A hole too deep for the sky to have made

The expansion rate measured with Cepheids and supernovae comes out near 73 kilometres a second per megaparsec. The rate inferred from the microwave background, through the standard cosmological model, comes out at 67.4. Both are quoted with errors of about a kilometre a second, and the gap between them — 8.3 per cent — has survived a decade of attempts to close it from either side.

Most proposed resolutions change something: an error in a calibration, a new component of the early universe that shortens the ruler the background measures, a different behaviour of dark energy. One proposal changes nothing at all. It notes that the two numbers measure different things — one the expansion rate of the universe as a whole, the other the expansion rate of the region within a few hundred megaparsecs of the Milky Way — and asks whether the region could simply be unusual.

The physics is not in doubt, and it is the same physics that makes a redshift survey’s map squashed along the line of sight: matter falling toward overdensities and away from underdensities adds a velocity to every galaxy, and the velocity field around a void is an outflow. An underdense region expands faster than the mean, because it has less matter decelerating it; an overdense region expands slower. In linear theory the local expansion rate inside a sphere of mean density contrast $\delta$ differs from the global rate by

$$\frac{\delta H}{H} = -\frac{f\,\delta}{3},$$

where $f$, the growth rate of structure, is about $\Omega_m^{0.55} \approx 0.53$ today. An observer at the centre of a region underdense by $\delta = -0.47$ would measure a Hubble constant 8.3 per cent high, and would be right about the region and wrong about the universe.

The question is not whether such a void would work. It is how likely it is that one exists, at the right size, centred close enough to the Milky Way. And that question has an answer in the same cosmology the tension is about.

How deep holes are #

The standard model predicts not only the mean density of the universe but the statistics of its departures from the mean. The density field is described by a power spectrum, fixed in shape by the matter and baryon content and in amplitude by the fluctuations measured in the microwave background and in galaxy clustering. From the power spectrum follows the variance of the density averaged over a sphere of any radius — the typical size, $\sigma(R)$, of the density contrast a randomly placed sphere contains.

The hero figure draws $\sigma(R)$. At twenty megaparsecs it is about a half: spheres that size are routinely half empty or half again overdense, which is why the nearby universe looks lumpy. It falls steeply with radius, because large spheres average over many independent fluctuations. At a hundred megaparsecs it is a tenth; at three hundred, 0.022; at six hundred, under one per cent. Large volumes are close to average, and that, measured, is what homogeneity above a hundred megaparsecs means.

The void the tension needs sits as a horizontal line across the figure. It must be underdense by 0.47 over the volume the supernovae sample — the supernovae used for the local measurement lie between about a hundred and six hundred megaparsecs away, deliberately beyond the nearby region where individual motions dominate. Across that range the required contrast is four to sixty-seven standard deviations. Even at the near edge, where fluctuations are largest, the void is a four-sigma event; at the scales that dominate the fit it is beyond any probability worth stating.

The scatter that is allowed #

The same calculation can be turned into a statement about the local Hubble constant itself. Any observer, anywhere, measuring the expansion rate inside a sphere of radius $R$ will find a value that differs from the global one by a random amount whose standard deviation is $f\sigma(R)/3$. This is cosmic variance for the Hubble constant: not a measurement error but a genuine difference between the local and the global universe, set by where the observer happens to be.

The curve explains a design choice of the local measurement that is easy to read past. The local measurement deliberately excludes supernovae closer than a redshift of about 0.023, roughly a hundred megaparsecs. Inside that distance the cosmic variance alone is several per cent, comparable to the tension, and the motions of individual galaxies add scatter on top. Beyond it the cosmic variance falls below the measurement’s own error, and by the time the sample reaches its median distance it is a few tenths of a per cent — too small by more than an order of magnitude to supply the gap.

That is the first and simplest result: an ordinary local fluctuation, of the size the standard model expects, changes the local Hubble constant by about half a per cent. The tension is sixteen times larger. Whatever it is, it is not the universe’s ordinary unevenness.

The step a void would leave #

A void deep enough to supply the whole gap would also have to have an edge, and the edge would show.

Inside the void, every supernova reports the inflated rate. Outside it, the rate a supernova reports is an average over its whole sightline — inflated rate inside, global rate beyond — and it declines toward the global value as the void becomes a smaller share of the path. The result is a step in the supernova Hubble diagram at the void’s edge, a change of several per cent in the inferred rate between supernovae just inside and well outside.

