iCosmos › CMB worksheet

CMB worksheet: five investigations with the power spectrum

For use with the iCosmos CMB lab (icosmos.co.uk/cmb/). Open the lab, choose the task under “Guided tasks” and use its slider; the read-out under the slider, the model panel and the spectrum plot give the numbers asked for here. Each task takes about ten minutes. No installation or account is needed.

1. Weigh the baryons

This universe has too much ordinary matter. Bring the second peak back to its measured height relative to the first.

1. With Ω_bh² = 0.030, what is the height of the second peak divided by the height of the first? What is it for the Planck 2018 model?
2. Find the range of Ω_bh² for which the ratio is within 2 % of the Planck value. Give the lower and upper ends.
3. For the Planck value of Ω_bh² and H₀ = 67.66 km s⁻¹ Mpc⁻¹, what fraction Ω_b of the critical density is ordinary matter?

Answers

1. 0.364 with Ω_bh² = 0.030; 0.452 for the Planck 2018 model.
2. About 0.0218 to 0.0231.
3. Ω_b = 0.02242 ÷ 0.6766² = 0.049: about 5 % of the critical density.

Before the CMB was released, light and ordinary matter formed one fluid that oscillated in the gravitational wells of the dark matter. Baryons give that fluid weight without adding pressure, so it falls deeper into each well and compresses more than it rebounds. Odd peaks (first, third) are compressions; even peaks (second) are rarefactions. More baryons therefore raise the odd peaks relative to the even ones. The height of the second peak relative to the first is the cleanest measure of the baryon density, and the value it gives, Ω_bh² ≈ 0.0224, agrees with the amount of deuterium made in the first minutes: two independent weighings of the same ordinary matter, about 5 % of the universe.

2. Find the dark matter

This universe has too little dark matter. Use the third peak to find how much there really is.

1. With Ω_ch² = 0.075, what is the height of the third peak divided by the height of the first? What is it for the Planck 2018 model?
2. Find the range of Ω_ch² for which the ratio is within 3 % of the Planck value.
3. For the Planck values, how many times more dark matter than ordinary matter is there (Ω_ch² ÷ Ω_bh²)?

Answers

1. 0.374 with Ω_ch² = 0.075; 0.443 for the Planck 2018 model.
2. About 0.1104 to 0.1294.
3. 5.32 times.

With less matter, radiation is still an important part of the density when the CMB is released. Radiation’s gravity decays as the waves oscillate, and the decaying wells give each wave a push: the peaks are boosted, the first ones most. Dark matter does not feel the radiation pressure, so its wells survive, and the more of it there is, the smaller the boost. A universe containing only the baryons measured in the first task would have a third peak far lower than the second; the measured third peak is almost as high as the second. That is direct evidence, from the CMB alone, that most matter is not ordinary matter: about five times more dark matter than baryons.

3. Measure the curvature

This universe is open (Ω_k = +0.08). Change the curvature until the first peak is where Planck sees it.

1. At what multipole (to the nearest whole number) is the first peak for Ω_k = +0.08, for Ω_k = 0 and for Ω_k = −0.08 (Ω_m and H₀ at the Planck values)?
2. Find the range of Ω_k that puts the first peak within ±3 multipoles of the Planck position.
3. By how many multipoles does the first peak move for a change of 0.01 in Ω_k near Ω_k = 0?

Answers

1. About ℓ = 244 (open), 220 (flat) and 195 (closed).
2. About −0.009 to +0.009.
3. About 3 multipoles: to higher ℓ for an open universe, to lower ℓ for a closed one.

The peaks mark a fixed length: the distance sound could travel before the CMB was released, about 145 Mpc today. What we measure is the angle that length covers on the sky, θ_* = r_*/D_M(z_*). Curvature changes D_M, the distance that converts a length into an angle. In an open universe the same length looks smaller, so the peaks move to higher ℓ; in a closed universe it looks larger and they move to lower ℓ. The first peak near ℓ = 220 is what a nearly flat universe gives for these densities. As the last task shows, this conclusion needs care: the position of the peaks fixes one combination of the parameters, not the curvature alone.

4. Tilt and the damping tail

The primordial tilt n_s is too low here. Use the points beyond ℓ = 1000 as well as those below ℓ = 500 to set it.

