Polarised only where the glow was lopsided

The microwave background is polarised at about a tenth of the size of its own anisotropy, which makes the polarised signal a few microkelvin on a sky whose total brightness is 2.7 kelvin. Martin Rees predicted it in 1968, and the first detection came thirty-four years later, in 2002, of a signal smaller than the temperature fluctuations by a factor of ten.

That the light is polarised at all is not obvious. The background was released by a plasma becoming transparent, and the last thing each photon did was scatter off a free electron. Scattering off electrons polarises light, so a polarised background might seem the expected thing. It is not, and the reason it is not is the whole of this essay: almost every radiation field an electron can sit in scatters into light with no net polarisation at all. The exception is narrow, it existed only for a short time, and its existence is a statement about what the plasma was doing when it let the light go.

What one scattering does #

Thomson scattering is the elastic scattering of light by a free electron, and its polarisation rule is simple. An electron driven by an incoming wave oscillates along the wave’s electric field and re-radiates like a small antenna. An antenna does not radiate along its own axis. So light scattered through ninety degrees comes out polarised perpendicular to the plane containing the incoming and outgoing rays, because the component of the electron’s motion that would have produced the other polarisation points straight at the observer and radiates nothing in that direction.

For one incoming beam the result is strongly polarised light. An electron in the early universe is not lit by one beam. It is bathed in radiation arriving from every direction at once, and the light it sends towards an observer is the sum of contributions from all of them. Each contribution is polarised perpendicular to its own scattering plane, and the planes point every way round the line of sight.

Write the incoming brightness as a function on the sphere of directions around the electron, and the scattered polarisation as a weighted sum over that sphere. The weight, for light arriving from polar angle $\theta$ and azimuth $\phi$ about the outgoing direction, is $\sin^2\theta\,e^{2i\phi}$ — the projection of each contribution’s polarisation onto a pair of fixed axes, with the factor of two in the exponent coming from the fact that a polarisation direction is a line and not an arrow, so it repeats after half a turn.

That weight is a pure quadrupole. On a sphere, functions of different multipole order are orthogonal, so integrating a quadrupole weight against any incoming brightness picks out the quadrupole part of the brightness and discards everything else. A uniform glow, the monopole, contributes nothing: every polarisation is cancelled by one at right angles to it. A glow brighter on one side than the other, the dipole, contributes nothing either: the excess from the bright side and the deficit from the dim side scatter into the same polarisation with opposite signs. Orders three, four and higher vanish for the same reason as the dipole.

The bars in the opening figure are that integral done numerically rather than by appeal to orthogonality. The dipole’s residual is at the level of rounding error, and the quadrupole’s is fifteen per cent of its own amplitude. A polarised photon from the early universe is therefore a record of exactly one thing: that the electron which last scattered it saw more light along one axis than across it.

Why the bath was nearly perfect #

Deep in the plasma before recombination, the electrons saw almost no quadrupole at all, and the reason is the same tight coupling that makes the plasma behave as a fluid.

A photon in the hot plasma travelled a very short distance between scatterings. Any anisotropy in the radiation around a given electron could only be built from photons that had arrived from within about one mean free path, and over one mean free path the plasma was nearly uniform. The radiation field at each point was isotropic in the plasma’s own rest frame, except for a dipole: the plasma was moving, sloshing in and out of the potential wells that the acoustic oscillations describe, and an electron moving through radiation sees it hotter ahead and cooler behind. A dipole, as the integral has just shown, polarises nothing.

A quadrupole requires something more. The plasma’s velocity has to differ across the distance the photons have come from. A photon arriving from one side has come from fluid moving at one speed, a photon from the other side from fluid moving at another, and the difference between them at the electron is a gradient of velocity across a free path. That gradient is quadrupolar in shape — the radiation looks hotter along the axis of the gradient, from both directions, than across it — and its size is proportional to how far the photons have streamed:

$$\Theta_2 \sim k\,\lambda\,v,$$

with $k$ the wavenumber of the fluid’s oscillation, $\lambda$ the photons’ free path and $v$ the fluid speed. While the plasma was opaque, $k\lambda$ was tiny for every wave of interest, and so was the quadrupole. The polarisation is generated only when the free path becomes long enough to let photons carry information about the fluid’s motion from one place to another, and that happens only as the plasma recombines.

