Why one distance does not determine a cosmic history

An integral can hide different rate functions

A cosmological distance can depend on an integral of the expansion history. One number can therefore combine information from many epochs without identifying the rate at each epoch uniquely.

A parameter degeneracy means that different allowed parameter choices can give the same or similar values for the particular observables being used. Its extent depends on the model family, calibration, redshift range and measurement precision.

To make one such ambiguity concrete, compare two separate ideal spatially flat general-relativistic backgrounds. Both are homogeneous, expanding, normalized to a_ref = 1 and omit radiation. Their matter and vacuum contents and reference rates differ.

The example deliberately matches only one chosen luminosity distance. It does not claim that the models fit actual observations or remain indistinguishable over a complete distance–redshift dataset.

Give both histories enough information to calculate

Model A contains only pressureless matter: Ω_m = 1, no cosmological constant and reference rate H₀,A = H_star, with H_star an arbitrary positive rate. Its dimensionless expansion function is E_A(z) = (1 + z)³ᐟ².

Model B has reference ratios Ω_m = 1/4 and Ω_vac = 3/4, with vacuum included once. Its dimensionless function is E_B(z) = √[(1/4)(1 + z)³ + 3/4], and its reference rate is H₀,B = sH_star.

For each flat model, write I(z) = ∫ from 0 to z of dz′/E(z′). Then luminosity distance is D_L = (c/H₀)(1 + z)I(z). Model A has the analytic result I_A(z) = 2[1 − 1/√(1 + z)]; I_B can be evaluated numerically from its supplied smooth integrand.

Use c/H_star as the common length unit for both models. In that unit the predictions are (1 + z)I_A(z) for A and (1 + z)I_B(z)/s for B. Using a different model’s own Hubble length as the plotting unit would conceal an absolute-scale comparison.

Choose the rate normalization to force one match

Set s = I_B(1)/I_A(1) ≈ 1.358446. At z = 1 this makes I_B(1)/s = I_A(1), so both luminosity distances equal 1.171573 times c/H_star by construction.

At z = 0.5, however, model A predicts about 0.55051 times c/H_star and B predicts about 0.49568 times that same unit. At z = 0.1 their predictions are about 0.10238 and 0.07942. Matching the selected endpoint did not make the full curves identical.

Even the expansion rates at the matched z = 1 differ: H_A/H_star ≈ 2.82843 and H_B/H_star ≈ 2.25273. A distance integral can match while its underlying local rate functions do not.

The table contains calculated predictions from chosen model parameters, not observed points or error bars. The exact matching condition is the integral ratio defining s; the displayed decimal values are rounded.

Flux also needs an absolute luminosity scale

A separate ambiguity appears if the intrinsic luminosity normalization is unknown. The flux equation F = L_em/(4πD_L²) remains unchanged under D_L → qD_L and L_em → q²L_em for any positive q.

For a deterministic example choose q = 1.1. Multiplying all distances by 1.1 and the luminosity normalization by 1.21 leaves the predicted fluxes unchanged. This is an algebraic scale symmetry of the stated account, not a measured calibration uncertainty.

Within a fixed dimensionless expansion family, distances scale as 1/H₀. The same distance multiplication therefore corresponds to H₀ → H₀/1.1 when the other dimensionless parameters are held fixed. An unknown absolute luminosity can then limit inference of an absolute rate from fluxes alone.

This calibration symmetry differs from the two-history comparison above. The latter changes the dimensionless shape and matches one selected endpoint; the former keeps a fixed dimensionless shape while changing an uncalibrated overall scale.

Use observations that test different remaining possibilities

Additional calibrated distances at other redshifts would test the different curves in the table. Independently inferred expansion rates or other geometric probes can constrain combinations not fixed by the one selected distance.

An absolute calibration can address the luminosity-scale ambiguity, but its dependence on shared anchors or ruler assumptions must remain visible. Combining outputs that inherit the same calibration does not automatically add independent absolute-scale information.

Real comparisons also require an uncertainty and covariance model, population and selection treatment, and a stated parameter family. Whether a remaining difference can be resolved depends on those details; the toy assigns no significance level.

A defensible conclusion states what the evidence constrains and which alternatives remain. One matching distance can be informative without uniquely reconstructing every epoch, component property or global feature of the Universe.

Worked reasoning

Try the reasoning

1. Do the matching z = 1 luminosity distances require identical H(z = 1) in the example?

   Show the answer and reasoning

   No.

   The calculated rates are about 2.82843H_star and 2.25273H_star. The distance summarizes an integral of each history.

2. If D_L is multiplied by 1.1, by what factor must L_em change to leave the stated flux unchanged?

   Show the answer and reasoning

   1.21.

   Flux depends on L_em/D_L², so the luminosity factor must be 1.1².

3. Does this one-point construction establish that the two models fit all actual distance data equally well?

   Show the answer and reasoning

   No.

   It uses synthetic inputs and forces only one match. Other predictions, calibration and uncertainties require their own comparison.

What to carry forward

• A distance integral can agree for histories with different contents and rates.
• An unknown absolute luminosity introduces an additional scale ambiguity.
• New constraints help when their calibration, assumptions and independence are established.