S8 Tension

The clumpiness contest

Imagine two ways of weighing how lumpy the universe has become. The first looks at the universe as a baby — the cosmic microwave background, the light released 380,000 years after the Big Bang, when the cosmos was a nearly uniform plasma with density ripples of just one part in 100,000. From those tiny seeds, and the known laws of gravity, you can run the clock forward 13.8 billion years and predict exactly how clustered matter should be today. The second way measures the grown-up universe directly: you map where the dark matter actually sits now by watching how its gravity distorts the shapes of distant galaxies.

These two answers ought to match. The whole point of the standard model of cosmology — six numbers, a cosmological constant, cold dark matter, known as ΛCDM — is that one set of parameters fit to the infant universe should describe everything that follows. And for most observables they agree beautifully. But on this one question — how clumpy is the universe right now? — the grown-up keeps coming back about 8 percent smoother than the baby predicted. That persistent, modest mismatch is the S8 tension. It is not yet a crisis. It is a splinter that refuses to come out.

What the number S8 actually means

Cosmologists quantify clumpiness with two parameters. The first is σ8 (sigma-eight): the root-mean-square fluctuation in the matter density, smoothed over spheres of radius 8 h⁻¹ megaparsecs (roughly 35 million light-years for h ≈ 0.7, a scale chosen historically because it gives σ8 ≈ 1 for galaxies). The second is Ωm, the fraction of the universe's energy density that is matter (dark plus baryonic), about 0.31 today.

Weak lensing is sensitive to a particular blend of the two, because both a denser universe and a more clustered one bend light more. The degeneracy traces a banana-shaped contour in the σ8–Ωm plane. The combination that lies across the short axis of that banana — the thing lensing measures crisply — is

S8 = σ8 √(Ωm / 0.3)

So S8 is not a new physical quantity; it is the projection of clumpiness onto the direction a lensing survey can actually pin down. Reporting S8 instead of σ8 is the difference between quoting a number with a 1 percent error bar and one with a 10 percent error bar. Every cosmic-shear paper leads with S8 for exactly this reason.

The two camps, in numbers

Here is the heart of the disagreement — early-universe predictions versus late-universe measurements:

Probe	Survey / dataset	S8	Epoch probed	Method
CMB (primary)	Planck 2018 (TT,TE,EE+lowE)	0.834 ± 0.016	z ≈ 1100	Acoustic peaks + ΛCDM growth
CMB lensing	ACT DR6 + Planck	0.83 ± 0.02	z ≈ 0.5–5	Lensing of the CMB itself
Cosmic shear	KiDS-1000	0.766 (+0.020/−0.014)	z ≈ 0.1–1.2	Galaxy shape distortion
Cosmic shear	DES Year 3	0.759 (+0.025/−0.023)	z ≈ 0.2–1.3	Galaxy shape distortion
Cosmic shear	HSC Year 3	0.78 ± 0.03	z ≈ 0.3–1.5	Galaxy shape distortion
3×2pt	DES Y3 (shear+clustering+GGL)	0.776 ± 0.017	z ≲ 1	Joint lensing + galaxy clustering

The pattern is unmistakable: every direct measurement of late-time structure lands near 0.76–0.78, while the early-universe extrapolation sits at 0.83. The CMB-lensing entries are interesting outliers — they are late-time probes too, yet they side with the high value, a clue we return to below.

How weak lensing weighs the smoothness

The late-universe number comes from cosmic shear. As light from a distant galaxy travels to us, it passes through the lumpy cosmic web. Each over-dense region bends the light slightly, so the galaxy's observed image is stretched — typically by about 1 percent, far too small to see in any single galaxy whose intrinsic ellipticity is ~30 percent. The trick is statistics: average the shapes of millions of galaxies, and the random intrinsic orientations cancel while the coherent lensing distortion survives. The result is a map of the foreground mass, dark matter included.

The amplitude of those correlated distortions encodes how much structure lies between us and the source galaxies — directly measuring S8. The key relation linking the lensing convergence κ to the matter overdensity δ along the line of sight is

κ(θ) = (3 H₀² Ωm / 2c²) ∫₀^χs  [ (χs − χ)χ / χs ]  δ(χθ, χ) / a(χ)  dχ

where χ is comoving distance, χs the distance to the source, and a the scale factor. The shear two-point correlation function ξ±(θ) — how aligned the distortions of two galaxies separated by angle θ are — is the observable. Its overall amplitude scales roughly as σ8² Ωm^1.5, which is why the well-measured combination is S8.

Why this is really a test of how gravity grows structure

Underneath the bookkeeping, the S8 tension is a test of structure growth. In a matter-dominated universe, density perturbations grow under their own gravity, with the linear growth factor D(a) governed by

δ'' + 2H δ' − 4πG ρ_m δ = 0
growth rate:  f = d ln D / d ln a ≈ Ωm(a)^0.55   (for ΛCDM)

The exponent γ ≈ 0.55 is the "growth index" of general relativity. Anything that slows growth — modified gravity with γ > 0.55, dark energy switching on earlier, free-streaming neutrinos, or dark matter that decays or self-interacts — leaves the late universe smoother than the CMB seeds plus standard gravity would give. So a genuine low S8 would be telling us that perturbations grew more slowly than ΛCDM predicts somewhere between z ≈ 1100 and z ≈ 0. That is why theorists treat S8 as a window onto the growth side of cosmology, complementary to the expansion side probed by H0.

Worked example: how much structure is "missing"?

Take the gap at face value: S8(lensing) ≈ 0.766 versus S8(Planck) ≈ 0.834. Since lensing power scales as S8², the ratio of clustering amplitudes is

(0.766 / 0.834)² = 0.918² ≈ 0.84
→ the late-universe lensing signal is ~16% lower in power,
  or ~8% lower in S8 amplitude.

