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CMB acoustic scale calculator

The acoustic angle θ_* is the angle the sound horizon at last scattering subtends on the sky, the best-measured of the cosmological parameters (Planck: 0.03 %).

The interactive calculator loads here: change the redshift and the cosmological parameters (Ω_m, Ω_Λ, Ω_r, w₀, w_a, H₀) and the result and plot update instantly.

What it is

About 370 000 years after the Big Bang, at a redshift of about 1090, the universe had cooled enough for electrons and protons to combine into neutral hydrogen. Photons stopped scattering and have travelled freely ever since: this is the cosmic microwave background (CMB), and the moment they last scattered defines a spherical shell around us, the last-scattering surface, at redshift z_*.

The acoustic angle

The CMB is not perfectly smooth. Sound waves in the earlier photon–baryon plasma left a characteristic size imprinted on its hot and cold spots: the sound horizon r_*, the comoving distance sound could travel before decoupling, about 145 Mpc. Seen from our position at the distance D_M(z_*) of the last-scattering surface, about 14 000 Mpc, it subtends the angle θ_* = r_*/D_M(z_*), close to 0.6°. iCosmos computes all three for your model.

The sound horizon as a standard ruler

Before decoupling, photons, electrons and baryons were tightly coupled into a single plasma. Gravity pulled it into denser regions and photon pressure pushed it back out, so every small disturbance launched a spherical sound wave. The sound speed was c/√(3(1 + R)), where R measures the inertia of the baryons: about c/√3 early on, but about 20 % lower by decoupling. At decoupling the photons were released and the waves stopped. The distance a wave had travelled by then, the sound horizon r_*, is fixed by well-understood physics and depends only on the physical densities of baryons and matter, Ω_bh² and Ω_mh². That makes it a standard ruler: an object of known size that can be seen at a known redshift.

The same ruler appears again in the distribution of galaxies as baryon acoustic oscillations. There the relevant scale is r_d at the slightly later drag epoch, when the baryons themselves were released; see the BAO calculator.

How θ* depends on curvature, dark energy and H₀

Because r_* is set by the early universe and D_M(z_*) by everything that happened since, θ_* ties the two together. Curvature has the largest effect: in a closed universe light rays from the last-scattering surface converge and the spots look larger (smaller ℓ_A); in an open universe they look smaller. Dark energy and H₀ change the distance through the late expansion history.

At fixed Ω_bh² and Ω_mh², many combinations of curvature, dark energy and H₀ give the same θ_*. This is the geometric degeneracy: the CMB alone cannot pin down curvature, and measurements of the late universe, such as BAO and H₀, are needed to break it. You can explore it here by raising H₀ while lowering Ω_m to keep Ω_mh² fixed, adjusting Ω_Λ to compensate, and watching θ_*.

From ℓ_A to the acoustic peaks

The CMB power spectrum shows a series of peaks at multipoles ℓ_m ≈ ℓ_A(m − φ_m). Odd peaks are compressions of the plasma in potential wells, even peaks are rarefactions. The phase shifts φ_m come from the driving of the oscillations by the decaying gravitational potential and from the way a three-dimensional pattern projects onto the sky; they are a few tenths and depend weakly on the matter and baryon densities (Hu, Fukugita, Zaldarriaga & Tegmark 2001, ApJ 549, 669). For Planck-like models the first peak lies at ℓ ≈ 220, about 0.73 ℓ_A.

Silk damping: why the pattern fades at small scales

Before decoupling, photons and baryons formed a single fluid, but not a perfect one. Photons scatter off electrons a finite number of times, so they random-walk a certain distance in the time available. On scales smaller than this diffusion length the photons leak out of hot, dense regions into cool, thin ones and smooth them away, carrying the baryons with them. The fluctuations on these small scales are therefore damped, roughly exponentially in (k/k_D)², where k_D is the damping wavenumber at decoupling. In the CMB spectrum this produces the damping tail: the acoustic peaks fade steadily towards high multipoles, and by the damping multipole ℓ_D the power is suppressed by roughly e⁻². The effect was predicted by Joseph Silk in 1968 (ApJ 151, 459), long before it was measured, and is now one of the cleanest tests of the physics of recombination. More baryons make the photons scatter more often, so they diffuse less far and the damping sets in at smaller scales; a larger distance to the last-scattering surface makes the same physical scale appear smaller on the sky, so ℓ_D grows. iCosmos computes k_D with a fit to CAMB and gives the damping length λ_D = 2π/k_D, the damping angle θ_D and ℓ_D = k_D D_M(z_*).

