Cosmology · Recombination, last scattering and the CMB

Acoustic oscillations and temperature peaks

Compare coherent ideal acoustic mode powers with an exact phase average, while keeping the result distinct from an observed angular temperature spectrum.

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

Connect a common acoustic phase to a sequence of ideal power maxima.
Distinguish signed displacement from its nonnegative power.
Explain what additional physics is needed to predict observed CMB angular peaks.

Pressure and gravity connect fluctuations to waves

Before photons propagate relatively freely, tight coupling and radiation pressure allow acoustic behavior in the photon–baryon system. Compression, rarefaction, velocities and gravitational effects contribute to the later temperature pattern.

Different spatial modes can be at different stages of oscillation when visibility is largest. A common initial phase relates those stages to the mode’s wavelength and the sound-travel scale.

The observed sky is a projected combination of several sources rather than a direct graph of one fluid displacement. It is useful first to isolate the phase logic in a much simpler oscillator.

The example below uses the unforced tightly coupled photon-fluid limit with negligible dynamical baryon inertia and constant sound speed. Gravity, baryon loading, damping, finite visibility and angular projection are omitted.

Normalize a small perturbation without making it physically large

Let k be a comoving wave number and r_s = ∫c_s dη the comoving sound-travel distance, using compatible units. Define κ = kr_s. The displayed domain is 0 ≤ κ ≤ 3π.

Choose an initially displaced oscillator with zero initial velocity. Its normalized displacement is D = Θ/Θ_initial = cos κ. The initial amplitude is a normalization of an unspecified small linear perturbation, not an actual order-unity radiation contrast.

With the compatible photon-fluid velocity convention, define V = v_γ/(3c_sΘ_initial) = sin κ. The displacement and velocity quadrature are a quarter cycle apart. Their normalization does not state that a physical material velocity equals this dimensionless number times c.

Define the ideal mode powers P_D = cos²κ and P_V = sin²κ. They are nonnegative and satisfy P_D + P_V = 1 in this normalized oscillator. They are not observed angular multipole powers C_l.

Both compression and rarefaction extrema contribute power

The displacement is +1 at κ = 0 and 2π, and −1 at κ = π and 3π. Squaring gives P_D = 1 at each of those extrema, so positive and negative displacements both produce power maxima.

At κ = π/2, 3π/2 and 5π/2, D vanishes while |V| = 1. The velocity quadrature power peaks between the displacement-power maxima. A zero in one ideal contribution does not imply that every observed temperature source vanishes.

The repeated density-power maxima are separated by π in κ, or by one unit on a κ/π axis. κ = 0 is the zero-wave-number limit of this teaching plot, not a separately observed sky peak.

Reading a signed oscillation as though it were its power would miss every other positive power maximum. The sign still carries phase information even though the square removes it.

Normalized coherent ideal mode and exact four-phase mean; these dimensionless powers are not CMB C_l data
κ/π	D = cos κ	V = sin κ	P_D	P_V	Four-phase mean P_D
0	1	0	1	0	1/2
1/4	1/√2	1/√2	1/2	1/2	1/2
1/2	0	1	0	1	1/2
3/4	−1/√2	1/√2	1/2	1/2	1/2
1	−1	0	1	0	1/2
3/2	0	−1	0	1	1/2
2	1	0	1	0	1/2
3	−1	0	1	0	1/2

Averaging incompatible phases removes this peak sequence

For a deliberately different construction, average displacement power equally over phase offsets φ = 0, π/2, π and 3π/2. The exact mean is [Σ cos²(κ + φ)]/4 = 1/2 for every κ.

The cosine-squared and sine-squared contributions complement each other, making the mean flat. This four-phase average is an exact algebraic illustration, rather than a fit to an incoherent physical source model.

Comparing it with the coherent cos²κ curve shows why phase relations matter for preserving a sequence of peaks. Merely having oscillations in individual modes is insufficient to guarantee the same pattern after a phase average.

A real cosmological model must predict its initial conditions and temporal coherence. A peak pattern can test those predictions, but this ideal calculation does not uniquely establish inflation or another origin mechanism.

Original normalized ideal acoustic oscillator, not observed angular C_l. In an unforced tightly coupled photon-fluid limit with negligible baryon inertia and constant sound speed, k and r_s have compatible comoving inverse-length and length units, so κ = k r_s is dimensionless. A small initially displaced perturbation with zero initial velocity has D = Θ/Θ_initial = cos κ and compatible velocity quadrature V = v_γ/(3 c_s Θ_initial) = sin κ. The physical amplitude remains unspecified and small. Both panels use κ/π from 0 to 3. Upper blue solid/circle displacement power cos²κ and orange dashed/square velocity power sin²κ sum to 1. The dotted guide is the exact equal mean of cos²(κ + φ) for φ = 0, π/2, π and 3π/2, namely 1/2 for every κ. The lower green solid/diamond curve shows signed D. At κ/π = 0, 0.5, 1, 1.5, 2 and 3, D is respectively 1, 0, −1, 0, 1 and −1: both displacement signs give positive power maxima. Quarter-cycle samples also show complementary half powers. κ = 0 is a formal zero-wave-number limit, not an observed peak. Gravity, baryon loading, driving, diffusion, finite visibility and angular projection are omitted. No exact observed peak positions or unique inflation inference follows. Original Ark teaching model.

Projection and driving change the simple oscillator

Baryon inertia changes the sound speed and the relative response of compressions and rarefactions. Gravitational potentials can drive oscillations, while diffusion and the finite visibility width alter the surviving small-scale signals.

Temperature observations combine intrinsic temperature, velocities and gravitational terms along the line of sight. Converting three-dimensional wave modes into angular multipoles adds a projection that this κ-space plot has not performed.

The ideal κ maxima therefore cannot be relabelled as exact observed l peak locations. Nor do these normalized powers supply measured amplitudes, a fitted baryon density or an instrumental transfer function.

A useful first interpretation identifies the physical sound scale, phase relation and source contributions, then turns to a complete radiation-transfer model for quantitative comparison with the angular sky.

Worked reasoning

Compare the two powers at κ = π/2

Which ideal power is maximal at κ = π/2?

cos(π/2) = 0, so P_D = 0.
sin(π/2) = 1, so P_V = 1.
The equal four-phase mean displacement power remains 1/2, independent of κ.

The velocity quadrature power is maximal; the coherent displacement power vanishes in this ideal mode.

Try the reasoning

Does D = −1 imply negative power?
Show the answer and reasoning
No; P_D = 1.
Power is the square of the signed normalized displacement.
Are the plotted κ maxima exact observed angular peak locations?
Show the answer and reasoning
No.
A C_l prediction requires gravitational and baryon physics, source combination, projection and other transfer effects.
What is the supplied equal four-phase displacement-power mean at κ = π?
Show the answer and reasoning
1/2.
The same identity gives 1/2 at every κ, despite the coherent curve’s maximum there.

What to carry forward

A shared phase can preserve a sequence of ideal acoustic power maxima.
Signed displacement and nonnegative power contain different information.
An ideal κ-space oscillator is a first step toward, rather than a calculation of, the observed angular CMB spectrum.

References

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