Luminosity Distance

The luminosity distance is the number you get when you take the inverse-square law seriously in a universe that refuses to sit still. In flat, static space a source of known power $L$ betrays its distance through how faint it looks: $F=L/(4\pi d^2)$ , and inverting the flux gives $d$ directly. Cosmologists keep that beloved equation by decree — they define a distance $d_L$ through $F\equiv L/(4\pi d_L^2)$ and pour all the complications of an expanding, curved spacetime into it. What comes out is neither the proper nor the comoving distance, but a hybrid carrying the fingerprints of the whole expansion history. Three separate effects reshape the flux. The photons spread over a comoving sphere whose area, $4\pi S_k(\chi)^2$ , is bent by spatial curvature. Each photon then loses a factor $(1+z)$ of energy as its wavelength stretches with the metric, and the photons arrive $(1+z)$ times less often, because the interval between them is dilated too. The energy tax and the rate tax multiply, dimming the flux by $(1+z)^2$ ; taking the square root to solve for a distance leaves exactly one factor of $(1+z)$ standing, and the master formula $d_L=(1+z)\,S_k(\chi)$ falls out. Because $S_k(\chi)$ is an integral over $H(z)$ , the shape of $d_L(z)$ is a direct readout of $\Omega_m$ , $\Omega_\Lambda$ , and $H_0$ — which is precisely how Type Ia supernovae revealed, in 1998, that the expansion is accelerating.

Key takeaways

The luminosity distance $d_L$ is defined so the ordinary inverse-square law $F=L/(4\pi d_L^2)$ survives in an expanding, curved universe — all the cosmology hides inside $d_L$ .
It is built from the comoving distance and lifted by one factor of redshift: $d_L=(1+z)\,S_k(\chi)$ , where that factor packages two distinct effects — each photon's energy loss and the dilation of the arrival rate.
In a flat universe $d_L=(1+z)\,\chi$ with $\chi=\int_0^z c\,\mathrm{d}z'/H(z')$ , so measuring $d_L(z)$ constrains $\Omega_m$ , $\Omega_\Lambda$ , and $H_0$ .
Type Ia supernovae are standard candles; their $d_L(z)$ Hubble diagram, read through the distance modulus $\mu=5\log_{10}(d_L/10\,\mathrm{pc})$ , revealed cosmic acceleration in 1998.
At low redshift $d_L\approx (cz/H_0)[1+\tfrac12(1-q_0)z]$ : the leading term is Hubble's law and the correction isolates $q_0=\Omega_m/2-\Omega_\Lambda$ ; the three cosmological distances lock together as $d_L=(1+z)^2 d_A=(1+z)\,S_k(\chi)$ , the Etherington relation.
Key idea: preserve the inverse-square law by absorbing all expansion and redshift effects into a single defined distance, $d_L=(1+z)\,S_k(\chi)$ , where one factor of $(1+z)$ dims each photon's energy and the other dilutes the arrival rate.

Where this lives today

The luminosity distance is the beating heart of observational cosmology. Type Ia supernovae remain the workhorse: surveys like the Dark Energy Survey, Pantheon+, and the Vera Rubin Observatory's LSST assemble Hubble diagrams of tens of thousands of standard candles to trace $d_L(z)$ finely enough to test whether dark energy's equation of state $w$ deviates from $-1$ . A second front is the standard siren: a compact-binary merger's gravitational-wave strain scales as $1/d_L$ , delivering the luminosity distance with no calibration ladder — GW170817, with its electromagnetic counterpart supplying a redshift, gave the first such $H_0$ , and future networks aim to arbitrate the Hubble tension between the local ladder ( $H_0\approx73$ ) and the CMB value ( $H_0\approx67$ ). A third front weaponizes the Etherington relation: because $d_L$ depends on photon energy and number while $d_A$ does not, pairing supernovae against baryon-acoustic-oscillation rulers tests distance duality to the percent level, hunting for cosmic dust, photon-axion conversion, or breakdowns of metric gravity.

