A constant that is an angle divided by a length

The microwave background’s power spectrum has peaks at particular angular scales, and the position of the first peak is measured to a part in three thousand — the best-determined number in cosmology. It is often described as a measurement of the expansion rate. It is not.

What it measures is an angle: how large the sound horizon at recombination appears on the sky. An angle is a length divided by a distance, and extracting a distance — and from it an expansion rate — requires the length.

Two measurements of one constant sit five sigma apart, and the disagreement is usually described as being about the expansion rate. It is more accurately described as being about a length that nobody has measured.

Where the length comes from #

The sound horizon is the distance a pressure wave travelled in the photon–baryon fluid between the end of inflation and recombination. It is an integral of the sound speed over time,

$$r_s = \int_0^{t_*} c_s\,\frac{dt}{a},$$

and every ingredient is a calculation rather than an observation. The sound speed depends on the ratio of baryons to photons; the expansion rate during that era depends on the radiation density, which depends on the photon temperature and on how many relativistic species there are; and the endpoint depends on when recombination happened, which is atomic physics.

None of that is unreasonable — the ingredients are individually well constrained, and the calculation is standard — and none of it is a measurement of $r_s$. The standard model gives 147.1 megaparsecs with an uncertainty under a per cent, and that uncertainty is a propagation of parameter errors within an assumed model rather than a measurement of a length.

It is worth noticing how well determined the ratio is compared with either quantity. The angle is known to a part in three thousand and the horizon to a part in two hundred, so the Hubble constant from the microwave background inherits the second — its half-per-cent uncertainty is almost entirely the horizon’s, and hardly at all the angle’s. That is an unusual situation: the celebrated precision of the peak position contributes almost nothing to the error on the number it is famous for producing. One number that is an age, a size and a density is derived here from a quantity that is not an observation at all.

The other side of the ratio is not simple either #

The angle is a length divided by a distance, and the previous section is about the length. The distance deserves a paragraph of its own, because it is what turns the ratio into an expansion rate and it is not a single number.

The distance to last scattering is an integral of the inverse expansion rate over redshift from zero to eleven hundred, so it depends on the matter density and on the Hubble constant together. To a good approximation it depends on the product of the matter density parameter and the square of the Hubble constant — the physical matter density — and on the Hubble constant separately, in a combination that the angle alone cannot untangle. That is the geometric degeneracy: a family of models with different matter densities and different Hubble constants all produce the same angular scale, and the peak positions cannot tell them apart.

What breaks it inside the microwave data itself is the peak heights. The ratio of the odd to the even peaks measures the baryon density, and the height of the third peak relative to the first measures the physical matter density, both from the physics of the oscillation rather than from geometry. With those two fixed, the angle determines the Hubble constant — and the chain of inference now runs through three measured quantities and one computed length rather than through one angle.

So the quoted precision belongs to the fit, not to the peak. The half-per-cent uncertainty on the microwave background’s Hubble constant is a marginalised uncertainty within a six-parameter model, and it is dominated by how well the physical densities are measured and by how well the horizon is computed. This is worth being explicit about, because the phrase “the best-measured number in cosmology” attaches to the angle and is routinely transferred to the constant derived from it.

The arithmetic connecting the two ends of the argument is nonetheless simple, which is why the locus in the figures is a straight hyperbola. Holding the angle and the physical densities fixed, the distance to last scattering is inversely proportional to the Hubble constant to a good approximation, so a horizon shorter by one per cent gives a Hubble constant larger by one per cent. Eight per cent in the length is eight per cent in the rate, exactly the size of the disagreement. That one-to-one exchange is the reason the tension can be stated as a statement about a length at all — and it is also why a partial explanation is of little use: something that shortens the horizon by three per cent leaves five per cent unexplained rather than reducing the problem to a manageable one. The amplitude and the optical depth arrive multiplied in the same data set and have the same structure, and the same warning applies to reading either one alone.

What a shorter horizon would take #

If the late-universe Hubble constant is right, the sound horizon must be about eight per cent shorter than the standard calculation gives. Shortening it means either making the sound speed lower, making the expansion during that era faster, or making recombination happen earlier.

