One number that is an age, a size and a density

The constant of proportionality in the velocity–distance law is usually written in a mongrel unit — kilometres per second per megaparsec — and that unit hides what it is. Kilometres and megaparsecs are both lengths, so they cancel, and what remains is one over a time.

That is unusual enough to be worth stopping on. A measurement of how fast galaxies recede, made entirely with a spectrograph and a distance ladder, produces a quantity with the dimensions of a clock. Divide one by it and the answer is 14.5 billion years. Multiply it into the speed of light and the answer is 14.5 billion light years. Square it, multiply by $3/8\pi G$, and the answer is a density: about nine hydrogen atoms per cubic metre, or rather five and a half, depending on how the arithmetic is rounded.

Each of those three numbers is quoted in the subject, each is close to something real, and each differs from the thing it is usually taken to be — by 5 per cent, by a factor of three, and by not being a measurement at all.

The Hubble time is the age of a universe with nothing in it #

Reading $1/H_0$ as the age assumes the expansion rate has always been what it is now. That is the definition of the empty model, and it is the top curve in the figure. Any universe with matter in it has been decelerating, so it was expanding faster in the past, so it took less time to get here: the age is smaller than $1/H_0$. The Einstein–de Sitter case, with exactly the critical density in matter and nothing else, gives $2/3$ of the Hubble time, or 9.7 billion years.

The actual answer is 13.80 billion years, which is 95 per cent of the Hubble time rather than 67 per cent, and the reason is that the universe has spent the last four billion years accelerating. Deceleration early and acceleration late very nearly cancel, and the coincidence is close enough to be misleading: the Hubble time is a good estimate of the age for a reason that has nothing to do with the reason it is usually given. It is not that the expansion has been steady. It is that it has been two opposite things for comparable stretches.

The general statement is an integral rather than a division:

$$t_0 = \int_0^1 \frac{da}{a H(a)} = \frac{1}{H_0}\int_0^1 \frac{da}{a E(a)},$$

with $E(a) = H(a)/H_0$ carrying the whole content of the model. The Hubble time factors out and everything else is a dimensionless number between about 0.5 and 1, decided by what the universe is made of.

The Hubble distance is not the size of anything #

Multiplying by $c$ gives the Hubble distance, 4,451 megaparsecs or 14.5 billion light years, and this is the reading that goes wrong worst. It is frequently glossed as the size of the observable universe, and the observable universe is three times larger.

The Hubble distance is the radius at which the recession rate reaches $c$ — the Hubble sphere — and that is a real surface with real consequences, but it is not a horizon. Light emitted from beyond it can still arrive here, because the Hubble sphere itself grows in comoving terms, and a photon that starts outside it can be overtaken by it and then make progress inward.

The factor of three between $c/H_0$ and the particle horizon is the same integral as the age, run with a different weight, and it goes the other way: the age integral divides by $aH$ and comes out slightly less than $1/H_0$, while the horizon integral divides by $a^2H$ and comes out three times $c/H_0$. Both are dominated by the early universe, where $a$ was small; one has a single power of $a$ to fight and the other has two.

The critical density is a unit, not a measurement #

The third reading squares the constant. Setting the kinetic energy of the expansion against the gravitational potential energy of a sphere — exactly the escape-velocity calculation, applied to a shell of the universe rather than to a rocket — gives the density at which the two balance:

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

which is five hydrogen atoms per cubic metre. That is a spectacularly small number. The best laboratory vacuum is ten orders of magnitude denser; interstellar space inside the Galaxy is five orders denser than that again. The universe as a whole is, by the standards of anywhere in it, empty.

But $\rho_{\rm c}$ is not the density of the universe. It is the density that would make the universe spatially flat, and it is used as a unit: every density in cosmology is quoted as a fraction of it, and those fractions are what is written $\Omega$. The measured total happens to be $1.000 \pm 0.002$, so the actual density and the critical density are the same to a part in five hundred — and that agreement is a measurement with a long history, not a definition.

