Two distances that must agree if light is conserved

Past a redshift of about a tenth, the word distance has four meanings, and they disagree by large factors. The luminosity distance is what an inverse-square law gives from a source’s brightness; the angular-diameter distance is what a known size gives from the angle it subtends. At redshift one the first is four times the second. That is not a discrepancy to be explained; it is a geometric identity, $D_L = (1 + z)^2 D_A$, which Ivor Etherington derived in 1933 from nothing but the requirement that light travel along the geometry of spacetime and be conserved along the way.

Because the identity does not depend on what the universe contains — not on the matter density, not on the dark energy, not on the curvature — it is a test of something more basic than any of them. Measure a luminosity distance and an angular-diameter distance to the same redshift, and if their ratio differs from $(1 + z)^2$, either photons have been lost or gained in transit, or light is not travelling along the geometry that general relativity describes. The test was a curiosity for most of the twentieth century. It stopped being one in 1999, when it became the cleanest way to decide whether the accelerating expansion was real.

Where the two factors of 1 + z come from #

The identity is worth deriving, because each of its factors has a physical meaning and the meanings say what a violation would have to do.

Imagine a source emitting light in all directions and an observer receiving some of it. A bundle of light rays leaving the source and reaching the observer’s telescope has a cross-section at each end. The angular-diameter distance is the ratio between the bundle’s cross-section at the source and the solid angle it subtends at the observer — how big the source looks. A second distance, the one the observer would infer from the cross-section of a bundle diverging from the observer and arriving at the source, is the one that sets how much of the source’s light enters the telescope. Etherington’s reciprocity theorem shows that in any spacetime, for light moving along null geodesics, these two areas-per-solid-angle differ by exactly $(1 + z)^2$ in area, one factor of $1 + z$ in distance. That is pure geometry: no assumption about the universe’s contents.

The other factor comes from the light itself. Each photon arrives with its energy reduced by $1 + z$, and the photons arrive at a rate reduced by another $1 + z$, because the interval between them is stretched as their wavelengths are. Together they reduce the received flux by $(1 + z)^2$, which in distance — the square root of the flux — is one more factor of $1 + z$. Combined with the reciprocity factor, the luminosity distance is $(1 + z)^2$ times the angular-diameter distance.

So the identity can fail in exactly two ways. The photon count can change in transit — absorption, scattering out of the beam, conversion into something else — which spoils the flux factor. Or light can fail to travel along null geodesics of a metric — which spoils the reciprocity. Grey dust is the first kind of failure; a modified theory of gravity in which photons do not follow the metric’s geodesics would be the second.

A dimming that could have been dust #

The acceleration was discovered as an excess faintness: supernovae at redshift a half were about a quarter of a magnitude fainter than a decelerating universe predicts. Faintness is a luminosity distance, and a luminosity distance can be inflated by anything that removes light between the source and the observer. The obvious candidate is dust.

Ordinary dust reddens what it dims, because small grains scatter blue light more than red, and dust makes everything look further away by an amount its colour reveals. The supernovae showed no such reddening. In 1999 Anthony Aguirre proposed a way round that: intergalactic dust made of grains large compared with optical wavelengths, which absorb all colours equally and so dim without reddening — grey dust. Grains too large to redden are known to exist in the Galaxy; whether galaxies expel enough of them into intergalactic space was an open question.

How much grey dust would be needed is a calculation. Take a universe of matter alone, decelerating, with the matter density at its critical value; compute the luminosity distance its supernovae would have; compare with the luminosity distance in a universe with a cosmological constant; and remove the constant offset, because the supernovae’s absolute magnitude is fitted and a uniform dimming at every redshift would be invisible. What remains is the dimming the dust must add as a function of redshift: 0.38 magnitudes by redshift a half, 0.57 by redshift one — an optical depth of a third to a half. A substantial amount of dust, but grey dust of that optical depth was not excluded by anything in 1999 except argument about where it could have come from.

What dust cannot do to an angle #

Dust dims a source. It does not change the angle the source subtends. That is the whole of the test.

A universe of matter alone with grey dust has its supernovae’s luminosity distances inflated by the dust and its angular-diameter distances untouched: rulers at a given redshift subtend the angles that universe’s geometry gives them. So the ratio $\eta = D_L / [(1 + z)^2 D_A]$, which is one in any universe that conserves its light, is larger than one in the dusty one, by exactly the dimming the dust adds.

