Luminosity and angular-diameter distances

Flux and angle define different distances

Suppose a source has known emitted bolometric luminosity L_em, meaning power summed over the relevant full spectrum. Define luminosity distance D_L by the received bolometric flux F = L_em/(4πD_L²), with the stated isotropic-luminosity convention.

Angular diameter distance instead relates a small angular size θ, in radians, to the corresponding transverse physical size at emission ℓ_em: D_A = ℓ_em/θ. The ruler is the proper transverse size at that emission epoch, not an unconverted comoving length.

These definitions attach distances to different observational relationships. They are not alternative units for one Euclidean separation. Inferring a numerical distance also requires a calibrated luminosity or physical-size model and the relevant measurement corrections.

Band-limited flux is not automatically bolometric flux. Spectral shape, redshift and instrument bandpass affect its interpretation. The ideal examples below use the bolometric definition and a small transverse angular extent explicitly.

Expansion connects the definitions under stated conditions

In the FLRW comparison, write transverse comoving distance as D_M. Angular diameter distance is D_A = D_M/(1 + z), while luminosity distance is D_L = (1 + z)D_M. Therefore D_L = (1 + z)²D_A.

For luminosity and flux, expansion reduces photon energies and spreads the arrival of successive photons in observer time. The resulting redshift factors belong in the luminosity-distance relation; treating received energy and arrival rate as unchanged would give the wrong flux account.

The distance-duality relation has physical conditions: light follows the relevant null geodesic bundles in a metric description and photon number is conserved along the propagation account. Unmodelled absorption can bias a flux-derived distance. A discrepancy is not automatically evidence for exotic physics.

Gravitational lensing does not by itself invalidate the infinitesimal-bundle reciprocity relation. Lensing and sample selection can still affect inferred source fluxes or averages and need their own treatment. The supplied toy contains no attenuation or unmodelled lensing.

Use the same expansion history for every curve

Choose the same kind of ideal background used in the preceding lesson: spatially flat, expanding, pressureless matter only, governed by general relativity, with no radiation or cosmological constant. Normalize a_ref = 1 and call its positive reference rate H_ref.

Define the length unit D_unit = c/H_ref without assigning any numerical or measured H_ref. In this ideal model, D_C/D_unit = D_M/D_unit = 2[1 − (1 + z)⁻¹ᐟ²]. Equality of radial and transverse comoving distances here uses spatial flatness.

Apply the two redshift factors to that common D_M. At z = 1, D_M ≈ 0.58579D_unit, D_A ≈ 0.29289D_unit and D_L ≈ 1.17157D_unit. The luminosity distance is four times the angular diameter distance, as (1 + z)² = 4 predicts.

Every table row uses the same background and length unit. The differences are consequences of the definitions and the supplied history, rather than inconsistent observations or changes in unit conversion.

Farther redshift need not mean larger D_A

In this particular ideal history, D_A reaches its maximum at z = 1.25 and then falls. D_L continues to rise over the displayed interval. The maximum is a property of the chosen model’s distance function, not a universal turnover redshift.

For a source of fixed emitted proper transverse size, θ = ℓ_em/D_A. A smaller D_A therefore means a larger angular size for that specified ruler. Real source sizes and populations may evolve, so raw angular sizes are not a direct geometric test without a ruler model.

For a concrete small-angle example at z = 1, supply ℓ_em = 0.001D_unit. Then θ = 0.001/0.292893… ≈ 0.003414 radians. The ratio uses the physical size at emission and the angular diameter distance in the same length unit.

This example does not assert that all galaxies have the same size or that a bound ruler follows the scale factor of the background. It illustrates how a supplied physical size is connected to an angle under a stated distance definition.

Separate a geometric relation from calibration

A known relation between distance definitions does not independently calibrate an unknown luminosity or ruler size. If the source’s absolute luminosity is uncertain, the flux alone does not determine an absolute D_L without additional information.

Likewise, an angular measurement constrains a physical-size-to-distance ratio. The next lessons examine how calibration chains and standardization turn suitable source populations or statistical rulers into distance information.

When comparing reported distances, retain redshift, physical versus comoving size, bolometric versus band-limited flux, propagation assumptions and the background model. Those details explain which relationship a number is meant to test.

The table provides a reusable check: at z = 3 the model has D_M = D_unit, D_A = D_unit/4 and D_L = 4D_unit. All are valid distances associated with the same redshift, with different observational meanings.

Worked reasoning

Try the reasoning

1. Under the stated distance-duality conditions, what is D_L/D_A at z = 1?

   Show the answer and reasoning

   4.

   The ratio is (1 + z)², which equals 4 at z = 1.

2. Should an unconverted comoving ruler length be inserted as ℓ_em in D_A = ℓ_em/θ?

   Show the answer and reasoning

   No.

   ℓ_em is the physical transverse size at emission; a comoving length needs the appropriate conversion.

3. Is z = 1.25 a universal angular-distance maximum for every cosmological model?

   Show the answer and reasoning

   No.

   It is the maximum of the particular ideal flat matter-only distance function supplied here.

What to carry forward

• Luminosity distance uses power and flux; angular diameter distance uses emitted physical size and angle.
• Distance duality states D_L = (1 + z)²D_A under its propagation and photon-conservation conditions.
• Distance curves and ruler interpretations require their model, calibration and size conventions.