Angular Diameter Distance

The angular diameter distance answers a question every observational astronomer meets: an object of known physical size $\ell$ shows an angular extent $\delta\theta$ on the sky — how far away is it? In flat, static space the answer is the schoolbook relation $d = \ell/\delta\theta$ , and $d_A$ is defined to keep exactly that form. The subtlety is that we live in an expanding, possibly curved universe, where a light ray from a distant galaxy climbs out of the past along a null geodesic of the FLRW metric while the scale factor grows underneath it. Tracing that ray shows the transverse proper size at emission is $a(t_e)\,S_k(\chi)\,\delta\theta$ , so preserving $\delta\theta = \ell/d_A$ forces $d_A = a(t_e)\,S_k(\chi) = S_k(\chi)/(1+z)$ — the comoving transverse distance dimmed by one power of $(1+z)$ . That single factor of $(1+z)$ produces the most counter-intuitive fact in cosmology: $d_A$ is non-monotonic. As redshift climbs, comoving distance $\chi$ keeps growing and spreads objects thinner, but the $(1+z)^{-1}$ compression grows without bound and eventually wins. The product peaks near $z \approx 1.6$ , then falls, so a galaxy at $z = 6$ subtends a larger angle than an identical galaxy at $z = 1$ — we see distant objects as they were when the whole universe was physically small and close. $d_A$ is the working currency of precision cosmology. It converts the sound horizon into the angular scale of the CMB acoustic peaks and of baryon acoustic oscillations, the two great standard rulers, and it is tied to the luminosity distance by Etherington's reciprocity theorem $d_L = (1+z)^2 d_A$ — a relation so general it tests photon physics itself.

Key takeaways

The angular diameter distance is defined so an object of proper transverse size $\ell$ subtends $\delta\theta = \ell/d_A$ , keeping the Euclidean size–angle relation exactly true in a curved, expanding universe.
In FLRW geometry $d_A = a(t_e)\,S_k(\chi) = S_k(\chi)/(1+z)$ : the comoving transverse distance $S_k(\chi)$ redshifted by one factor of $(1+z)$ , so $d_A$ is smaller than both the comoving and luminosity distances.
Uniquely among distance measures $d_A$ turns over — it rises, peaks near $z \approx 1.6$ for Planck ΛCDM (at exactly $z = 1.25$ for Einstein–de Sitter), then falls, so very distant objects can look angularly larger.
Etherington's reciprocity theorem $d_L = (1+z)^2 d_A$ ties luminosity and angular diameter distances together for any metric theory with conserved photon number — curvature and cosmology cancel out of the ratio.
$d_A(z_\ast)$ turns the CMB sound horizon $r_s \approx 147$ Mpc into the measured acoustic scale $\theta_s \approx 0.0105$ rad, making $d_A$ the central lever on $\Omega_m$ , $\Omega_\Lambda$ , and curvature.
Key idea: Define distance so that $\delta\theta = \ell/d_A$ still holds; in FLRW this forces $d_A = S_k(\chi)/(1+z)$ , which peaks at intermediate redshift because the $(1+z)^{-1}$ compression eventually overwhelms the growth of comoving distance.

Where this lives today

The angular diameter distance is the beating heart of standard-ruler cosmology. Baryon acoustic oscillation surveys — BOSS, eBOSS, and now DESI — measure the transverse imprint of the same sound horizon at $z \sim 0.3$ – $2.4$ , delivering $d_A(z)$ directly and charting the expansion history that constrains dark energy's equation of state. On the last-scattering surface, Planck's measurement of the acoustic scale $\theta_\ast$ to five significant figures fixes $d_A(z_\ast)$ and, through it, the tightest bound on spatial curvature ( $\Omega_k = 0.001 \pm 0.002$ ). The same physics drives the sound-horizon side of the Hubble tension: the CMB-plus-BAO inference of $H_0 \approx 67$ km/s/Mpc, resting entirely on $d_A$ and a calibrated $r_s$ , sits in $5\sigma$ disagreement with the local distance-ladder value near $73$ . Strong gravitational lensing exploits ratios of angular diameter distances $d_{LS}/(d_L d_S)$ — with the non-additivity $d_{LS} \neq d_S - d_L$ in curved space — to weigh cluster mass and, via time-delay cosmography of lensed quasars (H0LiCOW, TDCOSMO), to measure $H_0$ . And JWST now resolves galaxies past $z = 10$ whose angular sizes only make sense once the $d_A$ turnover is folded in.