The local measurement fits a single rate to supernovae spread across the whole range, and the residuals from that fit have been examined for exactly this signature. They show no step at the per cent level, and the inferred Hubble constant does not drift with the redshift cut applied. A void whose edge lay inside the sample would have pulled the fitted rate down toward the global value and left a visible trend in the residuals; neither is seen.

The two figures together close a loop. A small void is cheap in probability — a sphere of 150 megaparsecs can be underdense by a few tenths at a few sigma — but most of the supernovae lie outside it and measure the global rate. A large void contains the sample but is unaffordable in probability. There is no radius at which the void is both large enough to contain the measurement and shallow enough to be plausible.

The edge test is sharpened by how the calibration is arranged, too. The local measurement calibrates the supernovae on Cepheids and on the tip of the red giant branch in galaxies within forty megaparsecs, and measures the rate from supernovae beyond a hundred. A void containing both would affect the calibrating galaxies’ redshifts — which are not used — and the Hubble-flow supernovae’ redshifts — which are. The comparison of calibrated luminosities across that range is immune to a void at all, and the rate inferred from it is what carries the 73.

The bubble that was seen once #

The step has been looked for before, and once it appeared to be found. In 1998, with the first few dozen well-measured nearby supernovae, a group including Idit Zehavi and Adam Riess reported that supernovae within about seventy megaparsecs per unit of the dimensionless Hubble constant seemed to imply an expansion rate six or seven per cent higher than those beyond — a “Hubble bubble”, an edge of exactly the kind drawn above, at a distance where a modest underdensity would be plausible.

It did not survive. Later analyses with larger samples traced most of the apparent step to the way supernova colours had been converted into dust corrections: the nearby and distant supernovae had been drawn from different surveys with different filters, and a correction applied consistently across both removed the jump. A second, independent check came from the supernovae themselves once there were hundreds of them. Split by redshift, by sky position and by host galaxy, they give the same rate to within their errors, and the redshift cut used for the local measurement can be moved from a hundred megaparsecs to two hundred without changing the answer by more than its uncertainty.

The episode is a useful calibration for the argument here. A local void of the size needed would produce an edge in the same diagram that once produced a false one, and the method for finding it has been tested, in both directions, on the real data. What it finds is an expansion rate that does not depend on distance within the sample — which is what the global rate measured from the right place looks like, and what a void enclosing the sample would also look like, and what a void with its edge inside the sample would not.

That leaves only the enclosing void, the one that contains every supernova used, and it is the one the hero figure prices at tens of standard deviations. The data cannot see its edge because its edge would lie beyond them. The statistics of structure see it instead.

The odds of a partial answer #

If a void cannot supply the whole gap, it might supply part of it, and a partial void is not a fringe possibility. The local universe within a few hundred megaparsecs is known from galaxy counts to be somewhat underdense — several surveys of near-infrared galaxies have reported a deficit of ten to thirty per cent out to about three hundred megaparsecs, though the size and even the existence of the deficit depend on how the counts are normalised.

The figure gives the answer in its useful form. A fluctuation at the three-sigma level — rare, but the kind of thing an observer could find themselves in without anyone being surprised — supplies about a tenth of the gap on the scales the supernovae sample. That moves the local measurement by under a kilometre a second, which is comparable to its quoted uncertainty and smaller than the tension by a factor of about eight.

That puts a bound on the void hypothesis that does not depend on any particular survey of the local galaxies. If the deficit in galaxy counts is real and as large as twenty per cent across three hundred megaparsecs, it is a six-sigma fluctuation by the standard model’s own statistics — a tension of its own — and it still supplies only about a third of the gap. If it is smaller, as some re-analyses suggest, it supplies correspondingly less.

There is one more reason the probability is a reasonable guide here rather than a formality. The Milky Way’s position is not special in any way that would select for voids: observers are found where galaxies are, and galaxies are more common in denser regions, so if anything an observer is more likely to sit in an overdensity than in a void. The prior a void would need is not neutral; it leans the wrong way.