More relativistic species speed up the expansion before recombination, so the wave has less time to travel. The required number is well above the three neutrino species the standard model has, and it is constrained by the shape of the damping tail — so a pure addition of relativistic energy is disfavoured by the same data that motivates it.

Early dark energy — a component that contributes briefly around matter–radiation equality and then dilutes away — does the same job in a more targeted way, and the proposals of that kind are the most-discussed class.

Earlier recombination would also work, and it requires changing atomic physics or the primordial helium abundance, both of which are constrained elsewhere.

Every proposal has to shorten the horizon by eight per cent while leaving the peak positions, the peak height ratios and the damping tail as measured — which is a demanding requirement, and it is why none of them is comfortable.

Notice which of the three routes is the most demanding. Making recombination earlier requires changing atomic physics, which is constrained by laboratory measurements to a precision nothing cosmological approaches. Adding relativistic species is constrained by the damping tail and by the light-element abundances. Early dark energy is constrained by nothing directly — its whole appeal is that it acts briefly at an epoch nothing else probes — which is simultaneously why it survives and why it is unsatisfying: a component invented to have exactly the effect required, at exactly the time nothing can see it, is not much of an explanation.

The inverse distance ladder #

There is a route that inverts the whole argument and it is worth following because it isolates the assumption.

Baryon acoustic oscillations imprint the same sound horizon on the distribution of galaxies, so measuring their apparent scale at several redshifts gives a set of distances in units of the sound horizon. Calibrating those distances with a supernova sample then gives the expansion history in the same units.

Feeding in the sound horizon from the standard model gives a Hubble constant in agreement with the microwave background. Feeding in a value calibrated from the local distance ladder instead gives a sound horizon several per cent shorter than the standard calculation.

So the disagreement can be stated in two equivalent ways, and the second is more useful: either the local ladder is wrong, or the sound horizon is not what the early-universe model says. Nothing in between the two epochs — no change to dark energy, no change to the expansion history since recombination — resolves it, because the inverse ladder measures that history directly and finds it standard.

There is one more way to state the same conclusion and it is the most compact. Three quantities are involved: the sound horizon, the expansion history since recombination, and the Hubble constant today. The microwave background measures the first two in combination through one angle; the inverse ladder measures the second directly; the distance ladder measures the third. Any two of them determine the third, and the three measurements are mutually inconsistent by about eight per cent. Since the middle one is measured most robustly and agrees with the standard model, the inconsistency is between the first and the last — which is to say between a length computed from the early universe and a rate measured in the local one. The same ruler measured twice, ten billion years apart is what makes that decomposition possible.

There is a fourth possibility that the inverse ladder also constrains and that is worth naming because it is the one most often proposed casually: a change in the late expansion history — a different dark-energy equation of state, or a transition at low redshift. Such a change would alter the distance to last scattering and therefore the inferred Hubble constant, so it looks like a candidate. The inverse ladder measures that distance directly, from baryon oscillations and supernovae, and finds it standard to a per cent. So the late-universe route is closed by a measurement rather than by an argument, and the closure is the single most useful thing the inverse ladder has established.

What was actually measured #

Three numbers, and the third is the one that narrows the possibilities.

The angular scale, to a part in three thousand. $\theta_s = 1.04109\times10^{-2}$ radians, or about 0.6 degrees. It is the best-measured quantity in cosmology and it is not in dispute.

The sound horizon from the model, to under a per cent. 147.1 megaparsecs, with an uncertainty dominated by the baryon density.

And the expansion history between the two epochs, measured independently. The inverse ladder measures how distance grows with redshift from zero to about two and a half, and it agrees with the standard model. That is what excludes a late-universe explanation, and it is why the discussion has concentrated on the first four hundred thousand years.

There is a fifth measurement that deserves a line because it is the only genuinely independent check on the horizon’s length: big bang nucleosynthesis. The light-element abundances measure the baryon-to-photon ratio, which is one of the two main ingredients in the sound speed, and they do so with physics from the first three minutes rather than from four hundred thousand years. The two determinations of the baryon density — from the peak height ratios and from the deuterium abundance — agree to a per cent, which is a genuine consistency test of the era in which the horizon was set. It does not measure the horizon, and it removes one of the ways it could have been wrong.

