Angular Diameter Distance

The distance that runs backwards

Everyone has the same intuition about distance and size: the farther away something is, the smaller it looks. Hold a coin at arm's length and it covers the Moon; walk it across the room and it shrinks to a dot. That relationship — angular size inversely proportional to distance — is so reliable that it is the basis of how surveyors, photographers, and astronomers have estimated distances for centuries. In an expanding universe, it breaks. Push an object far enough away and it stops shrinking. Push it farther still, and it begins to grow. The quantity that captures this strange behaviour is the angular diameter distance, written d_A, and it is one of the most counter-intuitive constructs in observational cosmology.

The definition is deliberately built so that the familiar flat-space geometry still holds. If an object has true transverse physical size ℓ and subtends an angle θ on your sky, the angular diameter distance is whatever number d_A makes the small-angle relation θ = ℓ / d_A come out correct. In static Euclidean space d_A would simply equal the real distance. In an expanding Friedmann-Lemaître-Robertson-Walker (FLRW) universe it does not — and the gap between d_A and your intuition is where all the interesting physics lives.

How it works

In a spatially flat expanding universe, the angular diameter distance to an object at redshift z is

d_A = d_C / (1 + z)

where d_C is the line-of-sight comoving distance — the distance between us and the object measured on a slice of constant cosmic time, with the expansion of space factored out. The comoving distance grows monotonically with redshift; it never decreases. So all the strange behaviour of d_A comes from the division by (1 + z).

The reason the (1 + z) divisor belongs there is physical, not a bookkeeping trick. The light we receive from a galaxy at redshift z left it when the universe's scale factor was a = 1/(1+z) — that is, when everything was (1+z) times closer together. At that earlier epoch the galaxy was much nearer to us in proper distance, so it subtended a larger angle. That angle is locked into the geometry of the light rays and travels to us unchanged. The angular size we measure today reflects how close the object was when the light was emitted, not how far away it is now. Dividing the comoving distance by (1+z) is exactly the conversion from "where it is now" to "the angle it presented then."

Now the competition is visible. As you go to higher redshift, d_C keeps rising but flattens — there is only a finite comoving distance to the cosmic horizon. Meanwhile (1+z) keeps climbing without limit. At low z the comoving distance wins and d_A increases; at some redshift the two effects balance and d_A reaches a maximum; beyond that the (1+z) factor dominates and d_A decreases. For the standard Planck ΛCDM cosmology that turnover sits near z ≈ 1.6.

Worked example: a 10 kpc galaxy at three redshifts

Take a galaxy with a fixed physical diameter of ℓ = 10 kpc — a typical mid-size disk — and place it at three redshifts. Using flat Planck 2018 parameters (H₀ = 67.4 km/s/Mpc, Ω_m = 0.315, Ω_Λ = 0.685), numerical integration of the comoving distance gives the following angular diameter distances and the angle each galaxy subtends, via θ = ℓ / d_A:

z = 0.5 :  d_C ≈ 1888 Mpc   d_A = d_C/1.5 ≈ 1259 Mpc   θ ≈ 1.64"
z = 1.6 :  d_C ≈ 4654 Mpc   d_A = d_C/2.6 ≈ 1790 Mpc   θ ≈ 1.15"   ← smallest angle
z = 7.0 :  d_C ≈ 7365 Mpc   d_A = d_C/8.0 ≈  921 Mpc   θ ≈ 2.24"

Read the angles. As the galaxy recedes from z = 0.5 to z = 1.6, it shrinks from 1.64 to 1.15 arcseconds — ordinary behaviour. But push it on to z = 7 and it does not keep shrinking; it grows back to 2.24 arcseconds, almost twice the angular size it had at z = 1.6, and larger than it had at z = 0.5. The same physical galaxy, twice as big on the sky despite being far more distant in lookback time. The angular diameter distance has fallen from its peak of 1790 Mpc down to 921 Mpc, and angular size scales as 1/d_A.

