Cosmology · Evidence for accelerated expansion

Supernova distance diagrams and acceleration

Connect calibrated brightness with an integrated expansion history, using two supplied flat cosmologies to distinguish an observable from its interpretation.

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

Explain how standardized brightness supplies a conditional luminosity distance.
Compare two supplied distance-redshift laws with the same present expansion rate.
Distinguish an inference of acceleration from a direct measurement of its physical cause.

Measure light, then state the luminosity assumption

A source of known bolometric luminosity L produces bolometric flux F = L/(4πD_L²), defining its luminosity distance D_L. Bolometric means integrated over photon frequencies.

A measured flux alone therefore does not fix distance. The luminosity estimate is a separate part of the inference. For supernovae, light-curve and color information can help standardize the comparison; the events are not assumed to have identical uncorrected brightness.

Two historical supernova programs, the High-Z Supernova Search Team and the Supernova Cosmology Project, used such distance-redshift comparisons to infer accelerating expansion under their stated cosmological and calibration assumptions.

The numerical example here contains ideal negligible standard sources, not either historical dataset. In particular, a mathematical reference geometry does not supply a physical model of supernova formation.

Compare histories with the same present H₀

Set the present scale factor to a₀ = 1 and supply the same positive H₀ in two spatially flat reference cosmologies. Use cosmological redshift 1+z = 1/a and D_H = c/H₀ as a length unit.

The matter-only Einstein–de Sitter reference has H(z)/H₀ = (1+z)³ᐟ². The de Sitter reference has constant H(z)/H₀ = 1. Their scale factors decelerate and accelerate respectively.

For transparent metric photon propagation in a flat geometry, D_L = (1+z)c∫ from zero to z of dz′/H(z′). This distance integrates the intervening history; it is not simply cz/H₀ at arbitrary redshift.

The two integrals give D_L/D_H = 2(1+z)[1−(1+z)⁻¹ᐟ²] for the matter reference and D_L/D_H = z(1+z) for de Sitter. Both approach z at sufficiently small positive z.

Two stipulated flat geometries, with the same H₀ and ideal source luminosity
z	Matter D_L/D_H	de Sitter D_L/D_H	de Sitter/matter distance	Relative magnitude
0.1	0.102382	0.110000	1.074404	0.155839
0.5	0.550510	0.750000	1.362372	0.671479
1	1.171573	2.000000	1.707107	1.161303
1.5	1.837722	3.750000	2.040569	1.548757

Greater luminosity distance means lower flux

At z = 0.5, the de Sitter luminosity distance is about 1.362372 times the matter-reference value. At the same supplied luminosity, its flux is about 0.538776 of the matter-reference flux.

The magnitude difference is Δm = 5log₁₀(D_L,deSitter/D_L,matter), about +0.671479 at that redshift. A positive relative magnitude means the source appears fainter in this comparison.

The difference arises from the supplied full expansion laws. It is not an observed supernova residual and is not a fitted size of the historical acceleration evidence.

If luminosity, dust attenuation or calibration were changed instead, apparent brightness could also change. A real inference compares these possibilities across redshift and against independent information.

