A foreground drawn by the field it traces

An interstellar dust grain is elongated, and in the diffuse gas of the Galaxy it spins with its long axis across the local magnetic field. Starlight passing through a cloud of such grains is absorbed more strongly along the grains’ long axes, so it emerges polarised along the field. The grains also glow. They are heated by the starlight they absorb to about twenty kelvin, and they return that energy in the far infrared and millimetre, radiating most efficiently along their long axes — so the glow is polarised across the field, at right angles to the starlight’s. One grain population gives two perpendicular polarisations, and the direction of either one maps the field.

The glow is also the largest obstacle to one of the most sought measurements in cosmology. A background of gravitational waves from the first instant of the universe would leave, in the surface the background comes from, a faint twist in the polarisation of the microwave background — a curl-like pattern, called B-modes, that density fluctuations alone cannot make. Its amplitude is measured by the tensor-to-scalar ratio $r$, and that ratio is an energy scale for inflation. The searches now under way aim at $r$ of a few thousandths. The Galaxy’s polarised dust, in the cleanest part of the sky anyone has measured, imitates about $r = 0.05$ at the frequency most experiments use.

The figure is the shape of the whole problem. At low frequencies the polarised sky is dominated by synchrotron radiation from cosmic-ray electrons spiralling in the same magnetic field, which falls steeply with frequency. At high frequencies it is dominated by dust, whose emission rises steeply toward its far-infrared peak. The microwave background, measured in its own temperature units, is flat across the whole range. So there is a window, between about seventy and a hundred gigahertz, where the foregrounds are faintest, and it is not a window through which the sky is clean. It is only the least bright of the frequencies at which the sky is too bright.

A spectrum measured to a few per cent #

The dust’s frequency dependence is what makes the problem tractable at all. A grain population in equilibrium with the starlight around it emits a modified blackbody: a Planck function at the dust temperature, multiplied by a power of frequency that describes how the grains’ emissivity falls toward long wavelengths. Planck’s all-sky maps, which measured polarisation at seven frequencies from 30 to 353 GHz, fixed the polarised dust’s emissivity index at 1.53 and its temperature at 19.6 K, averaged over the high-latitude sky, with uncertainties of a few per cent.

That spectrum is steep. Between 150 and 353 GHz the dust’s polarised amplitude, in microwave-background temperature units, rises by a factor of about twenty. So a map at 353 GHz is almost pure dust: the primordial signal and the lensing B-modes — the same shear a million galaxies are averaged to see — are negligible there, and the dust is measured with a high signal-to-noise ratio. Scaled down by the known spectrum, it predicts the dust at 150 GHz, where the background is observed, and subtracting the prediction leaves the background. That is the method, and every current limit on $r$ relies on it, with the synchrotron removed the same way from maps at 23 to 40 GHz.

The retracted detection of 2014 was a measurement made at one frequency, without this lever. The polarised pattern at 150 GHz was real; its cause was dust, and only a second frequency could have said so. The combination of a ground-based 150 GHz map with Planck’s 353 GHz map showed that the whole signal was consistent with dust, and every result since has carried the dust in its model with its amplitude as a fitted parameter.

No patch is clean enough #

The obvious remedy is to look where there is less dust, and experiments already do. The ground-based B-mode searches observe a field of a few hundred square degrees chosen for having some of the faintest dust emission on the sky. The question is how much fainter a field would need to be.

The power 1.9 is itself a measurement. Planck compared the polarised dust power in many patches of high-latitude sky with each patch’s mean total intensity and found that one scales nearly as the square of the other, which is what would be expected if the polarisation fraction were roughly constant and the power went as the square of the amplitude. That scaling is what lets the chart be drawn: dust fainter by a factor of two gives B-mode power fainter by nearly four. It also says that the cleanest field is not an unusually quiet region of an otherwise bright sky. It is the faint end of a smooth distribution, and there is little room beyond it.

So the measurement at $r$ of a few thousandths will not be made around the dust. It will be made through it, with the dust subtracted to a precision of better than a tenth of itself. The precision of that subtraction, rather than the sensitivity of the detectors, is what the next decade of these experiments is designed around.

