, in a direction consistent with the dipole’s. Against the dipole’s own 370 kilometres a second, measured to a fraction of a per cent, that is agreement within the uncertainties. The statistical significance was about five standard deviations, and the systematic allowance — the noise model, the beam’s asymmetries, the masking — reduced the overall confidence considerably.

The sign was right. The spectrum is shifted towards higher multipoles in the direction the dipole is hottest, which is the direction of motion, and towards lower multipoles behind. An intrinsic temperature gradient would not have done that.

The measurement was published under the title “Eppur si muove” — the phrase attributed to Galileo after his recantation — and the title was fair. The solar system’s motion through the background had been inferred from the dipole since the 1970s. This was the first time it was seen in something other than the dipole.

Kinematic or intrinsic #

The question the measurement exists to answer is how much of the dipole could be something other than motion.

If the dipole contains an intrinsic part, the speed inferred from its amplitude is wrong by that part, while the speed inferred from the aberration is not, because aberration depends only on the observer’s velocity. The difference between the two speeds, as a fraction of the dipole, is therefore a direct measurement of the intrinsic fraction — and the precision of the aberration measurement is the precision of the bound.

By this counting, a Planck-like map leaves the question nearly open — an intrinsic part as large as most of the dipole is allowed at two standard deviations from the aberration alone — and even a perfect map to multipole 2500 bounds it only to about a third. Smaller scales help, because the damping tail goes on and its modes keep coming, but cosmic variance sets a floor: the sky contains a finite number of modes at any scale, and no instrument can measure more of them than exist.

That is a sobering bound for something so routinely assumed. The kinematic reading of the dipole is supported by much else: the dipole’s direction agrees with the motion expected from the Local Group’s infall towards nearby concentrations of galaxies, the velocity reconstructed from the surrounding galaxy distribution converges on roughly the right amplitude, and an intrinsic dipole of the observed size would require a large-scale fluctuation of a kind nothing else in the background shows. None of those is a measurement of the same velocity by the same means, and the aberration is.

It matters because another measurement of the same velocity disagrees. The dipole in the number counts of distant quasars and radio galaxies — which should be produced by the same motion, through aberration and Doppler boosting of the sources — comes out about twice as large as the microwave background’s dipole implies, at a significance several samples now support. Either the matter dipole is partly intrinsic, or the radiation dipole is, or the two frames are not the same frame. The aberration of the anisotropies is the one measurement that asks the radiation side of that question without assuming the answer, and its current precision is not enough to answer it.

Where the estimate stops #

The spectrum is an interpolation through measured peaks and troughs. Its slope, which is the lever, is therefore known only at the resolution of the measured extrema, and between them it is a representation of the measurement rather than the measurement itself. The figures’ numbers would move by tens of per cent with a better-resolved spectrum and their shapes would not.

The forecast counts modes on a whole sky, weighted by each map’s noise and beam, with no mask and no coupling between multipoles. A real mask removes a third of the sky and couples every multipole to its neighbours — the very coupling the estimator looks for — so the analysis must model the mask’s coupling and subtract it, which costs precision.

Only aberration is computed. The amplitude modulation adds signal of the same order whose size depends on frequency, and the polarisation, whose spectrum has steeper flanks than the temperature’s, adds more. A combined forecast would be somewhat better than any curve drawn.

The noise models are representative, not those of any particular instrument.

Seeing motion without anything to see moving against #

The general shape is that a velocity can be measured from a pattern with no fixed points in it, so long as the pattern’s statistics are known to be the same everywhere.

A star’s aberration is measured by comparing a position with itself six months later. The background’s is measured by comparing the statistics of one hemisphere with those of the other, at one instant, with no individual feature followed and no reference position known. What makes it possible is the assumption — itself tested to high precision, above a hundred megaparsecs and across the whole sky — that the true pattern is statistically isotropic, so any systematic difference between directions must come from the observer.

A symmetry that is assumed becomes a measuring instrument the moment something is known to break it. Statistical isotropy is the reason the anisotropies can be described by a single power spectrum; the observer’s motion is the one thing guaranteed to break it, by a calculable amount; and the breaking, measured, is a velocity. The same argument, run the other way, is the test of isotropy itself: once the motion is removed, any remaining difference between directions is a property of the universe.

The other reading is about where information lives. The velocity is measured mostly from scales of a few arcminutes, where each spot is barely above the noise and the spectrum is falling steeply, rather than from the bright degree-scale peaks that carry most of the background’s cosmology. A measurement’s best data are not always its loudest; here they are its most numerous and its steepest.

Still open: the frame the background is at rest in #

The aberration measurement leaves the dipole’s interpretation where it has been since the 1970s — kinematic, consistent with everything — but bounded far less tightly than the dipole’s own precision suggests. The next maps, deeper and sharper and at more frequencies, will push the bound down towards the cosmic-variance floor, and the polarisation will add a second, independent version of the same measurement with different systematic errors.

What they are aiming at is a question about frames. The background defines a rest frame, the one in which its dipole vanishes, and the distant matter of the universe defines another, the one in which the number counts of the most distant sources are isotropic. The cosmological principle says the two are the same frame. The counts say they differ by a velocity comparable to the one being measured, and whether that difference is in the matter, in the radiation, or in the counting is the question the finest spots in the map are now being asked.