A depth that belongs to the fit as much as the sky

The shadow cast on the microwave background by the first stars is predicted to be a trough in the sky’s radio spectrum, somewhere between 50 and 110 megahertz, a tenth of a kelvin or two deep, ten to thirty megahertz wide. It sits under the Galaxy’s synchrotron glow, which at those frequencies is a thousand to five thousand kelvin. The only reason the measurement is attempted at all is that the two differ in shape: the foreground is smooth in frequency and the trough is not, so a smooth function fitted to the spectrum and subtracted should leave the trough behind.

That sentence hides an assumption, and the assumption is what this essay is about. “The trough is not smooth” is a matter of degree. A trough twenty megahertz wide in a band fifty megahertz wide is, over most of the band, a gentle curve — gentler, in places, than the foreground’s own curvature. A smooth function given enough freedom to remove the foreground has enough freedom to remove a good part of the trough too, and it has no way to know which part of the spectrum it is fitting is which.

Why anyone measures an average #

The global signal is the sky-averaged 21-centimetre brightness at each frequency, and averaging the whole sky into one number per frequency looks wasteful when the gas that produced it was patchy. The reason is cost and directness. A single dipole antenna a few metres across, with a well-characterised receiver, can reach a noise level of a few millikelvin in weeks of integration, because the signal it is after is the same in every direction and the antenna collects all of them at once. The fluctuations from place to place are a hundredth to a tenth of the average, and measuring them needs an interferometer of hundreds of elements and years of data.

The average also answers a question the fluctuations answer only indirectly: when. The trough’s frequency is a redshift, and the redshift is the epoch at which the first sources became numerous enough to couple the line to the gas temperature. The fluctuations measure how clustered the sources were at each epoch; the average measures how many there were. Both are needed, and only one of them can be had cheaply — which is why a decade of experiments have chased a tenth of a kelvin under a thousand with a single antenna, and why the difficulty below has been met so many times by so many different instruments.

How much freedom the foreground needs #

The synchrotron sky is not a single power law. The spectral index — the slope of brightness against frequency on logarithmic axes — varies across the sky, from about −2.4 in some directions to −2.7 in others, because the cosmic-ray electrons that produce the emission have different energy spectra in different parts of the Galaxy. A synchrotron spectrum is set by the electrons, not by any temperature, so there is no physical reason for the index to be the same everywhere. A global experiment uses an antenna with a beam tens of degrees wide, and the spectrum it records is the beam-weighted sum of many such power laws. A sum of power laws with different indices is not a power law; on logarithmic axes it is curved.

The antenna adds structure of its own. Its beam is a pattern whose shape changes with frequency — wider at low frequency, narrower at high — so the patch of sky it averages over, and therefore the mix of spectral indices it sums, changes across the band. This beam chromaticity is modelled from electromagnetic simulations and measurements and corrected, but not perfectly, and a residual of a part in a thousand on a sky of two thousand kelvin is two kelvin of structure.

The sequence is the first half of the problem. A trough of 150 millikelvin cannot be seen through a residual of 170, so the fit needs at least five terms, and to get below the noise of a few months of integration it needs six. Every published global measurement has used between five and seven, or a physically motivated function of comparable flexibility.

What each term takes #

The second half is what those terms do to the signal. The fit is a projection: it finds the combination of its basis functions closest to the measured spectrum and subtracts it. Whatever part of the trough lies within the space the basis spans is removed with the foreground. What survives is the trough’s component orthogonal to that space, and as the basis grows, the orthogonal component can only shrink.

In the small-signal limit the arithmetic is exact. The trough divided by the foreground is a small function of frequency; the fit in log temperature projects it onto polynomials in log frequency; the fraction of its norm left outside the projection is the fraction of its signal-to-noise that survives. A trough fitted together with the foreground — the trough’s shape assumed, its depth a free parameter beside the foreground’s coefficients — is measured with the precision that surviving fraction allows, and no better. Its error is inflated by one over that fraction.

That empty window is the measurement’s central difficulty, and it is not a matter of collecting more data. Integration time lowers the noise; it does not change either curve. Below five terms the foreground swamps the signal. Above five, what survives of the trough is smaller than three times the noise floor even when the foreground has gone. A longer integration moves the dotted line down and opens a narrow window near six or seven terms, where the trough is worth twenty-odd millikelvin of signal against single millikelvin of foreground — an experiment with a noise of a few millikelvin can detect a trough of a hundred and fifty, but only by measuring a sixth of it.

Wide troughs and narrow ones #

The surviving share depends sharply on one property of the trough that the first stars, not the instrument, decide: its width.

