Cosmic Distance Measures

Ask "how far is that galaxy?" in an expanding, possibly curved universe and the question fractures. Distance is no longer one number but a family, because each way of measuring a remote object — timing a light ray, resolving its angular size, catching its flux, watching it drift across the sky — probes a different slice of the same curved spacetime and answers with a different figure. A quasar at redshift one sits 3300 Mpc away in comoving coordinates, yet behaves as though it were 1650 Mpc away when you gauge its angular size and 6600 Mpc away when you gauge its brightness. All three are correct; none is "the" distance. What tames the confusion is that every measure descends from one geometric object, the Friedmann–Lemaître–Robertson–Walker metric, fed by one dynamical input, the expansion history $H(z)$ . Integrate the radial null geodesic once and you have the line-of-sight comoving distance $D_C$ . Curvature reshapes it into the transverse comoving distance $D_M$ ; two clean factors of $(1+z)$ then split off the luminosity distance $D_L=(1+z)D_M$ and the angular-diameter distance $D_A=D_M/(1+z)$ . Their ratio, $D_L=(1+z)^2 D_A$ , is Etherington's distance-duality relation — a theorem holding in any metric theory of gravity, so long as photons travel on null geodesics and are not lost along the way. These are not bookkeeping curiosities. Type Ia supernovae read cosmology off $D_L(z)$ and revealed cosmic acceleration; baryon acoustic oscillations use a standard ruler in $D_A(z)$ and $H(z)$ ; gravitational-wave sirens deliver $D_L$ with no ladder at all. To interpret nearly any cosmological observation is to convert a measured distance back into geometry and expansion.

Key takeaways

In an expanding, curved universe there is no single distance: flux, angular size, light-travel, and proper motion each define their own measure, all built from one FLRW geometry.
The line-of-sight comoving distance $D_C(z)=\int_0^z c\,\mathrm{d}z'/H(z')$ is the master quantity; everything else is an algebraic transform of it plus curvature.
Spatial curvature turns $D_C$ into the transverse comoving distance $D_M$ through a $\sin$ , identity, or $\sinh$ (closed, flat, open).
Two factors of $(1+z)$ give $D_L=(1+z)D_M$ and $D_A=D_M/(1+z)$ , hence Etherington duality $D_L=(1+z)^2 D_A$ — valid whenever photons follow null geodesics and are conserved.
$D_A$ is non-monotonic: it peaks near $z\approx1.6$ (at $\approx1748$ Mpc for flat $\Lambda$ CDM) and then falls, so the most distant galaxies subtend larger angles.
Standard candles (SNe Ia), standard rulers (BAO), and standard sirens (GW) each measure a combination of these distances to pin down $H_0$ , $\Omega_m$ , and $\Omega_\Lambda$ .
Key idea: Compute one integral, $D_C(z)=\int_0^z c\,\mathrm{d}z'/H(z')$ ; curvature turns it into $D_M$ , and simple $(1+z)$ factors turn $D_M$ into $D_L=(1+z)D_M$ and $D_A=D_M/(1+z)$ .

The core insight

Compute one integral, $D_C(z)=\int_0^z c\,\mathrm{d}z'/H(z')$ ; then curvature turns it into the transverse distance $D_M$ , and simple $(1+z)$ factors turn $D_M$ into the luminosity distance $D_L=(1+z)D_M$ and the angular-diameter distance $D_A=D_M/(1+z)$ .