The physical picture

Hold a ruler of proper length $\ell$ at a fixed comoving location and ask how large it looked at the instant its light left. Back then the universe was smaller by $a(t_e) = 1/(1+z)$ , so the ruler filled a larger fraction of the sky. Trace the two edge rays back to the observer: they converge at the eye separated by the angle $\delta\theta$ , and the geometry fixes $\ell = a(t_e)\,S_k(\chi)\,\delta\theta$ . Divide by the angle and the angular diameter distance drops out, $d_A = a(t_e)\,S_k(\chi)$ . The first figure makes this bundle-of-rays definition concrete: physical size at emission, angle at the observer, and the geometry connecting them.

Figure

A source of proper size ℓ at emission subtends angle δθ at the observer; the angular diameter distance is defined so d_A = ℓ/δθ, which in FLRW equals a(t_e)·S_k(χ) = S_k(χ)/(1+z) — the object is seen as it was when the universe was smaller by the factor a(t_e)

Two effects now race as we look further out. Comoving distance $\chi$ grows and spreads a fixed physical size thinner across the sky; the shrinking emission-epoch scale factor magnifies it. Nearby, the thinning wins and angles fall with distance as usual; far away, the $(1+z)^{-1}$ magnification takes over, so the angle bottoms out and grows again. The result is a $d_A(z)$ that rises to a maximum and then declines — the sky at high redshift acts like a fisheye lens onto a small early universe. The second figure is the full turnover curve, with the real ΛCDM peak marked in gigaparsecs.

Figure

Angular diameter distance versus redshift for Planck ΛCDM (Ω_m = 0.315, H₀ = 67.4 km/s/Mpc) and Einstein–de Sitter: d_A rises, peaks, then falls — ΛCDM peaks at z ≈ 1.6 with d_A ≈ 1.79 Gpc, EdS at z = 1.25 with d_A ≈ 1.32 Gpc, so a galaxy at z = 6 subtends a larger angle than its twin at z = 0.5

The core equations

Everything begins with the operational definition: the angular diameter distance is whatever number makes the flat-space size–angle relation come out right — proper transverse size divided by observed angle.

d_A \equiv \frac{\ell}{\delta\theta} \qquad (1)

where: $\ell$ — proper transverse size of the source at emission; $\delta\theta$ — observed angular extent; $d_A$ — angular diameter distance. Reading: fix the definition and let the geometry of the universe decide what $d_A$ actually equals. Trace a light bundle through the FLRW metric and that geometry hands back a clean result. The transverse proper length at emission is the comoving transverse distance $S_k(\chi)$ times the emission scale factor times the angle, so $d_A$ carries exactly one factor of the scale factor.

d_A = a(t_e)\, S_k(\chi) = \frac{S_k(\chi)}{1+z} \qquad (2)

where: $a(t_e)$ — scale factor at emission, $= 1/(1+z)$ with $a(t_0) = 1$ ; $\chi$ — comoving radial distance; $S_k(\chi)$ — comoving transverse distance, curvature-dependent; $z$ — cosmological redshift. Reading: the lone $(1+z)^{-1}$ is the seed of the turnover. The curvature of space enters only through the transverse function $S_k$ , linear in a flat universe and trigonometric or hyperbolic otherwise.

S_k(\chi) = \begin{cases} R_0 \sin(\chi/R_0) & k = +1 \\ \chi & k = 0 \\ R_0 \sinh(\chi/R_0) & k = -1 \end{cases} \qquad (3)

where: $k$ — sign of spatial curvature ( $+1$ closed, $0$ flat, $-1$ open); $R_0$ — present curvature radius, $c/(H_0\sqrt{|\Omega_{k,0}|})$ . Reading: closed space focuses bundles ( $\sin$ ), open space defocuses them ( $\sinh$ ), flat space neither. The radial coordinate is an integral of the inverse expansion rate along the line of sight — comoving distance accumulates as light climbs out of the past.

