A wind the sound waves left behind

The baryon acoustic scale is the same ruler measured twice: once in the microwave background, as the spacing of the acoustic peaks, and once in the galaxies, as a faint preference for pairs separated by about 150 megaparsecs. Both are fossils of sound waves that ran through the hot plasma of the early universe and froze when the plasma became neutral gas, 380,000 years after the Big Bang. The galaxies remember the waves because the dark matter, which was not part of the plasma, fell into the shells of gas the waves left behind.

There is a third fossil of the same waves, and it is not a pattern in density at all. Until decoupling the ordinary matter — the baryons — was locked to the light, oscillating with it, while the dark matter simply fell under gravity. At the moment of release the two were moving differently. In every region of the universe the gas was streaming past the dark matter at some speed, in some direction, set by where in its oscillation the local sound wave had been caught. Tseliakhovich and Hirata pointed out in 2010 that this relative velocity was large — tens of kilometres a second — and that it had been left out of every calculation of how the first structures formed.

This essay follows that wind: how fast it was, how it decayed, why it mattered for the first stars and why, because it came from the sound waves, it carries the acoustic ruler into an era before there were any galaxies to measure it with.

A stream that slows, and a gas that cools faster #

At decoupling the root-mean-square relative velocity between baryons and dark matter is about thirty kilometres a second. After that nothing pushes on it. The gas and the dark matter both coast, their peculiar velocities decaying as the universe expands, inversely with the scale factor — so the stream falls in proportion to $1+z$.

What matters for the gas is not the stream’s speed alone but its speed compared with the speed of sound, because a gas flowing faster than sound past a dark-matter clump cannot settle into it: pressure waves that would let it adjust cannot keep up. The sound speed depends on the gas’s temperature. For about the first hundred million years after decoupling, the gas is kept at the temperature of the background radiation by the few free electrons that remain, scattering off photons; its temperature falls as $1+z$, its sound speed as the square root of that. After about $z = 150$ the coupling fails, and the gas cools adiabatically, its temperature falling as $(1+z)^2$ and its sound speed as $1+z$ — at the same rate as the stream.

Supersonic for a hundred million years #

The ratio of the two is the Mach number.

At decoupling the stream is five times faster than sound. While the gas is warm it falls, because the sound speed falls more slowly than the stream; after the gas decouples from the light both fall together, and the Mach number freezes at about two. It is still two at a redshift of twenty, two hundred million years after the Big Bang, when the first stars are forming in dark-matter haloes of a few hundred thousand to a million solar masses.

That is the coincidence that makes the wind important. The haloes in which the first stars form are small, with escape speeds of a few kilometres a second. Gas streaming past them at twice the speed of sound, with a velocity comparable to their escape speed, is not captured efficiently, and much of what is captured is displaced from the halo’s centre and cannot cool. A wind that has decayed to well under a kilometre a second would be irrelevant to a galaxy. To a minihalo it is a gale. The gas that does fall in arrives offset from the halo’s centre, downstream, and has to settle against the flow before it can reach the densities at which molecular hydrogen forms and cooling begins — a delay that, in the windiest regions, postpones the first star by tens of millions of years.

Every region has its own wind #

The thirty kilometres a second is a root-mean-square value. Each component of the relative velocity is a Gaussian random field, so the speed varies from place to place.

Almost nowhere is there no stream at all: the Maxwell distribution vanishes at zero speed because three independent components would all have to vanish together. Two-fifths of the universe streams faster than the rms; a few regions in a thousand stream at more than twice it. The velocity is coherent over scales of a few comoving megaparsecs — the scale on which the sound waves’ velocity field varies — so the wind blows steadily across a region holding thousands of minihaloes, all of which feel the same suppression. Neighbouring regions a few tens of megaparsecs away may feel a quite different one.

The smallest halo that can still make a star #

The effect on star formation can be expressed as a raised threshold. Without a stream, a halo needs a minimum mass to collect its share of gas and cool it by molecular hydrogen; with a stream, it needs more, because the gas’s effective random motion includes the bulk flow.

