Angular-Diameter and Luminosity Distances

An angle, a received flux, and a radial coordinate define three different measurement problems. Angular-diameter distance converts an observed solid angle into a physical source-plane area; luminosity distance converts emitted power into received bolometric flux. In an FLRW spacetime both reduce to the same transverse comoving scale with different redshift factors. This page derives those factors for an unlensed background beam, then checks flat, low-redshift, and turnover limits. Instrument calibration and current distance-ladder results remain outside its scope.

Required background. Affine and Variational Geodesics supplies the Geodesics and Jacobi Fields capability used to describe a null beam and its transverse separation vectors.

Helpful background. The Einstein Equation supplies the Einstein Equations behind any chosen $H(z)$; Relativistic Matter and Compact Objects supplies the Relativistic Matter and Compact Objects source models for that expansion; Relativistic Cosmology orients the FLRW and redshift conventions used below; and The Wave Equation and Characteristics supplies the Waves, Green Functions, and Geometric Optics approximation in which rays and infinitesimal beams are meaningful.

Area distance from a null beam

At the observer, two neighboring rays subtend a small screen-space angle $\boldsymbol\theta$. At the source, their physical transverse separation $\boldsymbol\xi$ is linear to first order:

$$\xi_A=\mathcal D_{AB}\theta_B.$$

The $2\times2$ Jacobi map $\mathcal D$ propagates the beam through spacetime curvature. For an infinitesimal observed solid angle $d\Omega_{\rm o}$ and physical source area $dA_{\rm e}$ orthogonal to both the source four-velocity and ray direction, define

$$\boxed{ D_A^2\equiv \frac{dA_{\rm e}}{d\Omega_{\rm o}} =\left|\det\mathcal D\right| }.$$

The absolute value gives the ordinary positive area distance. A signed determinant retains image parity and becomes important at a caustic. The Jacobi-map construction and its observer initial conditions are developed in Bartelmann and Schneider 2001, §3.2.1, printed pp. 52–53, eqs. (3.21)–(3.26) (Open PDF).

In the FLRW background, isotropy makes the Jacobi map proportional to the screen identity. Use $a_0=1$, a dimensionful reference curvature $K$, and

$$ds^2=-dt^2+a^2(t) \left[d\chi^2+S_K^2(\chi)d\Omega^2\right].$$

Here $K$ has dimensions of inverse length squared, with positive, zero, and negative values denoting spherical, flat, and hyperbolic reference slices, and

$$S_K(\chi)= \begin{cases} \sin(\sqrt K\,\chi)/\sqrt K, & K\gt0,\\ \chi, & K=0,\\ \sinh(\sqrt{-K}\,\chi)/\sqrt{-K}, & K\lt0. \end{cases}$$

The scalar-distance formulas below refer to the primary nonconjugate branch, where $S_K(\chi)\geq0$. In a closed model the antipode is a caustic with $D_A=0$; beyond a caustic, $|S_K|$ gives the positive scalar distance while the sign of the Jacobi determinant separately records image parity.

Two rays separated by $d\theta$ at the observer have physical transverse separation $a_{\rm e}S_K(\chi)d\theta$ at emission. Consequently,

$$\boxed{ D_A(z) =a_{\rm e}S_K[\chi(z)] =\frac{S_K[\chi(z)]}{1+z} }, \qquad \chi(z)=\int_0^z\frac{dz'}{H(z')}.$$

It is useful to name the transverse comoving distance $D_M=S_K(\chi)$. Then $D_A=D_M/(1+z)$. This result refers to the source rest frame and an infinitesimal resolved image. Peculiar velocities, lensing perturbations, finite source size, and source-frame orientation require additional transformations rather than a redefinition of $D_A$.

