A wave pulled ahead by particles that outran it

Every acoustic measurement so far has used the feature as a ruler. The sound horizon is a length fixed before recombination, the galaxies inherited it as a slight preference for separations of about a hundred and fifty megaparsecs, and comparing that preferred separation with what it should be gives a distance. Measured along and across the line of sight, it gives two distances; sharpened by moving the galaxies back along their own flows, it gives them better; read in the hydrogen between quasars, it reaches redshifts no galaxy survey does. In every case the only thing extracted from the feature is its position, which is to say its scale.

A ruler has only a scale. A wave has a scale and a phase, and the acoustic feature is a wave: in the power spectrum it is a train of oscillations in wavenumber, decaying toward small scales, whose crests sit at multiples of $2\pi$ over the sound horizon. Changing the length of the ruler moves every crest by the same fraction. Moving every crest by the same angle is a different operation, and one thing in the early universe performed it.

Particles that would not wait for the sound #

For the first few hundred thousand years the baryons and photons were one fluid, and a fluid carries sound. A density excess launched a spherical pressure wave that travelled outward at the fluid’s sound speed, a little below $c/\sqrt{3}$, until recombination released the photons and the wave stopped where it was. The distance it had reached is the sound horizon, and the shell at that distance is the excess the galaxies still remember. That is the picture the microwave background’s peaks record from the photon side.

The picture leaves out three species of neutrino. They decoupled from everything else about a second after the Big Bang, and from then on they did not scatter: they streamed through the plasma at essentially the speed of light, carrying with them about forty per cent of the universe’s radiation energy. They were not part of the fluid and they did not carry the sound, but they had energy, and energy gravitates.

So the same initial excess that launched the sound wave also launched a shell of neutrinos — moving at $c$ rather than $c/\sqrt{3}$, and therefore immediately ahead of the sound. Their gravity is outward of the acoustic shell, and it pulls on it. The wave is not changed in speed; it is displaced slightly forward of where it would have been without them, by an amount set during the first few oscillations, when the neutrinos were still a large share of the total energy and close enough to matter.

Daniel Bashinsky and Uroš Seljak worked the consequence out in 2004. In the oscillation’s argument the displacement appears as an added phase, $\sin(k r_s + \varphi)$ rather than $\sin(k r_s)$, and in the small-scale limit the phase is

$$\varphi_\infty \approx 0.191\pi\,\varepsilon_\nu,$$

with $\varepsilon_\nu$ the fraction of the radiation density carried by free-streaming species. For three neutrinos $\varepsilon_\nu = 0.41$ and the phase is a quarter of a radian.

Why it has to be a phase #

The phase is not an accident of approximation, and the reason is causal. A shift of the whole wave by a constant angle at small scales means the wave’s pattern began, in some sense, ahead of its own front — the gravitational pull that displaced it arrived before the sound did. Nothing that travels at or below the sound speed can do that. Daniel Baumann, Daniel Green, Joel Meyers and Benjamin Wallisch made the argument general in 2016: in the linear equations, a constant phase shift at large wavenumber requires a perturbation that propagates faster than sound, and in the standard universe the only candidates are free-streaming relativistic particles.

That gives the shift two properties that matter.

It counts something a total cannot. The amount of radiation in the early universe is measured in other ways — by the helium a faster early expansion would have made, and by the damping tail of the microwave background, which depends on how fast the universe expanded at recombination. Those respond to the radiation’s energy, whatever it does. The phase responds only to the part of it that streams. A hypothetical extra species that interacted with itself strongly enough to behave as a fluid would raise the expansion rate exactly as a free-streaming one would and add almost no phase, because a fluid’s pressure waves travel no faster than the plasma’s.

It sits in the oscillation, not the envelope. Changing the matter density, the baryon fraction or the amount of radiation changes the shape and amplitude of the smooth power spectrum, and analyses remove that shape before fitting the oscillation. A phase lives in the positions of the crests themselves. It is in the part of the data the acoustic measurement already uses, and it costs nothing to look for except the willingness to let the crests do something other than slide uniformly.

Why the largest scales escape it #

The phase is not the same at every wavenumber, and the way it varies is a record of when the pull acted.

A mode of a given wavelength starts to oscillate only when the horizon grows large enough to contain it. Small-scale modes entered early, in the radiation era, when the neutrinos were forty per cent of everything that gravitated and the gravitational potential was set largely by radiation. Those modes felt the full pull during their first oscillations, when it mattered most, and they carry the full asymptotic phase.

Large-scale modes entered later. By the time a mode of a few hundred megaparsecs began to oscillate, the universe was approaching matter domination: the dark matter, which does not stream and does not oscillate, was supplying most of the gravity, and the neutrinos’ share of the potential had fallen. Those modes felt a weaker pull and carry a smaller phase. The scale at which the phase reaches half its final value, $0.032\,h\,\mathrm{Mpc^{-1}}$, is within a factor of two of the scale that entered the horizon at matter–radiation equality, and that is not a coincidence: it is the boundary between modes that grew up among radiation and modes that grew up among matter.