What a one-per-cent correction costs #

The figures so far price the whole gap. It is worth pricing a correction of the size that does matter in practice, because the local measurement now quotes its total uncertainty at about 1.4 per cent and a systematic shift of one per cent would be a real fraction of its error budget.

A one-per-cent shift is affordable near the inner edge of the sample, where a sphere of a hundred megaparsecs is underdense by six per cent about one time in six. It becomes improbable further out, where the averaging is over much more volume. Since the supernova sample’s weight is spread across the whole range, the realistic expectation is a local offset of a few tenths of a per cent — which is why the error budget of the local measurement already carries a cosmic-variance term of that size and why it is not where the tension hides.

That accounting has a second consequence that is easy to miss. The microwave-background value is also an inference about one region, the last-scattering surface, but that surface is a shell at a distance of fourteen billion parsecs, enclosing a volume so large that its cosmic variance for the expansion rate is negligible. The asymmetry in the two measurements’ exposure to local structure is total: only the local one can be fooled by where the observer stands, and the figures say by how much.

A tension the void would only rename #

Suppose, against the odds, the void existed. It would not dissolve the tension. It would move it.

A void of that depth and size is a fluctuation the standard model says should not exist anywhere in the observable universe, let alone around the Milky Way. Finding it would be a contradiction with the power spectrum as sharp as the original disagreement about the expansion rate — sharper, in sigma. The two measurements of the Hubble constant would agree, and the amplitude of structure on scales of hundreds of megaparsecs would disagree with the microwave background by many standard deviations. The cosmology would be in trouble either way; it would only have changed which number was in trouble.

That is why the argument against the void is so strong despite needing no new data. The probability of the void is computed with the same parameters, from the same measurements, that set the global rate of 67.4. A resolution of the tension that requires the standard model’s parameters to be wrong about structure in order to save them from being wrong about expansion has not saved them.

Where the linear estimate stops #

The density contrast of 0.47 is not small. Linear theory relates density to velocity accurately for contrasts of a few tenths; for a half-empty region the exact spherical relation gives a somewhat larger expansion excess for the same contrast, and the contrast needed for the full gap falls toward 0.4. The sigma counts fall by a corresponding fraction and remain far beyond plausibility.

The void is spherical and the observer is at its centre. An off-centre observer would see a dipole in the supernova rates as well as a monopole — a direction-dependent Hubble constant — which is a further test, and which the supernova samples do not show at the level a void large enough to matter would produce.

The supernovae are treated as sampling a sphere. Real samples are concentrated toward the sky the surveys covered and toward the redshifts they reached, so the effective volume is smaller and lopsided, and the cosmic variance of the rate they measure is somewhat larger than the spherical figure — by tens of per cent of a quantity that is itself a few tenths of a per cent. The error budgets of the local measurement compute it for the actual sample geometry and arrive at the same order.

Only density enters. A void is drawn as a region of lower matter density expanding faster in proportion. In the standard model that is the whole effect; a void whose outflow had been amplified by something else — a modification of gravity strengthening growth on large scales — would need a smaller density contrast for the same offset, and would be a different hypothesis with its own tests.

The fluctuations are Gaussian. On scales of hundreds of megaparsecs the density field is very nearly Gaussian, as its origin in the primordial fluctuations requires and as the background confirms, so the tail probabilities are reliable within the standard model — which is exactly the model being tested.

The power spectrum has no acoustic wiggles in it, and is normalised to a fluctuation amplitude of 0.81 on eight-megaparsec scales. The wiggles change $\sigma(R)$ by a few per cent; the normalisation is itself mildly contested, and a lower value would make the void less likely still.

Still open: what the local universe actually weighs #

The void is excluded as the whole explanation by its own improbability and by the absence of its edge. What remains genuinely open is the smaller question it leaves behind: how far the local region departs from the mean, and in which direction, to the precision the local measurement now needs.

That is being measured by the peculiar velocities of galaxies — motions that answer for structure larger than any survey can map — and by the next generation of near-infrared galaxy counts over the whole sky. If the local region turns out to be underdense by a few per cent, it will move the local measurement’s answer by a few tenths of a kilometre a second, and both sides of the tension will have to quote that correction. If it turns out to be underdense by twenty per cent over three hundred megaparsecs, the standard model will have a second problem on its hands, larger than the first, and the Hubble tension will have become a statement about how lumpy the universe is.