Because it is a ratio to a quantity containing $H_0^2$, an $\Omega$ carries a hidden dependence on the expansion rate. A ten per cent error in $H_0$ is a twenty per cent error in $\rho_{\rm c}$ and therefore a twenty per cent error in $\Omega_{\rm m}$ derived from any absolute mass measurement. This is why several results in the field are quoted as $\Omega_{\rm m}h^2$ rather than $\Omega_{\rm m}$ — the combination $\Omega h^2$ is a physical density with the $H_0$ dependence removed, and it is what the microwave background actually constrains.

That convention leaks outward, and it is the reason for the otherwise baffling factor of $h$ attached to so many quantities in extragalactic astronomy. A galaxy’s luminosity is inferred from its flux and its distance, the distance scales as $1/H_0$, so the luminosity scales as $h^{-2}$ and is quoted in units of $h^{-2}L_\odot$. A mass inferred from a rotation curve scales as $h^{-1}$; a separation between galaxies scales as $h^{-1}$ too, which is why the correlation function is plotted in $h^{-1},$Mpc. The convention exists so that a result does not go out of date when the distance scale is revised, and it has survived the revisions that would otherwise have invalidated forty years of catalogues.

What was actually measured #

None of the three quantities above is measured. What is measured is a slope on a plot of recession velocity against distance, and, as the previous essay sets out, one axis of that plot is easy and the other is a decade of work. There is also a floor under how well the slope can ever be known from nearby galaxies, and it is not instrumental. Below about 20 Mpc the local flow is not the Hubble flow at all: the Local Group is falling towards the Virgo cluster at some 200 km/s, and a brightness turned into a distance in that region measures the infall as much as the expansion. Every local determination therefore begins by throwing away the galaxies whose distances are best known.

The conversion from a slope to an age brings in the second problem, which is that the integral needs $E(a)$, and $E(a)$ needs the densities. So the age is not a measurement either — it is the output of a fit. The quoted $13.797 \pm 0.023$ billion years has an error bar of two parts in a thousand, which is a good deal tighter than $H_0$ is known to, and that should be a warning: the precision comes from the model, not from a clock. Change the model — allow the dark energy to vary with time, add a fourth neutrino — and the age moves by more than that error bar.

There is one age measurement in astronomy that owes nothing to any of this, and it is worth the comparison.

That consistency is not a small thing, and for most of the twentieth century it did not hold. With Hubble’s original constant of 500 km/s/Mpc the Hubble time was two billion years, less than the then-known age of the Earth. Even after the distance scale was repaired, the 1990s had a standing “age crisis”: a matter-dominated universe with $H_0 = 70$ gives $2/3 \times 14 = 9.3$ billion years, and the globular clusters were coming out at thirteen or more. The crisis was real, it was quantitative, and it was resolved by the same discovery that resolved the supernova residuals — a universe that accelerates is older than a decelerating one at the same $H_0$, and the extra four billion years is exactly what was missing.

What the picture cannot show #

The hero figure draws four models and the real one is a fit, not a choice from a list. The parameter space is continuous and several combinations of matter density and dark-energy behaviour produce almost identical $a(t)$ over the observable range. A figure showing four curves suggests the question is which of four, and it is not.

None of these figures can distinguish a change in $H_0$ from a change in the distance scale. Everything on the horizontal axis of a Hubble diagram is a distance, so a systematic error in the ladder moves every point in the same direction and rotates the fitted line. That is not noise and no amount of data reduces it. It is the reason the local determination of $H_0$ has been reported as $50$, as $100$, and as most values in between, by careful people using good data.

And the age is a coordinate, not an elapsed time for anybody. The 13.8 billion years is proper time along a worldline that has always been at rest with respect to the cosmic rest frame. A clock carried on any other trajectory records less, and while the difference is negligible for anything in the Local Group, the statement “the universe is 13.8 billion years old” carries a hidden choice of who is doing the timing.

The rate at other times #

Everything above concerns one number, and the word constant in its name is doing damage. The expansion rate is not constant in time — it is a function, and the quantity everyone quotes is its value now.

The distinction matters because the function is measurable at other epochs, and measuring it there is a different experiment with different systematics.

