Cosmology · Evidence for accelerated expansion

Acceleration evidence versus its physical cause

Show how a constant geometric term and an equivalent effective stress give the same expansion equation, while the physical origin remains a separate question.

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 a measured expansion pattern from a proposed physical explanation.
Translate a positive constant between geometric and effective-fluid conventions.
Interpret a supplied accelerating model without identifying a microscopic mechanism.

An expansion inference does not finish the explanation

Evidence for acceleration concerns the history of a(t) under stated distance, calibration and propagation assumptions. Explaining that history requires a gravitational model and its sources.

A cosmological constant is one specified term in the gravitational equations. A dynamical field or a modified gravitational description requires further equations and predictions. Merely fitting an expansion curve does not make those explanations equivalent in every observable.

There is, however, an exact equivalence of notation between a geometric cosmological constant and a particular constant effective stress-energy. It illustrates why assigning a name to a fitted term is not the same as calculating its physical origin.

Use a supplied flat dust-plus-constant construction below. Its numerical density fractions are chosen mathematical inputs, not current cosmological estimates or fits to the historical supernova data.

Specify matter, constant contribution and units

Set a₀ = 1 and choose H₀ > 0. Supply present pressureless-matter fraction Ω_m0 = 0.4 and positive constant fraction Ω_X0 = 0.6, with their sum one and no radiation in this construction.

The matter mass density is ρ_m(a) = [3H₀²/(8πG)]0.4a⁻³. Its units are mass per volume. The specified background obeys H²/H₀² = 0.4a⁻³+0.6.

In geometric notation, choose Λ = 3H₀²(0.6)/c², with inverse-length-squared units. Then the flat Friedmann equation is H² = (8πG/3)ρ_m + Λc²/3.

Alternatively place the constant contribution in an effective source with energy density ε_X = 3H₀²c²(0.6)/(8πG) and pressure p_X = −ε_X. Both ε_X and p_X have energy-per-volume units. The density ε_X/c² is the corresponding mass-density equivalent.

Two descriptions of one specified term

The effective-source description gives H² = (8πG/3)[ρ_m+ε_X/c²]. Substituting ε_X produces the same constant 0.6H₀² as the geometric Λ term.

These are alternative accounts of the same contribution. Adding both the chosen Λ and that chosen ε_X as separate components would double count it.

At the background level, the effective acceleration equation is ä/a = −(4πG/3)[ρ_m+(ε_X+3p_X)/c²]. Since p_X = −ε_X, the constant contribution produces positive ä/a when it outweighs the matter term.

This algebra does not calculate a microscopic vacuum energy, explain why the supplied constant has that value or provide a new particle model. A constant effective stress is an exact rewriting of the stipulated gravitational term.

Accelerating a can coexist with declining H

The deceleration parameter for this supplied background is q(a) = [0.2a⁻³−0.6]/[0.4a⁻³+0.6]. It changes sign at a = (1/3)¹ᐟ³ ≈ 0.693361.

For larger a in the displayed interval, q < 0 and ä > 0. At a = 1, q = −0.4, while H = H₀.

Nevertheless Ḣ/H₀² = −0.6a⁻³ remains negative. The scale factor’s expansion accelerates even as its fractional expansion rate H = ȧ/a decreases toward a positive constant.

Both equivalent descriptions predict the same table and, with the same photon propagation, the same background distances. Agreement between these two accounts is a mathematical identity, not two independent confirmations from data.

