Distance measures and distance duality

Assume a spatially flat Friedmann–Lemaître–Robertson–Walker background with present scale factor one. Redshift from expansion obeys one plus z equal to inverse emission scale factor. Let H be the expansion rate and retain the speed of light c explicitly. Integrating a radial null ray gives the line-of-sight comoving distance chi. In this flat case chi also equals transverse comoving distance D M; spatial curvature would change that last identification.

The angular-diameter distance D A converts an observed small angle in radians into a proper transverse size at emission. A comoving ruler must first be divided by one plus redshift before using that definition. Luminosity distance D L is instead defined by a bolometric inverse-square flux law. Neither is generally equal to lookback time multiplied by c. Peculiar velocities and gravitational shifts also contribute to measured redshifts and require separate treatment when estimating the background relation.

A photon loses an energy factor one plus redshift between emission and observation. The arrival interval for successive wave packets is also longer by that factor. At the observer the radiation is distributed over a sphere with area determined by the transverse comoving distance, since the present scale factor is one. Combining energy loss and arrival-time dilation gives two factors in flux, and hence one factor between luminosity and transverse comoving distances.

Distance duality then relates luminosity and angular-diameter distances with two powers of one plus redshift. More generally reciprocity holds for metric light propagation along null geodesics, with photon conservation needed for its brightness interpretation. It does not require choosing numerical matter or dark-energy densities. Absorption, photon conversion, inconsistent source definitions, or calibration errors can make observational estimates fail the relation without demonstrating a failure of the background metric itself.

A telescope measures flux through finite passbands rather than integrating every photon frequency. Comparing an observed passband to an emitted rest-frame luminosity therefore requires a spectral correction and a convention for extinction. Galaxy sizes also evolve, so an apparent angular size is not automatically a calibrated standard ruler. A useful duality analysis must propagate these source and instrument uncertainties, including correlations when the same observations enter both inferred distance estimates.

Consider uniform attenuation with optical depth tau. Flux is multiplied by its exponential suppression, while a distance inferred by assuming transparent propagation increases by the exponential of half tau. The geometric angular-diameter distance need not change. This example shows why the definition of an inferred distance matters: a flux deficit can be expressed as an anomalous brightness distance, but it does not by itself determine whether the cause is absorption, evolution, selection, or exotic photon physics.