Two techniques do it directly. The first uses galaxies that have stopped forming stars and are simply ageing: comparing the spectra of two such populations at slightly different redshifts gives the difference in their ages, and the rate of change of age with redshift is the reciprocal of the expansion rate at that epoch. The method needs no distances at all — it is a differential age measurement, and its difficulty is entirely in whether the two populations really are the same kind of object seen at two times.

The second uses the acoustic scale imprinted on the matter distribution before recombination. Measured along the line of sight, that scale is a length converted into a redshift interval, and the conversion factor is the expansion rate at that redshift. Measured across the line of sight, the same scale gives an angular-diameter distance instead. One feature, two measurements, two different quantities.

What both return is the same function the age integral needs, so measuring it directly at several redshifts is a way of testing the model rather than assuming it. The results so far are consistent with the standard expansion history, and their precision is well below what the fits achieve — but they are honest measurements of a rate rather than outputs of a model, and they would show a departure from that history if one existed.

The Hubble constant is a boundary condition and the Hubble parameter is the function, and almost every disagreement in this subject is about whether a measurement of the first is consistent with an extrapolation of the second.

A measurement with no ladder and no background #

There is a third way to get the present rate, and it belongs to neither of the two families that disagree. It uses one lensed object and a clock.

A quasar behind a massive galaxy is imaged several times, and the light forming each image has taken a different path. The paths differ in geometric length and in how much time they lose passing through the lens’s potential, so a variation in the source’s brightness appears in the several images at different times — days to years apart.

That delay has the dimensions of a time, and the geometry that produces it is a set of angles. Angles times a distance give a length, and a length over a speed gives a time, so the measured delay is proportional to a distance — which is proportional to one over the expansion rate.

The method needs no distance ladder, because the angles are measured directly, and it needs no assumption about the early universe. What it needs instead is a model of the lens’s mass distribution, and that is where all the difficulty sits: the delay depends on the mass profile, and a profile that is steeper or shallower produces the same image positions with a different delay.

That degeneracy is a known one and has a name. Adding a uniform sheet of mass in front of the lens rescales the delays without changing anything about the images, so the source’s true position and the lens’s total mass cannot be separated by the imaging alone. Breaking it requires measuring the velocity dispersion of the lensing galaxy’s stars, or counting the mass in the line of sight, and the published uncertainties depend on how much freedom the mass model was given.

A third family of measurements is worth a great deal when two disagree, and this one currently sits between them with an uncertainty large enough to be consistent with either.

The generalisation #

The move made three times over in this essay — take a measured quantity, look at its dimensions, and read off the scales it implies — is one of the most productive habits in physics, and it is worth naming what makes it work and what makes it fail.

It works because a dimensionally correct combination of measured constants must be the right answer up to a dimensionless factor, and dimensionless factors in physics are usually of order one. It fails when they are not. Here they are $0.95$, $3.26$ and $1.00$, and the middle one is large enough to make the naive reading wrong by a factor of three.

The same habit produces the escape speed from a mass, which is a $\sqrt{2GM/r}$ built from nothing but the available constants; it produces the free-fall time of a cloud, which is $(G\rho)^{-1/2}$ and appears in every argument about how a star holds itself up; and it produces the Jeans length, the Schwarzschild radius and the Chandrasekhar mass. The lesson this essay adds is the one that is easy to skip: the dimensional estimate tells what the answer is proportional to, and the physics is entirely in the number in front.

Two of the three readings of the same constant are worth drawing over a longer span, since the constant’s meaning as an age and as a size both depend on how the expansion runs.

Where the ladder goes next #

Everything above assumed a single value of $H_0$ to build on. There is not one. Two families of determinations, each internally consistent and each with error bars of about one per cent, disagree by five — and the disagreement is the next essay.

Later rungs on this anchor: the deceleration parameter as the second term in the expansion of $a(t)$, and why it was the target for forty years before anything measured it; the $h$ convention, and why so many quantities are quoted with a factor of $h$ attached; the relation between $H_0$ and the sound horizon that makes the early-universe determination possible; time-delay cosmography, which measures $H_0$ from a single lensed quasar with no ladder at all; and gravitational-wave standard sirens, which measure a luminosity distance directly from a waveform and are the first genuinely new rung in fifty years.