To gauge significance, combine the error bars in quadrature. Planck's σ ≈ 0.016, KiDS's ≈ 0.017:

ΔS8 = 0.834 − 0.766 = 0.068
σ_combined = √(0.016² + 0.017²) = √(0.000256 + 0.000289)
           = √0.000545 ≈ 0.0233
significance = 0.068 / 0.0233 ≈ 2.9σ

So the headline KiDS-1000 versus Planck comparison sits near 2.9σ — a roughly 1-in-260 chance of being a fluke if the model is right. That is enough to keep cosmologists awake but well short of the 5σ (1-in-3.5-million) bar for claiming a discovery. The 3×2pt combinations and the latest reanalyses push it down toward 1.5–2σ, which is precisely why the tension is described as "soft."

Three families of explanation

• It's a systematic (the dull but likely answer). Baryonic feedback from supernovae and AGN blows gas out of halos, suppressing the matter power spectrum by 10–20 percent on small scales (k > 1 h Mpc⁻¹) — exactly where shear is most sensitive. Add photometric-redshift errors, intrinsic galaxy alignments, and shape-measurement bias, and a spurious low S8 is easy to manufacture. When surveys discard small scales or marginalise over feedback, the tension shrinks.
• It's a statistical fluctuation. At 2–3σ, the tension is not improbable enough to rule out simple bad luck, especially given the "look-elsewhere" effect of comparing many datasets. The fact that CMB lensing (a late-time probe) agrees with Planck rather than with cosmic shear is a hint that the galaxy-shape measurements, not the cosmology, may be the issue.
• It's new physics. If the smoothness is real, candidates include massive neutrinos (Σmν suppresses small-scale growth), decaying or self-interacting dark matter, modified gravity with a growth index > 0.55, or dynamical dark energy. The catch: many of these worsen the Hubble tension, and none yet fits all data better than ΛCDM at high significance.

S8 tension vs the Hubble tension

The S8 tension is often paired with the more famous Hubble tension. They are cousins — both pit the early universe against the late universe — but they probe different physics and have very different temperatures.

Feature	S8 tension	Hubble tension
Physical quantity	Clumpiness S8 = σ8√(Ωm/0.3)	Expansion rate H0
What it tests	Growth of structure	Expansion history / distance scale
Early-universe value	S8 ≈ 0.834 (Planck)	H0 ≈ 67.4 km/s/Mpc (Planck)
Late-universe value	S8 ≈ 0.76 (lensing)	H0 ≈ 73 km/s/Mpc (Cepheids+SNe)
Significance	≈ 2–3σ (and softening)	≈ 4–6σ (and firming)
Direction of mismatch	Late universe smoother	Late universe expands faster
Easy systematic culprit	Baryonic feedback, photo-z	Cepheid calibration, dust
Joint fix exists?	Rarely — early dark energy fixes H0 but worsens S8

This anti-correlation is the deepest reason the field is stuck. Early dark energy, the most popular Hubble-tension fix, shrinks the sound horizon and raises H0 — but it also boosts early structure growth, pushing predicted S8 up, away from the lensing data. A model that calms both tensions at once would be a genuine theoretical coup, and none has yet been found.

The surveys behind the measurements

• KiDS (Kilo-Degree Survey). A VST optical survey of ~1000 deg², famous for its KiDS-1000 cosmic-shear analysis (S8 ≈ 0.766) that has been the flagship low-S8 result. The 2024 KiDS-Legacy release, with improved redshift calibration over ~1350 deg², moved the value up and the tension down to ~1σ.
• DES (Dark Energy Survey). A 5000 deg² survey from the Blanco 4-m at CTIO. Its Year-3 "3×2pt" analysis (cosmic shear + galaxy clustering + galaxy-galaxy lensing) over ~100 million galaxies gives S8 ≈ 0.776, only mildly below Planck.
• HSC (Hyper Suprime-Cam). A deep Subaru survey reaching fainter, higher-redshift galaxies; its Year-3 shear gives S8 ≈ 0.78, consistent with KiDS and DES.
• Planck and ACT. The early-universe anchors. Planck's primary CMB gives S8 ≈ 0.834; ACT DR6 + Planck CMB lensing — a late-time probe of the same field — gives ≈ 0.83, siding with the high value.
• The next generation. The Vera C. Rubin Observatory (LSST), ESA's Euclid (launched July 2023, mapping ~1.5 billion galaxies over 14,000 deg²), and NASA's Roman Space Telescope will shrink the S8 error by several-fold in the late 2020s and early 2030s — the data that will decide whether the tension is physics or systematics.

Common misconceptions and edge cases

• "S8 measures the total amount of dark matter." No — that is Ωm. S8 measures how that matter is arranged: the amplitude of its clustering on ~30-million-light-year scales. You can have the right amount of dark matter distributed too smoothly.
• "A low S8 means there is less structure than we see." The universe plainly has galaxies, clusters, and a cosmic web. The tension is a few-percent difference in the statistical amplitude of clustering, not a missing-galaxies problem.
• "It's basically the same thing as the Hubble tension." They are independent. One is about expansion (H0), the other about growth (S8). A model can solve one and aggravate the other — which is exactly what early dark energy does.
• "CMB lensing should agree with cosmic shear since both are lensing." They lens different sources (the CMB versus galaxies) over different redshift ranges and with different systematics. CMB lensing currently agrees with Planck's high S8, which is itself a strong hint that galaxy-shear systematics may drive the low values.
• "At 3σ it's already a discovery." Particle-physics and cosmology convention reserves "discovery" for 5σ. A 2–3σ tension has, historically, often melted away with better data — and the recent reanalyses are trending that way.