A fit to simulations of gas in streaming haloes adds the stream to the cooling threshold in quadrature, with a weight of about four. At a redshift of twenty, a region streaming at the rms needs haloes about 1.7 times as massive to form stars as it would without the stream; at twice the rms, four times as massive. At earlier times the stream is faster relative to the haloes and the effect larger: at a redshift of thirty, the factors are about three and ten. Since small haloes vastly outnumber large ones, raising the threshold by a factor of a few removes most of the potential star-forming sites in a region.

The suppression is therefore patchy. Half the universe has its threshold raised by fifty per cent or more, a quarter has it doubled, a twentieth tripled. The first stars form preferentially in the calm regions, where the stream happens to be slow, and are delayed in the windy ones. By a redshift of fifteen or so, the haloes forming stars are massive enough that the wind no longer matters, and the pattern fades — but not before the first starlight has lit the universe unevenly.

The acoustic scale in the first light #

The pattern of that unevenness is not random. The stream velocity in each region is the velocity the sound waves had when they froze, and the sound waves had a characteristic length.

A Fourier mode of the density whose wavelength fitted a whole number of half-oscillations into the time before decoupling was caught at the turning point of its swing, its velocity zero; one caught halfway was moving fastest. The relative velocity is thus a sinusoidal function of wavenumber, oscillating with the sound horizon as its period — the same oscillation that appears in the density of galaxies, but shifted by a quarter-cycle, because velocity leads density in a sound wave. The density oscillations in the galaxy distribution are a few per cent of the smooth power. The velocity oscillations are of order one: the stream is made entirely of acoustic velocity, with nothing smooth underneath it.

So when the first stars form preferentially where the wind is calm, the pattern of where they form inherits strong acoustic oscillations. The first starlight couples the twenty-one-centimetre line of neutral hydrogen to the gas temperature, and the twenty-one-centimetre signal from that era — the trough whose depth depends on the fit as much as the sky — fluctuates across the sky with the acoustic scale stamped into it at an amplitude far larger than any galaxy survey sees. These velocity-induced acoustic oscillations were proposed in 2019 as a standard ruler for cosmic dawn: the same 147 megaparsecs, measured at a redshift of fifteen or twenty, in an era from which no galaxy redshift survey can ever return data.

Why it was left out for so long #

It is fair to ask how an effect this large went unnoticed for thirty years of calculations of the first structures. The answer lies in how those calculations were set up. Linear perturbation theory computes the evolution of each component of the universe — photons, baryons, dark matter, neutrinos — mode by mode, and it always contained the relative velocity: the baryon and dark-matter velocities in each Fourier mode are computed separately, and they differ. But in linear theory each mode evolves independently, and a relative velocity coherent over megaparsecs has no effect on the growth of a mode of the same scale.

Its effect is on small scales, in the formation of minihaloes a thousand times smaller than the coherence length of the wind, and that is a nonlinear coupling between large-scale velocity and small-scale density — the product of two perturbations, which linear theory drops by construction. Simulations of the first stars, meanwhile, typically began from initial conditions in a box a megaparsec across, small enough that the stream is uniform across it, and in the frame of the box’s mean motion the uniform stream looks like nothing at all unless it is put in deliberately. The effect was invisible both to the large-scale calculation, which had it but could not use it, and to the small-scale one, which could use it but had subtracted it.

That is a recurring pattern in cosmology. The bulk flows that blur the acoustic peak in the galaxies, and the lensing that smooths the background’s peaks, are also second-order couplings between large and small scales, and each was understood only when somebody asked how a large-scale field changes small-scale physics locally.

The gas temperature and the trough #

The same moment that freezes the Mach number — the end of the gas’s thermal coupling to the radiation near a redshift of 150 — also sets the temperature that the twenty-one-centimetre signal measures. After decoupling from the light, the gas cools faster than the background, as the square of $1+z$ rather than the first power, and by a redshift of twenty it is at about seven kelvin against the background’s fifty-seven. The hydrogen’s spin temperature, once starlight couples it to the gas, is dragged towards the gas’s temperature, a temperature that belongs to none of the gas’s radiation, and the line appears in absorption against the warmer background.