\chi(z) = c\int_0^z \frac{\mathrm{d}z'}{H(z')} \qquad (4)

where: $H(z)$ — Hubble parameter at redshift $z$ ; $c$ — speed of light. Reading: every distance measure traces back to this one integral over expansion history. The expansion rate follows from the Friedmann equation, each species diluting with its own power of $(1+z)$ .

H(z) = H_0\sqrt{\Omega_{m,0}(1+z)^3 + \Omega_{r,0}(1+z)^4 + \Omega_{k,0}(1+z)^2 + \Omega_{\Lambda,0}} \qquad (5)

where: $\Omega_{m,0}, \Omega_{r,0}, \Omega_{\Lambda,0}$ — present density parameters of matter, radiation, dark energy; $\Omega_{k,0} = 1 - \Omega_{m,0} - \Omega_{r,0} - \Omega_{\Lambda,0}$ — curvature. Reading: the cosmological model enters $d_A$ entirely through this function. The luminosity distance, built from the same $S_k(\chi)$ but carrying two extra $(1+z)$ factors from photon-energy and arrival-rate dimming, is locked to $d_A$ by a pure kinematic identity.

d_L = (1+z)^2\, d_A \qquad (6)

where: $d_L$ — luminosity distance, defined so $F = L/(4\pi d_L^2)$ . Reading: Etherington's reciprocity theorem — no cosmology, no curvature, only null geodesics and photon conservation. For a flat, matter-only (Einstein–de Sitter) universe the integral is elementary and $d_A$ closes in elementary functions, the standard analytic testbed.

d_A^{\,\mathrm{EdS}} = \frac{2c}{H_0}\,\frac{1}{1+z}\left(1 - \frac{1}{\sqrt{1+z}}\right) \qquad (7)

where: $H_0$ — Hubble constant. Reading: differentiating this closed form gives a maximum at exactly $z = 5/4 = 1.25$ . Finally, the reason $d_A$ matters observationally: it turns a known physical length — a standard ruler — into an angle on the sky.

\theta_s = \frac{r_s}{d_A(z_\ast)} \qquad (8)

where: $r_s$ — proper sound horizon at recombination (the standard ruler); $z_\ast \approx 1090$ — redshift of last scattering; $\theta_s$ — angular scale of the acoustic peak. Reading: measure $\theta_s$ , know $r_s$ , and you have measured $d_A(z_\ast)$ — an integral over the whole expansion history.

Derivation AFrom the FLRW metric to the angular diameter distance

The goal is to derive the central result $d_A = a(t_e)\,S_k(\chi) = S_k(\chi)/(1+z)$ from first principles, starting only from the FLRW line element and the definition $d_A = \ell/\delta\theta$ , and to see explicitly where the single factor of $(1+z)$ comes from — it is not inserted by hand but falls out of measuring a proper transverse length at the emission epoch, when the scale factor was $a(t_e)$ rather than its present value of one.

Step 1. Write the FLRW metric in comoving spherical coordinates, $\mathrm{d}s^2 = -c^2\mathrm{d}t^2 + a^2(t)\big[\mathrm{d}\chi^2 + S_k^2(\chi)\,\mathrm{d}\Omega^2\big]$ , where $\mathrm{d}\Omega^2 = \mathrm{d}\vartheta^2 + \sin^2\!\vartheta\,\mathrm{d}\phi^2$ . The source sits at comoving radius $\chi$ and the observer at the origin. Why: the metric is where geometry lives — the coefficient of the angular part, $a^2 S_k^2$ , is exactly the machine that turns an angle into a proper transverse length.

Step 2. Consider the two ends of the source separated by a small angle $\delta\theta = \delta\vartheta$ at fixed comoving radius $\chi$ and fixed emission time $t_e$ . Along that purely transverse displacement $\mathrm{d}t = \mathrm{d}\chi = 0$ , so the proper separation from the metric is $\ell = a(t_e)\,S_k(\chi)\,\delta\theta$ . Why: this is the physical size the source actually had when it emitted the light — measured with the scale factor frozen at its emission-epoch value, not today's.

Step 3. Apply the definition $d_A \equiv \ell/\delta\theta$ and divide out the angle, giving $d_A = a(t_e)\,S_k(\chi)$ . Why: the angle cancels by construction — that is the whole point of defining $d_A$ to preserve the Euclidean form $\delta\theta = \ell/d_A$ .

