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Matter power spectrum calculator

The power spectrum P(k) tells how strongly matter is clumped on each scale; σ₈ sums it up in one number.

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

Split the density fluctuations of the universe into waves of wavenumber k (wavelength 2π/k). P(k) is the variance of their amplitude: large where matter is strongly clumped on that scale. On the largest scales it follows the primordial spectrum, P ∝ k^n_{s}; it turns over at k_eq, the scale that entered the horizon at matter–radiation equality, because smaller scales could hardly grow while radiation dominated; and it carries the faint baryon acoustic wiggles left by sound waves before decoupling.

From the primordial spectrum to P(k)

Inflation, or whatever set the initial conditions, left curvature fluctuations with an almost scale-invariant spectrum: an amplitude A_s at the pivot scale k = 0.05 Mpc⁻¹ and a slight tilt n_s, measured by Planck as about 0.965. The transfer function T(k) then describes how each scale was processed while it entered the horizon: large scales entered late and kept their primordial shape, while small scales entered during radiation domination and grew far less. The growth factor D(z) finally scales every linear mode by the same amount, so the shape of P(k) is fixed early and only its height changes with redshift.

Change Ω_mh² and the turnover moves, because equality happens earlier or later; change Ω_bh² and the wiggles grow stronger while the small-scale power drops; change n_s and the whole spectrum tilts. Dark energy and curvature act only through D(z) in this linear picture.

The baryon acoustic wiggles

Before decoupling the baryons oscillated with the photons; afterwards they fell into the dark-matter wells, but they left a ripple in P(k) with a spacing set by the sound horizon r_d. The ripples are only a few per cent deep, so the second plot divides P(k) by the smooth no-wiggle form of Eisenstein & Hu: what remains are the oscillations. In real space they are the single bump at about 150 Mpc that galaxy surveys use as a standard ruler (see the BAO calculator).

Silk damping and the small-scale cutoff

On scales smaller than the photon diffusion length the baryon fluctuations were erased before decoupling (Silk damping). In P(k) this suppresses the wiggles beyond the damping wavenumber k_D, marked on the plot; dark matter, which does not feel photon pressure, keeps its fluctuations, so P(k) falls smoothly instead of being cut off; but because the baryons’ share was erased and later had to catch up, small-scale power is about half what it would be if all the matter were dark. The same physics produces the damping tail of the CMB spectrum.

σ₈ as one number

σ₈ is the rms fluctuation of matter in spheres of 8 h⁻¹ Mpc, an integral of P(k) weighted by the sphere’s window function. You can enter it, as many surveys quote it, or calculate it from the primordial amplitude A_s and tilt n_s, as the CMB measures them, under “Survey & CMB parameters”. The calculation uses the Eisenstein & Hu transfer function and your model’s growth factor, then a correction fitted to 1000 runs of the CAMB Boltzmann code that brings σ₈ to within 0.2 % of CAMB.

σ₈ from A_s

With ln(10¹⁰A_s) = 3.047, n_s = 0.9665 and Ω_bh² = 0.02242 (Planck 2018) and standard radiation (which changes σ₈ by only 0.04 % compared with Ω_r = 0), iCosmos calculates σ₈ = Classic concordance (iCosmos default): 0.8508; Planck 2018 flat ΛCDM: 0.8247; DESI DR2 w₀wₐ (DESI+CMB+DESY5): 0.8209; DESI DR2 w₀wₐ (DESI+CMB+Pantheon+): 0.8258; DESI DR2 w₀wₐ (DESI+CMB+Union3): 0.8150; DESI DR2 w₀wₐ (DESI+CMB): 0.7982; DESI DR2 + CMB flat ΛCDM: 0.8185; Open, matter only (Ωm = 0.3): 0.5008; Closed ΛCDM (Ωm = 0.3, ΩΛ = 0.9): 1.0623. Models without a Big Bang, without matter or outside the calibrated range are left out. Planck itself quotes σ₈ = 0.8102. Its model includes one neutrino of 0.06 eV, which does not cluster on small scales. With the same A_s, n_s, Ω_bh² and Ω_ch², CAMB gives 0.8103 with that neutrino and 0.8219 with massless neutrinos, 1.4 % higher (about 3 % in power). The Planck preset here also counts the neutrinos’ small density (Ω_νh² ≈ 0.0006) as clustering matter, which adds another 0.3 %.

Where linear theory stops

The plot is linear theory: every Fourier mode grows by the same factor D(z). Once fluctuations approach unity, here beyond k ≈ 0.2 h Mpc⁻¹ today, modes couple, structures collapse and the real P(k) rises above the linear one. Surveys model this with simulations or perturbation theory; iCosmos marks the scale and does not attempt it. At higher redshift the linear range extends to smaller scales.

Formula

• P(k, z) = (2π²/k³) (4/25) A_s (k/k_p)^n_{s−1} (k c/H₀)⁴ T(k)² (D(z)/Ω_m)², k_p = 0.05 Mpc⁻¹
• σ₈² = ∫ P(k) W²(kR) k³/(2π²) d ln k, R = 8 h⁻¹ Mpc, W(x) = 3(sin x − x cos x)/x³

Worked example

For the Planck 2018 model, matter–radiation equality is at z_eq = 3,404, which puts the turnover of P(k) near k_eq ≈ 0.015 h Mpc⁻¹, and the Silk damping wavenumber is k_D = 0.141 Mpc⁻¹. The section “σ₈ from A_s” below gives σ₈ calculated from Planck’s A_s and n_s.

Common questions

Why does P(k) turn over?

Fluctuations smaller than the horizon at matter–radiation equality entered it while radiation dominated, when matter could barely grow (the Mészáros effect). Larger scales entered later and kept growing, so the spectrum peaks near k_eq.

Is the plotted P(k) exact?

No. It uses the Eisenstein & Hu (1998) fitting formula, within 1.5 % of the CAMB Boltzmann code on average and 6 % at worst for Planck-like parameters (up to about 10 % for very low Ω_mh² or high baryon fractions), and linear theory, which fails beyond k ≈ 0.2 h Mpc⁻¹ today. The σ₈ number is corrected to CAMB to 0.2 %.

Should I enter σ₈ or calculate it?

Either. Surveys often quote σ₈ directly; the CMB measures A_s. “Calculate from A_s and n_s” links the two through your model’s transfer function and growth.

Related calculators

• σ₈, fσ₈ and S₈ calculator
• BAO distance calculator
• CMB acoustic scale calculator
• Linear growth factor 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 4.7.0 · updated 10 November 2026