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The cosmic-distances generator

Four distances to the same galaxy
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 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.

2 essays call cosmic-distances. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Every figure listed here is the same construction drawn at different numbers, so a correction to one is a correction to all of them.

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. Cosmology

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.

 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. Starlight

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.

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