A flatness that needs a second distance

The inventory of the universe has four entries that are measured and one that is inferred. Ordinary matter, dark matter and the cosmological constant are each weighed by some effect they have — on the abundances of light elements, on the heights of the acoustic peaks, on the expansion history. The fourth measured quantity is their sum, and it is usually quoted as one: the total density equals the critical density to two parts in a thousand, so space is flat. The budget that states it gives the credit to the position of the first acoustic peak in the microwave background — a ruler of known length, seen at a known distance, whose apparent size depends on whether light rays converge or diverge on their way across the universe.

That account is correct and incomplete, and the incompleteness has a precise shape. The peak position measures an angle. An angle is a length divided by a distance, and the length — the sound horizon at recombination — is fixed by the physics of the early plasma. What is left to be measured is therefore one distance: the transverse comoving distance to the last-scattering surface. Curvature enters that distance, but so does everything else about the expansion since recombination, and in particular the Hubble constant. A curved universe with a different expansion rate can have exactly the same distance to recombination as a flat one. The peaks then fall at exactly the same angles, and the background cannot tell them apart.

A line of universes that look identical #

The argument can be made into a calculation. Hold the physical densities of baryons, matter and radiation — $\omega_b = \Omega_b h^2$, $\omega_m = \Omega_m h^2$ and the radiation density — at the values the peak heights and the damping tail fix. These determine the sound horizon, 144.4 megaparsecs, and the redshift of the last-scattering surface, 1090, because both are set by the plasma’s own physics before recombination and depend on nothing that happened afterwards. Then choose a curvature $\Omega_k$ and ask which Hubble constant puts the last-scattering surface at the same transverse distance as in the flat model. Once $h$ is chosen the densities follow, $\Omega_m = \omega_m/h^2$ and $\Omega_\Lambda = 1 - \Omega_m - \Omega_k$, and there is exactly one $h$ that works.

The resulting line is the geometric degeneracy, and it is not a narrow sliver. A closed universe with four per cent curvature — the sign that makes light rays converge, and so makes the ruler look larger — compensates by expanding more slowly now, so that the extra distance light had to cover at a lower rate brings the last-scattering surface back to the same apparent distance. It needs a Hubble constant of 53.5 and a matter density of half the critical density. An open universe runs the other way: at one per cent open curvature the Hubble constant reaches 73, the value the local distance scale measures.

The angle is measured to three parts in ten thousand, and the line in the figure is exact to that precision all along its length. No refinement of the peak positions shortens it. That is the sense in which the flatness of the universe is not a measurement the background makes on its own.

What the background said by itself #

In practice the peaks do carry a little information about curvature, through two effects the angle does not capture. On the largest angular scales, the gravitational potentials change with time in a curved or Λ-dominated universe, and photons crossing them gain or lose a little energy; that integrated Sachs–Wolfe effect raises the lowest multipoles by an amount that depends on the expansion history. And on small scales the peaks are smoothed by gravitational lensing by all the structure between the observer and the last-scattering surface, by an amount that depends on how much matter there is and how clustered it has become. Both effects break the degeneracy slightly, the first weakly because the largest scales have few independent modes, the second more strongly.

The second one went the wrong way. The measured peaks are smoothed more than the standard flat model predicts — as if lensing were about fifteen or twenty per cent stronger than it should be. A closed member of the degeneracy has more matter and therefore more lensing, so the background’s own data, fitted with curvature free, moved along the line towards the closed end. The fit preferred $\Omega_k = -0.044$, with the flat universe excluded at a little over two standard deviations, and a Hubble constant in the mid-fifties.

That result was taken seriously for a while and argued over at length, because a closed universe is not a small change: it is finite, it has a different fate, and it would have made the discrepancy with the local Hubble constant three times worse. It did not survive the addition of other data, for reasons the next figures make plain. But it is worth remembering what it showed about the flatness figure. The figure of two parts in a thousand does not belong to the microwave background, and when the background was asked alone, it gave a different answer.

Distances nearer than recombination #

The members of the degeneracy agree about one distance and disagree about every other.

All three curves converge on zero at redshift 1090, where the construction pins them. Nearer, they separate, and by redshift one the closed member is fifteen per cent further away than the flat one at the same redshift and the open member twelve per cent nearer. That is an enormous difference by the standards of modern distance measurement. The acoustic scale imprinted in the galaxy distribution is the same ruler the background uses, now seen at redshifts from 0.1 to 2.3, and each redshift bin gives the ratio of the transverse distance to the ruler to about one per cent. Since the ruler’s length is fixed by the same physical densities, the acoustic distance at redshift 0.5 decides between members of the family that differ by eighteen per cent.

