The S8 / σ8 Tension

The concordance cosmology tells a two-act story about how the Universe grew clumpy. In the first act, at recombination, the CMB anisotropy power spectrum fixes the primordial amplitude of curvature perturbations $A_{\mathrm{s}}$ and the matter density $\Omega_{\mathrm{m}}$ ; in the second, the linear growth of structure and the transfer function play those seeds forward under $\Lambda$ CDM to a definite prediction for how strongly matter should cluster today. The number summarising "today's clumpiness" is $\sigma_8$ , the rms linear density contrast smoothed on spheres of comoving radius $8\,h^{-1}\,\mathrm{Mpc}$ . The $S_8$ tension is the finding that low-redshift probes — above all weak gravitational lensing, or cosmic shear — measure a value slightly below what the CMB, extrapolated forward, predicts. The parameter that makes the comparison sharp is not $\sigma_8$ itself but $S_8 \equiv \sigma_8(\Omega_{\mathrm{m}}/0.3)^{0.5}$ . Weak lensing cannot separate "how much matter" from "how clumpy" — a lot of smooth matter deflects light much like a little clumpy matter — so its constraints run along a banana-shaped degeneracy of nearly constant $\sigma_8\,\Omega_{\mathrm{m}}^{0.5}$ , and collapsing that ridge onto its well-measured direction is exactly what $S_8$ does. Planck 2018 gives $S_8 = 0.834 \pm 0.016$ ; KiDS-1000 and DES Y3 cluster near $S_8 \approx 0.76$ – $0.78$ . That $\sim 0.06$ offset is a $2$ – $3\sigma$ discrepancy — modest, systematics-limited, unlike the sharper Hubble tension — yet persistent enough to be treated as a real crack worth watching in the otherwise triumphant $\Lambda$ CDM picture.

Key takeaways

The $S_8$ tension is a mild ( $\sim 2$ – $3\sigma$ ) disagreement in the amplitude of matter clustering: the CMB extrapolated to $z=0$ under $\Lambda$ CDM predicts more clumping than low-redshift weak lensing measures.
The clean comparison variable is $S_8 \equiv \sigma_8(\Omega_{\mathrm{m}}/0.3)^{0.5}$ — the combination lensing constrains best, orthogonal to the $\sigma_8$ – $\Omega_{\mathrm{m}}$ degeneracy.
Numbers: Planck 2018 gives $S_8 = 0.834 \pm 0.016$ ; KiDS-1000 gives $0.766^{+0.020}_{-0.014}$ and DES Y3 gives $\approx 0.776$ — a $\sim 2.9\sigma$ pull.
The significance is systematics-limited: baryonic feedback, intrinsic alignments, photometric-redshift errors, and non-linear modelling each move $S_8$ at the percent level, comparable to the tension itself.
Physically, a low $S_8$ at fixed early-Universe seeds means late-time growth $D(a)$ ran slow — inviting modified gravity, interacting or free-streaming dark matter, or dynamical dark energy, though none yet beats $\Lambda$ CDM.
$S_8$ is the late-time counterpart of the primordial amplitude $A_{\mathrm{s}}$ : it tests the integrated growth of structure across nearly the whole history of the Universe.
Key idea: the CMB fixes the primordial amplitude and lets $\Lambda$ CDM predict today's clustering; weak lensing measures it directly, and their values of $S_8 = \sigma_8(\Omega_{\mathrm{m}}/0.3)^{0.5}$ differ mildly, hinting that either late-time growth is suppressed or lensing systematics are underestimated.

The core insight

The CMB fixes the primordial amplitude and lets $\Lambda$ CDM predict today's clustering; weak lensing measures it directly. Their preferred values of $S_8 = \sigma_8(\Omega_{\mathrm{m}}/0.3)^{0.5}$ differ mildly, hinting that either late-time growth is suppressed or lensing systematics are underestimated.