The physical picture

Picture a lightbulb of known wattage drifting away from you as space itself stretches. In flat static space you would read its distance straight off its brightness — halve the flux and it is $\sqrt2$ farther. In an expanding universe the same bulb looks fainter than raw geometry predicts, so if you stubbornly invert the inverse-square law you infer a distance that is too large. That inflated number is the luminosity distance, and the inflation is not error but physics: the expansion has taxed the light twice on its way to you. The first tax is on energy. Every photon rides the stretching metric, and by the time it arrives its wavelength has grown by $(1+z)$ and its energy $h\nu$ has shrunk by the same factor — the cosmological redshift. The second tax is on timing. Two photons emitted a hair apart arrive an interval $(1+z)$ times longer apart, because the light-travel path keeps lengthening beneath them; the stream of photons is diluted in exactly the way the wave crests are stretched. Energy per photon down by $(1+z)$ , photons per second down by $(1+z)$ : the received power falls by $(1+z)^2$ . The first figure follows a single burst of photons through both taxes and reads off the flux.

Figure

Two taxes of one-plus-z: energy per photon and arrival rate each drop by a factor 1+z, so the received flux is dimmed by (1+z)². For a source at z = 1 that is a factor 4 of dimming beyond geometry alone

Now let the source sit at a whole ladder of redshifts and the luminosity distance becomes a curve, $d_L(z)$ , whose shape is dictated by how fast the universe expanded at each epoch. A matter-filled universe decelerates, keeps distant sources relatively close, and makes them look bright; a $\Lambda$ -dominated universe accelerates, pushes sources farther for the same redshift, and makes them look faint. The second figure is the working diagram of observational cosmology: the same standard candle in three universes traces three separating curves, and a real supernova's brightness picks which one we live in.

Figure

Luminosity distance versus redshift for three universes at H₀ = 67.4 km s⁻¹ Mpc⁻¹. The concordance ΛCDM curve reaches dₗ = 6.80 Gpc at z = 1, while a decelerating matter-only Einstein–de Sitter universe gives only 5.21 Gpc — the extra 0.58 mag of dimming is the 1998 acceleration signal; the empty Milne universe is drawn for reference

The takeaway is that $d_L$ , the angular-diameter distance $d_A$ , and the comoving distance $\chi$ are three lenses on one bundle of null geodesics. Flux sees $d_L$ ; the apparent angular size of a ruler sees $d_A$ ; the underlying map coordinate is $\chi$ . They coincide at $z=0$ and peel apart as fixed powers of $(1+z)$ — pure kinematic bookkeeping whose exact ratios are guaranteed by a conservation law we meet below.

The core equations

Everything begins with a definition rather than a derivation. We insist that the inverse-square law hold and let a single symbol, $d_L$ , soak up whatever the universe does to the light. This is the equation that gives the quantity its name.

F=\frac{L}{4\pi d_L^2} \qquad (1)

where: $F$ — observed bolometric flux, energy per unit area per unit time; $L$ — intrinsic bolometric luminosity of the source; $d_L$ — luminosity distance, defined so this relation is exact. Reading: measure $F$ , know $L$ , and you own $d_L=\sqrt{L/(4\pi F)}$ — no cosmology required to report it, only to interpret it. The interpretation is the master formula: the luminosity distance is the curvature-corrected comoving distance lifted by one power of redshift, the lone survivor of the double dimming after the square root.

d_L=(1+z)\,S_k(\chi) \qquad (2)

where: $z$ — redshift of the source; $\chi$ — radial comoving distance; $S_k(\chi)$ — curvature-dependent transverse comoving distance. Reading: at $z=0$ , $d_L=\chi$ and every distance agrees; as $z$ grows, the $(1+z)$ prefactor inflates $d_L$ above the geometry, encoding the two taxes. The curvature function converts the radial comoving distance into the transverse one that sets the wavefront area — the only place spatial curvature enters, reducing to the identity when flat.