One supplied dust-plus-positive-constant background, expressed in either convention
a	H²/H₀²	H/H₀	q
0.5	3.800000	1.949359	0.263158
0.75	1.548148	1.244246	−0.081340
1	1.000000	1.000000	−0.400000
1.5	0.718519	0.847655	−0.752577
2	0.650000	0.806226	−0.884615

This original chosen flat dust-plus-positive-constant example supplies Ω_m0 = 0.4, Ω_X0 = 0.6, a₀ = 1 and arbitrary positive H₀, with no radiation. These fractions are mathematical inputs, not current estimates, historical fitted values or observations. Both panels use a linear scale-factor axis from 0.5 to 2. Panel A has a zero-based linear H/H₀ ordinate from 0 to 2. Solid blue circles follow H/H₀ = √(0.4a^(−3)+0.6). At a = 0.5, 0.75, 1, 1.5, 2, values are approximately 1.949359, 1.244246, 1.000000, 0.847655, 0.806226. Panel B uses a separate linear q ordinate from −1 to 0.5 and dashed orange squares for q(a) = [0.2a^(−3)−0.6]/[0.4a^(−3)+0.6]. At the same five scale factors, q is approximately 0.263158, −0.081340, −0.400000, −0.752577, −0.884615. The q ordinate includes negative values and is not a magnitude or Hubble-rate axis. A dotted green vertical reference and diamond mark q = 0 at a = (1/3)^(1/3) ≈ 0.693361. Larger scale factors within this domain have q < 0 and positive scale-factor acceleration ä. Nevertheless Ḣ/H₀² = −0.6a^(−3) is negative at every finite positive a in the displayed interval. At a = 1, q = −0.4 and Ḣ/H₀² = −0.6. Accelerating a and increasing fractional rate H are different statements. Shapes, dash patterns, explicit axis labels and the named boundary provide noncolor interpretation. The same constant can be expressed geometrically as Λ = 3H₀²(0.6)/c², in m^(−2), or as one effective constant source with energy density ε_X = 3H₀²c²(0.6)/(8πG), pressure p_X = −ε_X, both in J m^(−3). Matter density ρ_m = [3H₀²/(8πG)]0.4a^(−3) is mass per volume, and ε_X/c² is its constant mass-density equivalent. The identity Λc²/3 = (8πG/3c²)ε_X = 0.6H₀² rewrites one contribution; adding both as independent components would double count it. Both notational accounts predict these same curves and, under the same propagation assumptions, the same background distances. Their identity is not independent empirical confirmation, a distinct competing microscopic theory or a calculation of vacuum-energy origin. No dynamical alternative, coupling, perturbation prescription, current cosmological fit or future observational forecast is supplied. Original Ark teaching model.

A mechanism must do more than rename the term

A proposed dynamical explanation must specify how its state evolves and how it perturbs, couples and affects other tests. Matching one background history is only part of that task.

Conversely, two notational descriptions that yield exactly the same equations need not be observational competitors at the stated level. No background-distance measurement can distinguish a term from its exact algebraic rewriting.

Actual acceleration studies compare distances and other responses with uncertainty, calibration and selection retained. Historical evidence for an accelerating fit should not be presented as a direct microscopic measurement of vacuum physics.

The construction separates three statements: what expansion behavior is inferred, which effective equations describe it, and what physical theory explains those equations and their parameters. Each statement requires its own evidence or derivation.

Worked reasoning

Translate the constant contribution at a = 1

Why do the geometric and effective-source descriptions give the same H₀?

The geometric contribution is Λc²/3 = 0.6H₀².
The effective energy contribution is (8πG/3c²)ε_X = 0.6H₀².
Adding the supplied matter contribution 0.4H₀² gives H² = H₀² in either account.

They are two conventions for the same specified constant, which must be included only once.

Try the reasoning

What are the units of Λ in the stated convention?
Show the answer and reasoning
Inverse length squared.
Λc² must have inverse-time-squared units in the Friedmann equation.
At a = 1, can ä be positive while Ḣ is negative in this model?
Show the answer and reasoning
Yes.
q = −0.4 denotes acceleration, while Ḣ/H₀² = −0.6.
Does the exact constant-stress rewriting calculate the physical origin or value of the constant?
Show the answer and reasoning
No.
It establishes an equation-level equivalence under supplied inputs, leaving the microscopic explanation separate.

What to carry forward

Acceleration evidence constrains an expansion history under stated assumptions.
A constant geometric term has an equivalent effective-source description with p = −ε.
An exact rewriting does not supply an independent observation or a microscopic explanation.

References

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