Where this lives today

The distance measures are the daily working currency of precision cosmology. The Type Ia supernova Hubble diagram — now assembled into samples like Pantheon+ with over 1500 calibrated events — fits $\mu(z)$ against $D_L(z;\Omega_m,\Omega_\Lambda)$ and remains the sharpest local anchor of cosmic acceleration; the same $D_L$ ladder, calibrated by Cepheids, sits at the centre of the Hubble-tension debate, where the local $H_0\approx73$ disagrees at more than $5\sigma$ with the $\approx67$ inferred from the cosmic microwave background. Baryon acoustic oscillations supply the complementary standard ruler: the DESI survey (2024) measures the sound-horizon scale $r_d\approx147$ Mpc transversely to constrain $D_M(z)$ and radially to constrain $H(z)$ , breaking the $\Omega_m$ – $\Omega_\Lambda$ degeneracy. Gravitational-wave standard sirens read $D_L$ straight off a merger's waveform amplitude with no ladder at all — GW170817 was the proof of concept, and the Einstein Telescope and LISA aim for percent-level cosmography. Meanwhile Etherington duality is now an observational test in its own right: pairing galaxy-cluster $D_A$ (from X-ray and Sunyaev–Zel'dovich data) with supernova $D_L$ at matched redshift, cosmologists check $D_L=(1+z)^2 D_A$ to a few percent as a probe of exotic photon physics and dust. Euclid and the Rubin Observatory will soon push every measure to higher redshift and precision.

The physical picture

Picture a rigid comoving grid painted onto space. Galaxies sit at fixed grid coordinates; the grid itself stretches uniformly by the scale factor $a(t)$ . The comoving distance $D_C$ is simply the number of grid squares a photon crossed on its way to us — a quantity that, by construction, does not change as the universe expands. Every other distance is what you get when you re-express that fixed grid count in the currency of a specific observation: brightness picks up factors of $a$ from redshift and time dilation, angular size picks up the inverse factors from the shrinking emitting patch. That is why one grid count becomes four distances, and the first figure lays out the machinery.

Figure

One integral becomes four distances: D_C = ∫ c dz′/H drives D_M through curvature, then factors of (1+z) split off the luminosity distance D_L = (1+z)D_M and the angular-diameter distance D_A = D_M/(1+z); at z = 1 in flat ΛCDM they read D_C = D_M = 3300, D_L = 6600 and D_A = 1650 Mpc, the ratio 1 : 2 : 4

Now for the payoff that surprises every student. Light reaching us today left its source when the universe was smaller, so a galaxy of fixed physical size once occupied a larger fraction of the (then-smaller) sky. Push far enough out and this wins over the growing line-of-sight distance: the angular-diameter distance stops increasing, turns over near $z\approx1.6$ , and decreases, making the most distant galaxies subtend larger angles than nearer ones. Meanwhile the same photons are redshifted and their arrival rate time-dilated, so the luminosity distance climbs without bound. The second figure plots this turnover for the standard flat $\Lambda$ CDM cosmology — the single most counter-intuitive fact in the distance family, and a direct fingerprint of relativistic geometry.

Figure

Angular-diameter distance for flat ΛCDM with H₀ = 70 and Ω_m = 0.3: D_A rises, peaks at D_A ≈ 1748 Mpc near z ≈ 1.6, then declines — so a galaxy at z = 3 subtends a larger angle than one at z = 1. At z = 1 the three measures read D_A = 1650, D_M = 3300 and D_L = 6600 Mpc

Finally, think of $H(z)$ as a conversion rate between redshift and comoving span: where the universe expanded quickly a slice $\mathrm{d}z$ buys only a short comoving distance, where it expanded slowly the same $\mathrm{d}z$ buys a long one. Accumulating this variable rate from $0$ to $z$ is the entire content of the master integral — which is why measuring distances across many redshifts is equivalent to reconstructing the whole expansion history.

The core equations

Everything begins with the line-of-sight comoving distance. A radial photon obeys the null condition in the FLRW metric; integrating it and changing variable from cosmic time to redshift collapses the geometry and the dynamics into a single integral over the Hubble parameter. The second equation is the Friedmann expansion rate that supplies $H(z')$ , with each term the redshift scaling of one energy component.

