Four distances to the same galaxy

The distance ladder is a chain built to answer one question: how far away is it. Every rung of it — parallax, main-sequence fitting, Cepheids, supernovae — is a different technique for the same quantity, and the whole design of the ladder assumes that the quantity is the same at every rung, so that two techniques overlapping in range can be compared.

That assumption is exactly true out to a few tens of megaparsecs and quietly false beyond. In an expanding universe the operations that a nearby measurement performs interchangeably — dividing a flux by an inverse square, dividing a size by an angle, multiplying a light-travel time by $c$ — stop giving the same answer, and the disagreement grows without limit.

The four, and what each one is the answer to #

The place to start is the one quantity that is not a measurement at all, because everything else is built from it.

Comoving distance is the separation between here and there now, with the expansion divided out. It is obtained by integrating $c\,dt/a(t)$ along the light path, which converts each step of the journey into the distance that step corresponds to today. It is the coordinate a map of the universe is drawn in, galaxies sit still in it, and it is not directly observable — nothing arriving here carries information about the present, only about the past.

Light-travel distance is the lookback time multiplied by $c$. It is the one a reader is most likely to have in mind, because “13 billion light years away” is a phrase built out of it, and it is the least useful of the four: it corresponds to no measurement and appears in no equation. Its one virtue is that it is bounded — nothing can exceed the age of the universe times $c$ — and its one vice is that this makes it monotonic and undramatic in exactly the range where the other three are doing something.

Luminosity distance is defined so that the inverse-square law still works: $D_L \equiv \sqrt{L/4\pi F}$. It has to absorb two separate losses that Euclidean space does not have. Each photon arrives with its energy reduced by $(1+z)$, and the photons arrive spread out in time by a further $(1+z)$, so the flux is down by $(1+z)^2$ relative to the geometric expectation and $D_L = (1+z)\,D_M$.

Angular-diameter distance is defined so that small-angle trigonometry still works: $D_A \equiv \ell/\theta$. It has to absorb the fact that the light left when the universe was smaller, so the object was closer then than its comoving distance suggests, and $D_A = D_M/(1+z)$. The vertical-versus-horizontal reading is the cleanest way to see why the two disagree so badly. The vertical gap cannot exceed the age of the universe; the horizontal one is not bounded by anything, because it counts a separation that has been growing throughout the journey. Whether $c/H_0$ is called a size at all turns on the same distinction.

Those two definitions bracket the comoving distance, one factor of $(1+z)$ above and one below, and therefore

$$\frac{D_L}{D_A} = (1+z)^2$$

exactly, in any metric theory of gravity in which photons travel on null geodesics and are not destroyed on the way. That relation is called distance duality, and unlike almost everything else in this essay it is not model-dependent — it is a theorem, and it is a testable one.

The turnover, which is the surprising part #

The turnover is worth restating because it sounds like a mistake. A galaxy at $z = 8$ is at a comoving distance of 9.3 gigaparsecs and subtends the same angle as one at $z = 0.5$, which is at 1.9 gigaparsecs. The far one is five times further away in every sense that a map would recognise, and it looks the same size.

The consequence is practical. A telescope’s angular resolution stops being the limiting factor on high-redshift morphology somewhere around $z = 2$: past there, resolving a galaxy’s structure gets no harder, and what gets harder is only collecting enough photons. The searches for the earliest galaxies are photon-starved, not resolution-starved, and the two limits behave very differently as telescopes grow — aperture buys photons linearly in area and resolution only linearly in diameter, so the balance shifts further towards photons with every generation.

The same turnover has a much older cousin, which is worth naming because it makes the effect feel less exotic. The angular-diameter distance is small at high redshift for exactly the reason that a wide-angle lens makes a nearby wall fill the frame: the object was close when the light set out. What is unfamiliar is not the geometry but the fact that “when the light set out” is a different place from “where it is now”, and that the two differ by a factor that is itself the thing being measured.