The back of a thin shell #

The surface of last scattering is not a surface. Recombination took a finite time, and the probability that a photon arriving today last scattered at a given redshift — the visibility — is spread over a range of about two hundred in redshift around $z \approx 1080$. The polarisation is made inside that range, and not uniformly through it.

The source of polarisation at each redshift is the chance that a photon last scattered there multiplied by the quadrupole it would have seen when it did, and the quadrupole grows with the free path. Early in recombination the visibility is rising but the free path is still a few megaparsecs, comoving, and little polarisation is made. As the electrons recombine, the free path lengthens rapidly, roughly doubling between the visibility’s peak and the polarisation source’s, and the quadrupole grows with it. Late in recombination the free path is long but almost no photon is still scattering, so the visibility is falling away.

The product peaks later than the visibility. Computed with a three-level hydrogen atom and the free path capped once it exceeds the wavelength of a wave at multipole 500 — beyond which streaming further adds no more gradient — the source peaks near $z \approx 1030$, against the visibility’s 1080. The polarisation is made towards the back of the shell, in the last part of the transition, by electrons that were about to stop scattering anything.

This is the sense in which polarisation is a purer measure of the shell’s thickness than the damping of the small-scale anisotropies. The damping depends on the thickness because photons random-walk through it and smear small scales away. The polarisation is caused by it. An instantaneous recombination, with the plasma opaque one moment and transparent the next, would leave the photons no time in which their free path was long enough to see a gradient and short enough to still be scattering. It would produce a sky with acoustic peaks in its temperature and no polarisation at all.

Polarisation follows the velocity, not the density #

The quadrupole is built from the fluid’s velocity gradient, not from its density. That one fact fixes where the polarised signal sits in the angular power spectrum, and it does so in a way that makes the polarisation an independent test of the whole acoustic picture.

The temperature anisotropy is dominated by the fluid’s compression at the moment of last scattering: a region caught at maximum compression is hot, one caught at maximum rarefaction is cold, and either extreme gives a peak in the temperature power spectrum. A fluid at the extreme of an oscillation is momentarily at rest. Its velocity is largest a quarter of a cycle later, when it is passing through its mean density on the way from one extreme to the other.

So the polarisation peaks should fall between the temperature peaks. In the tight-coupling sketch drawn here, with the acoustic scale computed from the sound horizon and the distance to last scattering and the measured phase of the temperature peaks put in by hand, the temperature peaks sit near $\ell \approx 220$, 515, 810, 1110 and 1410, and the polarisation peaks near 130, 390, 680, 975 and 1270 — interleaved, each roughly midway.

The sketch is crude. It has no projection from three dimensions onto the sky, no decay of the gravitational potentials as the universe stops being radiation dominated, no thickness of last scattering, and it leaves out the baryons’ weight, which changes the heights of the temperature peaks and not where they are. It is drawn for one purpose, which is to show that the positions follow from the phase relation between a displacement and a velocity, and it serves that purpose because the positions do not depend on what it leaves out.

The measured spectra agree. The polarisation power spectrum measured by Planck and by ground-based experiments has its peaks at $\ell$ of about 140, 400, 700, 1000 and 1300 — between the temperature peaks, out of phase with them, and as sharply defined. The peaks in the polarisation are in fact sharper than those in the temperature, because the temperature is a sum of the compression term and a Doppler term from the velocity, a quarter-cycle apart, and the Doppler term partly fills in the temperature troughs. The polarisation has only one source and no second term to blur it.

A sign that changes twice a cycle #

The two signals come from the same oscillating fluid, so they are correlated with each other, and the correlation carries information neither carries alone.

A compression and a velocity a quarter-cycle apart multiply into a product that changes sign every quarter-cycle: it passes through zero whenever either factor does. The cross-spectrum of temperature against polarisation therefore oscillates twice as fast as either auto-spectrum, crossing zero about twice for every temperature peak.

This pattern is a test the temperature spectrum alone could never set. A temperature power spectrum with peaks at the right places is evidence of oscillation, but a sufficiently ingenious alternative could arrange peaks at those places without an oscillating fluid behind them. Several did, before 2000: defect models and other non-inflationary seeds produced temperature spectra with bumps, and were ruled out by the bumps’ positions and spacing rather than by their existence. The cross-spectrum requires more. It requires that the temperature and the polarisation come from one oscillator with one phase, and that the polarisation be generated by its velocity.