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, closed $(+1)$ , flat $(0)$ , open $(-1)$ ; $R_0=(c/H_0)/\sqrt{|\Omega_k|}$ — present-day radius of curvature. Reading: for $\chi\ll R_0$ all three branches agree, so curvature is a small correction until $\chi$ nears $R_0$ . The comoving distance itself is an integral over the expansion rate — where the whole history of the universe enters $d_L$ .

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

where: $H(z')$ — Hubble parameter at redshift $z'$ ; $c$ — speed of light. Reading: a faster past expansion (larger $H$ ) shrinks $\chi$ for a given $z$ , so more matter means nearer, brighter sources. The Friedmann equation supplies $H(z)$ as a sum over the cosmic energy budget, each component diluting with its own power of $(1+z)$ .

H(z)=H_0\sqrt{\Omega_r(1+z)^4+\Omega_m(1+z)^3+\Omega_k(1+z)^2+\Omega_\Lambda} \qquad (5)

where: $\Omega_r,\Omega_m,\Omega_k,\Omega_\Lambda$ — present density parameters for radiation, matter, curvature, and dark energy, summing to $1$ ; $H_0$ — Hubble constant today. Reading: the exponents $4,3,2,0$ are the equation-of-state signatures $d_L(z)$ ultimately measures. Observers do not quote $d_L$ in megaparsecs; they quote a logarithmic brightness, the distance modulus, which is the difference between apparent and absolute magnitude.

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

where: $\mu$ — distance modulus; $m$ — apparent magnitude; $M$ — absolute magnitude. Reading: five magnitudes are a factor of ten in distance and a hundred in flux, so $\mu$ against $z$ is the Hubble diagram in observers' units. Finally, the luminosity distance sits in a family with the transverse comoving distance $d_M=S_k(\chi)$ and the angular-diameter distance $d_A$ , all locked together by powers of redshift.

d_L=(1+z)\,d_M=(1+z)^2\,d_A \qquad (7)

where: $d_M=S_k(\chi)$ — transverse comoving distance; $d_A$ — angular-diameter distance, physical size over observed angle. Reading: the ratios are exact and model-independent — the Etherington relation derived below.

Derivation ATwo factors of one plus z and why the flux carries the square

The goal is to build the received flux from first principles — geometric spreading, energy loss, and rate dilution — and to show that although the flux is suppressed by $(1+z)^2$ , the luminosity distance carries only a single power of $(1+z)$ above the comoving geometry. We follow one burst of $N$ photons emitted isotropically at time $t_e$ with total energy $E_{\rm em}=N h\nu_e$ , track what happens to it on the way to the detector, and match the result to the defining equation $F\equiv L/(4\pi d_L^2)$ so that the master formula reads off directly.

Step 1. At reception the photons are spread over a sphere whose proper area, evaluated at the present scale factor $a_0=1$ , is $A_{\rm obs}=4\pi S_k(\chi)^2$ . The curvature function $S_k$ appears — not the raw radial distance $\chi$ — because it is the transverse comoving distance that sets how a bundle fans out. Why: only the transverse geometry controls the area of a wavefront, and in a curved space that transverse distance is $\sin$ or $\sinh$ of the radial one, not the radial one itself.

Step 2. Each photon is redshifted, $\nu_{\rm obs}=\nu_e/(1+z)$ , so its energy $h\nu$ drops by one factor of $(1+z)$ ; the burst's arriving energy is $E_{\rm obs}=E_{\rm em}/(1+z)$ . Why: the wavelength is stretched by the metric in lockstep with the scale factor, and energy per photon is inversely proportional to wavelength — this is the pure cosmological redshift, the same effect that reddens the whole spectrum.

Step 3. The arrival interval is dilated, $\Delta t_{\rm obs}=(1+z)\,\Delta t_e$ , so the photons land $(1+z)$ times less frequently. Why: the first and last photon travel along null paths that lengthen as space expands, and the delay between them grows by exactly $(1+z)$ — a rate dilution completely independent of the per-photon energy loss in Step 2.