What is actually measured, and what is inferred #

The distinction this whole essay turns on is the one the field’s obligation demands, so it is worth being blunt about it. None of the four distances is measured. What is measured is a flux, an angle, and a redshift. A distance is what those become after a cosmology is chosen. A supernova at $z = 1$ has a measured apparent magnitude. Turning that into a distance requires the luminosity distance formula, which requires $\Omega_{\rm m}$ and $\Omega_\Lambda$ — the very things the supernova was observed to measure. The circularity is only apparent: the fit is over the whole sample at once, and what is being fitted is the shape of the magnitude–redshift relation rather than any single point. There is a further correction that has to be applied before any of this can be done, and it is the one that most often bites. A magnitude is defined through a filter, and a filter that measures visible light from a nearby galaxy measures ultraviolet light from a distant one, because the spectrum has been shifted through the passband. Correcting for that — the K-correction — needs a model of what the source’s spectrum looks like in the band it has been shifted out of, and for a supernova it needs a model of how that spectrum evolves through the explosion. The correction is often larger than the cosmological signal being extracted, and it is the single largest reason that supernova cosmology waited for infrared detectors.

The size of it is worth a number. At $z = 0.5$ a filter centred at 550 nanometres is collecting light emitted at 370 nanometres, which for a type Ia supernova near maximum is on the steep ultraviolet side of the spectrum where the flux changes by tens of per cent across the band. The correction there is several tenths of a magnitude; the acceleration signal at the same redshift is 0.25 magnitudes. There is no sense in which the second is measurable without the first being modelled, and the modelling rests on nearby supernovae observed spectroscopically — which is to say that the high-redshift measurement inherits an assumption that distant supernovae are the same objects as nearby ones. That assumption has been tested, by comparing spectra rather than brightnesses, and it holds; but it is an assumption, and it is the one a sceptic should press.

The one relation that does not depend on the model #

Distance duality is a theorem, so it can be tested, and testing it tests the framework rather than the parameters. The cleanest version compares a luminosity distance and an angular-diameter distance to the same objects. Galaxy clusters supply both: the X-ray emission from the hot gas goes as the square of the electron density and the distortion the same gas imprints on the microwave background goes as the first power, so the two together fix the physical depth of the cluster and, against its measured angular size, give $D_A$ — while a supernova in the same cluster gives $D_L$. Current tests find the ratio consistent with $(1+z)^2$ to a few per cent out to $z \approx 1$. That $(1+z)^4$ is called Tolman dimming, and it is one of the few genuinely decisive observations in cosmology, because a static universe with tired light predicts $(1+z)^1$ instead. The observed dimming, once galaxy evolution is allowed for, follows the fourth power. It rules out the whole family of static alternatives without reference to any parameter, and it does so using only surface brightnesses — no ladder, no candle, no calibration.

What the pictures cannot show #

Every curve here assumes the Planck 2018 cosmology and would move if it changed. The turnover redshift of the angular-diameter distance is 1.59 in that model and would be 1.25 in an Einstein–de Sitter universe. Nothing in the figures indicates which parts are geometry and which are parameters, so it is worth stating: the existence of the turnover follows from expansion alone, and its location is a measurement.

None of these figures shows a distance to anything real. They are curves in a parameter, and a real galaxy contributes one point on one of them. The impression a smooth curve gives — that the distance to a given redshift is a settled number — is the impression this field most needs to resist.

And the hero figure’s logarithmic axes hide how bad the disagreement is at moderate redshift. At $z = 1$ the luminosity distance is four times the angular-diameter distance, which on a log plot is a modest vertical gap and in a calculation is a factor of four. A great many popular accounts quote one where they mean another, and the error is invisible to any check that only looks at the order of magnitude.

How the distinctions arrived #

The formalism is older than the observations that needed it. Howard Robertson and Arthur Walker had the metric by 1935, and the distance definitions follow from it directly; Richard Tolman derived the surface-brightness law in 1930, explicitly as a test to distinguish an expanding universe from a static one, at a time when the expansion was six years old and thoroughly disputed.

What is instructive is that the test he proposed was not performed convincingly for seventy years. The reason is the one this collection keeps meeting: the effect being sought is entangled with the evolution of the objects used to seek it. Galaxies at $z = 1$ are not the same galaxies as those nearby — they are younger, bluer, more compact and more actively forming stars — and separating a $(1+z)^4$ geometric dimming from an evolutionary brightening required a class of object whose evolution could be modelled. The measurement was not limited by the instrument. It was limited by having to know what the source was.