E over B is a measurement of the field #

Any pattern of headless vectors on the sky can be split into two parts. E-modes are patterns in which the polarisation runs parallel or perpendicular to the direction in which its strength changes — radial and tangential patterns around a spot. B-modes are patterns in which the polarisation runs at forty-five degrees to that direction — pinwheels. Density fluctuations in the early universe produce only E-modes; gravitational waves produce both. That is why the search is for B.

Dust produces both too, and in a ratio that turns out to be a statement about the interstellar medium. Planck measured the dust’s power in E-modes to be about twice its power in B-modes, over the whole high-latitude sky and at every angular scale it could measure, from several degrees to about a third of a degree. The same analysis found the dust’s temperature anisotropy and its E-mode polarisation positively correlated, and B-mode power only slightly below half of E at every scale — a ratio stable enough to be used as a constant in foreground models. Nothing about dust grains requires that. A random pattern of headless vectors has equal E and B.

The explanation is structural. Maps of neutral hydrogen at high resolution show the diffuse interstellar medium as a web of thin filaments, and the filaments are preferentially aligned with the magnetic field that the dust polarisation traces. A filament along the field, with its emission polarised across the field and therefore across the filament, is an E-mode pattern: the polarisation is perpendicular to the direction along which the brightness changes most steeply. If every filament followed the field exactly, the dust would have no B-modes at all, and the foreground problem would be far easier. It follows loosely, with a scatter the ratio measures.

This is the same kind of inference the Davis–Chandrasekhar–Fermi method makes. There, the scatter of polarisation angles around a mean field — produced by turbulence bending the field lines — was turned into a field strength, because a stiffer field is bent less. Here, the scatter of filament directions around the field direction is turned into the foreground’s B-mode power, because a filament at an angle to the field has a component that is not E. The same physics, a turbulent medium threaded by a field stiff enough to organise it, sets both. The foreground that obscures the early universe carries in its E/B ratio a measurement of how strongly the Galaxy’s field shapes its gas, and the same maps show a positive correlation between dust temperature and E-mode polarisation that the filament picture predicts and a random field does not.

The bet in the extrapolation #

The subtraction scales a 353 GHz map to 150 GHz by one spectrum. That assumes the dust on each line of sight has one spectrum, and it need not. A sight line out of the Galaxy crosses several clouds, and they can differ in temperature — a cloud near a bright star is warmer — and in the direction of their magnetic field. Their polarised emissions add as vectors, through the Stokes parameters $Q$ and $U$, so the observed polarisation angle is a weighted average of the clouds’ field directions. The weights depend on frequency, because the warmer cloud’s emission peaks at a higher frequency and fades more slowly toward the millimetre.

A turn of a degree sounds negligible, and whether it is depends on the target. A map scaled from 353 GHz points in the 353 GHz direction; subtracting it from the 150 GHz sky leaves a residual whose size is twice the sine of the angle turned, whatever amplitude is used. The name for the effect is frequency decorrelation: the dust maps at two frequencies are not perfectly correlated, and the missing correlation is a residual no amplitude scaling can remove.

The figure’s honest reading is reassuring for current experiments and a warning for the next ones. For clouds whose temperatures differ by a few kelvin and whose fields differ by tens of degrees, the residual is a few per cent of the dust, which at $r$ of a few hundredths is harmless. It matters at the level the next experiments are designed for, and it grows steeply for sight lines where two bright clouds happen to have nearly perpendicular fields. Planck’s own tests found the correlation between its dust maps at 217 and 353 GHz consistent with perfect over most of the high-latitude sky, within uncertainties of about a per cent; studies of sight lines that cross two distinct hydrogen clouds at different velocities have found angles that differ with frequency in the way the two-cloud sum predicts. The extrapolation is a bet on the structure of the interstellar medium, and so far the medium has mostly paid out.

The defence the experiments are building is to stop relying on one lever. Measuring at many frequencies between 90 and 400 GHz lets the dust spectrum be fitted pixel by pixel, including a variation of temperature and emissivity index across the sky, and the residual from a real two-cloud sight line then shows up as a failure of the single-spectrum fit rather than as a false signal.

The same sky at the next target #

The present limit is $r$ below about 0.036, and it was reached with the dust in the cleanest field imitating a signal slightly above that — a subtraction to about a factor of two. The experiments now observing aim at $r$ near 0.001, which is where the simplest inflation models with a large field predict a signal and below which a class of them is excluded. Redrawn for that target, the frequency picture changes in one important way and not in the obvious one.

