Comoving, Luminosity, and Angular-Diameter Distances

Derivation

Comoving, Luminosity, and Angular-Diameter Distances

D-341 Home PU-308 Threads light · symmetry Depends on Cosmological Redshift from the Scale Factor, The Friedmann Equations from Einstein's Equations

Statement

In a homogeneous, isotropic (FRW) universe, the line-of-sight comoving distance to redshift \(z\) is fixed entirely by the expansion history, \(D_C(z)=\int_0^z \frac{c\,dz'}{H(z')}\). The transverse comoving distance \(D_M=S_k(D_C)\) then determines the two operational distances through pure factors of \((1+z)\): the angular-diameter distance \(D_A=D_M/(1+z)\) and the luminosity distance \(D_L=(1+z)\,D_M\). Consequently \(D_L=(1+z)^2 D_A\) (the Etherington reciprocity relation).

Why it matters

There is no single "distance" in an expanding spacetime. A standard ruler and a standard candle placed at the same redshift return numerically different distances, and the ratio between them is not an experimental accident but a rigorous consequence of the FRW metric and photon conservation. Every measurement of dark energy from supernovae (\(D_L\)), baryon acoustic oscillations (\(D_M\)), and cluster or CMB angular scales (\(D_A\)) is anchored to these relations.

The reciprocity \(D_L=(1+z)^2 D_A\) is also a powerful null test: because it follows from metric geometry alone, an observed violation would signal photon non-conservation (dust, axion mixing) or a breakdown of the metric description, independent of the cosmological model.

Assumptions

Spatial homogeneity and isotropy (FRW metric with a single scale factor \(a(t)\)).Without a Robertson–Walker form there is no global scale factor, so "distance" becomes direction- and observer-dependent and the integral \(\int dz/H\) loses meaning.

Photons travel on radial null geodesics from source to observer.If light did not follow \(ds^2=0\) geodesics (e.g. strong local inhomogeneities, refractive media) the mapping between coordinate separation and redshift breaks and \(D_C\) is no longer the geodesic length.

Photon number is conserved along the beam (no true emission or absorption, metric theory of gravity).This is the physical content of the Etherington theorem; drop it and \(D_L=(1+z)^2 D_A\) fails even though each distance is still separately defined.

Constant spatial curvature \(k\) and a single connected FRW patch.If curvature varies or the topology is nontrivial, the transverse map \(S_k\) is not the simple \(\sin/\sinh\) and \(D_M\ne S_k(D_C)\).

Geometrical-optics limit (wavelength \(\ll\) curvature and expansion scales).Diffraction and wave-optics corrections would modify how a beam's cross-section and specific intensity scale, altering the flux–distance link.

Derivation

\[ ds^2=-c^2\,dt^2+a(t)^2\!\left[\frac{dr^2}{1-kr^2}+r^2\big(d\theta^2+\sin^2\!\theta\,d\phi^2\big)\right] \]

Robertson–Walker line element; sole geometry consistent with homogeneity and isotropy. Set \(a(t_0)=1\) today. A

\[ d\chi\equiv\frac{dr}{\sqrt{1-kr^2}},\qquad ds^2=-c^2\,dt^2+a^2\,d\chi^2 \]

Define the comoving radial coordinate \(\chi\), absorbing curvature into the radial measure. For a radial ray \(d\theta=d\phi=0\). A

\[ 0=-c^2\,dt^2+a^2\,d\chi^2\;\Longrightarrow\; c\,dt=a\,d\chi \]

Light moves on a null geodesic, \(ds^2=0\); take the outgoing branch \(d\chi>0\). A

\[ D_C\equiv\chi=\int_{t_e}^{t_0}\frac{c\,dt}{a(t)} \]

Integrate along the path; \(D_C\) is the line-of-sight comoving distance (coordinate separation between source and observer). A

\[ 1+z=\frac{a_0}{a}=\frac1a,\qquad dz=-\frac{da}{a^2},\qquad \dot a=aH\Rightarrow \frac{dt}{a}=\frac{da}{a^2H}=-\frac{dz}{H} \]

