Cosmology · The evidence motivating dark matter

CMB patterns and the growth of structure

Explain different sensitivities to photon pressure, baryon inertia and gravitational forcing, using a frozen acoustic example and a separate ideal matter-growth limit.

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

Describe why the coupled photon–baryon fluid has an acoustic response.
Distinguish equal acoustic phase from the same time variable.
Explain why a joint CMB and growth analysis requires a full common model.

Different components enter the response in different ways

Before photon drag becomes ineffective, photons and baryons can behave approximately as a tightly coupled fluid. Compression raises the photon pressure, which supplies a restoring response, while baryons add inertia.

Gravity also enters through the potential generated by component perturbations. A component that does not share the same photon pressure and drag can have a different perturbation history, changing the gravitational forcing of the fluid.

CMB patterns therefore contain more information than a direct image of the total mass. Interpreting their angular correlations requires initial perturbations, component responses, projection and the scattering history.

Use two distinct ideal limits to understand mechanisms. A frozen-coefficient acoustic calculation and a pressureless matter-growth calculation are not joined snapshots of one cosmic history or a fit to a measured angular spectrum.

Change inertia while prescribing the forcing

Define R = 3ε_b/(4ε_γ), with ε_b the baryon rest-energy density and ε_γ the photon energy density, neglecting baryon pressure. Compare supplied constant R = 0 and R = 1. The former is a mathematical photon-only inertia limit.

The sound speed is c_s = c/√[3(1+R)]. It is c/√3 for R = 0 and c/√6 for R = 1. These values describe the imposed coupled-fluid limits, not a measured sound-speed history.

Let conformal time be η = ∫dt/a in seconds, k a comoving inverse-length wavenumber and u = kcΔη/√3. Supply a constant dimensionless metric potential Ψ = Φ_grav/c² = −ψ_*, with ψ_* = 0.001, and a constant curvature potential.

For a fractional temperature perturbation Θ with Θ(0) = 0 and zero initial derivative, the frozen oscillator is Θ_uu + Θ/(1+R) = −Ψ. Tight coupling and negligible diffusion are imposed. The source of the prescribed potential is not solved by this equation.

Inertia changes the offset and oscillation period

Define the signed effective response y = (Θ+Ψ)/ψ_*. The supplied solution is y = R−(1+R)cos[u/√(1+R)], beginning at y = −1 in both cases.

The table compares the same u. At u = π, y = 1 for R = 0 and about 2.211400 for R = 1. These are signed perturbation responses, not total positive temperatures or observed CMB peak heights.

Equal acoustic phase instead means equal τ = u/√(1+R). At the compression phase τ = π the responses are 1 and 3, but the corresponding u values are π and √2π. Those equal-phase values are not simultaneous in the common u coordinate.

The frozen calculation isolates inertia and forcing. An evolving baryon loading, changing potentials, diffusion, the visibility distribution and angular projection would be needed to predict an actual CMB spectrum.

Prescribed acoustic responses at common u; Ψ and R are frozen, and y is dimensionless
u/π	y for R = 0	y for R = 1
0	−1	−1
1	1	2.211400
2	−1	1.532511
3	1	−0.856483
4	−1	2.716432

Matter can grow when pressure and drag are negligible

Separately, choose an Einstein–de Sitter background dominated by nonrelativistic pressureless matter. Let A = a/a_ref run from 1 to 8 and consider small perturbations on scales well below the Hubble length.

Writing N = ln A, the linear growth equation is δ_NN + δ_N/2−3δ/2 = 0. It has growing and decaying solutions. Supply δ = 0.01 and δ_N = 0.01 initially, selecting the growing solution δ = 0.01A.

The contrast then rises from 0.01 to 0.08. The ratio δ/A stays 0.01 and is a potential proxy within this separate sub-Hubble limit; it is not the numerical value of the earlier imposed Ψ.

Ordinary matter can also grow gravitationally once pressure and photon drag are negligible. The existence of growth alone therefore does not identify a nonbaryonic component. The discriminating question concerns a complete predicted history under common assumptions.