The dusty universe’s ratio rises from one at the start: 1.19 at redshift a half, 1.30 at redshift one. The universe with a cosmological constant, or any universe whose photons are conserved, sits at exactly one at every redshift — whatever its expansion history. The test does not ask which history is right. It asks whether the brightness and the angle are telling the same story, and a dimming that is not geometry makes them tell different ones.

Measured with rulers that do not care about brightness #

An angular-diameter distance needs a ruler of known size at a known redshift, and the acoustic scale imprinted on the distribution of galaxies is one: a preferred separation of about 150 megaparsecs between pairs of galaxies, seen at a measured angle across the sky at each redshift a survey covers. Its size is set by the physics of the early universe and is the same at every redshift, and it is measured from galaxy positions, which no dust can move.

The comparison has to be made carefully, because neither distance is absolute. The supernovae’s absolute magnitude is fitted along with everything else, so their luminosity distances are known only up to a constant factor. The acoustic scale’s true size depends on the sound horizon, which is computed rather than measured, so its angular-diameter distances are also known only up to a constant. What survives is the shape: how the ratio of the two distances changes with redshift.

The figure sets up the comparison with mock errors of the present size: supernova distances to two per cent in each redshift bin, acoustic-scale distances to a little over one per cent, at six redshifts from 0.15 to 1.5. Fitted as $\eta \propto 1 + \varepsilon z$, with each set’s constant removed, a universe that conserves light gives $\varepsilon = 0 \pm 0.023$ and the dusty one $\varepsilon = 0.18$, eight standard deviations away. Published analyses of the real data reach a similar precision and find $\varepsilon$ consistent with zero to within a few hundredths. Grey dust in the amount needed to fake the acceleration is excluded — not by an argument about where dust could come from, but by geometry.

That is a stronger conclusion than it might appear. The test does not assume the dark energy exists, does not assume general relativity’s expansion law, and does not depend on the matter density. It assumes only that photons reach the observer along the geometry and are conserved, and it finds that they do, to a few per cent out to redshift one and a half. Whatever made distant supernovae faint did it by making them farther away.

Where the dust would have had to come from, and where its light would go #

The geometric test is decisive, but it was not the only argument against grey dust, and the others say why the hypothesis was always strained.

Dust that absorbs starlight does not destroy its energy. Absorbed light is re-emitted, by grains warmed to a few kelvin above the background, at far-infrared and submillimetre wavelengths. An intergalactic optical depth of a third to a half would absorb that fraction of all the light emitted by galaxies since redshift one and re-radiate it, adding a diffuse glow to the cosmic infrared background — the sum of all far-infrared emission across cosmic time, measured by satellites since the 1990s. The measured background leaves room for some intergalactic dust emission, but very little, and the grey dust needed would have to be both large-grained and efficiently expelled from galaxies to sit within it.

Large grains in intergalactic space would also scatter X-rays: a distant X-ray source seen through such dust would be surrounded by a faint halo of scattered X-rays, as Galactic X-ray sources seen through the Milky Way’s dust are. Searches for such halos around distant quasars found none at the level the grey dust would require.

And there is some intergalactic dust. Correlating the colours of nearly a hundred thousand distant quasars with the positions of foreground galaxies, a study in 2010 found a slight reddening that extends to megaparsecs from galaxies, implying an average extinction of a few hundredths of a magnitude in the visual band out to redshift a half — about a tenth of what faking the acceleration would need, and red rather than grey. It matters at the level at which the dark energy’s equation of state is now measured, and supernova analyses correct for it; it does not matter for whether the acceleration exists.

What lensing does not break #

One thing that changes a supernova’s apparent brightness does not break duality, and it is worth seeing why. Matter along the line of sight bends light, and a supernova seen through a slight overdensity is magnified, one seen through a void slightly demagnified. Lensing changes the luminosity distance a supernova appears to have by a few per cent at redshift one, randomly from one supernova to the next.