Use the redshift–scale-factor relation (prior result) and \(H\equiv\dot a/a\) to trade \(t\) for \(z\); the sign flips because increasing \(t\) is decreasing \(z\). B

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

Substitute step 5 into step 4 and insert the Friedmann expansion history \(H(z)\) (prior result). Limits: observer \(z=0\), source \(z\). B

\[ D_M=S_k(D_C)=\begin{cases}\dfrac{c}{H_0\sqrt{\Omega_k}}\sinh\!\Big(\sqrt{\Omega_k}\,\dfrac{H_0 D_C}{c}\Big), & \Omega_k>0\\[4pt] D_C, & \Omega_k=0\\[4pt] \dfrac{c}{H_0\sqrt{|\Omega_k|}}\sin\!\Big(\sqrt{|\Omega_k|}\,\dfrac{H_0 D_C}{c}\Big), & \Omega_k<0\end{cases} \]

Transverse comoving distance: solve \(d\chi=dr/\sqrt{1-kr^2}\) for \(r=S_k(\chi)\); \(D_M=r\) is the comoving radius of the 2-sphere on which the source sits. B

\[ \ell = a(t_e)\,r\,\delta\theta = \frac{D_M\,\delta\theta}{1+z}\;\Longrightarrow\; D_A\equiv\frac{\ell}{\delta\theta}=\frac{D_M}{1+z} \]

A source of proper transverse size \(\ell\) at emission subtends angle \(\delta\theta\). Its proper size is the physical transverse separation \(a(t_e)\,r\,\delta\theta\) on the sphere of comoving radius \(r=D_M\). Definition \(D_A\equiv\ell/\delta\theta\). C

\[ F=\frac{L}{4\pi D_M^2}\cdot\frac{1}{(1+z)^2},\qquad D_L\equiv\sqrt{\frac{L}{4\pi F}}=(1+z)\,D_M \]

Emitted photons spread over a present-day sphere of proper area \(4\pi D_M^2\). Two \((1+z)\) factors dim the flux: each photon's energy redshifts by \(1/(1+z)\), and time dilation stretches the arrival rate by \(1/(1+z)\). Define \(D_L\) so Euclidean inverse-square holds. C

\[ \frac{D_L}{D_A}=\frac{(1+z)D_M}{D_M/(1+z)}=(1+z)^2\;\Longrightarrow\; D_L=(1+z)^2\,D_A \]

Divide step 9 by step 8; \(D_M\) cancels, leaving the distance-duality (Etherington) relation independent of geometry and dynamics. B

Result

\[ D_C=\int_0^z\frac{c\,dz'}{H(z')},\qquad D_A=\frac{D_M}{1+z},\qquad D_L=(1+z)D_M,\qquad D_L=(1+z)^2 D_A \]

Reading. One integral over the expansion history sets the comoving distance; curvature bends it into the transverse comoving distance \(D_M\); and the two observable distances then follow by dividing or multiplying by \((1+z)\). The angular-diameter distance is smaller (nearby objects look big), the luminosity distance larger (distant candles look faint), and their ratio is locked to \((1+z)^2\) by geometry alone.

Units check. \(c/H\) has units \(\mathrm{(m\,s^{-1})/(s^{-1})=m}\), and \(dz\) is dimensionless, so \(D_C\) is a length; \(S_k\) preserves length; multiplying or dividing by the dimensionless \((1+z)\) keeps \(D_A,D_L\) as lengths. The duality ratio \((1+z)^2\) is dimensionless, consistent with \([D_L]=[D_A]\).