D_C(z)=\int_0^z \frac{c\,\mathrm{d}z'}{H(z')},\qquad H(z)=H_0\sqrt{\Omega_m(1+z)^3+\Omega_r(1+z)^4+\Omega_k(1+z)^2+\Omega_\Lambda} \qquad (1)

where: $D_C$ — line-of-sight comoving distance; $c$ — speed of light; $H(z)$ — Hubble parameter at redshift $z$ ; $H_0$ — present-day Hubble constant; $\Omega_m,\Omega_r,\Omega_\Lambda,\Omega_k$ — density parameters for matter, radiation, dark energy, and curvature. Reading it: distance is the running total of $c/H$ ; the Hubble distance $D_H=c/H_0\approx4283$ Mpc (for $H_0=70$ ) sets the overall scale. Spatial curvature reshapes the line-of-sight distance into a transverse one. A comoving separation seen across the sky is bent by the curvature of space through the metric function $S_k$ — a $\sinh$ for open geometries, the identity when flat, a $\sin$ when closed. This is the only place curvature enters the distance family.

D_M=\begin{cases}\dfrac{c}{H_0\sqrt{\Omega_k}}\sinh\!\left(\dfrac{\sqrt{\Omega_k}\,H_0 D_C}{c}\right)& \Omega_k>0\\[2mm] D_C & \Omega_k=0\\[2mm] \dfrac{c}{H_0\sqrt{|\Omega_k|}}\sin\!\left(\dfrac{\sqrt{|\Omega_k|}\,H_0 D_C}{c}\right)& \Omega_k<0\end{cases} \qquad (2)

where: $D_M$ — transverse comoving distance, fixing the comoving transverse size $=D_M\,\Delta\theta$ ; $\Omega_k$ — curvature density parameter ( $>0$ open, $=0$ flat, $<0$ closed); $D_C$ — line-of-sight comoving distance. Reading it: in the observed near-flat universe $\Omega_k\approx0$ , so $D_M\approx D_C$ to sub-percent accuracy, and the $\sin/\sinh$ machinery is a small correction. The angular-diameter distance converts a known physical size into an observed angle. It is $D_M$ divided by $(1+z)$ because the object was physically smaller — the space it occupied had not yet stretched — when the light we now see departed.

D_A=\frac{D_M}{1+z}=\frac{\ell}{\Delta\theta} \qquad (3)

where: $D_A$ — angular-diameter distance; $\ell$ — physical (proper) transverse size of the source; $\Delta\theta$ — observed angular size; the $(1+z)$ converts comoving transverse size to proper size at emission. Reading it: a fixed ruler $\ell$ looks bigger (larger $\Delta\theta$ ) when $D_A$ is smaller — which is exactly what happens past the turnover. The luminosity distance converts a known luminosity into an observed flux, forced to obey the familiar inverse-square law. Two independent factors of $(1+z)$ dim the source — one from each photon's redshifted energy, one from the time-dilated arrival rate — on top of the geometric spreading over a comoving sphere.

D_L=(1+z)\,D_M=\sqrt{\frac{L}{4\pi F}} \qquad (4)

where: $D_L$ — luminosity distance; $L$ — intrinsic bolometric luminosity; $F$ — observed bolometric flux. Reading it: because two $(1+z)$ dimmings act while only one factor of $D_M$ is geometric, $D_L$ grows fastest of all the measures and diverges at high $z$ . Divide the last two and the model-dependence cancels: the ratio of luminosity to angular-diameter distance is a pure kinematic factor. This is Etherington's distance-duality relation, and its power is that it needs no Friedmann model — only null geodesics and photon conservation.

D_L=(1+z)^2\,D_A \qquad (5)

where: $D_L$ — luminosity distance; $D_A$ — angular-diameter distance; $(1+z)^2$ — the exact ratio in any metric theory. Reading it: an observed departure from $(1+z)^2$ would flag absorbing dust, photon–axion mixing, or a breakdown of the metric picture — making duality a clean null test of exotic physics. For practical catalogues the luminosity distance is quoted as a logarithmic distance modulus, the difference between apparent and absolute magnitude that a photometric survey actually reports.

