Astronomy & Celestial Mechanics
the sky, computed
Concept

Distance duality — where it appears

The rule that a source's luminosity distance exceeds its angular-diameter distance by exactly (1 + z)² wherever photons are conserved. It holds in any metric theory of gravity, so a measured violation would mean light is lost or gained on the way.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

Four distances to the same galaxy. Four quantities all called "the distance", against redshift, at a Hubble constant of 67.36 km/s/Mpc, a matter density parameter of 0.3153 and a dark-energy parameter of 0.6847. They agree below z ≈ 0.1 and then part company completely. Comoving distance is the separation now, and it is what a map of the universe is drawn in. Luminosity distance is what a brightness gives, and it is larger by (1+z) because the photons arrive both redshifted and spread out in time. Light-travel distance is the age difference times c, and it is bounded by the age of the universe. Angular-diameter distance is what an angle gives, and it is the odd one: it rises, turns over at z = 1.59 where it reaches 1.79 Gpc, and falls thereafter. Past that redshift a galaxy of fixed size looks bigger the further away it is, because the universe it is being seen across was smaller when the light left it. At z = 10 the four differ by a factor of 121 between the largest and the smallest, so a sentence quoting a cosmological distance without saying which one has not given a number.

Four distances to the same galaxy

Inside the Local Group the word "distance" has one meaning. Past a redshift of about a tenth it has four, they disagree by factors of a hundred by the time the light is old, and one of them stops increasing and starts coming back.

cosmology · Distance ladder The ratio of two distances, for a universe with dark energy and one with dust. The distance-duality ratio η — the luminosity distance divided by (1 + z)² times the angular-diameter distance — against redshift. In a universe with a cosmological constant, or any universe in which light travels on the geometry and is conserved, it is exactly one (solid). In a universe of matter alone, decelerating, with grey intergalactic dust dimming its supernovae just enough to make their distances match the accelerating one's, it rises with redshift: 1.19 at redshift 0.5, 1.30 at 1, because the dust makes supernovae look farther than the geometry puts them while leaving the acoustic-scale distances untouched. The shaded band is η = 1 ± 0.03z, about what supernova and acoustic-scale distances together allow. The dusty universe's η rises at the start with a slope of about 0.45 per unit redshift, more than ten times the band's, and leaves it at once.

Two distances that must agree if light is conserved

A supernova's distance comes from how faint it is; an acoustic ruler's distance comes from how small it looks. In any universe whose light follows the geometry and is neither created nor destroyed, the two are tied by an exact factor of (1 + z)², whatever the universe contains. Dust that dimmed distant supernovae without reddening them could have faked the accelerating expansion — and would have broken that tie by a fifth by redshift a half. The tie holds to a few per cent.

starlight · Distance ladder Four ways for a galaxy to fade with redshift. The dimming of a galaxy's surface brightness, in magnitudes per square arcsecond, against redshift, for the same galaxy placed at greater and greater distance. In an expanding universe the bolometric surface brightness falls as (1+z)⁴: 3.01 magnitudes by z = 1 and 7.8 by z = 5. Measured per unit frequency, as magnitudes in a filter are, one power is absorbed by the stretching of the band and the dimming is (1+z)³. If photons lost energy on a static journey ("tired light"), only one power would remain — 0.75 magnitudes at z = 1. In a static space with no loss there is no dimming at all, because flux and solid angle both fall as the inverse square of distance. At z = 1 expansion and tired light differ by 2.3 magnitudes, a factor of 8 in brightness.

A dimming short by what young stars repay

In an expanding universe a galaxy's surface brightness should fall as the fourth power of one plus its redshift; if light merely tired on its way through static space, as the first power. Measured on elliptical galaxies at redshift 0.85, it falls as about the two-and-a-half power — neither. The verdict on expansion rests on what the galaxies did in the meantime, and the decisive fact is not the size of the shortfall but its sign: expansion needs the galaxies to have been brighter when their stars were younger, and tired light needs them to have been fainter.

cosmology · Expansion

Named alongside it

The objects these essays reach for when they reach for this one.

Angular-diameter distanceLuminosity distanceK-correctionAccelerating expansionAcoustic scaleComoving distanceCosmic microwave backgroundCosmological distancesCosmological redshiftElliptical galaxyExpansion of spaceGrey dust

All concepts