That is why the curvature is always quoted from the background and the acoustic distances together. With both, the degeneracy line is cut at one point, and the curvature comes out as $\Omega_k = 0.0007 \pm 0.0019$ — the two parts in a thousand. Most of the constraining power is in the low-redshift distance, and the background’s role is to supply the line it cuts.

The acoustic scale measured along the line of sight gives a second, independent handle. Measured along and across, the same feature yields the transverse distance and the expansion rate at each redshift, and the members of the family differ in both. A universe with the wrong curvature cannot match both at once at several redshifts, which is a consistency test the acoustic data carry within themselves.

Three lines through one plane #

The degeneracy is easiest to see in the plane of the two densities, where each kind of measurement allows a line and the answer is where the lines cross.

The background’s line runs at a shallow angle to the flat diagonal, crossing it at the fitted point; along it the curvature changes slowly, and points well off the diagonal, with a matter density of a half, sit on it as comfortably as the flat point does. The supernova line — the shape of the distance–redshift relation, which is what supernovae standardised against one another measure without knowing their absolute brightness — climbs steeply across it: it constrains the difference between matter and dark energy, which sets the acceleration, much better than their sum. The acoustic distance falls through both at a third angle.

Any two of the three would give the crossing. That all three meet at one point, on the flat line, is the actual evidence for flatness, and it is a stronger statement than any one line with a small error bar, because the three are made with different physics at different epochs and have no systematic errors in common. A shift in the supernova calibration moves no line at all, because the supernova line used here depends only on relative brightnesses. A shift in the sound horizon moves the background’s and the acoustic line together. It is the geometry of the crossing, rather than the width of any line, that the flatness rests on.

The ages that cannot help #

There is one more independent quantity that obviously depends on curvature and the Hubble constant: the age of the universe, which must exceed the age of the oldest stars in it.

The age turns out to be almost useless against the degeneracy, and the direction of its failure is informative. Moving along the line towards a closed universe lowers the Hubble constant and raises the matter density. The first makes the universe older, since the Hubble time is longer; the second makes it younger, since more matter decelerates the expansion more. The first wins, so the closed members are older than the flat one — sixteen and a half billion years at ten per cent closure — and any lower limit on the age from the oldest stars is satisfied with room to spare. It is the open members that are too young, and they are cut off almost at once: a universe even slightly open, with the higher Hubble constant that would go with it, is younger than the oldest globular clusters.

That asymmetry is why the ages of stars have been a persistent argument against a high Hubble constant and never against a closed universe. It is also a reminder that the degeneracy is not an abstraction. Each point on the line is a complete, self-consistent universe with its own age, its own matter density, its own future, and its own sky that looks, at the resolution of the acoustic peaks, exactly like this one.

How big the bound says space is #

A curvature parameter translates into a length. A curved space has a radius of curvature, $R = (c/H_0)/\sqrt{|\Omega_k|}$, which for a closed universe is the radius of the three-sphere it lives on and for an open one is the scale on which triangles noticeably fail to add to two right angles.

The measured bound puts the radius of curvature beyond a hundred gigaparsecs, more than seven times the distance to the most distant light that can be seen. If space is closed, its total volume is at least seventeen hundred times the volume inside the horizon that bounds what can be observed. The observable universe is a small patch of whatever the whole is, and the flatness measured in it is the flatness of a patch — the statement that the patch is too small to show the curvature, not that there is none.

That is the correct reading of the result, and it is less than it is sometimes made to say. A closed universe with a radius of curvature of a thousand gigaparsecs is permitted, and so is an open one; nothing measured distinguishes them from each other or from exact flatness, and nothing will until the bound tightens by the factor needed to reach them.

A floor below which curvature is not a property #

It cannot tighten indefinitely. The curvature is inferred from how the observable patch expands, and the patch is not perfectly smooth: it sits inside fluctuations on the scale of the horizon itself, whose amplitude is the same $10^{-5}$ as every other fluctuation in the background. A fluctuation larger than the observable universe is invisible as a fluctuation — it has no edge within view — but it changes the local expansion rate, and the local expansion rate is what the curvature is inferred from. A flat universe with a horizon-sized ripple through the observer’s patch looks, from inside, like a curved one with an amplitude of about $10^{-4}$ in $\Omega_k$.