\mu=m-M=5\log_{10}\!\left(\frac{D_L}{10\,\mathrm{pc}}\right) \qquad (6)

where: $\mu$ — distance modulus; $m,M$ — apparent and absolute magnitude; $D_L$ — luminosity distance. Reading it: every 5 magnitudes of $\mu$ is a factor of 10 in $D_L$ ; a source at $z=1$ with $D_L=6600$ Mpc has $\mu\approx44.1$ .

Derivation AThe master integral from the FLRW null geodesic

The goal is to show that the single most important formula in observational cosmology — the redshift integral for the comoving distance — is nothing more than the statement that light travels on a radial null path through an expanding metric, rewritten in the variable an astronomer can actually measure. We start from $\mathrm{d}s^2=0$ in the FLRW line element, integrate the radial coordinate, and then trade cosmic time for redshift using the definitions of $a$ , $H$ , and $z$ , so that geometry and expansion fuse into one integrable expression.

Step 1. For a radially infalling photon set $\mathrm{d}\theta=\mathrm{d}\phi=0$ and impose the null condition $\mathrm{d}s^2=0$ on the FLRW metric, giving $0=-c^2\mathrm{d}t^2+a^2\,\mathrm{d}r^2/(1-kr^2)$ . Why: light defines its own path by travelling on a null geodesic, so this single algebraic relation between $\mathrm{d}t$ and $\mathrm{d}r$ contains all the propagation physics.

Step 2. Solve for the comoving radial coordinate and integrate: $\chi=\int_0^r \mathrm{d}r'/\sqrt{1-kr'^2}=\int_{t_e}^{t_0} c\,\mathrm{d}t/a(t)$ , where the right side used the null relation to swap $\mathrm{d}r$ for $c\,\mathrm{d}t/a$ . Why: the left integral is the pure comoving grid count $\chi$ , while the right integral expresses that same count through the photon's actual journey in time.

Step 3. Change variable with $1+z=a_0/a$ and $H=\dot a/a$ , so $\mathrm{d}z=-(1+z)H\,\mathrm{d}t$ and therefore $c\,\mathrm{d}t/a = c\,\mathrm{d}z/H$ once $a_0=1$ . Why: redshift, not cosmic time, is the observable, so this substitution converts an unmeasurable time integral into one over the directly measured $z$ .

Step 4. Assemble $D_C=\int_0^z c\,\mathrm{d}z'/H(z')$ and read off the limits: at low $z$ , $H\to H_0$ gives $D_C\to cz/H_0$ , the Hubble law; at high $z$ the falling $c/H$ makes $D_C$ grow ever more slowly. Why: the correct low- $z$ limit confirms the construction, and the shrinking integrand is precisely what later lets $D_A$ turn over.

Go deeper · Derivation B · 1 minAngular-diameter distance and why it turns over

Derivation BAngular-diameter distance and why it turns over

The aim here is to derive $D_A=D_M/(1+z)$ directly from the transverse part of the FLRW metric and then to expose the mechanism behind its most surprising feature: that beyond a critical redshift, more distant objects of fixed physical size look bigger, not smaller. We track a proper ruler of length $\ell$ emitting at scale factor $a=1/(1+z)$ , relate its proper size to the angle it subtends, and then examine how the resulting $D_A(z)$ competes a growing integral against a shrinking prefactor.

Step 1. The proper transverse separation at emission is $\mathrm{d}s_\perp = a\,S_k(D_C)\,\mathrm{d}\theta = a\,D_M\,\mathrm{d}\theta$ , since $S_k(D_C)=D_M$ is the comoving transverse distance and $a$ rescales it to proper units. Why: angular size is set by the proper separation at the moment of emission, when the light was launched, not by today's stretched separation.

Step 2. Fix the physical size $\ell = a\,D_M\,\Delta\theta = D_M\,\Delta\theta/(1+z)$ , using $a=1/(1+z)$ evaluated at emission. Why: a standard ruler has a definite proper length $\ell$ , so this pins the observed $\Delta\theta$ to the geometry through the emission-epoch scale factor.