Where this lives today

The $S_8$ tension is one of two headline anomalies driving observational cosmology into the mid-2020s, alongside the sharper Hubble tension. Stage-IV weak-lensing surveys are built to settle it: the Euclid satellite, launched in 2023, and the Vera C. Rubin Observatory's LSST each map cosmic shear over thousands of square degrees with tomographic redshift binning, shrinking the statistical error on $S_8$ well below the size of the tension and forcing the question onto systematics — feedback, intrinsic alignments, photometric-redshift calibration. In parallel, CMB-lensing from ACT and SPT, cross-correlated with galaxy surveys, localises where in cosmic history any growth suppression sets in; strikingly, the ACT DR6 and Planck CMB-lensing amplitudes come out consistent with the high, CMB-predicted $S_8$ , pushing the discrepancy toward the low-redshift shear regime and its astrophysics. Redshift-space distortions from DESI and Euclid deliver $f\sigma_8(z)$ , an independent growth-rate handle. Notably, the tension has been softening: joint KiDS-plus-DES reanalyses land near $S_8\approx0.79$ , and improved feedback and redshift modelling nudge shear values upward — so the honest verdict remains that $S_8$ is a suggestive crack, not yet a broken model.

The physical picture

Think of the primordial power spectrum as a catalogue of seed fluctuations, photographed by the CMB at $z\simeq1100$ . $\Lambda$ CDM plays those seeds forward with a growth rate $D(a)$ that linear theory computes exactly, and $\sigma_8$ is a snapshot of how tall the resulting weeds have grown today on $8\,h^{-1}\,\mathrm{Mpc}$ scales. The tension is that the CMB predicts slightly taller weeds than the lensing surveys see growing — the low-redshift ground looks a touch smoother than the early-Universe model insists. The trouble is that weak lensing does not measure $\sigma_8$ and $\Omega_{\mathrm{m}}$ separately: the shear imprinted on background galaxies depends on the total deflecting mass along the line of sight, and a smooth-but-abundant matter field mimics a clumpy-but-sparse one. So the constraint is a banana-shaped ridge of nearly constant $\sigma_8\,\Omega_{\mathrm{m}}^{0.5}$ , and only the direction across it — the combination $S_8$ — is pinned down. The first figure is that coordinate rotation made visible: the lensing banana, the Planck point sitting above it, and the $S_8$ axis along which the two disagree.

Figure

The weak-lensing degeneracy made visible: cosmic shear pins the combination S₈ = σ₈(Ωₘ/0.3)^0.5 along a banana of nearly constant S₈, while σ₈ and Ωₘ stay individually uncertain — Planck's CMB point sits above the lensing band, and that offset along the S₈ axis is the tension

The scale $8\,h^{-1}\,\mathrm{Mpc}$ is historical, not fundamental — it is where the rms fluctuation in galaxy counts was famously found to be of order unity, a convenience of the 1980s that stuck. It also sits awkwardly on the boundary between linear and mildly non-linear scales, which is exactly why baryonic feedback and non-linear modelling loom so large here: the $\sigma_8$ integral draws most of its weight from the wavenumbers where astrophysics is messiest. The second figure is the plot cosmologists actually draw — the ladder of measured $S_8$ values, Planck's prediction as a band and the lensing surveys sitting stubbornly to its left.

Figure

The tension plot: Planck 2018 predicts S₈ = 0.834 ± 0.016 from the CMB (amber band), while cosmic-shear surveys measure lower — KiDS-1000 0.766, DES Y3 0.776, HSC Y3 0.769 — a roughly 2 to 3σ pull that softens to about 0.79 in the joint KiDS plus DES reanalysis

The core equations

The whole subject rests on one definition. Weak lensing measures a projected, degenerate blend of density and clustering, so cosmologists rotate coordinates to the direction that lensing constrains sharply — and give it a name.

