Cosmology · Consistency, tensions and claims of new physics

Comparing expansion-rate estimates under common assumptions

Map one supplied radial response to a present-epoch rate under two declared late-time backgrounds and three imposed ruler calibrations.

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

Separate a selected-epoch expansion rate from a present-epoch rate.
Trace a length calibration and expansion-history assumption through a comparison.
Identify what is missing from a physical explanation of a discrepancy.

An expansion-rate comparison can depend on a conversion model

Expansion rate H(z) refers to a specified redshift. Comparing a response at nonzero redshift with a present-epoch reference H₀ can require an expansion-history model.

A calibrated radial clustering response also involves a length scale. The same dimensionless response need not yield the same absolute rate when the adopted calibration changes.

These dependencies need to be traced before interpreting agreement or disagreement between reported quantities. A changed inferred number alone does not identify a physical mechanism.

Use a supplied ideal response and separate late-time backgrounds below. No observed H₀, measured acoustic scale, error model, current tension or empirical likelihood is supplied.

The response fixes a product at the selected epoch

Choose arbitrary positive reference rate H_* in s⁻¹ and define length L = c/H_* in metres. At selected redshift z_* = 1, supply noiseless radial template summary Y = c/[H(1)r] = 4.

The calibration r is an imposed positive comoving length. Compare q = r/L equal to 1/10,1/8 and 3/20, holding Y fixed.

Then H(1)/H_* = 1/(4q). For q = 1/8, the selected-epoch rate is 2H_*. Y by itself does not fix the present-epoch H₀.

These ratios describe an ideal template response. They are not a literal finite-rod observation, a physical sound-horizon integral or a calculation of how the ruler was produced.

State the history used to convert the epoch

Separately consider two spatially flat, classical general-relativistic backgrounds over z = 0 to 1. Each contains one positive, separately conserved homogeneous fluid with constant equation-of-state ratio w = P/ε.

The positive expanding branch obeys H(z)/H₀ = (1+z)ⁿ, with n = 3(1+w)/2. Flat coasting fluid w = −1/3 gives n = 1; flat pressureless dust w = 0 gives n = 3/2.

The coasting case is a positive-fluid flat model. It is not the empty negatively curved Milne Universe, even though both have a linear scale-factor time dependence.

At z = 1, the conversion factors are 2 and 2√2. Thus H₀/H_* = 1/[4q2ⁿ], different for the two backgrounds even at fixed Y and q.

Supplied Y = 4 at z = 1; reference R = H_* is at z = 0 and has no assigned uncertainty
q = r/L	Flat background	H(1)/H_*	H(1)/H₀	H₀/H_*	(H₀−R)/H_*
1/10	Coasting w = −1/3	2.5	2	1.25	0.25
1/10	Dust w = 0	2.5	2√2	0.883883	−0.116117
1/8	Coasting w = −1/3	2	2	1	0
1/8	Dust w = 0	2	2√2	0.707107	−0.292893
3/20	Coasting w = −1/3	5/3	2	0.833333	−0.166667
3/20	Dust w = 0	5/3	2√2	0.589256	−0.410744

Calibration and late history are distinct assumptions

At q = 1/8, both backgrounds give H(1) = 2H_*. The coasting conversion gives H₀ = H_*, while the dust conversion gives H₀ = H_*/√2.

At fixed background, changing q instead rescales both H(1) and the converted H₀ reciprocally. A larger adopted length gives a smaller inferred rate under the same Y.

The stipulated present-epoch reference is R = H_*. The displayed residuals compare quantities at that same epoch, with no supplied standard error or significance assigned to their differences.

A physically derived calibration could share early-time assumptions with other probes. Merely selecting another r here supplies no consistent early-Universe mechanism or additional independent evidence.

This supplied hypothetical radial template summary Y = c/[H(1)r] = 4 at redshift z = 1 fixes H(1)r/c = 1/4. H_* is an arbitrary positive reference rate in seconds to the minus one, L = c/H_* is a length in metres, r is a comoving calibration length and q = r/L is dimensionless. The horizontal linear axis is q from 0.10 to 0.15; the vertical linear axis is the converted present-epoch H₀/H_* from 0.5 to 1.3. H(1)/H_* = 1/(4q). Two separately chosen flat positive-fluid late-time histories give H(z)/H₀ = (1+z)^n: solid blue circles show coasting w = −1/3, n = 1, H₀/H_* = 1/(8q); dashed orange squares show dust w = 0, n = 3/2, H₀/H_* = 1/(8√2 q). A dotted violet horizontal reference is the supplied present-epoch R/H_* = 1. It has no assigned uncertainty. Markers and the native table use q = 1/10, 1/8, 3/20. H(1)/H_* is respectively 2.5, 2, 5/3. Coasting H₀/H_* is respectively 1.25, 1, 0.833333; dust is approximately 0.883883, 0.707107, 0.589256. Converted-minus-reference residuals are these displayed rates minus 1, not comparisons of H(1) directly with a present rate. Each curve is sensitivity to an imposed common calibration and specified background, not an uncertainty distribution. The flat coasting fluid is not negatively curved Milne geometry. Positive homogeneous energy density and P = w epsilon can supply these conditional late backgrounds, but no early radiation history, tracer transfer, perturbative stability, calibration uncertainty, measured H₀, current tension, likelihood or viable physical resolution is supplied. Direct legend text, solid/dashed patterns, circle/square shapes and all six printed rates provide noncolor interpretation. Original Ark teaching model.

A consistent late background is not a complete explanation

Optional conservation gives ε(z) = ε₀(1+z)³⁽¹⁺ʷ⁾, with positive ε₀ = 3c²H₀²/(8πG) in J/m³ and P = wε in pascals. These relations reproduce the stated flat-fluid H(z).

This checks the supplied homogeneous expanding branch. It does not provide a matter-plus-radiation history, perturbation completion, tracer response, stability argument or survey selection model.

Comparing actual expansion estimates needs common definitions, epoch mapping, calibration dependencies and the appropriate joint uncertainties. Agreement under one set of assumptions and disagreement under another should be reported conditionally.

A proposed physical extension must explain the relevant calibration and expansion responses while meeting other tests. Adjusting an algebraic conversion in this table does not demonstrate a viable explanation of a real discrepancy.

Worked reasoning

Convert the same selected-epoch response

What does q = 1/8 imply for H₀ in the two supplied backgrounds?

Y = 4 gives H(1)/H_* = 1/[4(1/8)] = 2.
Coasting has H(1)/H₀ = 2, so H₀/H_* = 1.
Dust has H(1)/H₀ = 2√2, so H₀/H_* = 1/√2 ≈ 0.707107.

The same selected-epoch summary and calibration yield different present-epoch rates under different stated histories.

Try the reasoning

For q = 1/10 and flat coasting, find H₀/H_*.
Show the answer and reasoning
5/4 = 1.25.
The selected-epoch ratio is 2.5 and the conversion factor is 2.
For the same q and flat dust, find H₀/H_*.
Show the answer and reasoning
5/(4√2) ≈ 0.883883.
Divide the same selected-epoch ratio 2.5 by 2√2.
Does imposing another r establish a viable early-Universe model?
Show the answer and reasoning
No.
No mechanism producing that ruler calibration, early history or corresponding additional predictions has been supplied.

What to carry forward

Match the expansion quantity and epoch before comparing estimates.
Length calibration and expansion history introduce distinct dependencies.
An altered inferred rate is not by itself a physical explanation of a discrepancy.

References

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