The full march across redshift makes the turnover unmistakable. For flat Planck 2018 ΛCDM (H₀ = 67.4 km/s/Mpc, Ω_m = 0.315), here is the comoving distance, the angular diameter distance, the angle subtended by a fixed 10 kpc galaxy, and — by the Etherington relation — the luminosity distance, at six redshifts:

Read down the d_A column: it climbs to its maximum of ≈ 1790 Mpc at z ≈ 1.6, then falls. The θ column does the opposite of intuition — it bottoms out at the turnover and rebounds, so the z = 7 galaxy subtends nearly twice the angle of the z = 1.6 one. Meanwhile d_L = (1+z)² d_A keeps growing without limit, which is why the same objects are simultaneously larger in angle and far fainter in flux.

This is not a quirk of one parameter choice. The turnover is a generic feature of any decelerating-then-accelerating expansion; only its exact location shifts. In the simpler Einstein-de Sitter model (a flat, matter-only universe, Ω_m = 1), the maximum sits at z = 1.25 and d_A peaks at exactly (8/27)·(c/H₀) ≈ 0.296 c/H₀.

The Etherington relation: one object, two distances

If you instead measure the same galaxy's brightness, you compute a different distance — the luminosity distance d_L, defined so that the inverse-square law F = L / (4π d_L²) still works. The two are not independent. They are tied by the Etherington reciprocity theorem (1933), also called the distance-duality relation:

d_L = (1 + z)² d_A

This is one of the cleanest results in cosmology. It holds in any metric theory of gravity, not just general relativity, provided two conditions are met: photons travel on null geodesics in a Riemannian spacetime, and the number of photons is conserved along the way (no exotic absorption, no photon-to-axion conversion, no decay). The two factors of (1+z) have transparent origins: one comes from the redshifting of each photon's energy, and one from the time dilation that stretches the arrival rate of photons. Together with the geometric d_A, they produce d_L.

A direct corollary is the apparent surface brightness. Surface brightness is flux per unit solid angle, scaling as d_A² / d_L² = 1 / (1+z)⁴. This (1+z)⁻⁴ dimming is dramatic: a galaxy at z = 1 is dimmed by a factor of 16 in surface brightness, and at z = 7 by a factor of 4096. It is the deep reason that high-redshift galaxies, even when they subtend a respectable angle thanks to the small d_A, are agonisingly faint and demand instruments like JWST. The (1+z)⁻⁴ law is also the basis of the Tolman surface-brightness test, which distinguishes a genuinely expanding universe (where the law holds) from static "tired-light" alternatives (where it does not). Observations are decisively consistent with expansion.

Variants and regimes

There is not a single "distance" in cosmology but a family, each tailored to a different observable, all coinciding at z = 0 and diverging at high z:

• Comoving distance d_C. The expansion-removed separation; integrate c/H(z) along the line of sight. Monotonic in z. The raw ingredient for the others.
• Transverse comoving distance D_M. Equals d_C in a flat universe; in curved space it is (c/H₀)·(1/√|Ω_k|)·sin or sinh of the curvature-scaled comoving distance. The version that goes into d_A and d_L when Ω_k ≠ 0.
• Angular diameter distance d_A = D_M / (1+z). Non-monotonic; this article's subject.
• Luminosity distance d_L = (1+z) D_M = (1+z)² d_A. Grows fastest; what supernova cosmology measures.
• Angular diameter distance between two redshifts d_A(z₁, z₂). Needed for gravitational lensing, where the relevant geometry involves the lens at z₁ and the source at z₂. In a flat universe d_A(z₁,z₂) = [D_M(z₂) − D_M(z₁)] / (1+z₂). Note it is not symmetric: d_A(z₁,z₂) ≠ d_A(z₂,z₁).