S_8 \equiv \sigma_8 \left(\dfrac{\Omega_{\mathrm{m}}}{0.3}\right)^{0.5} \qquad (1)

where: $S_8$ — the clustering-amplitude parameter cosmic shear measures best; $\sigma_8$ — rms linear density contrast smoothed on $8\,h^{-1}\,\mathrm{Mpc}$ ; $\Omega_{\mathrm{m}}$ — present-day matter density, pivoted at $0.3$ so $S_8\approx\sigma_8$ near the fiducial cosmology. Reading it: $S_8$ is $\sigma_8$ tilted toward $\Omega_{\mathrm{m}}$ , so a lensing survey pins one number instead of a smear. The quantity $\sigma_8$ is itself an integral over the matter power spectrum, filtered by a real-space top-hat sphere of radius $R=8\,h^{-1}\,\mathrm{Mpc}$ . Its variance is the power spectrum weighted by the squared window transform.

\sigma_R^2 = \dfrac{1}{2\pi^2}\int_0^{\infty} P(k)\, W^2(kR)\, k^2\,\mathrm{d}k \qquad (2)

where: $\sigma_R^2$ — variance of $\delta$ smoothed on scale $R$ , with $R=8\,h^{-1}\,\mathrm{Mpc}$ giving $\sigma_8^2$ ; $P(k)$ — linear matter power spectrum today; $W(kR)$ — Fourier transform of the top-hat; $k$ — comoving wavenumber. Reading it: $\sigma_8$ counts power across all scales, but the window decides which scales contribute. The window is the transform of a uniform sphere: it passes long-wavelength modes intact and rolls off once the wavelength drops below the sphere.

W(x) = \dfrac{3\,[\sin x - x\cos x]}{x^3} \qquad (3)

where: $x=kR$ — dimensionless product of wavenumber and smoothing radius; $W$ — normalised to $W(0)=1$ , falling as $x^{-2}$ for $x\gg1$ . Reading it: $W\approx1$ for $kR\ll1$ and suppresses $kR\gg1$ , so $\sigma_8$ is dominated by $k\sim0.1$ – $0.4\,h\,\mathrm{Mpc}^{-1}$ — the awkward mildly-non-linear band. The power spectrum itself factorises into a primordial piece fixed by the CMB and a late-time growth piece computed by $\Lambda$ CDM — where the two acts of the story meet.

P(k) = A_{\mathrm{s}}\,k^{n_{\mathrm{s}}}\,T^2(k)\,D^2(a{=}1) \qquad (4)

where: $A_{\mathrm{s}}$ — primordial scalar amplitude, fixed by the CMB; $n_{\mathrm{s}}\approx0.965$ — scalar spectral index; $T(k)$ — transfer function encoding sub-horizon growth and radiation drag; $D(a)$ — linear growth factor, normalised to $D=1$ today. Reading it: the CMB sets $A_{\mathrm{s}}$ and $n_{\mathrm{s}}$ ; everything late-time lives in $D(a)$ . Because linear growth is scale-independent, every Fourier mode grows by the same factor, so $\sigma_8$ tracks $D(a)$ — the lever by which late-time physics moves the prediction.

\sigma_8(a) = \sigma_8(1)\,\dfrac{D(a)}{D(1)} \qquad (5)

where: $\sigma_8(a)$ — clustering amplitude at scale factor $a$ ; $D(a)$ — linear growth factor; $D(1)$ — its value today. Reading it: suppress growth at late times and you lower $D(1)$ , hence today's $\sigma_8$ and $S_8$ — the physical route to easing the tension. What lensing observes is a projected line-of-sight integral of this power spectrum weighted by a geometric efficiency kernel. In the Limber approximation each multipole $\ell$ samples one wavenumber per distance shell.

C_\ell^{\kappa\kappa} = \int_0^{\chi_H}\!\mathrm{d}\chi\;\dfrac{W^2(\chi)}{\chi^2}\,P\!\left(k=\tfrac{\ell+1/2}{\chi},\chi\right) \qquad (6)

where: $C_\ell^{\kappa\kappa}$ — angular power spectrum of the lensing convergence; $\chi$ — comoving distance ( $\chi_H$ the horizon); $W(\chi)$ — lensing efficiency kernel, $\propto\Omega_{\mathrm{m}}$ ; $\ell$ — multipole. Reading it: the signal scales as (kernel) $^2\times$ (power) $\propto\Omega_{\mathrm{m}}^2\,\sigma_8^2$ , which is where the degeneracy — and hence $S_8$ — comes from. Finally, the tension is quantified by a simple sigma metric comparing the CMB and lensing values against their combined error.