Two original stipulated flat FLRW reference geometries share a₀ = 1 and the same arbitrary positive H₀. The length unit is D_H = c/H₀. Ideal negligible standard sources and transparent metric photon propagation are supplied; no observed supernova dataset, fitted density or supernova formation model in empty de Sitter is implied. Matter-only Einstein–de Sitter uses H/H₀ = (1+z)^(3/2), whereas de Sitter has constant H/H₀ = 1. Panel A has linear redshift z from 0 to 1.5 and zero-based luminosity distance D_L/D_H from 0 to 4. Solid blue circles denote matter distance 2(1+z)[1−1/√(1+z)]; dashed orange squares denote de Sitter distance z(1+z). At z = 0.1, 0.5, 1 and 1.5, their values are approximately (0.102382, 0.110000), (0.550510, 0.750000), (1.171573, 2.000000), (1.837722, 3.750000). Both distances are zero at the origin and approach z in the nearby positive-redshift limit. Panel B has the same linear redshift domain and a separate zero-based magnitude-difference ordinate from 0 to 1.6. Dotted violet diamonds show Δm = 5 log₁₀(D_L, deSitter / D_L, matter), with samples approximately 0.155839, 0.671479, 1.161303 and 1.548757. The plotted value at z = 0 is the continuous relative-magnitude limit 0, not division of two zero distances. Equivalently the positive-redshift distance ratio extends continuously as [(1+z)+√(1+z)]/2, equal to 1 at zero. At z = 0.5 the distance ratio is about 1.362372; bolometric flux at common luminosity scales as inverse distance squared, giving flux ratio about 0.538776 and magnitude difference +0.671479. A positive relative magnitude denotes fainter light in this chosen comparison. The magnitude ordinate is a logarithmic brightness convention, not an independent distance datum. Dash patterns, distinct circle/square/diamond markers, legends and explicit formulas carry meaning without color. Luminosity is bolometric; real bandpass, dust, population, selection and calibration effects require their own response models and uncertainty. These are mathematical predictions, not actual historical residual sizes or proof of acceleration from one object. q = 1/2 for the matter power law and q = −1 for de Sitter; scale-factor acceleration and Hubble-parameter growth are different derivatives. Original Ark teaching model.

Name which rate is increasing

Expansion means ȧ > 0. Accelerating expansion means ä > 0. The deceleration parameter is q = −äa/ȧ², so q < 0 denotes acceleration.

The supplied matter-only solution a ∝ t²ᐟ³ has q = 1/2, while exponential de Sitter expansion has q = −1. These are properties of the mathematical solutions, not values estimated from the table’s nonexistent observations.

The Hubble parameter is H = ȧ/a. Its derivative obeys Ḣ = ä/a−H². Thus ä > 0 need not mean H itself increases: the two derivatives answer different questions.

Distances constrain acceleration through a reconstruction or a parameterized expansion model. A plot of brightness against redshift does not directly display a local accelerometer reading of ä.

Carry uncertainty into the interpretation

A supernova analysis must compare compatible rest-frame light, handle the observational selection, and account for luminosity calibration, color and dust. Band-limited flux requires spectral and redshift corrections beyond the bolometric definition used here.

Shared calibrations and overlapping data can correlate nominally separate analyses. Agreement is informative when those dependencies and the distinct methods are explicitly retained.

The historical acceleration inference motivates a physical question about the stress-energy content or gravitational dynamics. The distance relation by itself does not determine a microscopic mechanism.

This exercise establishes how two specified histories produce different ideal brightness predictions. Choosing between real histories requires observed data, a response model and an uncertainty calculation.

Worked reasoning

Compute a relative magnitude at z = 0.5

How much fainter is the ideal source in the de Sitter reference at the same supplied z and luminosity?

The two dimensionless luminosity distances are 0.750000 and about 0.550510.
Their ratio is about 1.362372, so the flux ratio is its inverse square, about 0.538776.
The relative magnitude is 5log₁₀(1.362372) ≈ +0.671479.

It is fainter by about 0.671479 magnitudes within these two supplied geometries.

Try the reasoning

Does one measured flux determine D_L without a luminosity estimate?
Show the answer and reasoning
No.
The defining relation contains both L and D_L.
Which sign of q denotes accelerating expansion?
Show the answer and reasoning
q < 0.
The minus sign in q = −äa/ȧ² makes positive ä correspond to negative q.
Must H increase whenever ä is positive?
Show the answer and reasoning
No.
Ḣ = ä/a−H² can remain negative during accelerated expansion.

What to carry forward

Calibrated luminosity and measured flux give a conditional luminosity distance.
A distance-redshift relation integrates an expansion history.
Acceleration, calibration and physical cause must remain distinct parts of the inference.

References

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