A ruler shortened by something already gone

The microwave background does not measure the Hubble constant. It measures an angle — the apparent size on the sky of the distance sound waves travelled in the early plasma before it became transparent — to a part in three thousand. Converting that angle into an expansion rate needs the physical length of the ruler, and the length is computed: the sound speed, integrated over the time between the beginning and recombination, four hundred thousand years later. The computation assumes the standard contents of the early universe, and with them it gives 144 megaparsecs and a Hubble constant of 67.4 kilometres per second per megaparsec.

The local distance scale gives 73. The two numbers are measured independently, each to about a per cent, and they differ by five standard deviations. One of the few ways to reconcile them without disturbing anything measured after recombination is to make the ruler shorter. A ruler that is shorter by seven or eight per cent, seen at the same angle, is nearer, and a universe in which the last-scattering surface is nearer, with everything else fixed, is expanding faster.

The ruler is an integral over time, so it can be shortened by making the time shorter — by making the universe expand faster while the sound is travelling. Any extra energy present before recombination does that. The difficulty is that almost every kind of extra energy that would do it would also do other things that are ruled out, and the idea called early dark energy is the attempt to find one that does only this.

Energy that behaves like a constant and then does not #

The requirements are strict. The component must be a significant fraction of the energy density — ten per cent or so — at the time most of the sound horizon is being built, which is in the tens of thousands of years before recombination. It must be negligible earlier, or it would change the abundances of the light elements made in the first minutes, which are measured. And it must be gone by recombination and afterwards, or it would change the distance from the last-scattering surface to the observer, the growth of structure and everything measured since, which is where the standard model already fits well.

A cosmological constant has the first part of that behaviour and not the second. Its energy density never changes, so it is negligible at early times, when radiation and matter are dense, and grows in relative importance as they thin out. Radiation dilutes as the fourth power of the scale factor, and nothing that is ordinary dilutes faster. The component needed has to be constant at first, like a cosmological constant, and then dilute faster than radiation — as the sixth power of the scale factor — so that it vanishes as quickly as it appeared.

A scalar field does that naturally. A field held in place by the expansion’s friction behaves like a cosmological constant; when the expansion slows enough, the field is released, rolls down its potential and oscillates, and for a steep enough potential its oscillating energy dilutes as the sixth power of the scale factor. The critical moment is set by the field’s mass. That is the origin of the parametrisation used here.

The figure shows why the component is sometimes called a bump. It is present for about one e-folding of expansion, peaks near the redshift at which matter and radiation had equal densities, and leaves. That narrowness is not an aesthetic choice. It is what the data force: a component that lingered would be seen in the late universe; one that arrived earlier would change what is already measured about the first minutes.

Why ordinary additions will not do #

The obvious way to add energy before recombination is to add radiation: another species of light particle, a fourth neutrino or something like it, moving close to the speed of light. It speeds the early expansion and shortens the sound horizon, and for a while it was the leading candidate. It fails for two reasons that are instructive about what early dark energy has to avoid.

Extra radiation is present from the start. It is there during the first three minutes, when the light elements are made, and a faster expansion then leaves more neutrons unconverted and more helium in the universe; the measured helium and deuterium abundances limit it to a fraction of one extra species. And it is present all the way to recombination, when it speeds the expansion in exactly the stretch where the damping length is being built, so it shrinks the ruler and the random walk together and the damping tail notices at once.

A component that is absent at the start and gone by the end avoids the first problem entirely and the second partly. That is the whole design brief. The energy has to arrive after the elements are made, dominate for less than an e-folding, and leave before the last scattering — a sequence ordinary matter and radiation cannot follow, because their densities only ever fall more slowly or at the same rate as the universe expands.

Where the ruler is built #

The sound horizon is not built evenly. Sound in the early plasma travels at about 58 per cent of the speed of light, reduced below $c/\sqrt{3}$ by the inertia of the baryons, and it covers comoving distance fastest when the expansion is slowest — which is the last stretch before recombination.