T \equiv \dfrac{\lvert S_8^{\mathrm{CMB}} - S_8^{\mathrm{lens}}\rvert}{\sqrt{\sigma_{\mathrm{CMB}}^2 + \sigma_{\mathrm{lens}}^2}} \qquad (7)

where: $T$ — number of sigma of discrepancy; $S_8^{\mathrm{CMB}},\,S_8^{\mathrm{lens}}$ — the two central values; $\sigma_{\mathrm{CMB}},\,\sigma_{\mathrm{lens}}$ — their $1\sigma$ errors. Reading it: for KiDS-1000 versus Planck this returns $T\approx2.9$ — real, but far short of the $5\sigma$ discovery bar.

Derivation AWhy the lensing-preferred combination is sigma-8 times Omega-m to the half power

The goal is to show, from the projected shear power spectrum, that weak lensing constrains not $\sigma_8$ and $\Omega_{\mathrm{m}}$ separately but a specific power-law blend of them close to $\sigma_8\,\Omega_{\mathrm{m}}^{0.5}$ — motivating $S_8$ as the one number a shear survey can quote and compare cleanly against the CMB. We track how the two parameters enter the observable, combine their scalings, and account for why the naive exponent flattens in a real survey.

Step 1. The lensing efficiency kernel scales as $W(\chi)\propto\Omega_{\mathrm{m}}\,(H_0^2/c^2)\,g(\chi)$ , where $g(\chi)$ is a purely geometric source-distance factor. Why: gravitational deflection is sourced by the matter density itself, so the strength of the shear signal is proportional to $\Omega_{\mathrm{m}}$ through the amount of deflecting mass along the line of sight.

Step 2. At fixed spectral shape, the amplitude of the matter power spectrum scales as $P(k)\propto\sigma_8^2$ , since $\sigma_8$ is by construction the rms of the linear field. Why: $\sigma_8$ is a normalisation of $P(k)$ , so doubling the clustering amplitude quadruples the power and hence the shear variance.

Step 3. Insert both into the Limber integral: $C_\ell^{\kappa\kappa}\propto W^2 P\propto\Omega_{\mathrm{m}}^2\,\sigma_8^2$ , so the constant-signal locus is $\sigma_8\,\Omega_{\mathrm{m}}=\text{const}$ . Why: the observable is quadratic in the kernel and linear in the power, so its amplitude depends only on the product $\sigma_8\,\Omega_{\mathrm{m}}$ to leading order, which is a degenerate ridge in the $\sigma_8$ – $\Omega_{\mathrm{m}}$ plane.

Step 4. A full likelihood integrates over many multipoles and tomographic redshift bins, where the shape of $P(k)$ , the growth history, and the geometric factor $g(\chi)$ each carry extra, weaker $\Omega_{\mathrm{m}}$ dependence, flattening the fitted exponent in $\sigma_8\,\Omega_{\mathrm{m}}^{\alpha}=\text{const}$ from $1$ toward $\alpha\approx0.5$ . Why: $\alpha=1$ holds only for a single scale and source plane; averaging over the survey's range dilutes the density dependence.

Step 5. Adopt $\alpha=0.5$ and pivot at $\Omega_{\mathrm{m}}=0.3$ to define $S_8\equiv\sigma_8(\Omega_{\mathrm{m}}/0.3)^{0.5}$ , measured to a few percent and orthogonal to the residual degeneracy. Why: this is the eigen-direction of the lensing likelihood, so quoting $S_8$ discards no constrained information while making cross-survey comparison unambiguous.