Separate supplied pressureless growing mode; these rows are not later states of the frozen acoustic calculation
A = a/a_ref	Density contrast δ	δ/δ_initial	Proxy δ/A
1	0.01	1	0.01
2	0.02	2	0.01
4	0.04	4	0.01
8	0.08	8	0.01

These panels are two explicitly distinct synthetic ideal limits, not successive epochs of one cosmic history and not a measured CMB angular spectrum. Panel A imposes tight coupling, negligible diffusion and frozen R = 3ε_b/(4ε_γ) equal to zero or one, negligible baryon pressure, constant dimensionless metric potential Ψ = −0.001 and constant curvature potential. Let conformal time η = ∫dt/a be in seconds, comoving k in inverse metres, and u = kcΔη/√3. With Θ(0) = 0 and Θ_u(0) = 0 the supplied oscillator Θ_uu+Θ/(1+R) = −Ψ has signed response y = (Θ+Ψ)/0.001 = R−(1+R)cos[u/√(1+R)]. The shared linear horizontal axis is u/π from zero to four. The separate linear signed vertical scale is y from −1 to 3; zero is explicitly gridded. Solid blue circles identify R = 0 and dashed orange squares R = 1. Both begin at y = −1, so their initial markers coincide exactly. At common u/π = 0, 1, 2, 3, 4, R = 0 gives −1, 1, −1, 1, −1, and R = 1 gives about −1, 2.211400, 1.532511, −0.856483, 2.716432. These are signed effective perturbations, not absolute positive temperatures or measured peak heights. The extra violet diamond marks R = 1 compression at u/π = √2 and y = 3. Equal acoustic phase τ = u/√(1+R) = π occurs for R = 0 at u/π = 1 and y = 1, but R = 1 at √2 and y = 3. It is an equal-phase comparison at different u, not a simultaneous common-u comparison or an artificial horizontal offset. Sound speeds c/√3 and c/√6 belong only to these frozen limits; the source of the prescribed potential is not solved. Panel B starts a separate matter-only Einstein–de Sitter sub-Hubble pressureless linear calculation. Its own horizontal axis is A = a/a_ref from one to eight and its zero-based vertical axis is density contrast δ from zero to 0.1. With N = ln A, the equation δ_NN+δ_N/2−3δ/2 = 0 and both initial conditions δ = δ_N = 0.01 select the growing solution δ = 0.01A. Solid green circle samples at A = 1, 2, 4, 8 give δ = 0.01, 0.02, 0.04, 0.08. The constant δ/A = 0.01 is a potential proxy in this distinct limit, not the numerical value of the earlier imposed Ψ. Ordinary matter can also grow after pressure and photon drag become negligible. No evolving R, diffusion, visibility, angular projection, radiation-to-matter transition, covariance, fitted abundance or particle identification is calculated. Real joint acoustic and structure evidence needs a complete common forward model and likelihood. Original Ark teaching model.

A common explanation predicts patterns and growth together

An early component affects the background expansion and the evolving potentials as well as later matter clustering. Its abundance and properties must be tested through those connected predictions, rather than by naming one acoustic feature.

CMB and structure measurements also have calibration, selection and nuisance assumptions. Shared theoretical inputs or observations can correlate their inferred quantities even when the physical responses differ.

A meaningful combined analysis calculates the observables, their covariance and the likelihood under a common model. It checks whether proposed changes improve the full account while retaining predictions already tested elsewhere.

These ideal examples explain why acoustic response and pressureless growth offer complementary sensitivities. They supply neither an empirical nonbaryonic abundance, a particle identity nor a present cosmological parameter estimate.

Worked reasoning

Compare the compression phase carefully

Why is the R = 1 response 3 at its compression phase but only about 2.211400 at u = π?

Its acoustic phase is τ = u/√2, so u = π gives τ = π/√2 rather than π.
The compression phase τ = π occurs at u = √2π, where y = 1−2cos π = 3.
The R = 0 compression is at u = π, so an equal-phase comparison uses different u values.

Changing the supplied inertia changes the period; equal phase and equal common time coordinate are different comparisons.

Try the reasoning

Which frozen case has the smaller sound speed, R = 0 or R = 1?
Show the answer and reasoning
R = 1.
The extra inertia changes c/√3 to c/√6 under the supplied tightly coupled assumptions.
What contrast does the separate growing dust mode have at A = 8?
Show the answer and reasoning
0.08.
Its supplied initial density and derivative select δ = 0.01A.
Does either toy calculation alone measure a nonbaryonic abundance or identify a particle?
Show the answer and reasoning
No.
A full common perturbation model, observables and statistical comparison remain required.

What to carry forward

Photon pressure, baryon inertia and gravitational forcing have distinct roles.
Equal acoustic phase need not mean the same time coordinate.
Joint acoustic and growth evidence is interpreted through a complete common model.

References

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