Below that level, then, the curvature of the patch is not a property of the universe but of where in it the observer happens to sit, and no measurement can separate the two. The present bound is twenty times above the floor. Inflation predicts a curvature far below it — the exponential expansion that solved the horizon problem drives any initial curvature towards zero by a factor of $e^{-2N}$ over $N$ e-folds — so if the universe inflated, the measurement will end at the floor without ever finding the curvature, and the result will be exactly what inflation predicts and exactly what a coincidence would have produced too. A detection of curvature above the floor, of either sign, would be much more informative: closed would argue against the simplest inflationary models, and open would point towards a universe born in a bubble, as some of the landscape-based scenarios suggest.

How the first flatness result was really made #

The history of the measurement shows the second distance at work before anyone called it that. The first acoustic peak was located by balloon-borne and ground-based experiments in 1999 and 2000, near a multipole of 200, and the result was announced as a demonstration that the universe is flat. The peak position alone could not have shown it. What the analyses did — stated in the papers, forgotten in the retelling — was to combine the peak with a prior on the Hubble constant from the local distance scale, which the Hubble Space Telescope’s key project had just put at about 72 with a ten per cent error, and with a prior that the universe is at least as old as its oldest stars. The Hubble constant was the second distance: a measurement of the expansion rate today that cuts the degeneracy line at one point, and the point it cut was close to flat.

That combination has a weakness that later became important. A prior on the Hubble constant is a single number with an error bar, and the degeneracy line is steep in that direction: across the whole range from 60 to 80 kilometres a second per megaparsec the curvature changes by only a couple of per cent. So the early flatness result was loose — a few per cent — and it depended on the local distance scale being right. Today the local scale and the background disagree about the Hubble constant by five standard deviations, and that disagreement would, if that scale were used as the second distance, pull the curvature towards the open side. It is not used that way now, because the acoustic distances are more precise and have nothing to do with Cepheids; but it is a measure of how much the flatness figure owes to which second distance is chosen.

The distance that does the work today is one of four distinct distances a cosmologist can mean — the transverse comoving distance, the one that converts a proper size into an angle and the only one in which the curvature appears directly. The background measures it once, at the farthest redshift there is; the galaxy surveys measure it again and again at redshifts below two. Flatness is the statement that the two sets of measurements are consistent with one smooth curve with no curvature term, and the curve is pinned at both ends.

What the drawn degeneracy leaves out #

The figures hold the physical densities exactly fixed, which is an idealisation; the background measures $\omega_m$ to about one per cent, and a real fit lets it move a little along with the curvature, which widens the line without changing its direction. They treat the dark energy as a cosmological constant. Letting its equation of state vary opens a second degeneracy that runs into the first — a dark energy that is not constant changes the late-time distance in ways that mimic curvature — and a fit with both free is much weaker than a fit with either, which is why curvature bounds are almost always quoted with the dark energy fixed. And they ignore neutrino masses, which change the late expansion rate slightly and the lensing more, and which are fitted jointly with curvature in any full analysis.

The acoustic distance is drawn as one number at one redshift. A real survey delivers five or ten such numbers with correlated errors, plus the expansion rate at each; the constraint on curvature comes from the whole set, and it depends on the sound horizon being what the background says it is. If some new physics before recombination changed the sound horizon — one of the proposals for reconciling the two values of the Hubble constant — the background line and the acoustic line would move together, and the crossing would move with them, possibly off the flat line.

Still open: whether the lensing excess was telling the truth #

The preference of the background alone for a closed universe came from peaks smoothed more than the standard model predicts, and the smoothing has never been fully explained. Later analyses with more careful treatment of the high-frequency foregrounds and of the likelihood reduced it; lensing reconstructed directly from the four-point statistics of the map shows no excess; ground-based measurements of the damping tail at finer resolution see a smaller one. Whether the excess was a statistical fluctuation, a subtle foreground residual, or a sign of something that also looks like extra lensing — a modification of gravity, or a small primordial feature — is not settled, and it matters for curvature only indirectly: the flat answer comes from the crossing with the low-redshift distances, which are unaffected. What would matter directly is a crossing that moved. The acoustic surveys now under way are measuring those distances to a few tenths of a per cent at a dozen redshifts, and if the three lines stop meeting at one point on the flat diagonal, it will be because one of them has changed — and the curvature is the parameter most likely to be asked to absorb the difference.