A budget whose familiar part is five per cent

The pie chart of the contents of the universe is the single most reproduced image in cosmology, and it has three properties that its usual presentation hides. Its three numbers come from three unrelated measurements and are not equally well known. Its denominator is not a measured density but a reference value built out of the expansion rate. And it is a snapshot of one moment, in a history where the same three components have traded places twice.

The denominator #

An $\Omega$ is a density divided by the critical density, and the critical density is

$$\rho_{\rm c} = \frac{3H_0^2}{8\pi G} = 8.5\times10^{-27}\ {\rm kg\,m^{-3}},$$

which is about five hydrogen atoms per cubic metre. It is the density at which the expansion is exactly marginal — the same balance as an escape velocity, applied to a shell of the universe — and it is a unit, not a measurement. Every fraction in the hero figure is a ratio to it.

That has a consequence which is easy to miss and which propagates into every number below. Because $\rho_{\rm c}$ contains $H_0^2$, a determination of $\Omega_{\rm m}$ from an absolute mass measurement inherits twice the fractional error on $H_0$. Different measurements handle this differently: some constrain $\Omega_{\rm m}$, some constrain the physical density $\Omega_{\rm m}h^2$, and comparing the two families requires a value of $h$ that is itself in dispute at the five per cent level.

It is worth being exact about which quantity each experiment returns, because the chart’s slices are not what any of them measure.

The microwave background does not measure $\Omega_{\rm b}$ or $\Omega_{\rm m}$. What the acoustic peaks fix are two physical densities, $\Omega_{\rm b}h^2$ and $\Omega_{\rm c}h^2$, together with one angle — the angular scale of the sound horizon, which is determined to about a part in three thousand and is the best-known number in the subject. Physical densities are what the plasma responds to: a peak height depends on how many baryons there are per photon, and not on how that compares with a critical density built from an expansion rate that has no meaning at that epoch.

Turning those into a pie chart needs the angle, the angle needs a distance, and the distance needs an expansion history for the thirteen billion years since. So $\Omega_{\rm m}$ and $H_0$ emerge from the microwave background jointly, as a derived pair, conditional on the history in between being the one the model asserts. They are anti-correlated along a narrow ridge: raise the matter density and the distance to the surface shortens, and the peak angle is restored by lowering the expansion rate.

That structure is why the disagreement over the expansion rate is also a disagreement about this chart. The physical densities are not contested — every party agrees about them to about a per cent. What is contested is where along the ridge the answer sits, and sliding along it moves $\Omega_{\rm m}$ between roughly 0.27 and 0.32 while changing nothing whatever about the plasma at recombination.

A discipline follows. Before comparing one quoted cosmological number with another, establish whether it is a physical density, a fraction of critical, or a distance-weighted combination of both — three quantities that share a notation and answer to entirely different measurements. The fraction of critical is the least fundamental of the three and by some distance the most quoted.

Where each number comes from #

The three fractions are not one measurement split three ways. They are four measurements, over-determined, and the over-determination is what makes the budget credible.

Ordinary matter, from deuterium and from the peaks.

Total matter, from things that orbit. Dark energy, from the expansion history.

And the total, from an angle. The position of the first acoustic peak is a ruler of known length seen at a known distance, and the angle it subtends depends on the curvature of the space in between. It gives $\Omega_{\rm total} = 1.000 \pm 0.002$, which is the fourth measurement and the one that closes the system. Three components and four constraints is one more than is needed, and the leftover is the consistency check that the whole model passes.

It is worth pausing on how differently these four are made, because the reflex reading of a pie chart is that its slices came off one instrument. One is a nuclear reaction network read off an absorption line in a quasar spectrum. One is a velocity field measured in twenty-one centimetre emission across a galactic disc, summed over a luminosity function. One is a residual of a fifth of a magnitude in the brightness of exploding white dwarfs. One is the angular position of a bump in the variance of a temperature map. The instruments have nothing in common, the systematics have nothing in common, and the epochs they probe span from three minutes to the present.

The over-determination is the argument #

It is worth laying the arithmetic out explicitly, because the individual numbers are quoted so often that the structure disappears behind them.