Where the picture stops #

Three of them, and the third is the honest state of the question.

The angular scale is not quite the peak position. The observed peak positions are shifted from the pure acoustic scale by a phase that depends on the driving of the oscillations, and the conversion carries a small model dependence of its own. It is well understood and it is not where the eight per cent could hide.

A shorter horizon is not free. Any modification that shortens it also changes the damping scale, the peak height ratios and the growth of structure, and the data constrain all of those. The proposals that survive are the ones tuned to change the horizon and little else, and their tuning is itself an argument against them.

And the alternative is that a measurement is wrong. The local distance ladder is a chain of calibrations, each with its own systematics, and the possibility that one of them is wrong by a few per cent has not been excluded — nor has the possibility that the microwave background analysis has an unrecognised systematic. The field’s honest position is that a five-sigma disagreement between two carefully done measurements is more often a systematic than new physics, and that nobody has found the systematic.

A fourth belongs with them, and it is about what “the standard model” is doing in the argument. The sound horizon’s calculation uses the same six-parameter model that the peak positions constrain, so the horizon and the angle are not independent: fitting the model to the data determines both together, and the quoted horizon uncertainty is a marginalised one within that fit. A modification that changes the pre-recombination physics changes the fit as well as the horizon, so the eight per cent cannot simply be inserted — the whole solution has to be recomputed, and the recomputation usually pushes some other parameter outside its measured range. That coupling is why the proposals are so constrained and it is invisible in the simple locus drawn here.

Why stating it as a length is more useful than as a rate #

The general point is worth pulling out, because it changes what would count as a solution.

A measurement quoted in the wrong variable hides where its assumptions are. Quoting the microwave background’s result as a Hubble constant makes it look like a direct measurement of the expansion rate, and invites comparisons with local measurements as though the two were the same kind of thing. Quoting it as an angle plus an assumed length makes the structure visible: one exquisitely measured observable, one computed quantity, and a conclusion that inherits both.

That reframing has been productive. It identified the sound horizon as the quantity the disagreement is about, showed that late-universe modifications cannot resolve it, and turned a vague question about cosmology into a specific one about the first four hundred thousand years. None of that followed from improving either measurement.

The general form is one this collection meets repeatedly: a rotation curve’s decomposition depends on a mass-to-light ratio, a time delay’s Hubble constant depends on a mass sheet, and a peak position’s Hubble constant depends on a sound horizon. In each case the useful move is to quote the thing measured and name the thing assumed — because a reader can then see which one to doubt.

End on what a resolution would look like, since the question has been open for a decade. If the local ladder has a systematic, it will show up as a discrepancy between the several independent routes through it — Cepheids, the tip of the red-giant branch, masers, Miras — and those routes currently agree with each other better than they agree with the microwave background, which is the main argument against a single systematic. If the early universe is non-standard, the modification will have to survive the constraints from the damping tail, the polarisation, the light-element abundances and the growth of structure simultaneously, and none proposed so far does comfortably. And if it is a statistical fluctuation, it will shrink as both measurements improve, which they are doing. The one thing the decade has established is where the answer cannot be: between the two epochs, where the expansion history is measured directly and is standard.

A final note on where the ruler came from, because it is what makes the whole comparison possible. The sound horizon is not an arbitrary length: it is the distance a wave travelled in a fluid whose properties are set by the photon and baryon densities, stopped at the moment the fluid ceased to exist. A standing wave frozen at one instant is the picture, and its consequence is that one length was laid down everywhere at once — which is why it appears both as an angle on the last-scattering surface and as a separation between galaxies today. A ruler that is a piece of physics rather than an object is the only kind that could be measured at both ends of the universe’s history, and that is precisely why an error in computing it propagates into everything.

Where the ladder goes next #

The obvious next rung is the calculation of the sound horizon itself: what goes into the integral, which ingredient dominates its uncertainty, and how much room the other data leave for changing each one. The rung after it is the inverse ladder in detail — how baryon acoustic oscillations and supernovae are combined into an expansion history in units of a ruler, and what that history rules out.