Flux distance and two redshift factors

For an isotropic source with bolometric luminosity $L_{\rm e}$, luminosity distance is defined operationally by

$$F_{\rm o}\equiv\frac{L_{\rm e}}{4\pi D_L^2}.$$

In transparent FLRW propagation, the photons spread over a present transverse area $4\pi D_M^2$. Each photon arrives with energy smaller by $(1+z)^{-1}$, and the observer’s arrival interval is longer than the source emission interval by another factor $1+z$. Photon-number conservation then gives

$$F_{\rm o} =\frac{L_{\rm e}} {4\pi D_M^2(1+z)^2}, \qquad \boxed{D_L=(1+z)D_M}.$$

Combining the two FLRW formulas yields

$$\boxed{D_L=(1+z)^2D_A}.$$

The next page proves the general reciprocity theorem and separates its geometric hypotheses from photon conservation. Here the flux argument is restricted to a resolved background image, isotropic bolometric emission, and transparent geometric-optics propagation. Dust opacity, photon conversion, or a finite observing band changes the flux inference; anisotropic emission requires a directional luminosity. Hogg’s distance dictionary gives the same relations in Hogg 2000, §§5–7, printed pp. 4–6, eqs. (16), (18), and (21) (Open PDF).

Low-redshift and curvature checks

Define $q_0=-\ddot a_0/(a_0H_0^2)$. Near the observer,

$$\chi(z) =\frac{1}{H_0} \left[z-\frac12(1+q_0)z^2+O(z^3)\right],$$

and $S_K(\chi)=\chi-K\chi^3/6+O(\chi^5)$. The explicit spatial-curvature mapping therefore first enters these kinematic series at cubic order. Curvature may also affect $H_0$ and $q_0$ when a dynamical model relates them to its components. To quadratic order,

$$\begin{aligned} H_0D_A &=z-\frac12(3+q_0)z^2+O(z^3),\\ H_0D_L &=z+\frac12(1-q_0)z^2+O(z^3). \end{aligned}$$

Both recover $D\simeq z/H_0$ locally, and their coefficients satisfy the duality relation. These expansions are kinematic: interpreting $q_0$ in terms of matter or dark energy requires a dynamical model.

Angular-diameter distance need not grow forever. Differentiating the exact expression gives

$$\frac{dD_A}{dz} =\frac{1}{(1+z)^2} \left[ \frac{(1+z)S_K'(\chi)}{H(z)} -S_K(\chi) \right].$$

A stationary point therefore obeys

$$\frac{1+z}{H(z)}S_K'(\chi)=S_K(\chi).$$

In the flat case this reduces to $\chi=(1+z)/H$. A turnover is common in big-bang cosmologies because a fixed transverse comoving separation occupied a smaller physical scale at emission, but it is not a theorem for every expansion history. For example, flat de Sitter space has $D_A=z/[H(1+z)]$, which approaches $H^{-1}$ monotonically.

Common pitfalls

Using radial distance where transverse distance is required. The angular formula contains $D_M=S_K(\chi)$, not $\chi$ unless $K=0$.

Losing one redshift factor in flux. Photon energy and arrival rate each contribute one factor. Since distance is inferred from the square root of inverse flux, together they produce $D_L=(1+z)D_M$.

Treating a background distance as a measured catalog entry. Calibration, selection, standardization, peculiar velocities, and lensing affect the likelihood that connects observations to these geometric functions.

Exercises

Find the angular-diameter-distance maximum in a flat, matter-only universe. Use

$$\chi(z)=\frac{2}{H_0} \left[1-(1+z)^{-1/2}\right].$$

Solution

Set $y=\sqrt{1+z}$. Then

$$D_A =\frac{\chi}{1+z} =\frac{2}{H_0}\left(y^{-2}-y^{-3}\right).$$

Because $dy/dz\gt0$, the maximum satisfies

$$\frac{dD_A}{dy} =\frac{2}{H_0}\left(-2y^{-3}+3y^{-4}\right)=0.$$

Thus $y=3/2$, so

$$z_{\rm max}=\frac54, \qquad D_{A,\rm max} =\frac{8}{27H_0}.$$

The derivative changes from positive to negative at this point. The luminosity distance does not turn over there: duality multiplies $D_A$ by $(1+z)^2$, illustrating why the two operational definitions must remain distinct.

References

• Bartelmann, Matthias, and Peter Schneider. “Weak Gravitational Lensing.” Physics Reports 340 (2001): 291–472. DOI; Open PDF.
• Hogg, David W. “Distance Measures in Cosmology.” arXiv:astro-ph/9905116v4, 16 December 2000. Open PDF.