Step 4. Received power is arriving energy over arrival time, $P_{\rm obs}=E_{\rm obs}/\Delta t_{\rm obs}=[E_{\rm em}/(1+z)]/[(1+z)\Delta t_e]=L/(1+z)^2$ . Why: the two taxes are levied on different quantities — one on the numerator, one on the denominator — so they multiply rather than cancel, and the observed luminosity falls by the product $(1+z)^2$ .

Step 5. Dividing power by area gives the flux, $F=P_{\rm obs}/A_{\rm obs}=L/[4\pi S_k(\chi)^2(1+z)^2]$ . This is the physical brightness the detector actually records for a source of luminosity $L$ . Why: flux is nothing but power spread over the wavefront, and both the geometric dilution and the redshift dimming now sit in one expression.

Step 6. Match to the definition $F\equiv L/(4\pi d_L^2)$ by comparing denominators: $d_L^2=S_k(\chi)^2(1+z)^2$ , hence $d_L=(1+z)\,S_k(\chi)$ . Why: because $d_L$ is defined through a squared distance, solving for it takes a square root of the $(1+z)^2$ suppression, and a single power of $(1+z)$ survives — this is the whole reason $d_L$ carries one redshift factor, not two. *Verified numerically: for $\Omega_m=0.315$ , $\Omega_\Lambda=0.685$ , $H_0=67.4$ , the assembled $d_L=(1+z)S_k(\chi)$ reproduces $6.80$ Gpc at $z=1$ , matching Hogg (astro-ph/9905116) Eqs. (15)–(21) and Ryden Eq. (6.35).*

Go deeper · Derivation B · 1 minThe low-redshift expansion and the deceleration parameter

Derivation BThe low-redshift expansion and the deceleration parameter

The aim is to Taylor-expand $d_L(z)$ about the present and show that the first two terms are model-independent bookkeeping: the leading term is Hubble's law, and the quadratic coefficient isolates the deceleration parameter $q_0$ , with curvature not appearing until third order. We expand the scale factor about today, invert to get redshift as a function of look-back time, integrate the null geodesic for the comoving distance, and apply the master formula, watching how the $(1+z)$ prefactor reshapes the coefficient.

Step 1. Expand the scale factor, $a(t)/a_0=1+H_0(t-t_0)-\tfrac12 q_0 H_0^2(t-t_0)^2+\cdots$ , where $q_0\equiv-\ddot a\,a/\dot a^2$ evaluated today is the deceleration parameter. Why: $q_0$ is by construction the dimensionless curvature of $a(t)$ at the present, so it is the first quantity beyond $H_0$ that any distance measurement can hope to constrain.

Step 2. Invert $1+z=a_0/a$ to get $z=H_0(t_0-t)+(1+\tfrac{q_0}{2})H_0^2(t_0-t)^2+\cdots$ . Why: observations are indexed by redshift, not by look-back time, so every distance must ultimately be written as a series in $z$ , and this inversion is what converts the kinematics into observable variables.

Step 3. Integrate the radial null geodesic, $\chi=c\int_t^{t_0}\mathrm{d}t'/a(t')=(c/H_0)[z-\tfrac12(1+q_0)z^2+\cdots]$ . Why: the comoving distance is the accumulated conformal time, and at this order the curvature corrections in $S_k$ are cubic in $\chi$ , so $S_k(\chi)\approx\chi$ and spatial curvature stays silent.

Step 4. Apply the master formula $d_L=(1+z)\,\chi$ and collect terms: the prefactor contributes $+z\cdot\tfrac{c}{H_0}z=+\tfrac{c}{H_0}z^2$ , which shifts the quadratic coefficient from $-\tfrac12(1+q_0)$ to $-\tfrac12(1+q_0)+1=\tfrac12(1-q_0)$ . Why: this is the crucial sign flip — the redshift prefactor turns the deceleration term inside out, so that an accelerating universe ( $q_0<0$ ) makes $d_L$ larger and distant candles fainter, exactly the observable signature of dark energy.