The measured cross-spectrum is the most direct evidence of that. The WMAP satellite’s first-year maps measured it at high significance well before the polarisation’s own spectrum could be measured as precisely, because it is the geometric mean of a large signal and a small one and so stands further above the noise than the small one alone. Its sign changes lie where the acoustic picture puts them, including one particular negative dip near $\ell \approx 150$ that only adiabatic fluctuations — perturbations of the total density with every component moving together — can produce on scales that were larger than the horizon at last scattering. That dip is the strongest observational statement that the fluctuations were already in place, coherently, before any causal process inside the horizon could have arranged them. It is the same conclusion inflation was invented to explain, reached by an entirely different route.

Why only a few per cent #

The polarisation is about a tenth of the temperature anisotropy in amplitude at its peaks, and the reason is in the formula for the quadrupole. The quadrupole is the velocity times $k\lambda$, the ratio of the free path to the wavelength, and at the epoch where polarisation is made that ratio is still less than one for the waves that make the acoustic peaks.

Two consequences follow, and both are measured.

The polarised fraction rises towards smaller scales. Larger $k$ means a larger $k\lambda$ and a bigger quadrupole for the same velocity, so the ratio of polarisation to temperature increases with multipole through the acoustic peaks — until the damping erases both at the same scale and the comparison stops.

On the largest scales there is almost nothing. A wave much larger than the free path sees a nearly uniform velocity across it, and produces almost no quadrupole. The polarisation spectrum from last scattering falls steeply towards low multipoles, and below $\ell$ of about thirty it would be nearly zero — were it not for a second source.

The second source is a second scattering. After the universe was reionised by the first stars and galaxies, hundreds of millions of years after recombination, free electrons filled intergalactic space again, and a small fraction of the background photons scattered off them. Those electrons were lit by the background radiation itself, and each of them saw, as its incoming radiation field, the last-scattering surface around it — a sphere of radius several gigaparsecs, with the temperature anisotropy written on it. That sphere has a quadrupole. The scattering polarised it.

The quadrupole a reionised electron sees is set by the size of its last-scattering sphere, and projected onto today’s sky from the distance of reionisation it lands near multipole three to five — a pattern of a few hot and cold patches covering the entire sky. The amplitude of that large-scale polarisation is proportional to the fraction of photons scattered, the optical depth $\tau$, and its position tells when the scattering happened.

That makes the lowest-multipole polarisation the only direct measurement of the optical depth, and therefore of when the universe was reionised. The measured value, $\tau = 0.054$, corresponds to a sudden reionisation near redshift 7.7 — and it is measured from a handful of modes at the scale of the whole sky, where the Galaxy’s own polarised emission is brighter than the signal at every frequency and where the instrument’s systematic errors are largest. It took three satellites to settle: the first published value, from WMAP’s first year, was about three times the present one, and it came down as the polarised foregrounds and the instruments’ own systematic errors were understood. It mattered beyond reionisation, because the temperature spectrum measures only the product of the fluctuations’ amplitude and the scattering’s damping, and the large-scale polarisation is what separates them.

What was actually measured #

The polarisation of the microwave background is measured as two Stokes parameters at every point of a map, and the step from those to anything physical is the same decomposition that turns any polarisation map into two independent fields.

Q and U at each point. A polarimeter measures the difference in power between two orthogonal linear polarisations, and a second set rotated by forty-five degrees. The two differences are $Q$ and $U$, which are not quantities on the sky but quantities relative to the instrument’s axes: rotate the instrument and they mix.

E and B over the sky. Because $Q$ and $U$ depend on axes, the physical description of a polarisation pattern is in terms of its geometry. A pattern can be decomposed into a part whose polarisation directions line up with or perpendicular to gradients of an underlying scalar — the E mode, with no handedness — and a part whose directions sit at forty-five degrees to such gradients, the B mode, which has a handedness. Scattering of a quadrupole sourced by density perturbations produces E modes only. That is why the essay has spoken of E-mode polarisation throughout: the acoustic oscillations, the reionisation bump and the cross-spectrum are all E.

