Cosmology · Evidence for accelerated expansion

Combining independent distance probes

Compare ideal luminosity, angular and radial responses in one supplied geometry, then identify which measurements and calibrations can add independent information.

Preparation: ratios, powers of ten and ordinary arithmetic. Physical quantities and notation are introduced as they are needed.

Back to the chapter

After reading, you can

Distinguish luminosity, angular-diameter and comoving distances.
Interpret a supplied transverse ruler and a local radial ruler response.
Avoid counting algebraically related estimates as independent observations.

Ask what each probe actually measures

A candle compares flux with an inferred luminosity. A transverse ruler compares an angle with an inferred length. A radial ruler compares a redshift separation with an inferred comoving separation.

These responses can constrain the same expansion history through different primitive observations and calibrations. Combining them is useful when their relationships and shared uncertainties are modeled.

The acoustic clustering feature is one example of a ruler whose physical calibration and late-time response require their own model. Its use does not remove assumptions about the early population or the survey.

Here supply a simple ideal comoving rod instead. No acoustic production calculation, measured feature, instrument or empirical ruler length is adopted.

Keep the distance convention explicit

Use the flat matter-only reference H(z)/H₀ = (1+z)³ᐟ², with a₀ = 1 and D_H = c/H₀. Its transverse comoving distance is D_M/D_H = 2[1−(1+z)⁻¹ᐟ²].

For the chosen transparent metric propagation, D_L = (1+z)D_M and D_A = D_M/(1+z). Thus D_L = (1+z)²D_A.

D_A converts a small observed angle to a transverse proper size at emission. D_M converts it to the corresponding comoving size. They answer different size questions at the same redshift.

The reciprocity relation is a propagation condition within this model. Measuring one distance and transforming it algebraically into the other does not create a second observation.

Exact distance functions in the supplied flat matter geometry
z	D_M/D_H	D_L/D_H	D_A/D_H
0.25	0.211146	0.263932	0.168916
0.5	0.367007	0.550510	0.244671
1	0.585786	1.171573	0.292893
2	0.845299	2.535898	0.281766

A ruler tests a distance and a local expansion rate

Supply a small comoving rod of length r_c = 0.01D_H. Its transverse proper length at emission is r_c/(1+z), so the leading small-angle response is θ ≈ r_c/D_M.

For an ideal radial rod aligned with the line of sight, the local small-baseline redshift response is Δz ≈ H(z)r_c/c. This linear relation neglects changes in H across the finite rod and assumes peculiar-velocity and measurement effects absent.

At z = 1 these leading responses are θ ≈ 0.017071 radians and Δz ≈ 0.028284. They depend differently on the integrated D_M and local H.

The table evaluates those stated leading formulas. It does not give exact finite-angle or finite-baseline observables to the displayed precision, and it is not a prediction for an observed acoustic peak.

Leading small-angle and small-baseline responses of the stipulated comoving rod
z	θ ≈ r_c/D_M, radians	Δz ≈ H r_c/c
0.25	0.047361	0.013975
0.5	0.027247	0.018371
1	0.017071	0.028284
2	0.011830	0.051962