Step 4. Convert the emission scale factor to redshift. With $a(t_0) = 1$ and $1 + z = 1/a(t_e)$ , we get $a(t_e) = 1/(1+z)$ and therefore $d_A = S_k(\chi)/(1+z)$ . Why: this exposes the lone $(1+z)^{-1}$ — one power, not two as in $d_L$ — which is what makes $d_A$ turn over instead of climbing forever. *Three quick checks confirm the result: dimensionally $[d_A] = [S_k] =$ length; as $z \to 0$ , $S_k(\chi) \to \chi \to cz/H_0$ and $d_A \to cz/H_0$ , recovering the Hubble law; and in flat space $S_k(\chi) = \chi$ gives $d_A = \chi/(1+z)$ directly.*

Go deeper · Derivation B · 1 minWhy the angular diameter distance turns over

Derivation BWhy the angular diameter distance turns over

The goal is to prove that $d_A(z)$ has a genuine interior maximum — that it is not monotonic like every other distance measure — by differentiating the flat-space form and locating where the slope vanishes. We work in a flat universe so $S_k(\chi) = \chi$ , use $\mathrm{d}\chi/\mathrm{d}z = c/H(z)$ from equation (4), and read off the competition between comoving growth and redshift compression that produces the peak.

Step 1. Start from the flat-space result $d_A(z) = \chi(z)/(1+z)$ . Both factors grow with $z$ : $\chi$ climbs toward the horizon while $(1+z)$ grows without bound. Why: a ratio of two increasing functions can peak — the outcome depends on which accelerates away, so we differentiate rather than guess.

Step 2. Differentiate with the quotient rule: $\dfrac{\mathrm{d}\,d_A}{\mathrm{d}z} = \dfrac{1}{1+z}\dfrac{\mathrm{d}\chi}{\mathrm{d}z} - \dfrac{\chi}{(1+z)^2}$ . Why: the two terms are the two competing effects made quantitative — the first is comoving growth spreading objects thinner, the second is the compression magnifying them.

Step 3. Substitute $\mathrm{d}\chi/\mathrm{d}z = c/H(z)$ and pull out the common positive factor, giving $\dfrac{\mathrm{d}\,d_A}{\mathrm{d}z} = \dfrac{1}{(1+z)^2}\left[\dfrac{(1+z)\,c}{H(z)} - \chi(z)\right]$ . Why: the prefactor $1/(1+z)^2$ is always positive, so the sign of the slope is set entirely by the bracket — the whole turnover is decided by one competition.

Step 4. Set the bracket to zero: the maximum sits where $\dfrac{(1+z)\,c}{H(z)} = \chi(z)$ . Below it the first term dominates and $d_A$ rises; above it $\chi$ has accumulated enough to win and $d_A$ falls. For Planck ΛCDM this crossing is at $z \approx 1.6$ ; for Einstein–de Sitter it is at exactly $z = 1.25$ . Why: this is the precise statement that the sky becomes a fisheye lens — beyond the peak, identical objects grow with distance. *The EdS case checks out explicitly: with $\chi = \frac{2c}{H_0}(1 - (1+z)^{-1/2})$ and $H = H_0(1+z)^{3/2}$ , the turnover condition $(1+z)c/H = \chi$ becomes $1/(1+z)^{1/2} = 2\big(1 - (1+z)^{-1/2}\big)$ , i.e. $3(1+z)^{-1/2} = 2$ , so $(1+z)^{1/2} = 3/2$ and $z = 5/4 = 1.25$ .*

Go deeper · Derivation C · 2 minThe Etherington reciprocity relation

Derivation CThe Etherington reciprocity relation

The goal is to derive $d_L = (1+z)^2 d_A$ and show why it survives independent of the cosmological model — no dependence on $\Omega_m$ , $\Omega_\Lambda$ , $k$ , or even the theory of gravity. We build both distances from the same comoving factor $S_k(\chi)$ , count the powers of $(1+z)$ that photon-energy and arrival-rate dimming attach to the flux, and take the ratio so the geometry cancels, leaving a pure kinematic number.

Step 1. From the previous derivation, the angular-diameter side is $d_A = S_k(\chi)/(1+z)$ : a solid angle $\delta\Omega_o$ at the observer maps to proper area $d_A^2\,\delta\Omega_o$ at the source. Why: this is the "size" half of reciprocity — one inverse power of $(1+z)$ from the emission-epoch scale factor.