Deuterium says baryons are 0.049. Clusters and galaxies say all matter is about 0.31. Subtracting, non-baryonic matter is about 0.26 — and that subtraction is the strongest statement in the subject that dark matter is not ordinary matter that happens to be dark, because the baryon number is fixed by nuclear physics in the first three minutes and cannot be raised by hiding baryons in faint stars or cold gas.

The peaks say the total is 1.000. Subtracting the matter, something with 0.69 of the critical density is present and is not matter. The supernovae say that something has negative pressure. Neither measurement needs the other, and each was made by people not primarily interested in the other’s result.

Any one of the four could be dropped and the remaining three would still fix the model. That is the difference between a fitted description and a measured one, and it is the reason the budget survived the arrival of two independent 1 per cent measurements in the 2000s rather than being adjusted by them.

The chart is about now #

The second crossing is the awkward one, and the awkwardness has a name: the coincidence problem. Matter and dark energy have densities that differ by twenty-nine orders of magnitude at nucleosynthesis and by a hundred and twenty at the Planck scale, and they happen to be within a factor of two of each other now. On a logarithmic time axis spanning the whole history, “now” is a very narrow window in which to find them comparable.

Whether that is a problem is genuinely contested, and it is worth stating both readings. Against: the crossing had to happen at some time, and structure formation requires a long matter-dominated era, so any observer capable of asking the question exists within a few e-folds of the crossing more or less by construction. For: that argument depends on an anthropic step that is hard to make quantitative, and the field has a poor record with coincidences it has waved away.

There is a second and less discussed coincidence in the same figure, and it is the one at the left. Matter–radiation equality happens at $z = 3419$ and recombination at $z = 1090$ — a factor of three apart, out of the sixty decades in scale factor the plot could have put them in. That proximity is not a curiosity either: it is why the acoustic peaks are as prominent as they are, since modes that entered the horizon during radiation domination were driven and boosted while later ones were not, and the boundary between the two families falls right in the middle of the observed range of multipoles. A universe with equality a hundred times earlier would have a visibly different power spectrum, which is precisely how the matter density is read off the third peak.

What is missing from the chart, and what is mislabelled #

Two entries deserve correction.

Radiation is not zero, it is small. Photons contribute $\Omega_\gamma = 5.4\times10^{-5}$ and neutrinos, now non-relativistic but with tiny masses, contribute somewhere between $10^{-3}$ and $10^{-2}$ depending on the mass sum. Neutrinos are matter today and were radiation before about $z = 200$, which means one entry in the chart changes category over cosmic history. The chart cannot show that, and the sum of neutrino masses is one of the things the microwave background and galaxy surveys jointly constrain — currently to less than about 0.12 electronvolts.

The baryons are mostly not in stars. Of the 4.9 per cent, stars account for about 0.25 per cent — a twentieth of the ordinary matter and a two-hundredth of everything. Most of the rest is diffuse ionised gas: in clusters, and in the warm–hot intergalactic medium between galaxies, which was itself missing from the inventory until the 2010s and was found by absorption against distant quasars and by the dispersion of fast radio bursts. The thing labelled “ordinary matter” in every pie chart is, to ninety-five per cent, invisible gas rather than anything that shines.

That is worth holding beside the statement the chart is usually used to make. The familiar contrast is between the five per cent that is understood and the ninety-five that is not; the sharper contrast is that the fraction of the universe made of anything that has ever emitted a photon is a quarter of one per cent. Everything in the eight fields of this collection before this one — every orbit, every star, every galaxy, every planet — is drawn from that quarter of a per cent, and it is not a representative sample of anything. The luminosity function counts what shines; the budget counts what is there; and the ratio between them is not a small correction.

What the picture cannot show #

The hero figure’s bars have no error bars. The four numbers are known to very different precision: the total to 0.2 per cent, the baryons to 0.7 per cent, the matter to about 2 per cent, and the dark-energy equation of state to about 3 per cent. Drawing them as flat coloured blocks implies a uniformity of confidence that does not exist.

Every bar assumes the components are what the labels say. The measurement that gives 0.69 gives the density of something with negative pressure; the measurement that gives 0.26 gives the density of something that clusters gravitationally and does not interact with light. Those are behavioural descriptions. Nothing in the budget identifies either substance, and the chart’s neat categories are considerably more confident than the evidence.