Limiting cases

Low redshift (\(z\ll1\)): \(D_C\to (c/H_0)z\) and \(D_A\approx D_L\approx D_C\); all distance measures collapse to the Hubble law \(cz=H_0 D\).
Flat universe (\(\Omega_k=0\)): \(D_M=D_C\) exactly, so \(D_A=D_C/(1+z)\), \(D_L=(1+z)D_C\).
Empty / Milne (\(\Omega_m=\Omega_\Lambda=0\), \(\Omega_k=1\)): \(H=H_0(1+z)\) gives \(D_C=(c/H_0)\ln(1+z)\), a closed-form check.
High-\(z\) turnover of \(D_A\): because \(D_A=D_M/(1+z)\) and \(D_M\) grows sub-linearly, \(D_A\) reaches a maximum (near \(z\sim1.5\)–\(2\) in \(\Lambda\)CDM) then decreases — very distant objects subtend larger angles.
de Sitter (\(\Omega_\Lambda=1\)): \(H=H_0\) constant, so \(D_C=(c/H_0)z\) even at large \(z\).

Breaks when

The medium is not transparent (intervening dust, or photon–axion/graviton mixing): photon number is not conserved, the flux picks up an extra factor, and \(D_L=(1+z)^2 D_A\) is violated even though each distance remains defined.
Strong inhomogeneity along the line of sight (voids, lensing by structure): the real beam is sheared and magnified relative to the smooth FRW beam, so measured \(D_A,D_L\) scatter about the FRW prediction (the "Dyer–Roeder" / lensing regime).
Peculiar velocities dominate the redshift (very low \(z\), or near massive structures): observed \(z\) is not the cosmological redshift, so \(\int dz/H\) mis-estimates distance.
Non-metric gravity or varying fundamental constants: the Etherington theorem assumes a metric theory with null geodesics; abandon that and the duality relation has no reason to hold.

Failure modes

Confusing \(D_C\) with \(D_M\): setting \(D_A=D_C/(1+z)\) in a curved universe. This is only valid when \(\Omega_k=0\); otherwise the \(\sin/\sinh\) map is required.
Dropping a \((1+z)\) in the flux: writing \(F=L/4\pi D_M^2(1+z)\) with one factor. There are two independent \((1+z)\)'s — energy per photon and photon arrival rate.
Integrating \(dt\) instead of \(dt/a\): forgetting that the comoving distance weights each interval by \(1/a\); this loses the entire expansion effect.
Using \(a_0\ne1\) inconsistently: mixing conventions so that \(1+z=a_0/a\) and the \(S_k\) prefactors disagree, corrupting the curvature term.
Assuming \(D_A\) grows monotonically: extrapolating "farther means smaller angle" past the \(D_A\) turnover, contradicting the actual high-\(z\) behaviour.
Sign error in \(dz=-da/a^2\): producing a negative distance or integrating from \(z\) to \(0\) with the wrong orientation.

Discussion

The deep point is that a single geometric object, the comoving distance \(D_C\), underlies every distance an observer can measure; the different "distances" differ only in how the observable is defined (an angle, a flux, a coordinate) and therefore in how many \((1+z)\) factors the definition drags along. The expansion enters exactly once, through \(H(z)\) inside the integral; everything downstream is kinematics of light in the smooth metric.

Curvature acts purely transversely. Along the line of sight, distance is the additive geodesic length \(D_C\); across the sky, the focusing or defocusing of geodesics by spatial curvature converts \(D_C\) into \(D_M\) via \(S_k\). This is why \(D_C\) appears in radial BAO measurements while \(D_M\) governs the transverse BAO scale, and why combining the two constrains \(\Omega_k\) directly.

The Etherington reciprocity theorem is the sharpest statement here: for any spacetime in which photons follow null geodesics and photon number is conserved, the ratio of solid angles subtended at source and observer forces \(D_L=(1+z)^2 D_A\), regardless of the Einstein equations or the matter content. It is a theorem about the geometry of geodesic bundles (via the reciprocity of the optical-scalar / Sachs equations), not about \(\Lambda\)CDM. That universality makes the duality relation a clean, model-independent probe: measure \(D_A\) (clusters, BAO) and \(D_L\) (supernovae) at the same \(z\) and test the ratio.

Common misconceptions. "Redshift is a Doppler shift of receding galaxies" — in FRW it is the integrated stretching of wavelength by expansion, \(1+z=1/a\), not a velocity in flat space. "The luminosity distance is how far the light travelled" — none of these distances equals the light-travel (lookback) distance \(c\int dt\); they are operational definitions built to make a specific Euclidean formula (inverse-square, small-angle) hold in a curved, expanding geometry.