Step 2. Now the flux. Photons emitted with luminosity $L$ spread over a comoving sphere of area $4\pi S_k^2(\chi)$ , and two separate $(1+z)$ factors dim them: one redshifts each photon's energy, the other dilates the arrival rate. So $F = \dfrac{L}{4\pi S_k^2(\chi)}\,\dfrac{1}{(1+z)^2}$ . Why: these are the same two factors that appear in surface-brightness dimming — energy per photon and photons per second each fall by $(1+z)$ .

Step 3. Match the definition $F \equiv L/(4\pi d_L^2)$ to read off $d_L = (1+z)\,S_k(\chi)$ . Why: the luminosity distance is the comoving distance dimmed by $(1+z)$ — one power, not two, because half of the flux suppression is bundled into the geometric $S_k$ and half into $d_L$ .

Step 4. Take the ratio: $\dfrac{d_L}{d_A} = \dfrac{(1+z)\,S_k(\chi)}{S_k(\chi)/(1+z)} = (1+z)^2$ ; the curvature function $S_k(\chi)$ cancels completely. Why: because $S_k$ carries all the cosmology and it cancels, the result holds for any FLRW model — reciprocity is a theorem, not a fit to data.

Step 5. State the generality: the relation holds in any metric theory of gravity provided photons travel on null geodesics with conserved number, so a measured violation signals photon non-conservation — dust extinction or exotic axion–photon mixing. Why: this promotes an algebraic identity into a laboratory — deviations from $(1+z)^2$ are a clean probe of new physics. *Two consistency checks: the ratio $d_L/d_A = (1+z)^2$ is dimensionless and independent of $H_0$ , the $\Omega$ 's, and $k$ ; and as $z \to 0$ both distances reduce to $cz/H_0$ , so $d_L/d_A \to 1$ consistently.*

Limiting cases

The behaviour of $d_A(z)$ across the instructive regimes, each recovering a familiar result or exposing the turnover:

Case	Result	Meaning
Low redshift	$d_A \approx \frac{cz}{H_0}\left[1 - \tfrac{1}{2}(3 + q_0)z + \cdots\right]$	recovers the Hubble law $d_A \to cz/H_0$ ; the deceleration parameter $q_0$ and curvature enter only at second order in $z$
Flat matter-only (EdS)	$d_A = \frac{2c}{H_0}\frac{1}{1+z}\left(1 - \frac{1}{\sqrt{1+z}}\right)$	the analytic testbed; $d_A$ peaks at exactly $z = 5/4 = 1.25$ where $\mathrm{d}d_A/\mathrm{d}z = 0$
High redshift	$d_A \to \frac{S_k(\chi_\infty)}{1+z} \sim \frac{1}{z}$	$\chi$ saturates at the finite comoving horizon $\chi_\infty$ , so $d_A$ falls off as $1/z$ — distant objects look ever larger
Turnover condition	$\frac{\mathrm{d}\,d_A}{\mathrm{d}z} = 0 \Longleftrightarrow (1+z)\frac{c}{H(z)} = \chi(z)$	in a flat universe the maximum sits where comoving distance equals $(1+z)c/H(z)$ ; for Planck ΛCDM this is $z \approx 1.6$

Common trapTreating d_A as just another distance that grows with redshift

The single most common error is to assume $d_A$ increases monotonically with distance — it does not. It rises, peaks near $z \approx 1.6$ , then decreases, so beyond the peak more distant objects subtend larger angles. A related trap is conflating the three distance measures: at $z = 1$ they stand in the ratio $S_k : d_L : d_A = 2 : 4 : 1$ . Using the wrong one corrupts an inferred size by tens of percent — a $z = 1$ galaxy's true size is $d_A\,\delta\theta$ , never $d_L\,\delta\theta$ or $\chi\,\delta\theta$ . And $d_A$ is not a distance you could ever traverse; it is a device defined solely to make $\delta\theta = \ell/d_A$ true.

Worked examples

1 · The Einstein–de Sitter turnover, fully worked

In a flat, matter-only universe with $H_0 = 67.4$ km/s/Mpc, find the redshift where $d_A$ is maximal, the value $d_A^{\max}$ , and the angle subtended there by a galaxy of proper diameter 20 kpc.