And the fractions are not conserved quantities. A pie chart implies a fixed pie being divided, and here the total is not fixed: the physical density of matter falls as the universe expands while the dark-energy density does not, so the slices are not being redistributed, they are separately shrinking at different rates. The bars in the hero figure are normalised at each epoch precisely so that this is visible as a change of proportion, and it is worth remembering that the absolute dark-energy density is identical in all four.

The budget is a snapshot, and the same four numbers at four other epochs say what kind of snapshot it is.

The same chart, run forward #

The hero figure runs backward from the present. Running it forward is quicker, because two of the three components are already committed.

Matter dilutes as the cube of the scale factor and the cosmological constant does not, so once the expansion is exponential the ratio between them falls exponentially too. A hundred billion years from now — seven times the present age — the matter fraction is of order a ten-thousandth, and the chart is one colour. Nothing in that projection is speculative: it requires no new physics and no parameter that is not already measured, only that the constant stays constant.

What makes it worth stating is what it does to the measurements rather than to the contents. Every constraint in this essay depends on there having been an epoch in which matter mattered. The acoustic peaks exist because the universe was once a plasma. The supernova residual is measurable because the expansion was once decelerating, so that there is a difference between histories to detect. The cluster baryon fraction requires clusters, and by then the clusters will have been carried beyond each other’s horizons. An observer at that epoch could measure an expansion rate and would have no way to decompose it, because everything that made the decomposition possible has diluted below detection or receded out of view.

So the present is not merely an unrepresentative moment in the history of the composition. It is a privileged moment for the measurement of it — late enough that the acceleration is visible, early enough that the evidence of everything preceding it survives. That is a coincidence of a different kind from the one above, and a more defensible one, since it says only that a question gets asked in the interval when it is answerable.

How the numbers settled #

For most of the twentieth century the expected answer was $\Omega_{\rm m} = 1$ with nothing else, and the expectation was theoretical rather than observational — inflation predicts flatness, and the only known way to be flat was to have critical matter density.

The observations disagreed for two decades and were not believed. Cluster baryon fractions gave 0.3 in the early 1990s; galaxy cluster counts gave 0.3; the shape of the galaxy power spectrum gave 0.3. Each result was resisted on the grounds that the local sample might be unrepresentative, and each was, in retrospect, right.

What changed was not a better matter measurement but the arrival of the other two. The supernovae in 1998 supplied a component that was not matter; the microwave background in 2000 supplied a total of exactly 1. Flatness was preserved, inflation’s prediction survived, and the 0.3 that everyone had been arguing about turned out to be the whole of the matter and only a third of the universe.

The lesson is uncomfortable and worth keeping. Three independent measurements agreeing on an unwelcome number were held off by a theoretical expectation for ten years, and the expectation was correct — about flatness — while the inference drawn from it was wrong.

The generalisation #

The habit that makes this budget trustworthy is over-determination, and it is the same habit that certifies a measurement anywhere in this collection: arrange for more independent constraints than there are unknowns, and treat the leftover as a test rather than as a nuisance.

A transit and a radial velocity give a planet’s density because two measurements over-determine a body that either alone would leave ambiguous. A cluster weighed three ways is the same argument at a different scale, and the agreement between the three routes is what makes its mass a measurement rather than a model. The four contact points of a transit fix three quantities, and the fourth is a consistency check that catches a blend.

The corresponding failure mode is the one this field has to guard against hardest. A model with as many parameters as observables always fits, and fitting then means nothing. ΛCDM has six parameters and constrains many more than six observables, which is the only reason its success counts for anything; a version with a free equation of state, free curvature and free neutrino masses fits better and says less.

Two more readings, of the history the budget implies and of the one number in it that is fitted rather than counted.

Where the ladder goes next #

Every number here rests on a distance or a density scale, and one of the four measurements — the acoustic peak position — has a partner at low redshift that has not yet been used. The next essay measures the same ruler in a galaxy survey ten billion years later.

Later rungs on this anchor: the cosmic baryon inventory in detail, and where the missing baryons were found; the neutrino mass as a cosmological observable; the growth of structure as a constraint separate from the expansion history, and the $S_8$ tension it has produced; $\Omega_k$ as a measurement rather than an assumption; and the argument about whether the coincidence problem is a problem at all.