Cosmology · Evidence for accelerated expansion

Expansion geometry and growth as complementary tests

Hold an expansion history fixed while changing a stipulated perturbation response to show why distances alone do not determine structure growth.

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 an expanding background from its small density perturbations.
Compare supplied linear-growth solutions with a common background.
State why a response comparison is not a completed alternative gravity theory.

Ask both how distances evolve and how inhomogeneities respond

An expansion history describes the homogeneous scale factor and the light paths associated with it. Structure growth describes how small departures from a homogeneous density evolve.

In a physical model these sectors are related by the gravitational equations and the properties of the matter. Specifying only H(a) does not state every perturbation equation, initial condition or tracer response.

Distance measurements can therefore be complemented by observations sensitive to density growth, motion or lensing. Their comparison must use a common model and the actual observation responses.

A controlled mathematical example below holds the background fixed and changes one stipulated perturbation coefficient. It isolates information carried by that coefficient rather than proposing a fully consistent modified theory.

Keep background gravity separate from the test coefficient

Supply the same Einstein–de Sitter background a(t) ∝ t²ᐟ³ for every branch. Write A = a/a_i, N = ln A and use the finite interval 1 ≤ A ≤ 8. Its deceleration parameter is q = 1/2.

For a weak, subhorizon, pressureless small mode in the standard reference, the contrast obeys δ_NN + (1/2)δ_N − (3/2)δ = 0. Subscripts N denote derivatives with respect to N.

Now define a separate phenomenological response family δ_NN + (1/2)δ_N − (3/2)gδ = 0 with a supplied constant g = 0, 1 or 2. The usual reference is g = 1.

The coefficient changes only this stipulated perturbation equation. It is not substituted into the background Friedmann equation, and no covariant gravity theory, screening mechanism or observational viability is supplied for g ≠ 1.

A growth law needs its initial motion

A trial solution δ ∝ A^p gives p² + p/2 − 3g/2 = 0. Choose the nondecaying branch p = [−1+√(1+24g)]/4, which is 0, 1 or 3/2 for the supplied g values.

Set δ(A = 1) = 0.001 and δ_N(A = 1) = 0.001p in each branch. These two initial conditions select the displayed solution δ = 0.001A^p.

The initial contrast is common but the initial derivative is not. The table therefore compares explicitly selected modes; it does not claim that one identical two-condition initial state evolved three different ways.

Every displayed contrast stays small over the stated interval, with the largest about 0.022627. This finite small-amplitude construction does not extend the linear solutions into nonlinear halo formation.

Same supplied background, explicitly selected nondecaying perturbation modes
A = a/a_i	δ for g = 0, p = 0	δ for g = 1, p = 1	δ for g = 2, p = 3/2
1	0.001000	0.001000	0.001000
2	0.001000	0.002000	0.002828
4	0.001000	0.004000	0.008000
8	0.001000	0.008000	0.022627

An identical geometry can leave another prediction undetermined

Because the background and photon-propagation assumptions are held common, its distance-redshift relations are identical in these stipulated comparisons. The perturbation amplitudes differ.

At A = 4, the g = 2 branch has twice the contrast of the g = 1 branch. Equal background distances alone would not tell these two supplied perturbation responses apart.

This logical comparison does not show that arbitrary background and perturbation equations can be joined into a viable fundamental theory. A complete alternative must satisfy its field equations, conservation and consistency conditions.

It does show which missing information a growth test can target: the gravitational response of inhomogeneities and its initial state, beyond the homogeneous geometry.

This original finite mathematical comparison holds the Einstein–de Sitter background a(t) proportional to t^(2/3), q = 1/2, and photon-propagation assumptions fixed. Its background distance laws are therefore common. A = a/a_i runs from 1 to 8 and N = ln A. A weak subhorizon pressureless contrast follows the separately stipulated response equation δ_NN+(1/2)δ_N−(3/2)gδ = 0 with chosen g = 0, 1, 2. Only the perturbation source coefficient changes; g is not substituted into the background Friedmann equation or background G. The graph has linear A from 1 to 8 and a zero-based linear dimensionless contrast ordinate from 0 to 0.024. Dotted green diamonds show g = 0, p = 0; solid blue circles show standard-reference g = 1, p = 1; dashed orange squares show g = 2, p = 3/2. The selected nondecaying solution is δ = 0.001A^p, where p = [−1+√(1+24g)]/4. At A = 1, all contrasts equal 0.001 and the three markers intentionally coincide. The separate native table lists both initial conditions: rows (g, p, initial δ, initial δ_N) are (0, 0, 0.001, 0), (1, 1, 0.001, 0.001) and (2, 3/2, 0.001, 0.0015). Thus equal initial amplitude does not mean an identical two-condition initial state. At A = 2, contrasts are 0.001, 0.002, approximately 0.002828. At A = 4, they are 0.001, 0.004, 0.008. At A = 8, they are 0.001, 0.008, approximately 0.022627. The selected g = 2 branch is exactly twice the g = 1 branch at A = 4. All remain small in the supplied finite domain; no continuation into nonlinear collapse, halo formation or arbitrary A is claimed. Shapes, dash patterns, formulas, initial-state table and explicit legends distinguish the three branches without color. The family isolates extra information in the perturbation response and its initial motion that a background distance law alone does not fix. It is not a covariant modified-gravity theory, a screening calculation or a statement of observed viability; a complete physical theory must consistently relate background, conservation and perturbation sectors. Source growth equations written with energy density require ρ_energy/c² when compared with mass-density notation. Galaxy catalogues, velocities and lensing require their own tracer and observation responses before they test total-matter δ. No measured growth amplitude, fit, uncertainty or preferred gravitational coefficient is supplied. Original Ark teaching model.

Predict the measured growth response

A real galaxy catalogue does not directly report this δ of total matter. Its relation to matter, selection and calibration must be modeled. Motions and lensing offer other responses with their own assumptions.

A useful joint comparison predicts those primitive observations through the same background, matter content and perturbation theory, including nuisance parameters and covariance.

If expansion and growth predictions disagree with data, the explanation could involve calibration, tracer modeling, initial conditions, matter physics or gravity. The discrepancy does not identify one of them automatically.

Complementary tests are valuable because they constrain additional parts of the explanation. They turn a fit to distances into a broader consistency test while keeping the physical theory’s obligations explicit.

Worked reasoning

Compare two selected modes at A = 4

How do the g = 1 and g = 2 contrasts compare at the same supplied background epoch?

For g = 1, p = 1, so δ = 0.001 × 4 = 0.004.
For g = 2, p = 3/2, so δ = 0.001 × 4³ᐟ² = 0.008.
Their ratio is two, while the stipulated background distance laws remain common.

The selected g = 2 mode is twice as large; this does not establish a consistent alternative gravity theory.

Try the reasoning

For the displayed g = 2 branch, what initial δ_N accompanies δ = 0.001?
Show the answer and reasoning
0.0015.
The selected p = 3/2 mode requires δ_N = pδ.
Does g = 0 in the supplied perturbation equation remove the background expansion?
Show the answer and reasoning
No.
The background is held fixed by construction; g alters only the separate perturbation response.
Can a detected galaxy contrast be substituted for total-matter δ without a tracer model?
Show the answer and reasoning
No.
Formation, bias, selection and observational response connect the tracer to the matter perturbation.

What to carry forward

Background geometry and perturbation growth constrain different parts of an explanation.
A linear mode needs both amplitude and initial derivative.
A complete physical theory must relate these sectors consistently before confronting observations.

References

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