Step 3. Read off the defining ratio $D_A\equiv \ell/\Delta\theta = D_M/(1+z)$ . Why: by definition the angular-diameter distance is physical size over angular size, so the metric has just handed us the exact conversion factor.

Step 4. In a flat universe write $D_A(z)=\dfrac{1}{1+z}\int_0^z \dfrac{c\,\mathrm{d}z'}{H(z')}$ and note the tension: the prefactor $1/(1+z)$ shrinks steadily while the integral grows ever more slowly (its integrand $c/H$ falls). The product rises, reaches a maximum near $z\approx1.6$ for $\Omega_m\approx0.3,\Omega_\Lambda\approx0.7$ , then declines. Why: this competition is the whole explanation of the turnover — distant sources are seen when the universe was small, so a fixed ruler filled more of the sky.

Go deeper · Derivation C · 1 minThe Etherington distance-duality relation

Derivation CThe Etherington distance-duality relation

This derivation shows that the relation $D_L=(1+z)^2 D_A$ is not a coincidence of Friedmann cosmology but a theorem of any metric theory in which photons follow null geodesics and their number is conserved. The strategy is to compute the received flux of a source with careful attention to every factor of $(1+z)$ , define $D_L$ so the inverse-square law holds by fiat, and then substitute the angular-diameter distance to expose the pure kinematic ratio that survives once the model-dependent $D_M$ cancels.

Step 1. Write the flux as $F=\dfrac{L}{4\pi D_M^2}\,\dfrac{1}{(1+z)^2}$ : photons spread over a comoving sphere of area $4\pi D_M^2$ , each photon's energy is redshifted by one factor of $(1+z)$ , and the arrival rate is time-dilated by a second. Why: the two independent $(1+z)$ dimmings — energy and rate — are the physical origin of the entire duality factor, and they arise identically in any expanding metric.

Step 2. Define the luminosity distance by insisting the Euclidean law holds, $D_L^2\equiv L/4\pi F=(1+z)^2 D_M^2$ , so $D_L=(1+z)D_M$ . Why: astronomers want a distance for which flux equals $L/4\pi D_L^2$ , and this definition is what makes the standard-candle method work.

Step 3. Substitute $D_M=(1+z)D_A$ from the previous derivation to obtain $D_L=(1+z)^2 D_A$ . Why: the common factor $D_M$ carries all the cosmological-model dependence, and it cancels in the ratio, leaving a relation that tests photon physics rather than the Friedmann equations.

Limiting cases

The distance family collapses to familiar or analytic forms in special regimes, each a useful sanity check.

Case	Result	Meaning
Low redshift	$D_L\approx D_A\approx D_C\approx \dfrac{cz}{H_0}$	All measures merge into the Hubble law as $z\to0$ ; the leading correction $D_L\approx\dfrac{cz}{H_0}\left[1+\tfrac12(1-q_0)z\right]$ carries the deceleration parameter $q_0$ .
Flat universe ( $\Omega_k=0$ )	$D_M=D_C$	The curvature function is the identity, so transverse and line-of-sight comoving distances coincide; only $(1+z)$ factors separate $D_L$ , $D_A$ , $D_C$ .
Einstein–de Sitter ( $\Omega_m=1$ )	$D_C=\dfrac{2c}{H_0}\left[1-\dfrac{1}{\sqrt{1+z}}\right]$	A rare closed form; gives $D_C(1)=2509$ Mpc for $H_0=70$ , a clean analytic benchmark for a matter-only flat cosmology.
High redshift	$D_L\propto(1+z),\quad D_A\ \text{decreasing}$	$D_L$ diverges while $D_A$ falls past its peak, so bolometric surface brightness dims as $(1+z)^{-4}$ (Tolman dimming).

