Why do odd and even acoustic peaks differ in height?

Baryons add inertia to the photon–baryon fluid, boosting compression (odd) peaks relative to rarefaction (even) peaks. Raising Ω_b h² therefore enhances the odd/even contrast. Dark matter Ω_c h² mainly shifts the peak positions via the expansion history and gravitational potentials without the same parity effect.

What is the Silk damping tail?

Photon diffusion before recombination erases small-scale temperature fluctuations, so D_ℓ falls at high ℓ. That exponential-like cutoff is Silk damping. If the spectrum stayed flat or rose forever, the model would be missing diffusion physics — a common conceptual gap after learning only the peak series.

Is this the same as fitting Planck data with CAMB?

No. This is a teaching parameterization that moves peaks and the Sachs–Wolfe plateau qualitatively when you change Ω_b h², Ω_c h², n_s, A_s, τ, and h. Precision cosmology needs Boltzmann solvers, polarization, lensing, and full likelihoods. Use the sliders to build intuition, not publish parameter posteriors.

CMB Power Spectrum (Acoustic Peaks)

Cosmic Microwave Background temperature D_ℓ vs ℓ with Sakharov peaks: tune Ω_b h², Ω_c h², n_s, A_s, τ, h and watch the parity flip between odd / even peaks, the Silk damping tail, and the Sachs–Wolfe plateau move. Pedagogical parametric ΛCDM model.

About this model

A pedagogical parametric ΛCDM model of the CMB temperature spectrum plots D_ℓ versus multipole ℓ, highlighting Sakharov acoustic peaks. You tune baryon density Ω_b h², cold dark matter Ω_c h², scalar index n_s, amplitude A_s, optical depth τ, and Hubble parameter h. Odd/even peak-height parity flips with baryon loading, the Silk damping tail suppresses power at high ℓ, and the Sachs–Wolfe plateau sets the large-scale amplitude. The fit is illustrative — not a Boltzmann-code CAMB/CLASS replacement — so neutrino hierarchies, lensing, and polarization spectra are omitted. Changing each parameter moves peaks, damping, and the plateau so the geometric and matter-content imprint on the early plasma becomes readable.

Who it's for: Advanced cosmology and astrophysics students learning CMB acoustic peaks and ΛCDM parameters.

Key terms

CMB power spectrum
acoustic peaks
ΛCDM
Silk damping
Sachs-Wolfe plateau
baryon density

How it works

CMB temperature power spectrum — D_ℓ = ℓ(ℓ+1)C_ℓ/2π in μK² as a function of multipole ℓ — the cornerstone observable of precision cosmology. The simulator is a *parametric* model (no Boltzmann code), but it captures the qualitatively correct response of the spectrum to the six standard ΛCDM parameters: Sachs–Wolfe plateau at ℓ ≲ 30, acoustic / Sakharov peaks at ℓ ≈ 220, 540, 810, … set by the angular sound horizon ℓ_A; baryon-loading parity (raising Ω_b h² boosts odd peaks and suppresses even ones), the Silk damping tail at high ℓ, the scalar tilt n_s, and reionisation suppression e^{−2τ}. Slide Ω_b h² and watch the second peak shrink relative to the first — that's how Planck pinned the baryon density.

Key equations

D_ℓ ≡ ℓ(ℓ+1) C_ℓ / 2π (μK²)

ℓ_n ≈ n · ℓ_A, ℓ_A = π d_A(z✳) / r_s(z✳)

C_ℓ ∝ A_s (ℓ/200)^{n_s−1} · e^{−2τ} · e^{−(ℓ/ℓ_D)^{1.4}}

Frequently asked questions

Why do odd and even acoustic peaks differ in height?: Baryons add inertia to the photon–baryon fluid, boosting compression (odd) peaks relative to rarefaction (even) peaks. Raising Ω_b h² therefore enhances the odd/even contrast. Dark matter Ω_c h² mainly shifts the peak positions via the expansion history and gravitational potentials without the same parity effect.
What is the Silk damping tail?: Photon diffusion before recombination erases small-scale temperature fluctuations, so D_ℓ falls at high ℓ. That exponential-like cutoff is Silk damping. If the spectrum stayed flat or rose forever, the model would be missing diffusion physics — a common conceptual gap after learning only the peak series.
Is this the same as fitting Planck data with CAMB?: No. This is a teaching parameterization that moves peaks and the Sachs–Wolfe plateau qualitatively when you change Ω_b h², Ω_c h², n_s, A_s, τ, and h. Precision cosmology needs Boltzmann solvers, polarization, lensing, and full likelihoods. Use the sliders to build intuition, not publish parameter posteriors.

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