Step 5. The result is $d_L=\dfrac{c}{H_0}\left[z+\tfrac12(1-q_0)z^2+\mathcal{O}(z^3)\right]$ , with $q_0=\Omega_m/2-\Omega_\Lambda$ for the concordance fluid. Why: the leading term is Hubble's law $d_L\approx cz/H_0$ recovered as a theorem, and the next term is a clean, curvature-free handle on $q_0$ — this is the cosmographic backbone of the supernova program. *Verified numerically: at $z=0.05$ with $q_0=-0.53$ the series gives $230.9$ Mpc against the exact integral $230.7$ Mpc, agreement to $0.1\%$ ; matches Weinberg, Cosmology, §1.4.*

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

Derivation CThe Etherington distance-duality relation

The target is the exact, model-independent identity $d_L=(1+z)^2 d_A$ , which ties the flux distance to the angular-size distance using nothing but the reciprocity of null geodesic bundles and the conservation of photon number. We write down each distance from its own operational definition, form the ratio, and watch the shared geometry cancel, leaving only the redshift factors — a result so general it holds in any metric theory of gravity, not just in Friedmann cosmology.

Step 1. The angular-diameter distance is $d_A=S_k(\chi)/(1+z)$ : a ruler of physical length $\ell$ subtends angle $\theta=\ell/d_A$ , and because the ruler was physically smaller by the scale factor $a=1/(1+z)$ at emission, the transverse comoving distance $S_k(\chi)$ is divided by $(1+z)$ . Why: angular size is fixed by the transverse geometry at the epoch of emission, which is why $d_A$ carries $(1+z)$ in the denominator where $d_L$ carries it in the numerator.

Step 2. The luminosity distance is $d_L=(1+z)\,S_k(\chi)$ from the master formula, with the same transverse comoving distance $S_k(\chi)$ appearing. Why: both distances are built on the identical null-geodesic bundle, so they must share the geometric factor $S_k(\chi)$ — only the redshift dressing differs.

Step 3. Take the ratio: $d_L/d_A=[(1+z)S_k(\chi)]/[S_k(\chi)/(1+z)]=(1+z)^2$ . Why: the common $S_k(\chi)$ cancels exactly, so the ratio knows nothing about curvature, matter content, or the Friedmann equations — it is pure kinematics, testable against data without assuming a cosmological model.

Step 4. The relation needs only three ingredients — a metric theory, photons on null geodesics, and conserved photon number — so any measured violation, $d_L\neq(1+z)^2 d_A$ , would signal cosmic opacity, photon-to-axion mixing, or non-metric gravity rather than a mistake. Why: because the derivation invokes no dynamics, distance duality becomes a null test of exotic physics, which is exactly how it is used today by combining supernova and baryon-acoustic-oscillation data. *Verified: Etherington (1933); confirmed observationally to the percent level by pairing SN Ia ( $d_L$ ) with BAO and cluster ( $d_A$ ) measurements.*

Limiting cases

Case	Result	Meaning
Low redshift	$d_L\approx\dfrac{cz}{H_0}\left[1+\tfrac12(1-q_0)z+\cdots\right]$	Leading term is Hubble's law; the correction isolates $q_0=\Omega_m/2-\Omega_\Lambda$ , curvature-free at this order.
Flat universe	$d_L=(1+z)\,\chi=(1+z)\displaystyle\int_0^z\dfrac{c\,\mathrm{d}z'}{H(z')}$	With $\Omega_k=0$ the curvature function collapses to $S_k(\chi)=\chi$ and drops out.
Empty (Milne)	$d_L=\dfrac{c}{H_0}\,z\!\left(1+\dfrac{z}{2}\right)$	Closed form for $\Omega=0$ ; gives $q_0=0$ and a clean analytic benchmark.
Matter-only (EdS)	$d_L=\dfrac{2c}{H_0}(1+z)\!\left[1-\dfrac{1}{\sqrt{1+z}}\right]$	Decelerating $q_0=\tfrac12$ ; distant candles look brighter than in $\Lambda$ CDM.
Distance duality	$d_L=(1+z)^2 d_A$	Exact in any metric theory whenever photon number is conserved.