The volume element, which every rate depends on #

There is a fifth quantity built out of these four, it appears in every occurrence rate anybody quotes, and it is the least examined step in the chain.

A survey counts objects. To convert a count into a density it has to know what volume was surveyed, and in an expanding universe the comoving volume in a shell between two redshifts is not what a naive calculation gives. It is the comoving distance squared, times the comoving distance increment, times the solid angle — and the increment involves the expansion rate at that redshift, so the volume element depends on the cosmology twice over.

The consequence is that the volume per unit redshift is not monotonic. It rises steeply at low redshift, because the shells are getting bigger, and then turns over and falls, because the increments in comoving distance per unit redshift are shrinking as the expansion rate rises. The peak is somewhere around a redshift of two and a half, depending on the parameters.

That non-monotonicity means a survey that counts objects per unit redshift sees a distribution whose shape is mostly geometry. A population with a genuinely constant comoving density produces counts that rise, peak and fall, and reading that peak as an epoch of formation would be an error of exactly the kind this essay exists to prevent.

Number counts were once proposed as a cosmological test in their own right: since the volume element depends on the parameters, counting galaxies as a function of magnitude would measure them. It does not work, and the reason is the one that defeated the surface-brightness test for seventy years. The population evolves. Galaxies at high redshift are more numerous, smaller and bluer than those nearby, and separating that from the volume element requires knowing the evolution — which is the thing nobody knows.

A rate is a count divided by a volume, and the volume is a model, which is why occurrence rates in cosmology are quoted with the assumed cosmology attached and rates in the solar system are not.

The horizon, which is none of the four #

The most commonly muddled number in the subject is the size of the observable universe, and the muddle is exactly the one this essay is about.

The universe is 13.8 billion years old, so light has been travelling for at most that long, so the observable universe is 13.8 billion light years across — which is the light-travel distance, and it is the wrong answer. The regions whose light is arriving now have been receding throughout the journey, and their present separation from here is the comoving distance, which comes to about 46 billion light years.

That is the particle horizon: the comoving distance to the furthest matter whose light has had time to arrive. It grows with time, and it grows faster than the light-travel distance does, because new regions come into view while the ones already visible recede further.

There is a second horizon that behaves in the opposite way and is a consequence of the acceleration. In a universe whose expansion is speeding up, there is a comoving distance beyond which light emitted now will never arrive at all, because the space between is growing faster than the light can cross it. That is the event horizon, it is finite, and it is currently about sixteen billion light years.

The two together give a slightly melancholy picture. The particle horizon grows, so more of the past becomes visible; the event horizon means that galaxies now beyond it are permanently out of contact, and as the acceleration continues more of them cross out. Nothing that is visible ever becomes invisible — its light is already on the way — but its later history does.

Three different lengths are all called the size of the universe, and they differ by factors of three; the quantity that has to be named is which operation produced it.

The generalisation #

The pattern here is one this collection has met in every field and never quite this starkly: a word that names one operation nearby names several once the geometry changes, and the operations only agree in the limit where the geometry is flat.

The mass of a galaxy stops being a number and becomes a profile for the same kind of reason — the quantity was always a function and the nearby case was a special one. An exoplanet’s mass is $m\sin i$ until a transit removes the inclination, which is the same structure again: one word, two operations, agreement in a limit. A magnitude is a flux until a distance is assumed, at which point it is a luminosity.

The habit worth carrying is the one that separates the three: ask what operation the number came out of. A distance derived from a brightness and a distance derived from an angle are different numbers about the same object, and in a curved, expanding spacetime the ratio between them is itself an observable — which is how a definitional nuisance became one of the field’s few model-independent tests.

One more cosmology makes the point that the four distances differ by ratios fixed by geometry rather than by the model.

Where the ladder goes next #

The next essay takes the luminosity distance and pushes it as far as the supernovae reach, and finds that the curve does something a universe full of matter cannot do.

Later rungs on this anchor: the K-correction as a measurement problem rather than a nuisance; standard sirens, which deliver a luminosity distance with no photometry at all; the maximum of the angular-diameter distance as a probe in its own right; distance duality as a test for photon number violation and intergalactic dust; and the comoving volume element, which is what turns any survey’s count of objects into a density and is the least examined step in every occurrence-rate calculation in astronomy.