Common pitfalls and misconceptions

• Treating d_A as "the real distance." It is not the proper distance to the object now (that is larger and keeps growing), nor the distance the light travelled (the lookback/light-travel distance). It is an effective distance engineered to make angular sizes obey flat-space geometry — nothing more.
• Thinking the galaxy is physically bigger at high z. The object's true size ℓ is fixed. What changes is the angle it subtends, because the angular diameter distance shrinks past the turnover. "Looks bigger" is purely an angular-size statement.
• Confusing "bigger" with "easier to see." A larger angular size at high z is fought by the brutal (1+z)⁻⁴ surface-brightness dimming. Past-turnover galaxies cover more sky but are vastly fainter per unit area, so total detectability still falls.
• Forgetting the turnover redshift is cosmology-dependent. z ≈ 1.6 is specific to Planck ΛCDM. A different Ω_m or a dynamical dark-energy equation of state w(z) moves the peak — which is precisely why d_A(z) is a useful cosmological probe.
• Assuming Etherington always holds. The duality relation requires photon-number conservation. Exotic physics — opacity from intergalactic dust, photon-axion mixing — would violate d_L = (1+z)² d_A, and searching for such violations is a live test of fundamental physics.
• Mixing up d_A(z₁,z₂) symmetry. The angular diameter distance between two redshifts is not symmetric in its arguments, unlike ordinary distance; lensing calculations must use the ordered (lens, source) form.

Observational status and applications

Because d_A converts a known physical length into an observed angle, it is the workhorse distance wherever a "standard ruler" is in play:

• Baryon acoustic oscillations (BAO). The sound horizon at the drag epoch (~150 Mpc comoving) is imprinted as a preferred separation in galaxy clustering. Measuring its angular size at various redshifts maps d_A(z) and constrains dark energy. Surveys like BOSS, eBOSS, and DESI use exactly this.
• The CMB acoustic peaks. The angular scale of the first acoustic peak (≈ 1°) fixes the angular diameter distance to the last-scattering surface at z ≈ 1090, where d_A is only ≈ 13 Mpc — deep past the turnover, an extreme example of the falling d_A.
• Strong gravitational lensing. Einstein-ring radii depend on the ratio of d_A values to the lens and the source; time-delay cosmography (e.g. the H0LiCOW/TDCOSMO programmes) turns lensed-quasar light curves into a measurement of H₀ through these distances.
• The Sunyaev-Zel'dovich + X-ray cluster method. Combining the SZ decrement and X-ray surface brightness of a galaxy cluster yields its physical size and hence d_A directly — a distance independent of the cosmic distance ladder.
• The Tolman surface-brightness test. The (1+z)⁻⁴ dimming that follows from the d_A/d_L split has been confirmed in galaxy samples, ruling out static-universe tired-light models and corroborating genuine cosmic expansion.

Quantitative analysis: computing d_A from the cosmology

The recipe is short. First build the dimensionless expansion rate. For a flat ΛCDM universe,

E(z) = H(z)/H₀ = √( Ω_r(1+z)⁴ + Ω_m(1+z)³ + Ω_Λ )

where Ω_r is the radiation density (≈ 9×10⁻⁵, negligible except at very high z), Ω_m the matter density, and Ω_Λ the cosmological-constant density, summing to 1 in the flat case. The line-of-sight comoving distance is the integral of the Hubble distance c/H₀ weighted by 1/E:

d_C(z) = (c/H₀) ∫₀ᶻ dz' / E(z')

The Hubble distance c/H₀ for H₀ = 67.4 km/s/Mpc is about 4450 Mpc, which sets the overall scale. The integral has no elementary closed form for general ΛCDM and is evaluated numerically (Ned Wright's calculator and astropy.cosmology both return it). The angular diameter distance is then simply

d_A(z) = d_C(z) / (1+z)        (flat universe)

To locate the turnover, differentiate: d_A is maximal where d(d_A)/dz = 0, i.e. where d(d_C)/dz · (1+z) = d_C. Since d(d_C)/dz = (c/H₀)/E(z), the condition becomes (1+z)·(c/H₀)/E(z) = d_C(z) — the comoving distance equals the local Hubble distance scaled by (1+z). Solving numerically for Planck parameters gives z_max ≈ 1.6, d_A,max ≈ 1790 Mpc. For Einstein-de Sitter the whole thing is analytic: d_C = (2c/H₀)[1 − 1/√(1+z)], so d_A = (2c/H₀)[1 − 1/√(1+z)]/(1+z), which a quick derivative shows peaks at z = 5/4 = 1.25 with value (8/27)(c/H₀).

Finally, the duality relation gives everything else for free: multiply by (1+z) to get the transverse comoving distance, by (1+z)² to get the luminosity distance, and divide flux by (1+z)⁴ to get the surface-brightness dimming. One numerical integral, and the whole distance family follows.