Common trapDoubling the redshift dimming by hand

The single most common error is to take the master formula $d_L=(1+z)S_k(\chi)$ , plug it into $F=L/(4\pi d_L^2)$ , and then also multiply the flux by an extra $(1+z)^2$ for good measure — double-counting the dimming. The two factors of $(1+z)$ are already baked into $d_L$ : one energy tax, one rate tax, and taking the square root to define a distance is what left a single power of $(1+z)$ in $d_L$ rather than two. A quieter trap is to write $d_L=(1+z)\chi$ in a curved universe; that holds only when $\Omega_k=0$ , because in general the transverse distance $S_k(\chi)$ — a $\sin$ or $\sinh$ of $\chi$ — is not the radial $\chi$ . Keep the roles straight: $\chi$ is radial, $S_k(\chi)$ sets areas, and $d_L$ is $S_k(\chi)$ lifted by exactly one $(1+z)$ .

Worked examples

1 · A Type Ia supernova at z = 1 (fully worked)

A Type Ia supernova with absolute magnitude $M_B\approx-19.3$ explodes at $z=1$ in the concordance universe ( $\Omega_m=0.315$ , $\Omega_\Lambda=0.685$ , $H_0=67.4\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ ). Find its luminosity distance, distance modulus, apparent magnitude, and angular-diameter distance — then say how much brighter it would look in a decelerating matter-only universe.

Step 1. Compute the comoving distance from equation (4). With the Hubble distance $c/H_0=299792.458/67.4=4448$ Mpc and $E(z')=\sqrt{0.315(1+z')^3+0.685}$ , numerically integrating $\int_0^1\mathrm{d}z'/E(z')=0.765$ gives $\chi=4448\times0.765=\mathbf{3401\ Mpc}$ . The universe is flat, so $S_k(\chi)=\chi$ .

Step 2. Lift by one factor of redshift, $d_L=(1+z)\,\chi=2\times3401=\mathbf{6803\ Mpc}=6.80\ \mathrm{Gpc}$ . This is the blue point marked on the Hubble diagram above.

Step 3. Convert to a distance modulus with equation (6): $\mu=5\log_{10}(6803\times10^6/10)=5\log_{10}(6.803\times10^8)=\mathbf{44.16}$ . The predicted brightness is $m=M+\mu=-19.3+44.16=\mathbf{24.86}$ , a 25th-magnitude speck — the regime that demanded 4-metre and Hubble-class telescopes in the 1990s.

Step 4. Get the angular-diameter distance from equation (7): $d_A=d_L/(1+z)^2=6803/4=\mathbf{1701\ Mpc}$ , a factor of four below $d_L$ and even below $\chi$ .

Step 5. In a matter-only Einstein–de Sitter universe the same candle sits at $d_L^{\rm EdS}=5211$ Mpc, modulus $\mu_{\rm EdS}=43.58$ , so the concordance supernova is $\Delta\mu=44.16-43.58=\mathbf{0.58\ mag}$ fainter than deceleration predicts. That excess dimming — candles farther than a slowing universe allows — is the acceleration signal that won the 2011 Nobel Prize.

2 · A nearby galaxy and the Hubble law (confidence builder)

A galaxy sits at $z=0.05$ . Estimate its luminosity distance two ways — the bare Hubble law and the cosmographic correction — and check them against the exact integral. Use $H_0=67.4$ and $q_0=-0.53$ .

Step 1. The leading Hubble term is $d_L\approx cz/H_0=(299792.458\times0.05)/67.4=\mathbf{222.4\ Mpc}$ . At this redshift the recession is barely relativistic and the inverse-square law is nearly Newtonian.

Step 2. Add the cosmographic correction from the low- $z$ expansion: $d_L=\dfrac{cz}{H_0}\left[1+\tfrac12(1-q_0)z\right]=222.4\times[1+\tfrac12(1.53)(0.05)]=222.4\times1.038=\mathbf{230.9\ Mpc}$ .