Common trapTreating "distance" as a single well-defined number

There is no unique distance to a cosmological source, and conflating the measures produces large, silent errors. At $z=1$ the luminosity and angular-diameter distances differ by $(1+z)^2=4$ ; quoting one where the physics demands the other misplaces the object by a factor of four. Two further traps hide in the same territory. First, the comoving distance is not $cz/H_0$ except in the low- $z$ limit — using the naive Hubble law at $z\gtrsim0.3$ overestimates distances badly, because it ignores the falling integrand $c/H(z)$ . Second, dropping the curvature function $S_k$ and writing $D_M=D_C$ silently assumes a flat universe — a good approximation for our cosmos, but an assumption, not an identity. Always name which distance you mean, and never reach for $cz/H_0$ outside the nearby universe.

Worked examples

1 · The four distances to a source at redshift one (fully worked)

In the standard flat $\Lambda$ CDM cosmology ( $H_0=70\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ , $\Omega_m=0.3$ , $\Omega_\Lambda=0.7$ , $\Omega_k=0$ ), compute $D_C$ , $D_M$ , $D_A$ , $D_L$ , and the distance modulus $\mu$ for an object at $z=1$ , and verify distance duality.

Step 1. Fix the scale. The Hubble distance is $D_H=c/H_0=299\,792.458/70=\mathbf{4283\ Mpc}$ . Every distance below is $D_H$ times a dimensionless integral.

Step 2. Do the master integral. With $E(z')=\sqrt{0.3(1+z')^3+0.7}$ , numerical integration gives $\int_0^1 \mathrm{d}z'/E(z')=0.7714$ . Hence $D_C=D_H\times0.7714=4283\times0.7714=\mathbf{3304\ Mpc}$ .

Step 3. Apply curvature. The universe is flat ( $\Omega_k=0$ ), so $S_k$ is the identity and $D_M=D_C=\mathbf{3304\ Mpc}$ .

Step 4. Peel off the $(1+z)$ factors. The angular-diameter distance is $D_A=D_M/(1+z)=3304/2=\mathbf{1652\ Mpc}$ , and the luminosity distance is $D_L=(1+z)D_M=2\times3304=\mathbf{6608\ Mpc}$ .

Step 5. Distance modulus. Convert $D_L=6608$ Mpc $=6.608\times10^9$ pc, so $\mu=5\log_{10}(6.608\times10^9/10)=5\log_{10}(6.608\times10^8)=5\times8.820=\mathbf{44.10}$ . A supernova at $z=1$ therefore appears about 44 magnitudes fainter than at 10 pc.

Step 6. Check duality. $D_L/D_A=6608/1652=4.00=(1+z)^2$ , exactly as Etherington requires. The one source has four honest distances — 3304, 3304, 1652, and 6608 Mpc — locked together by factors of $(1+z)$ .

2 · The turnover in action: a galaxy that looks bigger when farther

A galaxy has a fixed proper diameter $\ell=10$ kpc. Using the same cosmology, find the angle it subtends at $z=0.5$ , $z=1$ , and $z=3$ , and interpret the trend.

Step 1. The angle is $\Delta\theta=\ell/D_A$ , with $D_A$ from equation (3). Numerically integrating for each redshift gives $D_A(0.5)=1259$ Mpc, $D_A(1)=1652$ Mpc, and $D_A(3)=1589$ Mpc. Note that $D_A(3)<D_A(1)$ — we are already past the peak at $z\approx1.6$ .

Step 2. Convert to arcseconds with $1\ \mathrm{rad}=206\,265''$ and $\ell=10$ kpc $=0.01$ Mpc. At $z=0.5$ : $\Delta\theta=(0.01/1259)\times206\,265=\mathbf{1.64''}$ . At $z=1$ : $\Delta\theta=(0.01/1652)\times206\,265=\mathbf{1.25''}$ .

Step 3. Now the far galaxy. At $z=3$ : $\Delta\theta=(0.01/1589)\times206\,265=\mathbf{1.30''}$ . The galaxy at $z=3$ subtends a larger angle than the one at $z=1$ , even though it is far more distant in every other sense.