But lensing changes the angle a source subtends by exactly the same factor. A gravitational lens magnifies by enlarging: it stretches the image, conserving surface brightness, so the light per unit area is unchanged and the ratio of the two distances is untouched. That is Etherington’s reciprocity at work: lensing is a deflection of null geodesics, and the theorem holds for any spacetime, lumpy or smooth. A supernova magnified by lensing looks closer by both measures at once. This is why lensing adds scatter to the supernova Hubble diagram without biasing the duality test, and why the duality test can tell grey dust, which dims without shrinking, from the lumpy universe’s own focusing, which brightens and enlarges together.

The same identity as a surface brightness #

The duality relation has a second form that is much older than the supernovae. Surface brightness — light per unit area of sky — is luminosity divided by the square of the luminosity distance, divided by solid angle, which goes as the inverse square of the angular-diameter distance. Their ratio is $D_A^2/D_L^2 = (1 + z)^{-4}$.

That is Tolman’s surface-brightness test, proposed by Richard Tolman in 1930: in an expanding universe that conserves light, a galaxy’s surface brightness falls as the fourth power of $1 + z$ — three magnitudes per square arcsecond by redshift one — whatever the cosmology. Two factors of $1 + z$ come from the photons’ energy and arrival rate being stretched, and two from the solid angle and luminosity distance relation. If the redshift were instead a loss of energy by light travelling through a static universe — the “tired light” proposed in 1929 as an alternative to expansion — the fall would be only one power of $1 + z$, less than a magnitude at redshift one.

The test was carried out with elliptical galaxies out to redshift 0.9 by Lori Lubin and Allan Sandage in 2001. After correcting for the brightening of the galaxies’ younger stellar populations in the past, the surface brightness followed the fourth power and excluded tired light. The correction for evolution is large — a magnitude or more by redshift one — which is why the test is weaker than the comparison of supernovae with the acoustic scale. But it is the same identity, and it settles a different question: not whether photons are lost, but whether the redshift is a stretching of space at all.

The background’s own photons #

A third consequence of the same assumption is in the microwave background. If the background’s photons are conserved and their wavelengths stretch with the expansion, the background’s temperature must have been higher in the past by exactly a factor of $1 + z$.

The temperature at other redshifts can be measured two ways. A galaxy cluster’s hot gas scatters background photons and distorts their spectrum in a way whose shape depends on the background’s temperature at the cluster’s redshift — the shadow that does not get fainter with distance carries the temperature in its spectral shape. And molecules in distant gas clouds, seen in absorption against quasars, are kept in their excited levels partly by the background’s radiation, so the populations of their rotational levels are a thermometer: carbon monoxide seen at redshift 2.69 gives a temperature of about ten and a half kelvin, where $T_0(1 + z)$ predicts 10.1. Written as $T_0(1 + z)^{1-\beta}$, the measurements put $\beta$ within a few hundredths of zero.

Duality, Tolman’s test and the background’s temperature are three expressions of one assumption — that the universe’s light is conserved as it travels through an expanding geometry — tested on three different kinds of light. They agree.

What the comparison leaves out #

The dusty universe drawn here is one specific alternative: matter alone, at the critical density, with grey dust whose amount is set to match a cosmological constant’s distances. Dust could be added to other universes, and in a universe with some dark energy the dust needed would be less; but any amount of dust large enough to matter for the dark energy’s properties would break duality at a level the present data can test. The mock errors are of the right size but treat each redshift bin as independent, and the real comparison has to account for the supernovae’s systematic errors, which standardising their brightness introduces and which are correlated across redshift. The constraint on $\varepsilon$ also cannot see a violation of duality that is constant with redshift, because both sets of distances carry a free constant.

Still open: whether anything else can break the tie #

Grey dust is excluded as the explanation of the acceleration. Other ways of losing photons are constrained less well. Light could convert into hypothetical light particles, axions, in intergalactic magnetic fields, which would dim distant sources by an amount depending on the field and the particle’s coupling — and unlike grey dust, this dimming could depend on wavelength in a characteristic way. Photons could decay, if they have a tiny mass. And light might travel along paths that differ from the geometry if gravity is modified on large scales. Each of these predicts a specific redshift dependence of $\eta$, and the surveys now running will measure it to better than a per cent out to redshift two — using supernovae as before, and also gravitational-wave distances that involve no photons at all, which give a luminosity distance from the waves’ amplitude and so test whether light and gravity agree on how far away things are.