1. With n_s = 0.93, by what percentage does the spectrum differ from the Planck 2018 model at ℓ = 100 and at ℓ = 1500?
2. For the Planck 2018 model, what is the height of the spectrum at ℓ = 1500 as a fraction of the height of the first peak? (Read both from the spectrum plot.)
3. What is the damping multipole ℓ_D of the Planck 2018 model? (In the iCosmos calculator: choose the preset “Planck 2018 + BAO”, “Add CMB radiation”, then “Show more quantities”.)

Answers

1. +5.6 % at ℓ = 100 and −3.2 % at ℓ = 1500.
2. 0.12: at ℓ = 1500 the spectrum is about 12 % of the height of the first peak.
3. ℓ_D = 1956.

Beyond ℓ ≈ 1000 the peaks fade. In the plasma, photons wandered between hot and cold regions, further and further as it became transparent, and erased differences on small scales: this is diffusion (Silk) damping. Its scale is set by the densities already fixed in the earlier tasks, so the height of the damping tail compared with the first peaks measures something else: how the primordial fluctuations were shared between large and small scales. That is the tilt n_s. A value of 1 would mean equal strength on all scales; Planck finds 0.965, slightly more on large scales, as the simplest inflation models predicted. Without the damping tail, n_s would be several times less certain.

5. The geometric degeneracy

Can the CMB tell a flat universe from a curved one? Find curved universes that Planck’s spectrum cannot distinguish from the flat one, including one with H₀ = 73.

1. Curvature alone. Switch off “Keep the peaks fixed” and set Ω_k = −0.020. By how many multipoles has the first peak moved, and what is the χ² of Planck’s temperature points compared with the flat model’s?
2. On the line. Switch on “Keep the peaks fixed” and go to each end of the slider. At each end give H₀, the age of the universe, and the χ² of the CMB points and of the BAO distances. Which of the two measurements tells these universes from the flat one?
3. H₀ = 73. Find the universe on the line with H₀ = 73 km s⁻¹ Mpc⁻¹, the value measured with the distance ladder. What is its Ω_k? Which measurement rules it out, and what would have to be different for it to survive?

Answers

1. The first peak moves by −6 multipoles (to lower ℓ: this universe is closed). Planck’s spectrum no longer fits: χ² = 6331 for 65 points, against 58 for the flat model.
2. Ω_k = −0.030: H₀ = 57.0, age 14.9 billion years; CMB χ² = 51, BAO χ² = 1747. Ω_k = +0.020: H₀ = 81.1, age 12.6 billion years; CMB χ² = 76, BAO χ² = 1234. Flat model: H₀ = 67.7, 13.8 billion years, CMB χ² = 58, BAO χ² = 25 (13 points). The CMB fits all three (slightly better at the closed end, slightly worse at the open end; the peaks have not moved). The BAO distances rule out both ends.
3. Between Ω_k = +0.009 (H₀ = 72.6) and +0.010 (H₀ = 73.3): a slightly open universe. The CMB fits it (χ² = 64 and 65); the BAO distances rule it out (χ² = 142 and 189, against 25 for the flat model). To survive, the BAO sound horizon r_d would have to be shorter than the 147 Mpc that standard physics gives: about 142 Mpc for this universe (the BAO χ² would then be 22). For a flat universe with H₀ = 73, DESI’s H₀r_d gives about 139 Mpc. A shorter sound horizon needs new physics before the CMB was released. (The 144 Mpc in the task’s sketch is r_*, the same sound wave at the slightly earlier moment the CMB shows.)

The CMB measures an angle: the sound horizon divided by the distance to the CMB. Curvature changes that distance, but so does the Hubble constant, and the two can be traded so that the angle stays put. Planck’s spectrum fits a closed universe with H₀ = 57, the flat one with 67.7 and an open one with 81 almost equally well. The small differences are gravitational lensing, which tilts Planck slightly towards the closed side, and the late integrated Sachs–Wolfe effect at the largest scales, hidden by cosmic variance. So the CMB’s precise H₀ ≈ 67 assumes a flat universe. A second measurement breaks the tie: BAO surveys see the same sound horizon in the galaxy distribution and, with its standard length, rule out both ends of the line and the universe with H₀ = 73. Only a shorter ruler, new physics before the CMB was released, could change that.

The spectrum in the lab is computed by an emulator of the CAMB code and is accurate to a few tenths of a percent, so answers read from the lab may differ from the key in the last digit.

Open the CMB lab

Open the iCosmos cosmology calculator

iCosmos 4.11.1 · updated 12 November 2026