A trough’s width is set by how quickly the first stars switched the 21-centimetre line from tracking the microwave background to tracking the cold gas, and how quickly X-rays from their remnants then heated the gas back up. Rapid star formation and early heating make a narrow trough. A slow build-up of the first galaxies, or heating that lagged, makes a broad one. The measurement is therefore most sensitive to precisely the scenarios it has least reason to expect, and least sensitive to the slow, broad troughs a conservative model of the first galaxies produces.

This is why a global experiment cannot simply report a depth. “The trough is 150 millikelvin deep” means “a trough of the assumed shape, after projection through this foreground basis, is consistent with the data if its depth is 150 millikelvin” — and the same data, analysed with a basis of one more term, or a trough assumed five megahertz wider, gives a different number with a different error. The claimed detection of 2018 reported a flattened trough half a kelvin deep and nineteen megahertz wide, fitted jointly with a five-term foreground, and several re-analyses found that the depth, the width and the flattening all moved when the foreground function was changed.

What the width says about the first stars #

The two numbers the trough would deliver are its timing and its depth, but its width is where the astrophysics is densest, and it is the number the projection hides best.

The trough opens when ultraviolet light from the first stars, scattering resonantly in the Lyman-α line, drags the hydrogen’s spin temperature down to the temperature of the cold gas. How fast that happens depends on how quickly star-forming haloes appeared — which depends on the smallest halo mass in which gas could cool and form stars, and on how much of the gas became stars. A mass function weighted toward massive stars produces far more ultraviolet per unit of gas than one like the present day’s.

The trough closes when X-rays heat the gas above the microwave background’s temperature, and the X-rays come from the remnants of stars that lived only a few million years: high-mass X-ray binaries, accreting black holes. A population that formed its black holes quickly and accreted onto them efficiently closes the trough early and makes it narrow; a population whose X-ray output lagged makes it wide.

So a narrow trough says the first galaxies came on fast and heated their surroundings quickly, and a wide one says the opposite — and the fourth figure says that the narrow case is what a global experiment can measure well and the wide one is what it measures badly. The instrument’s sensitivity is correlated with the answer. A null result in a global experiment is weakest exactly where the slow, conservative scenarios lie, and a positive result from a narrow trough would be telling the truth about a population that is, for that reason, the easiest to see.

Recovering a depth by passing the model through the fit #

There is a correct way to compare a model with a residual that has passed through a foreground fit, and it is the way the forest analyses already work: pass the model through the same fit first.

Take a candidate trough — a shape, a depth, a centre — add it to a foreground model, fit the result with the identical basis used on the data, and compare what is left with what was left in the data. Every candidate is thereby projected exactly as the real signal was, and the comparison is like with like. Done this way the fit’s absorption is not an error at all: it is part of the model, and a likelihood built on it returns the true depth, with an uncertainty inflated by the factor the projection costs.

What it cannot return is a depth independent of the shape. Two candidate troughs with the same narrow core and different wings project to nearly the same residual, so the data constrain the core and leave the wings to the prior. A forward-modelled global measurement therefore reports the depth of a family of shapes, and its credibility depends on how many families were tried.

A ripple that survives looking like a trough #

The worst case is not a signal absorbed by the fit. It is structure that is not signal and is not absorbed either.

A reflection between the antenna and its receiver, a resonance in the ground plane, an impedance mismatch that varies with frequency — each puts a periodic modulation on the measured spectrum, and the period of the modulation is set by a length in the instrument. A ripple whose period is comparable to the trough’s width survives a smooth fit for the same reason the trough does: it is not smooth. The fit cannot tell a sky feature from an instrument feature, because to the fit both are simply the part of the spectrum its basis does not span.

The defence is the one the contradiction between the first two experiments illustrated. The instrument’s ripples have periods set by its own dimensions, and a second instrument has different dimensions, so its ripples fall at different frequencies. A trough in the sky appears at the same frequency in both. The experiment that failed to confirm the 2018 trough was built to have different systematics — a different antenna, floating on a lake so that the ground beneath it was uniform — and its absence of a trough says that at least one of the two spectra contained something that was not sky.

The same projection, in a different tracer #

This loss is not peculiar to radio astronomy. The hydrogen forest measured in quasar spectra has exactly the same structure: the quasar’s unabsorbed continuum must be fitted, the fit is a smooth function along each spectrum, and it removes the forest’s large-scale fluctuations along with any error in the continuum. That analysis does not pretend otherwise. It applies the same continuum fit to simulated spectra with a known signal, measures how much of the signal the fit removes at each scale, and divides the measured correlations by the same distortion. The loss is treated as part of the estimator rather than as a nuisance to be minimised.