The shape of that curve is the reason the critical redshift matters. A component peaking at a redshift of a hundred thousand would act when very little of the horizon is being built and change almost nothing. One peaking after recombination could not shorten the horizon at all, since the horizon is finished. The effect is largest for a component acting in the last few thousand years’ worth of expansion before recombination, where the curve is highest.

The published analyses find their best fits with the component peaking at a redshift of about three to five thousand — earlier than the purely geometric optimum in the figure, because a component too near recombination alters the relative heights of the acoustic peaks through its effect on the gravitational potentials the plasma oscillates in. The window is narrow in both directions, and its position, close to the time matter and radiation had equal densities, is a coincidence the model has to accept: the same kind of timing coincidence the late dark energy already carries, now repeated at an unrelated epoch.

The dark matter it costs #

A shorter ruler at the same angle requires a nearer surface, and the nearness is bought by raising the Hubble constant. But the Hubble constant enters the rest of the fit, and the rest of the fit pushes back. The matter density as a fraction of the total is measured independently at late times, by the baryon acoustic feature in the galaxy distribution and by supernovae, at about 0.31. Holding that fraction fixed while raising the Hubble constant from 67 to 73 means raising the physical matter density — the fraction times the square of the Hubble constant — by eighteen per cent. Almost all of the increase is dark matter.

The fits to the microwave background find the same thing by a different route. Early dark energy, while present, speeds the expansion and weakens the gravitational driving of the acoustic oscillations near the time it peaks, which changes the heights of the peaks. Restoring them requires more dark matter. The two requirements, from geometry and from the peaks, agree in direction and roughly in size, which is part of why the model fits the background as well as the standard one does.

The extra dark matter has a consequence that points the wrong way. More dark matter means structure grows faster, and the amplitude of matter clustering today — measured by weak lensing as a combination called $S_8$ — comes out higher than the standard model’s. The lensing surveys already find it lower than the standard model’s. So early dark energy, in reducing one tension, worsens another, and joint fits that include the lensing data settle on less early dark energy and a smaller shift in the Hubble constant than either would alone.

There is a subtler way to put the cost. The combination of measurements that fixes the Hubble constant from the background is not the angle alone but the angle together with the matter density in physical units, which the relative heights of the peaks measure. In the standard model the two are fixed, and the Hubble constant follows. Early dark energy works only because the peaks’ heights can be kept while the matter density rises — the component’s effect on the potentials compensates. The more precisely the peak heights are measured, the less room there is for that compensation, and the satellite and ground-based data together now pin the physical matter density to about one per cent in the standard model and to a few per cent with early dark energy allowed. The published fits with early dark energy settle on a rise of the dark-matter density of about ten per cent, and a Hubble constant of 71 to 72; the eighteen per cent the geometry asks for, to reach 73, is more than the peaks allow them to take. The two halves of the data are pulling on the same parameter from opposite sides, and the compromise lands between the two measured values of the Hubble constant rather than on the local one.

The damping tail as a test #

There is one more consequence, and it is the one that can decide the question. The microwave background carries two lengths from recombination, not one. The sound horizon sets the spacing of the acoustic peaks. The diffusion length of photons — the distance a photon random-walks by scattering before recombination — sets where the peaks are erased at small angles, the damping tail. The damping tail is a measure of the thickness of recombination, and it responds to the expansion rate differently.

The difference in scaling is general. Anything that shortens the time before recombination shortens a straight-line distance in proportion and a random-walk distance by its square root, so the ratio of the two is a direct test of whether the early universe expanded faster than the standard model says. With the angle of the acoustic peaks fixed by the data, a larger ratio puts the damping tail at larger angles — lower multipoles — than the standard model predicts, by an amount that the ground-based telescopes mapping the background at arcminute resolution can measure.

