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Home/Astronomy & The Sky/CMB Power Spectrum (Acoustic Peaks)

CMB Power Spectrum (Acoustic Peaks)

This interactive simulator explores CMB Power Spectrum (Acoustic Peaks) in Astronomy & The Sky. 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. Use the controls to change the scenario; watch the visualization and any graphs or readouts to connect the model with lectures, labs, and homework.

Who it's for: For learners comfortable with heavier math or second-level detail. Typical context: Astronomy & The Sky.

Key terms

  • cmb
  • power
  • spectrum
  • acoustic
  • peaks
  • cmb power spectrum
  • astronomy

ΛCDM cosmological parameters

0.0224
0.12
0.965
-8.67778070526608
0.054
0.674

Shortcuts

  • •Slide Ω_b h² and watch odd peaks rise / even peaks fall — this is how Planck weighed the baryons

Measured values

Ω_m h² (= Ω_b + Ω_c h²)0.1424
ℓ_A (sound-horizon angle)304
ℓ_D (Silk damping)1312
ℓ₁ (1st peak)222
ℓ₂ / ℓ₁2.47
D(ℓ₁)5734μK²
D(ℓ₂) / D(ℓ₁)0.61
D(ℓ₃) / D(ℓ₁)0.39

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}}