Worked examples

Low-redshift standard candle: a supernova at \(z=0.10\) in flat \(\Lambda\)CDM, \(H_0=70~\mathrm{km\,s^{-1}\,Mpc^{-1}}\). Find \(D_C,\ D_A,\ D_L\) and the distance modulus.

Symbols first, then numbers. A

\[ D_H\equiv\frac{c}{H_0}=\frac{2.998\times10^{5}~\mathrm{km/s}}{70~\mathrm{km\,s^{-1}\,Mpc^{-1}}}=4283~\mathrm{Mpc} \]

Hubble distance sets the overall scale. A

\[ D_C=\int_0^{0.1}\frac{c\,dz'}{H(z')}\approx D_H\Big[z-\tfrac12(1+q_0)z^2\Big],\quad q_0=\tfrac12\Omega_m-\Omega_\Lambda=-0.55 \]

Leading Taylor expansion of the integral; \(q_0\) is the deceleration parameter for \((\Omega_m,\Omega_\Lambda)=(0.3,0.7)\). B

\[ D_C\approx4283\big[0.10-\tfrac12(0.45)(0.01)\big]=4283(0.09775)=419~\mathrm{Mpc} \]

Insert numbers; the second-order term trims the naive \(D_H z=428\) Mpc by \(\sim2\%\). Flat, so \(D_M=D_C\). B

\[ D_A=\frac{419}{1.1}=381~\mathrm{Mpc},\qquad D_L=1.1\times419=461~\mathrm{Mpc} \]

Apply the \((1+z)\) factors. A

\[ D_C\approx419~\mathrm{Mpc},\quad D_A\approx381~\mathrm{Mpc},\quad D_L\approx461~\mathrm{Mpc},\quad \mu=5\log_{10}\!\frac{D_L}{10\,\mathrm{pc}}=38.3 \]

Reading. At \(z=0.1\) all three agree to within \(\sim10\%\); the distinctions are small but already measurable by precision supernova cosmology. Units check. \(\mu=5\log_{10}(461\times10^{6}\,\mathrm{pc}/10\,\mathrm{pc})=5\log_{10}(4.61\times10^{7})=38.3\) mag, dimensionless as required.

High-redshift standard ruler and candle: \(z=1.0\), flat \(\Lambda\)CDM \((\Omega_m,\Omega_\Lambda)=(0.3,0.7)\), \(H_0=70\). Find \(D_C, D_A, D_L,\mu\), and verify duality.

The linear approximation fails here; integrate. B

\[ D_C=D_H\!\int_0^1\frac{dz'}{\sqrt{0.3(1+z')^3+0.7}},\qquad E(z')\equiv\sqrt{0.3(1+z')^3+0.7} \]

Insert \(H(z)=H_0 E(z)\). B

\[ \begin{array}{c|ccccc} z' & 0 & 0.25 & 0.5 & 0.75 & 1\\\hline 1/E & 1.000 & 0.882 & 0.764 & 0.658 & 0.568\end{array} \]

Tabulate the integrand at five nodes for Simpson's rule, step \(h=0.25\). C

\[ \int_0^1\frac{dz'}{E}\approx\frac{h}{3}\big[f_0+4(f_1+f_3)+2f_2+f_4\big]=\frac{0.25}{3}(9.256)=0.771 \]

Simpson's rule. C

\[ D_C=4283\times0.771=3303~\mathrm{Mpc}=D_M\ \ (\Omega_k=0) \]

Multiply by the Hubble distance; flat, so \(D_M=D_C\). B

\[ D_A=\frac{3303}{2}=1652~\mathrm{Mpc},\qquad D_L=2\times3303=6606~\mathrm{Mpc} \]

Apply \((1+z)=2\). A

\[ D_C=D_M\approx3.30~\mathrm{Gpc},\quad D_A\approx1.65~\mathrm{Gpc},\quad D_L\approx6.61~\mathrm{Gpc},\quad \mu\approx44.1 \]

Reading. At \(z=1\) the luminosity and angular-diameter distances differ by a factor \((1+z)^2=4\); a galaxy is only \(1.65\) Gpc "big on the sky" yet \(6.6\) Gpc "faint as a candle." Duality check: \(D_L/D_A=6606/1652=4.00=(1+z)^2.\) Units check. \(\mu=5\log_{10}(6.606\times10^{9}\,\mathrm{pc}/10\,\mathrm{pc})=5\times8.82=44.1\) mag, matching observed SN Ia at \(z=1\).