Step 3. The exact numerical integral gives $230.7$ Mpc, so the two-term series is accurate to $\mathbf{0.1\%}$ at $z=0.05$ . The correction is a $3.8\%$ upward nudge, and its sign — pushing $d_L$ up because $q_0<0$ — is the same acceleration effect seen at $z=1$ , just far smaller. Even here $q_0$ leaves a measurable, few-percent fingerprint on the distance.

Physics of the portrait

The luminosity distance is where three centuries of measuring the sky converge: a brightness anyone can record, a geometry Friedmann and Lemaître made rigorous, and a conservation law that turns the whole construction into a test of fundamental physics.

Origins · 1 minluminosity distance and how this was discovered

luminosity distance

the standard-candle distance dₗ = (1+z) Sₖ(χ) · cosmological distance measures

The idea that brightness measures distance is as old as astronomy, but making it cosmological took a chain of people and a century. Henrietta Swan Leavitt supplied the first rung in 1912, finding that a Cepheid variable's pulsation period tracks its luminosity — the first standard candle, and the tool Edwin Hubble used in 1929 to read a linear distance–velocity law off the galaxies. The theoretical scaffold came from Georges Lemaître and Alexander Friedmann, whose solutions of Einstein's equations turned "distance" into a family of operational quantities, each weighted by a different power of redshift. In 1933 Richard Etherington proved the reciprocity theorem that binds them, $d_L=(1+z)^2 d_A$ , a result that needs no dynamics and so survives even where the cosmological model is unknown. The luminosity distance came into its own in 1998, when the High-z Supernova Search Team and the Supernova Cosmology Project pushed Type Ia candles out to $z\sim1$ and found them fainter than any decelerating universe allowed — the discovery of cosmic acceleration, recognized by the 2011 Nobel Prize. The modern quantity, $d_L(z)=(1+z)\int_0^z c\,\mathrm{d}z'/H(z')$ in a flat universe, completes what Leavitt began: a brightness is now a computed function of the entire expansion history of the cosmos.

History

1912

— Leavitt publishes the Cepheid period–luminosity relation, the first standard candle

1929

— Hubble reads a linear distance–velocity law off Cepheid distances

1933

— Etherington proves the distance-duality relation dₗ = (1+z)² d_A

1998

— the High-z and Supernova Cosmology Project teams find distant SN Ia too faint, signalling acceleration

2017

— GW170817 delivers the first standard-siren luminosity distance from a gravitational wave

Going further

Texts. Ryden, Introduction to Cosmology, 2nd ed., Ch. 6 (Measuring Cosmological Distances) · 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, §3.1–3.2 · Hogg, Distance Measures in Cosmology, arXiv:astro-ph/9905116.

Provenance. The master formula $d_L=(1+z)S_k(\chi)$ and the two- $(1+z)$ -factor accounting follow Ryden Ch. 6 and Hogg Eqs. (15)–(21); the low- $z$ coefficient $\tfrac12(1-q_0)$ is checkable against Weinberg, Cosmology, §1.4, and was verified numerically here to $0.1\%$ at $z=0.05$ . The concordance numbers ( $\Omega_m=0.315$ , $\Omega_\Lambda=0.685$ , $H_0=67.4$ ) are Planck 2018 values, giving $d_L=6.80$ Gpc, $\mu=44.16$ , and $d_A=1701$ Mpc at $z=1$ by direct integration; the EdS comparison ( $5.21$ Gpc, $0.58$ mag brighter) uses the closed form in the limiting-cases table. The distance-duality relation is Etherington (1933); the SN Ia absolute magnitude $M_B\approx-19.3$ is a standard calibrated value, and $q_0=\Omega_m/2-\Omega_\Lambda=-0.53$ follows from the concordance parameters.

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

Ask the tutor: ask me how the same supernova brightness pins down $\Omega_\Lambda$ , or why the angular-diameter distance turns over while the luminosity distance keeps climbing.

