iCosmos › σ₈, fσ₈ and S₈ calculator

σ₈, fσ₈ and S₈ calculator

Q: What is the S₈ tension?

Several weak-lensing surveys found S8 a few per cent below the value predicted from the Planck CMB in ΛCDM. The final Kilo-Degree Survey analysis (KiDS-Legacy, Wright et al. 2025, arXiv:2503.19441) agrees with the Planck prediction, so whether a real tension exists is still debated.

Q: Why is σ₈(z) not shown for my model?

σ8(z) and fσ8 need the growth factor, which iCosmos computes only for Ωm > 0, Ωr = 0 and a universe with a matter-dominated past. S8 needs only Ωm and is always shown.

Q: Why 8 h⁻¹ Mpc?

Historically, galaxy counts in spheres of 8 h−1 Mpc had a variance close to one, so it became the standard scale for quoting the amplitude of fluctuations.

σ₈ measures how clumpy matter is on scales of 8 h⁻¹ Mpc; fσ₈ and S₈ are the combinations galaxy surveys measure.

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

σ₈ is the root-mean-square fluctuation of the matter density today, smoothed in spheres of radius 8 h⁻¹ Mpc, in linear theory. It sets the overall amplitude of structure; you choose it under “Survey & CMB parameters” (default: Planck 2018, σ₈ = 0.8102).

Earlier in cosmic history fluctuations were smaller: σ₈(z) = σ₈ D(z)/D(0), where D is the linear growth factor. Redshift-space distortions in galaxy surveys measure fσ₈(z), the growth rate f = d ln D/d ln a times σ₈(z). Weak gravitational lensing best measures S₈ = σ₈ (Ω_m/0.3)^1/2.

iCosmos computes D and f by solving the linear growth equation, assuming dark energy does not cluster, for models without radiation (Ω_r = 0).

Formula

σ₈(z) = σ₈ D(z) / D(0)
fσ₈(z) = f(z) σ₈(z), f = d ln D / d ln a ≈ Ω_m(z)^0.55 in ΛCDM
S₈ = σ₈ (Ω_m / 0.3)^1/2

Worked example

For the Planck 2018 model (Ω_m = 0.3111, σ₈ = 0.8102) S₈ = 0.8251, matching Planck’s published 0.825.

At z = 0.5 the amplitude was σ₈(z) = 0.6239 and fσ₈ = 0.4729; at z = 1, σ₈(z) = 0.4926 and fσ₈ = 0.431. fσ₈ changes slowly with redshift because f rises as σ₈(z) falls.

Common questions

What is the S₈ tension?

Several weak-lensing surveys found S₈ a few per cent below the value predicted from the Planck CMB in ΛCDM. The final Kilo-Degree Survey analysis (KiDS-Legacy, Wright et al. 2025, arXiv:2503.19441) agrees with the Planck prediction, so whether a real tension exists is still debated.

Why is σ₈(z) not shown for my model?

σ₈(z) and fσ₈ need the growth factor, which iCosmos computes only for Ω_m > 0, Ω_r = 0 and a universe with a matter-dominated past. S₈ needs only Ω_m and is always shown.

Why 8 h⁻¹ Mpc?

Historically, galaxy counts in spheres of 8 h⁻¹ Mpc had a variance close to one, so it became the standard scale for quoting the amplitude of fluctuations.

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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.