Problems

Show that in the Milne (empty) universe \(H(z)=H_0(1+z)\), so that \(D_C=(c/H_0)\ln(1+z)\). Evaluate \(D_C,D_A,D_L\) at \(z=1\) for \(H_0=70\).
Solution
With \(\Omega_m=\Omega_\Lambda=0,\ \Omega_k=1\): \(H=H_0\sqrt{(1+z)^2}=H_0(1+z)\). Then \(D_C=\int_0^z c\,dz'/[H_0(1+z')]=(c/H_0)\ln(1+z)\). At \(z=1\): \(D_C=4283\ln2=4283(0.693)=2969\) Mpc. Milne is spatially open, but for a quick estimate \(D_M\approx D_C\) at this \(z\) (strictly \(D_M=(c/H_0)\sinh(\ln(1+z))\); with \(\sinh(\ln2)=0.75\), \(D_M=4283\times0.75=3212\) Mpc). Using \(D_M=3212\): \(D_A=3212/2=1606\) Mpc, \(D_L=2\times3212=6424\) Mpc.
A radio galaxy of proper physical size \(\ell=30~\mathrm{kpc}\) at \(z=1\) is observed with the \(D_A=1652\) Mpc of worked example 2. What angle does it subtend?
Solution
\(\delta\theta=\ell/D_A=(30\times10^{-3}~\mathrm{Mpc})/(1652~\mathrm{Mpc})=1.82\times10^{-5}\) rad \(=1.82\times10^{-5}\times206265''=3.7''\). Note we use \(D_A\), not \(D_L\): angular sizes always take the angular-diameter distance.
Two supernovae, identical intrinsic luminosity, sit at \(z=0.5\) and \(z=1.0\) with luminosity distances \(D_L=2900\) Mpc and \(6606\) Mpc. What is the ratio of their observed fluxes?
Solution
\(F\propto1/D_L^2\), so \(F(0.5)/F(1.0)=(6606/2900)^2=(2.278)^2=5.19\). The nearer supernova appears about \(5.2\times\) brighter. Equivalently \(\Delta\mu=5\log_{10}(6606/2900)=5(0.358)=1.79\) mag.
Verify the distance-duality relation numerically for the \(z=0.5\) point of the previous problem, given \(D_A=1289\) Mpc there.
Solution
\((1+z)^2 D_A=(1.5)^2\times1289=2.25\times1289=2900\) Mpc \(=D_L\). The relation holds exactly, as it must for photons on null geodesics with conserved number.
In a flat universe, expand \(D_L(z)\) to second order in \(z\) and identify the coefficient that measures cosmic acceleration. Use \(D_C=D_H[z-\tfrac12(1+q_0)z^2+\dots]\).
Solution
\(D_L=(1+z)D_C=D_H(1+z)[z-\tfrac12(1+q_0)z^2+\dots]=D_H[z+z^2-\tfrac12(1+q_0)z^2+\dots]=D_H\big[z+\tfrac12(1-q_0)z^2+\dots\big]\). The \(z^2\) coefficient is \(\tfrac12(1-q_0)\); a measured value above \(\tfrac12\) requires \(q_0<0\), i.e. accelerated expansion. For \((\Omega_m,\Omega_\Lambda)=(0.3,0.7)\), \(q_0=\tfrac12\Omega_m-\Omega_\Lambda=-0.55\), giving coefficient \(\tfrac12(1.55)=0.775\) — the excess brightness-vs-redshift curvature that supernova surveys detected as dark energy.