Why is the luminosity distance different from the "real" distance to a galaxy?

Because there is no single "real" distance in an expanding universe — there are several, each answering a different operational question. The luminosity distance answers "how faint does a known candle look?"; the angular-diameter distance answers "how big does a known ruler appear?"; the comoving distance answers "what is the fixed coordinate separation on the expanding grid?" They coincide only at $z=0$ . The luminosity distance is inflated above the comoving distance by $(1+z)$ because expansion taxes the light's energy and arrival rate, making sources fainter and so apparently farther. None is more real than the others; you simply pick the one matched to your measurement.

Where do the two factors of (1+z) actually come from?

From two physically independent effects, and it is worth keeping them separate. The first is energy: every photon's wavelength is stretched by the metric, so its energy $h\nu$ drops by $(1+z)$ — the ordinary cosmological redshift you would measure from a shifted spectral line. The second is rate: photons emitted a short interval apart arrive an interval $(1+z)$ times longer apart, because the light-travel path keeps lengthening beneath them, so the photon flux is diluted. Energy per photon down by $(1+z)$ , photons per second down by $(1+z)$ : the received power falls by the product $(1+z)^2$ . Taking the square root to define a distance leaves one factor in $d_L$ .

If the flux is dimmed by (1+z)², why does dₗ only carry one factor of (1+z)?

Because $d_L$ is defined through a squared distance. The definition is $F=L/(4\pi d_L^2)$ , so $d_L\propto F^{-1/2}$ — the luminosity distance responds to the square root of the flux. The physical flux is suppressed by $(1+z)^2$ relative to the geometric prediction, and the square root of $(1+z)^2$ is $(1+z)$ . So the double dimming becomes a single enhancement of $d_L$ over the transverse comoving distance $S_k(\chi)$ . This is not a coincidence to memorize but a direct consequence of distances being defined through inverse-square laws; the same square-root logic converts the $(1+z)^4$ surface-brightness dimming into the $(1+z)^2$ of the duality relation.

How can the angular-diameter distance decrease with redshift?

It genuinely does, and it is one of the strangest predictions of the geometry. The angular-diameter distance is $d_A=S_k(\chi)/(1+z)$ : the numerator, the transverse comoving distance, keeps growing with redshift, but the $(1+z)$ in the denominator grows faster once you are far enough out. In the concordance model $d_A$ peaks near $z\approx1.6$ and then falls, so a galaxy of fixed physical size subtends a larger angle the farther away it is beyond that turnover — you are seeing it as it was when the whole universe, and hence that transverse patch, was much smaller. The luminosity distance never turns over, because its $(1+z)$ sits in the numerator.

What is a "standard siren" and why is it exciting?

A standard siren is a gravitational-wave source — typically a merging neutron-star or black-hole binary — whose waveform encodes its luminosity distance directly, because the strain amplitude scales as $1/d_L$ and the waveform's shape fixes the source's intrinsic size. Unlike supernovae, sirens need no calibration ladder: general relativity supplies the absolute scale. If an electromagnetic counterpart pins the redshift, as with GW170817, you get a clean point on the Hubble diagram and an independent $H_0$ . Because sirens and supernovae have completely different systematics, they are a promising route to arbitrating the Hubble tension between local and CMB-based values of $H_0$ .

Does the distance-duality relation ever fail, and what would that mean?

Within standard physics it holds exactly, because it assumes only a metric theory of gravity, photons travelling on null geodesics, and conservation of photon number — no dynamics, no specific cosmology. A measured violation, $d_L\neq(1+z)^2 d_A$ , would therefore be a discovery, not a calibration error. Excess $d_L$ would point to photons being lost en route — cosmic dust dimming the flux, or photons oscillating into axion-like particles in intergalactic magnetic fields. A breakdown could also signal that gravity is non-metric or that fundamental constants drift. Current tests, pairing supernova $d_L$ with baryon-acoustic-oscillation and cluster $d_A$ , confirm duality to the percent level, tightly bounding all of these exotic possibilities.