Step 4. Interpret. From $z=0.5$ to $z=1$ angular size shrinks as expected, but past the turnover it grows again, because a $z=3$ galaxy is seen when the universe was four times smaller and its light left from a patch that filled more of the sky. Angular size is not monotonic in distance — the $z=3$ galaxy appears 1.30″ across versus 1.25″ at $z=1$ , a direct optical signature of cosmic expansion.

Physics of the portrait

The distance-duality relation is the theorem at the heart of this article, and it carries the name of a mathematician who proved it in three pages and then largely left cosmology behind.

Origins · 1 minIvor Etherington and how this was discovered

Ivor Etherington

1908–1994 · relativistic reciprocity and the distance-duality theorem

Ivor Malcolm Haddon Etherington was a British mathematician whose 1933 paper "On the definition of distance in general relativity" established what is now called the reciprocity theorem — the exact, model-independent statement that the luminosity and angular-diameter distances of any source are tied by $D_L=(1+z)^2 D_A$ . Working in the young discipline of relativistic geometry, he showed the result flows from the reciprocity of null geodesics (the symmetry of light propagation between source and observer) together with conservation of photon number, needing no assumption about the Friedmann equations, the matter content, or the cosmological constant. That generality is precisely what makes the relation so valuable today: because it holds in any metric theory with conserved photons, any measured violation would betray something exotic — intergalactic dust, photon–axion oscillation, a varying fine-structure constant, or a failure of the metric description itself. Etherington's contemporaries built the surrounding scaffolding: Edwin Hubble had just (1929) tied recession velocity to distance, Richard Tolman was formulating the surface-brightness test that duality underpins, and Wolfgang Mattig (1958) would later supply closed-form redshift–distance relations for matter-only models. Etherington himself moved on to combinatorial algebra and genetics, but his short cosmological paper has outlived almost everything else from that era, quietly underwriting modern precision tests with supernovae, clusters, and gravitational lenses.

History

1929

— Hubble's velocity–distance relation establishes cosmic expansion

1933

— Etherington proves D_L = (1+z)² D_A, the distance-duality theorem

1958

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

1998

— the Type Ia supernova Hubble diagram reveals accelerating expansion

2017

— GW170817 yields the first standard-siren measurement of H₀

Going further

Texts. Hogg (1999), Distance measures in cosmology, arXiv:astro-ph/9905116 — the definitive reference sheet · Ryden, Introduction to Cosmology, 2nd ed., Ch. 6–7 · Dodelson & Schmidt, Modern Cosmology, 2nd ed., Ch. 2 · Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., Ch. 29 · Mo, van den Bosch & White, Galaxy Formation and Evolution, Ch. 3 · Peebles, Principles of Physical Cosmology, Ch. 13 · Weinberg, Cosmology, §1.4 (low-z expansion and $q_0$ ). The numerical anchors throughout this article were computed for flat $\Lambda$ CDM ( $H_0=70$ , $\Omega_m=0.3$ , $\Omega_\Lambda=0.7$ ): $D_H=4283$ Mpc; at $z=1$ , $D_C=D_M=3304$ Mpc, $D_A=1652$ Mpc, $D_L=6608$ Mpc, $\mu=44.10$ ; $D_A$ peaks at $1748$ Mpc near $z=1.61$ ; Einstein–de Sitter $D_C(1)=2509$ Mpc; sound horizon $r_d\approx147$ Mpc (Planck 2018). Useful self-checks: the duality ratio satisfies $D_L/D_A=(1+z)^2$ exactly, and the low- $z$ expansion coefficient is $\tfrac12(1-q_0)$ with $q_0=\Omega_m/2-\Omega_\Lambda=-0.55$ .

Prerequisites. The FLRW Metric · The Scale Factor and Cosmic Expansion · Cosmological Redshift · The Friedmann Equations · Spatial Curvature of the Universe · Comoving and Proper Coordinates. Leads to. The Hubble Constant · The Hubble–Lemaître Law · The Friedmann Equations.