This original chosen flat matter-only reference has a₀ = 1, H/H₀ = (1+z)^(3/2) and arbitrary length unit D_H = c/H₀. It supplies a small known comoving rod r_c = 0.01D_H; no acoustic-production calculation, observed BAO scale or actual instrument is supplied. Panel A has linear redshift z from 0 to 2 and zero-based distance ordinate from 0 to 3 in units D_H. Solid blue circles denote transverse comoving distance D_M/D_H = 2[1−1/√(1+z)]. Dashed orange squares show D_L = (1+z)D_M; dotted green diamonds show D_A = D_M/(1+z). At z = 0.25, 0.5, 1, 2, the three distance rows are approximately (0.211146, 0.263932, 0.168916), (0.367007, 0.550510, 0.244671), (0.585786, 1.171573, 0.292893), (0.845299, 2.535898, 0.281766). Thus angular-diameter distance can turn over while luminosity distance increases. All are derived from one supplied geometry and are not independent observations. Transparent metric propagation gives D_L = (1+z)²D_A. Panel B has its own linear redshift domain from 0.25 to 2, deliberately excluding the vanishing-distance origin, and zero-based leading angle ordinate from 0 to 0.06 radians. Dashed violet squares show θ ≈ r_c/D_M; samples at z = 0.25, 0.5, 1, 2 are approximately 0.047361, 0.027247, 0.017071, 0.011830 radians. Panel C separately uses the same positive-redshift domain and a zero-based dimensionless separation ordinate from 0 to 0.06. Dotted orange diamonds show the local radial leading response Δz ≈ H(z)r_c/c = 0.01(1+z)^(3/2), with samples approximately 0.013975, 0.018371, 0.028284, 0.051962. These two panels evaluate stated leading small-angle and small-baseline formulas to displayed arithmetic precision; they do not claim exact finite-angle or finite-baseline observables at that precision. The radial approximation neglects H variation across the rod and removes peculiar velocities and measurement effects by assumption. The transverse proper rod at emission is r_c/(1+z): at z = 1, 0.005D_H divided by D_A ≈ 0.292893D_H gives the same leading angle as 0.01D_H divided by D_M ≈ 0.585786D_H. Distinct shapes, dash styles, panel labels, units and legends provide noncolor interpretation. A separately observed calibrated angle or redshift separation can add information; algebraically transforming one measured distance does not create a new datum. Real probes retain shared calibration, selection, physical-ruler and covariance assumptions. Original Ark teaching model.

Count independent inputs, not renamed outputs

Suppose a luminosity-distance estimate comes from a flux measurement and source calibration. Dividing it by (1+z)² gives an inferred D_A under reciprocity, but no independent angle has been measured.

A separately measured angle with a calibrated physical ruler can add information. Its ruler calibration may still share cosmological or observational assumptions with another probe.

Similarly, several catalogue summaries can contain the same events or calibration modes. A joint likelihood needs their covariance so that one source of information is not counted repeatedly.

The common-geometry table is a consistency construction. Its exact agreement cannot serve as independent empirical confirmation of that geometry.

Use complementary response and calibration tests

For real data, predict the primitive observables through a common expansion model and each probe’s physical response. Keep calibration, selection, peculiar motions and other relevant nuisance quantities explicit.

An absolute luminosity or ruler calibration can set a distance scale. If that calibration is not known, the data may constrain only a relative combination of scale and expansion parameters.

Transverse and radial information can help separate integrated geometry from a local expansion response. Growth or lensing observations can then test further physical assumptions rather than simply renaming the same distance.

Complementarity is therefore a property of the information and assumptions in the joint comparison. Different labels alone do not guarantee independent evidence for acceleration.

Worked reasoning

Use comoving and proper rod lengths consistently

At z = 1, what transverse angle follows from r_c = 0.01D_H in the supplied leading response?

The proper rod length at emission is r_c/2 = 0.005D_H.
The angular-diameter distance is about 0.292893D_H, giving θ ≈ 0.005/0.292893.
Equivalently, use the comoving distance: θ ≈ 0.01/0.585786 ≈ 0.017071 radians.

Both consistent distance-size conventions give the same leading small angle, about 0.017071 radians.

Try the reasoning

At z = 1, what is D_L/D_A under the stipulated transparent metric propagation?
Show the answer and reasoning
Four.
The ratio is (1+z)².
Does computing D_A = D_L/(1+z)² from one measured D_L add an independent observation?
Show the answer and reasoning
No.
It is an algebraic transformation under a propagation assumption.
Which appears directly in the local radial formula: H(z) or its distance integral?
Show the answer and reasoning
H(z).
The supplied leading response is Δz ≈ H(z)r_c/c; the transverse response instead uses D_M.

What to carry forward

Distance conventions must match the proper or comoving size being compared.
Radial and transverse responses depend differently on the expansion history.
A joint test needs independent inputs and their shared calibration and covariance.

References

Sources supporting the explanations, physical distinctions and examples in this lesson.