The power spectra. From the E field and the temperature field come three spectra: temperature, E, and their cross-correlation. All three are now measured from the largest scales to multipoles of several thousand, and the cosmological parameters fitted to all three together agree with those fitted to the temperature alone, at a precision the temperature alone does not reach.

Where the sketch stops #

The acoustic figures are a sketch. They take the tight-coupling solution for the fluid’s compression and velocity at one instant, multiply by a damping factor, and read the multipole off the wavenumber by $\ell = kD$. A real calculation integrates the coupled photon, baryon, neutrino and dark-matter perturbations through recombination, follows the polarisation hierarchy, and projects the result onto the sky with spherical Bessel functions that smear each wavenumber over a range of multipoles. The positions of the peaks survive the simplification; their shapes and heights do not, and none is claimed.

The source figure uses one wavelength. The cap on the free path’s effect was set for a wave at multipole 500. A wave at higher multipole saturates sooner, so its polarisation is made a little earlier, and the shift drawn is representative rather than universal.

Helium is left out of the recombination, which moves the visibility by a few per cent in redshift, and the hydrogen is a three-level atom. Neither affects the ordering of the two peaks.

The reionisation figure assumes a sudden transition. A real reionisation history is extended over hundreds of millions of years and patchy, and the optical depth constrains an integral over it rather than a single redshift. The multipole marked for each redshift is an estimate from the angle the last-scattering sphere subtends, not a computed spectrum.

The B modes nobody has found #

The decomposition into E and B is where the measurement turns into a search. Scalar density perturbations, scattered through any number of quadrupoles, produce no B modes at all. Two other things do.

Gravitational lensing does, by deflecting the E-mode pattern on its way here and shearing some of it into B. That lensing B mode has been measured, at the amplitude predicted from the matter between here and the last-scattering surface, and it is now a measurement of that matter in its own right.

And a background of gravitational waves does, because a gravitational wave stretches space in a quadrupolar pattern that produces a quadrupole in the radiation without any density perturbation behind it — and that quadrupole has a handedness. Such a background is the one prediction of inflation that measures the energy scale at which it happened. Its amplitude relative to the density perturbations is bounded below about four per cent and not detected, and the bound comes almost entirely from the large-angle B-mode polarisation, where the gravitational waves’ signal would peak at the same reionisation scale and at the recombination scale near multipole eighty.

The difficulty is not sensitivity alone. The Galaxy’s dust is polarised by its magnetic field, and its emission at the relevant frequencies is brighter than any plausible primordial B mode everywhere on the sky. One celebrated claim of a detection, in 2014, turned out to be that dust.

A rule about what scattering can remember #

The general statement is about symmetry. Thomson scattering is a filter that passes the quadrupole of the incoming light and blocks every other multipole. Whatever it produces is therefore a record of whatever made the light around each electron quadrupolar, and of nothing else.

In the early universe two things did. One was the velocity gradient of an oscillating fluid streaming across a free path — which exists only because recombination took time, only in the later part of the transition, and only in proportion to how far the photons could travel. The other was the anisotropy of the background itself, seen again by electrons freed some seven hundred million years later. Everything measured in the polarised sky so far is one of those two, and everything that would be new — gravitational waves from inflation, a primordial magnetic field, a rotation of the polarisation by a new field along the way — would show up as polarisation that neither can produce.

An observable that is blind to most of what happened is worth more than one that sees everything, provided what it is blind to is known exactly. The temperature of the background records densities, velocities, potentials and their evolution all added together. The polarisation records one of them, and its pattern on the sky is a statement of which one by its symmetry alone.

Still open: whether anything else polarised the light #

What comes next is the boost. Every map of the background, polarised or not, is made by an observer moving at three hundred and seventy kilometres a second relative to it, and the motion distorts the small-scale pattern as well as producing the dipole. The dipole was subtracted on the assumption that it is entirely the observer’s motion, and the distortion of the anisotropies is the independent check on that assumption — a velocity measured from the finest structure in the map rather than from its largest.

Beside it lies the rotation question. A field that coupled to light could rotate every photon’s polarisation by the same small angle on its way here, turning some E into B and correlating them in a way scattering never can. A rotation of a third of a degree has been reported, at the edge of significance, from the correlation between E and B in Planck’s maps — and its size is comparable to the uncertainty in the orientation of the polarimeters themselves, which is the other thing that would produce it.