---
title: "CMB Acoustic Peaks Interactive Lab | Sandbox Physics"
description: "How can different expanding universes leave almost the same acoustic pattern? The peaks depend strongly on the angle subtended by the sound horizon."
canonical_url: "https://sandboxphysics.com/labs/cmb-acoustic-peaks/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/cmb-acoustic-peaks/"
---

FR08 · Hearing the early universe

# CMB Acoustic Peaks

**How can different expanding universes leave almost the same acoustic pattern?** The peaks depend strongly on the angle subtended by the sound horizon. Different late expansion histories can preserve that angle and keep the peaks close, while changing the baryon density also changes their relative heights.

Change the baryon density and the late expansion. Compare the angular sound horizon with temperature spectra computed by CAMB, and find two universes whose peaks nearly coincide.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Model assumptions

The curves interpolate seventy-five offline CAMB 1.6.6 lensed scalar temperature spectra in flat constant-dark-energy models. Cold dark matter, primordial power, optical depth, neutrino mass and radiation content are fixed. This is a limited model grid, not observed Planck data or a likelihood fit. The circular wave is a projected sound-horizon schematic; its animated phase measures sound-travel distance, not cosmic time, and its colors do not predict a temperature map. The spectrum and distance readouts come from the same Boltzmann grid. Grid boundaries are enforced; its measured interpolation error and exact fixed parameters are listed below.

## The sound pattern in temperature power

Cyan: selected CAMB model. White: fixed reference cosmology. These are model predictions; no measured sky is plotted.

Lensed scalar temperature spectrum$D_\ell^{TT}=\ell(\ell+1)C_\ell^{TT}/(2\pi)\quad[\mu\mathrm K^2]$

> Temperature spectra interpolated from the frozen CAMB grid.

Grid, accuracy and fixed cosmology

CAMB 1.6.6, seventy-five grid nodes. Trilinear interpolation covers the physical baryon density, present expansion rate and constant dark-energy equation of state. The displayed multipoles are sampled every four.

$$
0.0204\leq\Omega_bh^2\leq0.0244,\quad60\leq H_0\leq76,\quad-1.2\leq w\leq-0.8
$$

$$
\Omega_ch^2=0.120,\quad\Omega_K=0,\quad\tau=0.0544,\quad A_s=2.1\times10^{-9},\quad n_s=0.9649
$$

$$
\sum m_\nu=0.06\,\mathrm{eV},\quad N_{\mathrm{eff}}=3.044,\quad T_{\mathrm{CMB}}=2.7255\,\mathrm K,\quad k_0=0.05\,\mathrm{Mpc}^{-1}
$$

Six withheld points recomputed at higher CAMB accuracy give a largest tested per-multipole relative error of 0.80% and a largest RMS relative error of 0.35% over multipoles thirty through seventeen ninety-eight. These tests do not bound every interior point. Tiny spectral differences can lie below the interpolation accuracy. Spectra are suitable for illustration, not precision inference.

The first peak location is read from the sampled spectrum, so its resolution is four multipoles. The angular sound scale uses the ratio of the interpolated comoving sound horizon and distance.

[Planck 2018 VI, version four: acoustic scale and geometric degeneracies](https://arxiv.org/abs/1807.06209v4)

Physics tutorial

## Why the acoustic peaks move

**Background** Before neutral atoms formed, photons repeatedly scattered from electrons and pushed the baryonic matter with them. Gravity pulled overdense regions inward; photon pressure pushed back. The coupled fluid carried sound waves.

**Why it matters** The early sound horizon supplies a length. The distance to the last-scattering surface turns that length into an angle. These two pieces let an early-universe pattern constrain later expansion.

Start with the essentials

- **Focus question**: How can two expansion histories give nearly the same peak positions?
- **One-sentence intuition**: Keeping the sound-horizon angle nearly fixed keeps the acoustic spacing nearly fixed.