How iCosmos computes these numbers

z_*, r_* and k_D come from fits to the CAMB Boltzmann code in Ω_bh² and Ω_mh², accurate to a few parts in 10⁵ over a wide range. D_M(z_*) comes from the calculator’s own distance integral, so θ_* responds to curvature, dark energy and H₀ exactly as every other distance on the site does. Against CAMB with the same physics, θ_* agrees to about 2 × 10⁻⁵. This is θ_* as defined in CAMB, not θ_MC, an approximation used to sample the Planck chains (100 θ_MC = 1.04101 for the same parameters). Matter–radiation equality is 1 + z_eq = Ω_m/Ω_r. The Reference page gives the details and the tests.

Formula

• θ_* = r_* / D_M(z_*), ℓ_A = π / θ_*
• r_* = ∫ c_s(z) dz / H(z), from z_* to ∞, with c_s = c / √(3(1 + R)) and R = 3ρ_b/(4ρ_γ)
• Peak positions: ℓ_m ≈ ℓ_A (m − φ_m), with small phase shifts φ_m
• Silk damping: θ_D = π / (k_D D_M(z_*)), ℓ_D = k_D D_M(z_*)

Worked example

For the Planck 2018 model (H₀ = 67.66, Ω_m = 0.3111, Ω_bh² = 0.02242) with standard radiation added, iCosmos gives z_* = 1,089.84, r_* = 144.41 Mpc and D_M(z_*) = 13,880 Mpc, so 100 θ_* = 1.04041 (θ_* = 0.5961°) and ℓ_A = 302.

Planck publishes 100 θ_* = 1.04119 ± 0.00029 for these parameters. The 0.075 % difference comes from the neutrino: Planck includes one neutrino of 0.06 eV, which was still relativistic at decoupling, while iCosmos counts its present-day density as ordinary matter in Ω_m at all times and treats the remaining neutrinos as massless radiation. The damping multipole is ℓ_D = 1,956, about 6.5 times ℓ_A.

Common questions

Why is θ* so precisely measured?

The acoustic peak positions fix it almost independently of everything else: Planck measured 100 θ_* = 1.04119 ± 0.00029, to 0.03 %, the best-measured of the cosmological parameters.

Why does iCosmos ask me to add radiation?

Radiation still matters at z ≈ 1090: leaving it out changes the distance to the last-scattering surface by about 0.5 %, more than ten times the precision of Planck’s θ_*, so θ_* is shown only with the standard radiation density.

Does the first peak sit exactly at ℓ_A?

No: phase shifts put it near 0.73 ℓ_A, about ℓ = 220, for Planck-like models (Hu, Fukugita, Zaldarriaga & Tegmark 2001, ApJ 549, 669).

Related calculators

• BAO distance calculator
• Curvature of the universe calculator
• The Hubble tension
• Angular diameter distance calculator
• The full iCosmos cosmology calculator

About iCosmos

iCosmos was created by Mihran Vardanyan. Mihran Vardanyan is a cosmologist who earned his DPhil in cosmology at the University of Oxford, supervised by Joseph Silk and Roberto Trotta. He developed iCosmos during his research years at Oxford Astrophysics, where his work applied Bayesian model selection to the curvature and size of the Universe.

How to cite: Vardanyan, Mihran. iCosmos V4: Cosmology Calculator. https://icosmos.co.uk/. Include your access date, model parameters and exported data when reporting calculations.

iCosmos 4.6.0 · updated 10 November 2026