This is the connection worth carrying. A smooth fit used to remove a bright foreground is a linear operator on the data, and it acts on the signal exactly as it acts on everything else. The forest analysis survives its continuum fit because the operator is known and inverted; the global 21-centimetre measurement has the same operator, and the same option of forward-modelling it — passing every candidate trough through the identical fit before comparing it with the residual. Comparing an unprojected model trough with a projected residual is how depths come out wrong.

The same lesson appears in the polarised dust foreground that stands between the microwave background and its primordial signal, where the extrapolation from the frequencies that measure the dust to those that measure the signal rests on a model of the dust’s spectrum, and every assumption in the model is a component the subtraction can absorb or leave behind.

Three responses, and what each assumes #

The experiments now running answer the problem in four different ways, each of which moves the difficulty rather than removing it.

Restricting the fit’s freedom. A polynomial with many terms can produce wiggles; the true foreground almost certainly cannot. Fitting with “maximally smooth” functions — polynomials constrained so that their higher derivatives never change sign — removes curvature without removing turning points, so a trough, which has two, cannot be absorbed by a function forbidden from having them. The price is that a real foreground with any structure of that kind, such as an imperfectly corrected beam ripple, is forbidden too, and survives into the residual as a feature.

Modelling the beam and the sky together. Rather than fitting a smooth function to the spectrum, the sky is modelled as a map with spatially varying spectral indices, multiplied by a simulated beam at every frequency and integrated. The foreground then has as many parameters as the sky model, all physical, and the trough is fitted alongside them with a Bayesian comparison deciding how many are needed. The radio experiment in South Africa’s Karoo was designed around this approach. The price is that the beam model must be right: a beam simulated to a part in a thousand is, as the second figure shows, a beam whose leftover is a kelvin of structure.

Leaving the ionosphere in the model. Below about 70 megahertz the Earth’s ionosphere absorbs a fraction of a per cent of the sky’s emission and emits a little of its own, both varying with the time of day and the solar cycle. A fraction of a per cent of two thousand kelvin is ten kelvin, varying by the hour. The ionospheric terms are smooth in frequency, roughly as the inverse square, so they add a term or two to the foreground basis — and therefore subtract a little more of the trough — and they differ between nights, so data averaged over many nights carry a foreground that is an average of different shapes.

Changing the instrument. Several experiments have moved to sites with less radio interference, to Arctic and desert locations, to islands in the Southern Ocean, and proposals exist for the far side of the Moon, where the ionosphere that distorts the low end of the band is absent. A better site lowers the noise and removes some of the foreground’s complications; it does not change the projection, which is a property of fitting a smooth function over a finite band.

What the figures assume #

The foreground is a sum of five power laws and one sinusoid. A real beam-weighted sky has a continuous distribution of spectral indices and a beam correction whose residual is not a single slow ripple; the number of terms needed and the residual at each are illustrative, chosen to resemble published residuals rather than computed from a sky map and a beam.

The trough is a Gaussian. The 2018 claim was flattened, and physical models give asymmetric troughs with a steeper low-frequency side. The surviving share depends on the shape as well as the width, and a flattened trough loses its flat bottom to the fit more readily than a peaked one.

The fit is unweighted in log temperature. Real analyses weight by the noise, which is proportional to the sky temperature and therefore larger at low frequencies, and some fit in temperature rather than its logarithm. Both change the numbers without changing the shape of the argument.

The noise is a single level. A real spectrum has noise rising toward low frequency, and an experiment chooses its band partly to keep the edges, where the foreground is brightest and the leftover largest, out of the fit.

Still open: the first trough confirmed by two instruments #

No global trough has been confirmed. The experiments under way differ deliberately in antenna, site and analysis, and when two of them report a trough at the same frequency with depths that agree after each has projected its model through its own foreground fit, the measurement will exist. Its frequency will date the first starlight that coupled the spin temperature; its depth, corrected for what each fit absorbed, will be the temperature of the gas a third of a billion years after the Big Bang.

What will remain uncertain even then is the shape. The surviving share of a broad trough is small, and different shapes with the same narrow core survive identically, so the global measurement will constrain the timing and the depth of the first stars’ effect far better than its duration. The duration is what the fluctuations across the sky, measured by interferometers that remove the foreground by its smoothness along each line of sight rather than across a band, are better placed to supply — with a projection of their own, which removes the largest scales along the line of sight for the same reason the global fit removes the broadest troughs. The forest that survives at lower redshift records the end of the same story, where none of this difficulty arises because the signal is a hundred per cent absorption rather than a tenth of a kelvin under a thousand.