The stream heats that gas a little where it is strongest, by sound waves and shocks as the stream flows past the minihaloes, and it delays the first starlight that turns the absorption on. Both effects make the trough’s onset and depth depend on the local wind. The global trough is an average over all the windy and calm regions; the fluctuations are where the acoustic pattern lives. A shadow cast on the microwave background by gas in this era is thus not uniform but patterned on the scale of the sound horizon — weak, but with a contrast set by the stream rather than by the density, and so larger than the density’s few per cent.

Does the wind bias the galaxies’ ruler? #

If the wind changed where the first stars formed, might it also have changed where the galaxies are, and so shifted the ruler measured in them? Galaxies today grew from many generations of mergers, and the minihaloes whose star formation the wind delayed were among their earliest building blocks. If a galaxy’s present brightness or colour remembered, even faintly, how windy its region was at cosmic dawn, the galaxy population would carry an extra pattern with the stream’s acoustic oscillation — shifted by a quarter-cycle from the density’s — and fitting the acoustic peak while ignoring it would shift the inferred ruler.

The effect was estimated soon after the stream was recognised, and the possible shift was found to be up to about a per cent for galaxies with a strong memory, comparable to the precision of the largest surveys. The surveys tested for it directly, by fitting the galaxy correlation function with an extra term of the stream’s shape. They found no evidence for it, and limited the memory to a level that shifts the ruler by much less than the survey errors. Galaxies at a few billion years old seem to have forgotten the wind of their earliest ancestors, which is reassuring for the distance measurements and itself a small fact about how thoroughly hierarchical assembly mixes its ingredients.

A third fossil of the same waves #

The microwave background records the sound waves at the instant they froze, as a standing wave caught at one moment. The galaxies record them billions of years later, in the dark matter that fell into the shells the waves left. The streaming velocity records them in a third way, through the difference in motion between two components that the waves treated differently — and it carries its record into the gap between the other two, the half-billion years when there was neither a background still being emitted nor galaxies yet formed.

The three records test one another. The ruler’s length is fixed by the physics before decoupling, which is why particles that streamed ahead of the waves leave the same phase shift in all of them. A ruler measured at cosmic dawn would pin the expansion rate at a redshift of twenty, testing the same cosmological model that fits the background and the galaxies at a time when dark energy was negligible and the universe was nearly pure matter. If the three agreed, the model would be confirmed across the whole of cosmic history; if the dawn measurement disagreed, the disagreement would be in the era least understood.

What the figures simplify #

The stream’s rms and its decay are robust, being set by the well-measured parameters of the standard cosmology; the 30 kilometres a second is the conventional value. The gas temperature uses a sharp switch from coupled to adiabatic evolution at $z = 150$, where the real transition is gradual over a range of redshifts; the Mach number’s plateau is correspondingly smoothed in detailed calculations. The minimum halo mass follows a fit to simulations with a single weighting factor; different simulations, with different treatments of cooling and of the gas’s response, give factors that differ by tens of per cent.

The mode figure is schematic: the relative velocity in a mode depends also on the growth of the dark-matter perturbation before decoupling and on a transfer function computed by integrating the coupled equations of photons, baryons and dark matter, and its envelope is not a simple Gaussian. The essential feature — oscillation at the sound horizon with order-unity amplitude — is what the full calculation also shows.

Still open: whether the dawn ruler can be read #

The velocity-induced oscillations are a prediction. Measuring them requires mapping fluctuations of the twenty-one-centimetre signal at redshifts of fifteen to twenty — radio frequencies of about seventy to ninety megahertz — through foregrounds from the Galaxy a hundred thousand times brighter. The radio interferometers designed to detect the end of reionisation at lower redshifts are pushing towards those frequencies, and their sensitivity forecasts say the acoustic feature at cosmic dawn could be detected at several standard deviations with years of integration, if the foregrounds can be removed as well as they must be.

There is a second, subtler question. The suppression by the stream depends on how the first stars form in minihaloes — on molecular cooling, on whether the first stars were massive and few or modest and many — and the amplitude of the oscillations depends on it. A measured amplitude would therefore be a measurement of how the first stars formed, while the measured wavelength would be a measurement of the expansion. One signal, two quantities: the ruler’s length testing cosmology and its contrast testing the astrophysics of the first light. Whether either can be separated from the other, or from the foregrounds, is what the next generation of low-frequency radio arrays has to show.