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.

Angular sound scale $100\theta_* = 1.04$

First temperature peak $\ell_1 = 222$

First / second peak height $2.21$

Comoving sound horizon $144\,\mathrm{Mpc}$

Change the cosmic ingredients

Physical baryon density

0.0224

\Omega_b h^2

Present expansion rate

67.4

H_0\;[\mathrm{km\,s^{-1}\,Mpc^{-1}}]

Dark-energy pressure / density

-1

w=p_{\mathrm{de}}/\rho_{\mathrm{de}}

Drag horizontally across the wave to change the present expansion rate. The white reference spectrum stays fixed.

What the wave represents

The circular impulse sketches how a sound horizon projects onto the sky. Its phase follows a fraction of the final sound-travel distance. The temperature spectrum is independently calculated from the same cosmology by CAMB.

Follow the cosmology course

CAMB power-spectrum and distance conventions

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.

Lensed scalar temperature spectrum

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

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

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 misconceptionThe first peak determines the present expansion rate by itself.

Better mental modelThe 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

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.
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.
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.

CMB Acoustic Peaks

The sound pattern in temperature power

Why the acoustic peaks move

Core mathematical model

The sound horizon

Projecting the ruler

Temperature power

Common difficulties

One angle and several parameters

Run the experiment

Change the mass carried by sound

Move the ruler on the sky

Keep the angle nearly fixed

Related experiments and articles

Expansion, sound, sky statistics and inference