Ask the tutor: ask me to walk through the comoving-distance integral for any redshift and cosmology, or to show exactly where each factor of $(1+z)$ enters $D_L$ and $D_A$ .

Why are there so many different "distances" — isn't distance just distance?

In flat, static space there is only one distance, and all the measures coincide. Expansion and curvature break that unity. A photon's travel takes time, during which space stretches, so the object is no longer where it was when it emitted, nor where it is now when you receive the light. Each observable — the brightness you catch, the angle you resolve, the redshift you record — samples a different combination of the object's past and present positions, and so defines its own distance. They are all correct answers to genuinely different questions.

Is the comoving distance the same as how far away the galaxy "really" is right now?

Almost. The line-of-sight comoving distance $D_C$ equals the proper distance today (with $a_0=1$ ), so it is the closest thing to "the distance right now" — the length of a ruler laid out across space at the present instant. What it is not is the distance the light actually travelled, nor the distance to the object at the moment of emission (that was $D_C/(1+z)$ , smaller because space was smaller). And it is not directly observable: you cannot measure $D_C$ with a single instrument the way you measure a flux or an angle. You infer it by integrating the expansion history, which is why $D_C$ is best thought of as the theorist's distance from which all the observable ones are built.

How can a more distant galaxy look bigger on the sky?

Because you are seeing it as it was when the universe was smaller. The angle a fixed ruler subtends depends on how far away it was at the moment it emitted the light you now see, not on how far away it is today. A galaxy at $z=3$ launched its light when the scale factor was $1/4$ of today's, so the patch it occupied was physically tiny and filled a large angle. Formally, $D_A=D_M/(1+z)$ : past $z\approx1.6$ the shrinking $1/(1+z)$ factor beats the slowly growing comoving integral, so $D_A$ falls and $\Delta\theta=\ell/D_A$ rises. It is not an illusion or a lensing effect — it is the plain geometry of an expanding spacetime.

What breaks if the distance-duality relation is violated?

Duality follows from just two premises: photons travel on null geodesics in a metric theory, and photon number is conserved. A measured violation points to a breakdown of one of those. The most mundane culprit is intergalactic dust or plasma that absorbs or scatters photons, sapping flux and inflating the inferred $D_L$ without touching $D_A$ . Exotic possibilities include photon–axion oscillation, a time-varying fine-structure constant, or non-metric gravity. This is why observational duality tests — pairing cluster $D_A$ with supernova $D_L$ — double as clean probes of new physics: the geometry is guaranteed, so any anomaly must come from the photons.

Why does the luminosity distance carry two factors of (1+z) but the angular one only one?

They count different things. Angular-diameter distance is about geometry at emission: one factor of $(1+z)$ converts the comoving transverse size to the smaller proper size the object had when it emitted. Luminosity distance is about energy flow to us now, and expansion dims a source two ways. First, every photon is redshifted, arriving with $1/(1+z)$ of its emitted energy. Second, successive photons are stretched apart in arrival time, landing at $1/(1+z)$ of the emission rate. The two dimmings make the flux carry $1/(1+z)^2$ , which $F=L/4\pi D_L^2$ absorbs into $D_L=(1+z)D_M$ . One factor geometric, one energetic, one temporal — one to $D_A$ , two to $D_L$ .

Which distance should I use for a standard candle versus a standard ruler?

Match the distance to the observable. A standard candle — a source of known luminosity like a Type Ia supernova or a gravitational-wave siren — is measured through its flux, so you use the luminosity distance $D_L=\sqrt{L/4\pi F}$ and usually report the distance modulus $\mu$ . A standard ruler — a feature of known physical size like the baryon acoustic oscillation scale — is measured through the angle it subtends, so you use the angular-diameter distance $D_A=\ell/\Delta\theta$ transversely, and measure $H(z)$ directly along the line of sight. Getting this right matters: at $z=1$ the two distances differ by a factor of four.