Those measurements, combined with the satellite data, have so far found the damping tail where the standard model puts it. They leave room for a few per cent of early dark energy and are in tension with the larger fractions needed to reach the local Hubble constant. The clumped-recombination idea, which shortens the ruler by making recombination happen earlier rather than by speeding the expansion, is tested by the same tail and constrained in the same way. The finest detail of the oldest light is where every proposal to move the ruler has to pass.

What the calculation holds fixed #

The figures compute the background — the expansion rate, the sound horizon, the diffusion length and the distance to last scattering — and solve for the Hubble constant that keeps one measured angle fixed, with the baryon density and the matter fraction held at measured values. They do not fit the full spectrum of the background’s anisotropies, which also depends on the component’s perturbations, on the spectral index and amplitude of the primordial fluctuations, and on the details of recombination that the diffusion integral here approximates by assuming full ionisation until the last moment. The published fits adjust all of those, and they find somewhat larger shifts in the Hubble constant for a given fraction than the purely geometric calculation — about 71 to 72 for ten per cent rather than 70.

The shape of the component is one parametrisation among several. Models with different potentials dilute at different rates after the critical redshift and give different trade-offs between the Hubble constant and the fit to the peaks. None has a theoretical motivation independent of the tension it addresses; the field’s mass is chosen so that it is released at the right time, and the fraction so that it does the right amount. That is a fair description of a phenomenological model, not a criticism of it, and it is why the damping tail and the lensing amplitude, which the model did not set out to fit, are the tests that matter.

What early means in years #

The redshifts translate into times that make the proposal’s delicacy vivid. A redshift of 3500, where the component peaks in the fits, is about fifty thousand years after the beginning. Recombination, at 1090, is about three hundred and seventy thousand years. The component would have risen from insignificance to a tenth of everything and fallen back again in something like a hundred thousand years, in a universe that was then tens of billions of times denser and three and a half thousand times hotter than today and in which nothing had yet formed but the light elements and the sound waves.

It would also have been utterly different from the dark energy that dominates now. The present dark energy is several billion times less dense than the early component would have been at its peak, and there is no known reason why a field should have been released at fifty thousand years rather than at any other time. The two share a name only because both behave, for part of their history, like a constant energy of empty space. If the early component exists, it is a second episode of that behaviour, at a density nine or ten orders of magnitude higher, with no connection to the first that anyone has found.

Three quantities and two epochs #

The Hubble tension can be stated through three lengths: the sound horizon at recombination, the expansion history since, and the present expansion rate. The background measures the first two in combination; the inverse route, calibrating the acoustic scale with supernovae, measures the second directly and finds it standard; the local distance scale measures the third. The inconsistency lies between a length computed from the early universe and a rate measured in the local one. Early dark energy is the answer that takes the local rate at face value and says the computed length was wrong because the early universe contained something that is no longer there.

It is an answer of an unusual kind. The component would have left no trace in the present universe at all: it existed for a few thousand years, three hundred and seventy thousand years before the first atoms, and decayed away as fast as anything can. Its only lasting effect is the length of a ruler, and the only evidence for it is that the ruler seems too long. A measurement that would confirm it has to come from the same epoch — from the damping tail, from the precise shape of the peaks, or from the polarisation of the same light.

Still open: whether the ruler or the local scale moves #

The two sides of the tension are being tested separately. On the local side, independent calibrations of the local distance scale — using the tip of the red giant branch, or carbon stars, instead of Cepheids — give values between 70 and 73, and whether the Cepheid calibration carries a systematic error of a few per cent is argued with new infrared data from space. On the early side, the damping-tail measurements are tightening each year, and the next generation of ground-based surveys will measure the tail and the polarisation to the precision at which a few per cent of early dark energy would be either detected or excluded. If the local value settles near 70, a small early component, or none, would suffice. If it stays at 73, the ruler will have to be shortened by more than the damping tail currently allows, and the explanation will have to be something that shortens a distance and a random walk in the same proportion — which no proposal yet does.