Step 1. Write $d_A^{\mathrm{EdS}} = \dfrac{2c}{H_0}\dfrac{1}{1+z}\left(1 - \dfrac{1}{\sqrt{1+z}}\right)$ and substitute $u = (1+z)^{-1/2}$ , so $d_A \propto u^2(1 - u) = u^2 - u^3$ . This turns a distance-versus-redshift extremum into an elementary polynomial maximization.

Step 2. Differentiate: $\dfrac{\mathrm{d}}{\mathrm{d}u}(u^2 - u^3) = 2u - 3u^2 = u(2 - 3u) = 0$ , so $u = 2/3$ . Then $(1+z)^{-1/2} = 2/3$ gives $(1+z)^{1/2} = 3/2$ , hence $1 + z = 9/4$ and $z_{\max} = 5/4 = 1.25$ .

Step 3. Evaluate the peak. The Hubble distance is $c/H_0 = 299792.458/67.4 = 4448$ Mpc. At $u = 2/3$ : $d_A^{\max} = \dfrac{2c}{H_0}\,u^2(1 - u) = \dfrac{2c}{H_0}\cdot\dfrac{4}{27} = 0.2963\,\dfrac{c}{H_0}$ , so $d_A^{\max} = 0.2963 \times 4448 = 1318$ Mpc $\approx$ 1.32 Gpc.

Step 4. The angle. A galaxy of proper diameter $\ell = 20$ kpc $= 0.020$ Mpc at $d_A = 1318$ Mpc subtends $\delta\theta = 0.020/1318 = 1.52\times10^{-5}$ rad $=$ 3.1 arcsec — the smallest angle such a galaxy can ever present, since $z = 1.25$ is the peak. Push it to $z = 5$ , where $d_A$ drops to $877$ Mpc, and the same galaxy grows back to $4.7$ arcsec.

2 · A more distant galaxy that looks bigger (Planck ΛCDM)

Two identical galaxies of proper diameter 15 kpc sit at $z = 1.6$ (the ΛCDM peak) and $z = 6$ . Using $d_A(1.6) = 1794$ Mpc and $d_A(6) = 1203$ Mpc, which subtends the larger angle?

Step 1. At the peak, $\delta\theta_{1.6} = \ell/d_A = 0.015/1794 = 8.36\times10^{-6}$ rad $=$ 1.72 arcsec. This is the minimum angle for a 15 kpc galaxy anywhere on the sky.

Step 2. At $z = 6$ , $d_A$ has fallen to $1203$ Mpc, so $\delta\theta_6 = 0.015/1203 = 1.25\times10^{-5}$ rad $=$ 2.57 arcsec.

Step 3. The galaxy nearly four times further in redshift appears 50% larger on the sky. This is the turnover made observable: high- $z$ galaxies are angularly magnified, partly offsetting the brutal $(1+z)^4$ dimming. A more distant object can subtend a larger angle — d_A, not comoving distance, controls apparent size.

3 · The CMB acoustic scale as a standard ruler

The comoving sound horizon at recombination is $r_s \approx 147$ Mpc, and for Planck ΛCDM the comoving distance to last scattering ( $z_\ast = 1090$ ) is $\chi(z_\ast) = 13\,932$ Mpc. Predict the angular scale of the first acoustic peak.

Step 1. Since $r_s$ is quoted comoving and $\chi$ is comoving, the cleanest route is $\theta_s = r_s^{\text{com}}/\chi(z_\ast) = 147/13\,932 =$ $0.01055$ rad $= 0.60°$ (equivalent to $r_s^{\text{phys}}/d_A(z_\ast)$ with both dimmed by the same $(1+z_\ast)$ ).

Step 2. Equivalently via $d_A$ : with $d_A(z_\ast) = \chi/(1 + z_\ast) = 12.8$ Mpc and physical horizon $r_s^{\text{phys}} = 147/1091 = 0.135$ Mpc, $\theta_s = 0.135/12.8 = 0.0105$ rad — identical.

Step 3. This matches the measured Planck value $\theta_\ast \approx 0.0104$ rad, the first acoustic peak near $\ell \approx 220$ . Because $d_A(z_\ast)$ integrates the whole expansion history, this one angle tightly constrains $\Omega_m$ , $\Omega_\Lambda$ , and curvature. A 147 Mpc ruler at z = 1090 subtends 0.6° — a cosmological measuring stick.