This is also why the effect is much harder to see in galaxies than in the microwave background. The photon peaks of the background sample wavenumbers well above that boundary, where the phase has saturated and the oscillations are many; the galaxy feature is a damped wave whose first few crests sit near the boundary, where the phase is still rising and the crests are few. A galaxy survey sees the phase over about one decade of wavenumber, in four or five crests, before the damping erases them.

How little room there is #

The saturation is worth dwelling on because it sets what the phase is good for. The share $\varepsilon_\nu = 0.227N/(1 + 0.227N)$ comes from the neutrinos’ energy relative to the photons’, each species contributing $\tfrac{7}{8}(4/11)^{4/3}$ of the photon density. The first species moves the share from nothing to eighteen per cent; the fourth moves it from forty-one to forty-eight. A phase measured to ten per cent of its standard value distinguishes three species from none with ease and three from four only marginally.

That is the opposite of how the question is usually posed. The quantity cosmologists conventionally quote, $N_{\rm eff}$, is centred on 3.044 and the interesting hypotheses are small additions to it — a light relic adding a few hundredths, a sterile neutrino adding up to one. For that question the phase is a weak instrument. For the question it answers well — whether the radiation that is known to be there streamed freely at all, as the standard model of particle physics says — it is the only instrument outside the microwave background, and its error does not depend on the background’s.

A shift that no ruler can copy #

An acoustic analysis fits the position of the feature by stretching a template. If the template’s ruler is too short by one per cent, every crest in the data sits one per cent closer together than in the template, and stretching the template by one per cent puts them back. The stretch is the measurement; converted through the sound horizon, it is the distance.

A phase cannot be removed that way, and the reason is the whole of what makes it measurable.

At low wavenumber, where the crests are few and far apart, a fixed angle is a large fraction of the spacing, and the phase looks like a ruler about three per cent too long. At high wavenumber it looks like a ruler under one per cent too long. No single stretch reproduces both, and the mismatch between the curve and its best horizontal line is the signal. An analysis that fitted only a stretch, against a template with no phase in it, would absorb most of the shift into the distance and report a ruler two per cent too long — which is why every real template already includes the standard neutrinos’ phase, and why a wrong number of species in the template would be a quiet systematic in every distance quoted from the feature.

This is the surprising connection. The acoustic feature is advertised as a ruler whose length is computed from the early universe and whose position is measured late, and the two meet in a distance. Part of what makes that computation trustworthy is a statement about particle physics — that three species of neutrino streamed freely from the first second — and the same feature that delivers the distance can check that statement without leaving the galaxy survey.

What the data can separate #

Treating the phase as a free parameter has a price, and the price is the distance.

The correlation between the two parameters in this toy fit is 0.92. Freeing the phase inflates the error on the ruler from about half a per cent to about one and a half, because a part of any apparent stretch could now be a phase. So the measurement of the phase is not free in the sense that matters to an analysis whose purpose is distances: it is made either in a separate fit, with the ruler free and the distance thrown away, or in a joint one with the ruler tied to the microwave background’s calibration. The published measurement used the first.

The other feature of the figure is less comfortable. With $\beta$ measured to $\pm 0.9$, the standard universe sits only about one standard deviation from a universe with no phase at all. A survey of that size, if the truth is three neutrinos, should expect to see the shift at about one sigma. Six times the volume brings it to between two and three. Detection is a question of volume, and the number of independent modes grows only as the volume does.

What was actually measured #

In 2019 Daniel Baumann, Florian Beutler and colleagues fitted the twelfth data release of the Baryon Oscillation Spectroscopic Survey — about a million galaxies in two redshift slices between 0.2 and 0.75 — with a template in which the neutrino phase was multiplied by a free amplitude $\beta$, and the ruler, the broadband shape and the damping were all left free.

The result was $\beta$ near two, with an uncertainty of about one, excluding zero at better than ninety-five per cent confidence. The value is larger than the standard prediction by about one standard deviation, which is what the figure above says a survey of that size should expect roughly a third of the time. It is the first detection of the cosmic neutrino background’s gravity in anything other than the microwave background, and like every first detection near two sigma it is mostly a demonstration that the method works.

Three things stand between the galaxy catalogue and that number.

The smooth spectrum must be removed without removing the phase. The oscillation is separated from the broadband power by fitting a smooth function, and a smooth function flexible enough to absorb the broadband shape could in principle absorb some of the phase at the lowest wavenumbers, where the phase is changing fastest. The analysis tested its templates on simulations with the phase switched on and off.

The damping must be modelled. Bulk flows smear the feature, and in Fourier space the smearing suppresses the oscillation’s amplitude at high wavenumber while leaving its phase alone — to first order. At second order, nonlinear evolution itself shifts the feature by a fraction of a per cent, and that shift is a small phase of its own. It is calculable and was included, and its size is a known limit on how precisely the neutrino phase can ever be read.