The obvious change is that the dust must be removed more precisely, by a factor of about fifty in power, which the multi-frequency fits are designed to deliver. The less obvious one is that the lensing B-modes, made when the background’s E-mode pattern is sheared by the matter in front of it, are now larger than the signal at every frequency, and they have exactly the background’s spectrum. No frequency lever touches them. They have to be subtracted by reconstructing the lensing from the same maps, or from the distribution of galaxies, and that reconstruction is degraded by the same dust — whose small-scale E and B modes leak into the lensing estimate. The foreground and the lensing are therefore not two separate problems at the next target. They share a residual, and the dust’s filamentary structure, which sets its small-scale power, is part of what decides how well the lensing can be removed.

That is where the E/B ratio stops being a curiosity. A foreground model that got the dust’s B-mode power right on large scales and wrong on small ones would pass every test at $r = 0.03$ and fail at $r = 0.001$, and the filament statistics are what fix the small-scale ratio. The interstellar medium’s geometry, measured from hydrogen and from starlight, enters the error budget of an experiment on inflation.

What the figures do not show #

The amplitudes in the first figure are for one angular scale, around a multipole of 80, where the primordial signal from recombination peaks. The dust’s power falls slowly with multipole — as the multipole to the power $-0.4$ to $-0.5$, measured — and the lensing B-modes rise, so the relative sizes of the three curves depend on which scale is meant. The primordial and lensing levels drawn are the approximate values in the standard model at that scale, not a fit.

The synchrotron level is drawn at about its measured upper limit in the cleanest field and could be several times lower. Where it is lower, the cleanest frequency moves downward, toward sixty gigahertz, and the minimum becomes fainter; the conclusion that dust dominates above a hundred gigahertz does not change.

The two-cloud model is the simplest case of a real distribution. Real sight lines contain a continuum of gas at different temperatures and field directions, and the decorrelation of a continuum is smaller than that of two discrete clouds of equal brightness, because the angles partly average. The two-cloud figures are therefore close to the worst case for given temperature and angle contrasts, and they are drawn to show where the effect comes from rather than how large it is on the sky.

And none of the figures shows grains. The emissivity index, the temperature and the polarisation fraction are properties of a grain population — its composition, its sizes, how well it is aligned — and they are measured here only as the parameters of a spectrum. Whether the polarised and total emission come from the same grains, and whether the index changes with frequency in a way a single modified blackbody would miss, are open questions about the dust that bear directly on the extrapolation.

One population, two uses #

Dust is always both instrument and nuisance. To a Galactic astronomer, the polarised glow is a map of the magnetic field threading every cloud in the sky, and its filaments, its E/B ratio and its correlation with hydrogen are measurements of how the field and the gas shape each other. To a cosmologist, the same glow is fifty times brighter than the signal, and every property that makes it informative — its steep spectrum, its filamentary structure, its dependence on the field’s direction along the line of sight — is a property that has to be modelled well enough to be removed.

The two readings are not in competition. The dust spectrum that makes the subtraction possible was measured by the same all-sky maps that measured the Galactic field, and the filament–field alignment that explains the E/B ratio was found by comparing those maps with hydrogen observed at radio wavelengths — the polarisation a photon count throws away, recovered at two ends of the spectrum at once. The better the Galaxy is understood as an object, the more precisely it can be removed as a foreground, and the B-mode searches have made polarised dust one of the best-measured components of the interstellar medium as a by-product of trying to see past it.

Still open: whether the grains that glow are the grains that align #

Every step above treats the polarised emission as a fixed fraction of the total, with the fraction set by the field geometry. Planck measured a maximum polarisation fraction of about twenty per cent, in the most favourable geometry, and that number is a statement about the grains: it requires that a large part of the emitting mass be both elongated and well aligned. Whether the silicate and carbonaceous grains are aligned equally, and whether one of them glows unpolarised, changes the dust’s polarised spectrum relative to its total — and the ratio of polarisation in emission to polarisation in absorption, measured along the same sight lines to background stars, is the observation that separates the possibilities. It currently favours a single aligned population with both materials mixed, which is not what the older two-component grain models assumed.