Physics of the portrait

The angular diameter distance is where the geometry of general relativity meets a telescope: a concept that began as Hubble's raw plot of nebular sizes and hardened, through Etherington and Mattig, into the precision ruler of modern cosmology.

Origins · 1 minangular diameter distance and how this was discovered

angular diameter distance

the distance d_A = ℓ/δθ = S_k(χ)/(1+z) that peaks at intermediate redshift · distance measures in relativistic cosmology

The angular diameter distance is one of a family of "distances" that only became necessary once the universe was known to expand — in a static Euclidean cosmos there is just one, but in an FLRW spacetime the comoving, luminosity, and angular diameter distances all differ, tied together by powers of $(1+z)$ . The geometric stage was set by Alexander Friedmann (1922) and Georges Lemaître (1927), whose expanding solutions made the scale factor $a(t)$ the master variable, and by Howard Robertson (1935) and Arthur Walker (1936), who fixed the standard FLRW line element with its curvature function $S_k(\chi)$ . Edwin Hubble had already been measuring the angular sizes of nebulae — the raw material of $d_A$ — without the relativistic apparatus to interpret them. The decisive step came from the British relativist Ivor Etherington, who in 1933 proved the reciprocity relation now bearing his name: $d_L = (1+z)^2 d_A$ in any spacetime, from photon conservation alone, with Richard Tolman's punishing $(1+z)^4$ surface-brightness dimming as its observational shadow. Wolfgang Mattig (1958) then produced the celebrated closed-form redshift–distance relation for dust cosmologies, and by the time David Hogg wrote his widely-used 1999 note "Distance Measures in Cosmology," the definitions were standardized into the toolkit every survey now uses to turn angles on the sky into constraints on dark energy.

History

1929

— Hubble publishes the velocity–distance relation for galaxies, the empirical seed of cosmological distances

1933

— Etherington proves the reciprocity relation d_L = (1+z)² d_A from photon conservation

1935

— Robertson and Walker fix the standard FLRW line element with its curvature function S_k(χ)

1958

— Mattig derives a closed-form redshift–distance relation for dust cosmologies

1999

— Hogg's note "Distance Measures in Cosmology" standardizes the modern definitions

Going further

Texts. Ryden, Introduction to Cosmology, 2nd ed., §6.4 (angular diameter distance) · Dodelson, Modern Cosmology, §2.4 (distances) · Hogg, Distance Measures in Cosmology, astro-ph/9905116 · Mo, van den Bosch & White, Galaxy Formation and Evolution, §3.2 · Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., Ch. 29 · Binney & Tremaine, Galactic Dynamics, 2nd ed. (geometry of null geodesics). The EdS comoving integral $\chi = \frac{2c}{H_0}(1 - (1+z)^{-1/2})$ reaches its maximum at $z = 5/4 = 1.25$ via the $u = (1+z)^{-1/2}$ substitution; the reciprocity ratio follows by cancelling $S_k(\chi)$ ; and the low- $z$ limit gives $d_A \to cz/H_0$ . The plotted turnover uses Planck 2018 parameters ( $\Omega_m = 0.315$ , $H_0 = 67.4$ ), giving the peak $z \approx 1.6$ , $d_A^{\max} \approx 1.79$ Gpc, and $d_A^{\mathrm{EdS,max}} \approx 1.32$ Gpc at $z = 1.25$ . CMB anchors used throughout are $z_\ast \approx 1090$ , $r_s \approx 147$ Mpc comoving, $\theta_\ast \approx 0.0104$ rad, and first peak $\ell \approx 220$ .

Prerequisites. The FLRW Metric · Comoving and Proper Coordinates · Cosmological Redshift · Spatial Curvature of the Universe · The Friedmann Equations. Leads to. Hubble–Lemaître Law · The Hubble Constant · Equation of State in Cosmology.

Ask the tutor: Ask me to walk through why a galaxy at $z = 6$ can look bigger than its twin at $z = 1$ , or to compute $d_A$ , $d_L$ , and the acoustic scale for any cosmology you name.

Why is the angular diameter distance smaller than the comoving distance?