The redshift-space distortions must be separated. The galaxies’ peculiar velocities change the spectrum’s amplitude along the line of sight; they do not, at the order that matters, change the phase.

The phase had been seen before, in the microwave background: Brent Follin, Lloyd Knox, Marius Millea and Zhen Pan found it in the Planck temperature spectrum in 2015, by the same reasoning applied to the acoustic peaks of the photons rather than of the galaxies. The galaxy measurement is not more precise. It is independent, made with a different tracer at a different epoch through a different chain of systematics, and its agreement is the point.

The same particles, counted three ways #

The early universe’s neutrinos are now measured in three ways that answer three different questions.

Big Bang nucleosynthesis measures the expansion rate at one second, when the neutron-to-proton ratio froze, through the helium it left. It counts every form of energy present then, streaming or not.

The microwave background’s damping tail measures the expansion rate at recombination, through how far photons diffused before the plasma became transparent. It too counts energy, streaming or not — and, separately, contains the phase in its own peaks.

The galaxy power spectrum contains the phase and nothing else about the neutrinos at this order. It says what fraction of the radiation streamed.

If all three agree on three species — and so far they do, within their errors — then the standard picture holds on three tests that could each have failed differently. If the energy measurements found an excess and the phase did not, the excess would be radiation that does not stream: a self-interacting dark sector, which is exactly the kind of addition proposed to ease the disagreement over the expansion rate. If the phase were larger than the energy allowed, something would be streaming that is not counted as radiation at all. The phase is the test that separates “more radiation” from “different radiation”, and no single measurement does that.

The neutrinos’ mass is a fourth number from the same particles, and a different one. Its effect on structure appears billions of years later, when the neutrinos have slowed enough to be held back from small clumps, and it suppresses the smooth power spectrum rather than shifting the oscillation. The phase was set in the first few hundred thousand years, when the neutrinos were still effectively massless; the mass suppression was set in the last few billion. The same particle writes on the same spectrum twice, in two different parts of it, at two different ages.

What the figures leave out #

The oscillation drawn here is a sinusoid with a Gaussian envelope, and the real one is not. Its amplitude and damping were chosen to look like the measured feature; only the phase is computed. The quantities read off the figures — the equivalent stretch, the correlation in the fit — depend on the damping scale chosen, and a sharper feature after reconstruction would measure the phase somewhat better and with a weaker correlation.

The template for the phase’s shape is a fit, from Baumann and colleagues, to the output of the full Boltzmann equations. Its asymptote is analytic; its turnover at $k_*$ is not, and a different number of species moves the turnover slightly, which the template ignores.

The fit is a Fisher estimate on noise-free data with independent bins of constant noise. The real covariance is correlated between bins and grows toward low wavenumber, where cosmic variance dominates — which is exactly where the phase is changing and the distinction from a stretch is sharpest. The width was set to match the published measurement rather than computed from a survey model, so the six-fold extrapolation is a scaling, not a forecast.

And the number of species is not the only thing that shifts the phase. Anything that streams at relativistic speed contributes, including any light particle beyond the standard model, and the nonlinear evolution of the galaxies adds its own small shift. The phase measures the sum, and only the second is predictable well enough to be removed.

A ruler that is also a clock #

The phase belongs to a more general observation. A periodic signal carries two numbers — its period and its phase — and most measurements throw away the second because the first is what they were designed to get. A ruler is a period. The phase records when, relative to the wave’s own start, something else acted on it, and in the acoustic case the something else is a set of particles that left the plasma at one second and outran the sound for the next four hundred thousand years.

That makes the galaxy power spectrum a record of the first second as well as of the expansion history. The record is faint — three per cent of a cycle, in a feature that is itself a few per cent of the power — and the first reading of it is barely above the noise. But it is legible with the same data that measure the distances, and every survey large enough to measure distances well will read it whether it intends to or not.

Still open: whether the phase and the energy agree #

The decisive comparison is the one the three measurements make together. The microwave background and nucleosynthesis now measure $N_{\rm eff}$ to about a fifth of a species and agree on three; the galaxy phase, measured to about one species-equivalent in its first reading, agrees with them inside its much larger error.

The surveys now running have several times the volume and a broader redshift range, and the phase measurement improves roughly as the square root of the number of modes. A reading at the level of a third of the standard value — the inner contour above — would test, independently of the microwave background, whether the radiation the expansion rate requires is radiation that streams. Proposals that add a self-interacting dark radiation to ease the tension in the local and early-universe expansion rates predict a deficit in exactly that comparison, and the galaxy phase is the measurement that could find one without trusting the background’s own model of the plasma.

What it cannot do is reach the light relics a few hundredths of a species large. For those the phase is saturated in the wrong direction, and the microwave background’s next generation — not the galaxies — is where the count will be settled.