### Core mathematical model

#### The sound horizon

$$
r_s(a_*)=\int_0^{a_*}\frac{c_s(a)}{a^2H(a)}\,da,\qquad c_s=\frac{c}{\sqrt{3(1+R)}},\quad R=\frac{3\rho_b}{4\rho_\gamma}
$$

The integral adds the comoving distance travelled by sound before last scattering. Baryons add inertia, lower the sound speed and alter compressions relative to rarefactions.

#### Projecting the ruler

$$
\theta_*=\frac{r_s(z_*)}{D_M(z_*)},\qquad\ell_A=\frac{\pi}{\theta_*}
$$

Both lengths are comoving. The acoustic spacing is set by the inverse angle. The first temperature peak has a phase shift and is not located exactly at this spacing.

#### Temperature power

$$
D_\ell^{TT}=\frac{\ell(\ell+1)}{2\pi}C_\ell^{TT}
$$

The temperature spectrum below is a lensed scalar prediction in microkelvin squared. The circular wave above only illustrates a projected sound horizon.

### Common difficulties

#### One angle and several parameters

**Typical misconception** The first peak determines the present expansion rate by itself.

**Better mental model** The angle is a ratio of two lengths. Different expansion histories can preserve that ratio. Other peak features and independent distance measurements help distinguish them.

### Run the experiment

1. 01

   #### Change the mass carried by sound

   Choose Heavier acoustic fluid and compare the cyan curve with the white reference.

   **What to observe:** The first peak grows relative to the second because baryons change the compression–rarefaction balance.

2. 02

   #### Move the ruler on the sky

   Choose Shift the angular scale, then drag horizontally across the circular wave or move Present expansion rate.

   **What to observe:** The projected horizon changes with the distance, and the temperature peaks move together.

3. 03

   #### Keep the angle nearly fixed

   Choose Nearly the same peak positions. Inspect the spectral residual below the stage.

   **What to observe:** Changing dark-energy pressure and the present expansion rate together preserves the angular scale closely. Low multipoles and lensing still retain differences.

Physical conclusion

Peak positions encode geometry; relative peak heights help identify the matter that oscillated. Combining these clues is more informative than reading one peak alone.

## Related experiments and articles

Cosmology from expansion to inference · Reading path

### Expansion, sound, sky statistics and inference

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **How small fluctuations become structure**](https://sandboxphysics.com/stories/cosmology/linear-perturbations/)
2. [Article **Why primordial sound leaves a series of peaks**](https://sandboxphysics.com/stories/cosmology/acoustic-oscillations/)
3. [Article **From three-dimensional waves to an angular spectrum**](https://sandboxphysics.com/stories/cosmology/transfer-and-degeneracy/)
4. [Article **What sky patterns imply about dark components**](https://sandboxphysics.com/stories/cosmology/dark-components-and-models/)
5. [Lab **Cosmology, Likelihoods and Priors** Separate estimation from model preference](https://sandboxphysics.com/labs/cosmology-likelihood-priors/)

Waves & optics · Starter path

## From one ray to interference measurement

Start with refraction, find where the ray picture stops working, then use interference to measure tiny distances.

[Start here → **Optical Boundary Explorer** Turn the incident angle and watch reflected and transmitted power trade off. Then switch to the critical-angle scene. Observe · 11 min](https://sandboxphysics.com/labs/optical-boundary-explorer/)

1. [1 Optical Boundary Explorer](https://sandboxphysics.com/labs/optical-boundary-explorer/)
2. [2 Rainbow Caustic Observatory](https://sandboxphysics.com/labs/rainbow-laboratory/)
3. [3 Camera Obscura](https://sandboxphysics.com/labs/camera-obscura/)
4. [4 Coherent Wave Composer](https://sandboxphysics.com/labs/coherent-wave-composer/)
5. [5 Young Double-Slit Bench](https://sandboxphysics.com/labs/young-double-slit-bench/)
6. [6 Michelson Precision Lab](https://sandboxphysics.com/labs/michelson-precision-lab/)

[View the full path →](https://sandboxphysics.com/topics/waves-and-optics/#starter-path)