Because $d_A = S_k(\chi)/(1+z)$ carries an extra factor of $1/(1+z)$ that the comoving distance $\chi$ does not. Physically, $d_A$ measures the geometry at the moment of emission, when the universe was smaller by $a(t_e) = 1/(1+z)$ , whereas the comoving distance is scaled to today. A fixed comoving separation corresponded to a smaller proper separation in the past, and dividing that by the observed angle gives a smaller distance. At $z = 1$ the factor is 2; at $z = 1090$ it is over a thousand, which is why $d_A$ to last scattering is only about 13 Mpc even though the comoving distance is nearly 14 Gpc.

How can a galaxy that is further away look bigger on the sky?

This is the turnover, and it is real. For nearby objects angular size falls with distance as intuition demands, but beyond the peak near $z \approx 1.6$ , $d_A$ starts decreasing with redshift, so the same physical size divided by a smaller $d_A$ gives a larger angle. The reason is that we see distant objects as they were when the universe was small and everything crammed close together — a fixed comoving patch subtended a huge angle back then, as through a fisheye lens. A 15 kpc galaxy subtends about 1.7 arcsec at $z = 1.6$ but 2.6 arcsec at $z = 6$ , and this angular magnification helps offset the severe surface-brightness dimming that would otherwise hide early galaxies.

Why does Etherington's reciprocity hold for essentially any cosmology?

Because its derivation uses only two ingredients: photons travel on null geodesics, and photon number is conserved (no absorption, no emission along the way). Both distances are built from the same comoving factor $S_k(\chi)$ , which carries all the dependence on $\Omega_m$ , $\Omega_\Lambda$ , curvature, and even the theory of gravity — and that factor cancels exactly when you form the ratio $d_L/d_A$ , leaving the pure kinematic number $(1+z)^2$ . This robustness is a gift: because the theorem is nearly assumption-free, any observed departure from $(1+z)^2$ is not a failure of the cosmological model but a signal that photons are being lost or created — cosmic dust, or exotic axion–photon oscillations — which turns reciprocity into a clean test of fundamental physics.

What is the practical difference between d_A, d_L, and comoving distance?

They answer three questions about the same source. Comoving distance $S_k(\chi)$ asks "how far apart are we in coordinates scaled to today?" Angular diameter distance $d_A = S_k/(1+z)$ asks "how do I turn this object's known size into the angle I measure?" Luminosity distance $d_L = (1+z)S_k$ asks "how do I turn its known brightness into the flux I receive?" They stand in the ratio $S_k : d_L : d_A = 2 : 4 : 1$ at $z = 1$ . Use $d_A$ for standard rulers (CMB peaks, BAO, galaxy sizes) and $d_L$ for standard candles (supernovae); mixing them corrupts an inferred size or luminosity by factors of $(1+z)$ .

Why is surface-brightness dimming so much steeper than either distance?

Surface brightness — flux per unit solid angle — falls as $(1+z)^{-4}$ , the harshest scaling in observational cosmology, and it is reciprocity's shadow. Two powers come from flux dimming ( $d_L^2 \propto (1+z)^2 S_k^2$ ) and two more from the solid angle set by $d_A^2 \propto S_k^2/(1+z)^2$ , which grows the apparent area: one power each from photon-energy redshift, arrival-rate dilation, and two from angular spreading. This is the Tolman dimming test, historically used to confirm that the universe genuinely expands — an expanding cosmos predicts $(1+z)^{-4}$ , tired light far less.

How does the CMB acoustic peak pin down spatial curvature?

The sound horizon $r_s$ at recombination is fixed almost entirely by pre-recombination physics — baryon and photon densities — so it is a standard ruler of known size. Its angular scale $\theta_s = r_s/d_A(z_\ast)$ depends on the geometry of the paths photons took across the observable universe, which is curvature-sensitive: in a closed universe the rays converge ( $S_k = R_0\sin$ ) and the ruler looks larger; in an open one they diverge and it looks smaller. Measuring $\theta_s \approx 0.0104$ rad and knowing $r_s$ reads off $d_A(z_\ast)$ and, given the matter and dark-energy content, the curvature. Planck's result is a spatially flat universe to about $0.2\%$ — one of the tightest measurements in cosmology, resting entirely on the angular diameter distance.