Go deeper · Derivation B · 1 minAssembling sigma-8 from the power spectrum and the top-hat window

Derivation BAssembling sigma-8 from the power spectrum and the top-hat window

The goal is to build $\sigma_8$ explicitly from the linear power spectrum and the top-hat filter, exposing why the fiducial scale $8\,h^{-1}\,\mathrm{Mpc}$ is delicate: the variance integral draws its weight from wavenumbers straddling the linear-to-non-linear transition, precisely where baryonic astrophysics is hardest to model. We reduce the variance to a one-dimensional integral by isotropy, examine the window cutoff, and read off the sensitivity.

Step 1. The smoothed density field has variance $\langle\delta_R^2\rangle=\int\dfrac{\mathrm{d}^3k}{(2\pi)^3}\,P(k)\,\lvert W(kR)\rvert^2$ , the power spectrum weighted by the squared window transform. Why: smoothing in real space is multiplication by $W$ in Fourier space, so the variance of the filtered field is $P(k)$ times $\lvert W\rvert^2$ integrated over all modes.

Step 2. Angular integration over the sphere in $k$ -space collapses the $\mathrm{d}^3k$ to a radial integral, giving $\sigma_R^2=\dfrac{1}{2\pi^2}\int_0^\infty P(k)\,W^2(kR)\,k^2\,\mathrm{d}k$ . Why: for a statistically isotropic field $P$ and $W$ depend only on $k=\lvert\mathbf{k}\rvert$ , so $\int\mathrm{d}^3k\to4\pi\int k^2\,\mathrm{d}k$ and $(2\pi)^{-3}\cdot4\pi=(2\pi^2)^{-1}$ .

Step 3. The top-hat window satisfies $W(kR)\approx1$ for $kR\ll1$ and falls as $(kR)^{-2}$ for $kR\gg1$ , so with $R=8\,h^{-1}\,\mathrm{Mpc}$ the integrand peaks around $k\sim0.1$ – $0.4\,h\,\mathrm{Mpc}^{-1}$ . Why: the filter passes fluctuations larger than the sphere and averages away those smaller than it, concentrating the variance in a band that unfortunately overlaps the onset of non-linear clustering.

Step 4. Substituting $P=A_{\mathrm{s}}k^{n_{\mathrm{s}}}T^2 D^2(1)$ factorises the result as $\sigma_8^2\propto A_{\mathrm{s}}\,D^2(1)\int T^2(k)\,k^{2+n_{\mathrm{s}}}\,W^2(8k)\,\mathrm{d}k$ . Why: this cleanly separates the CMB-fixed amplitude $A_{\mathrm{s}}$ and the late-time growth $D(1)$ from a pure shape integral, showing that $\sigma_8$ carries both primordial and growth information — exactly the two acts the tension compares.

Step 5. Because the dominant $k$ range is mildly non-linear, baryonic feedback that suppresses $P(k)$ there by up to $\sim10\%$ shifts the inferred $\sigma_8$ by a few percent. Why: $\sigma_8\propto\sqrt{\int P}$ , so a $\sim10\%$ suppression over the contributing band moves $\sigma_8$ by several percent — comparable to the tension itself, which is why feedback modelling is make-or-break.

Limiting cases

How $S_8$ and the tension behave in the regimes that bound the problem:

Case	Result	Meaning
Perfect early-late consistency	$S_8^{\mathrm{CMB}} = S_8^{\mathrm{lensing}}$	$\Lambda$ CDM passes: growth from $z\simeq1100$ to $z=0$ matches the CMB extrapolation. This is the null hypothesis the data mildly violate.
Suppressed late growth	$D(1)_{\mathrm{obs}} < D(1)_{\Lambda\mathrm{CDM}}\ \Rightarrow\ S_8^{\mathrm{lens}}\downarrow$	Modified gravity or interacting dark energy weakens clustering after matter- $\Lambda$ equality, lowering the observed $S_8$ without touching the CMB peaks.
Linear vs non-linear scale	$k \gtrsim 0.2\,h\,\mathrm{Mpc}^{-1}$	Here $P(k)$ is non-linear and baryonic feedback suppresses power up to $\sim10\%$ ; mis-modelling biases the inferred $S_8$ comparably to the tension.
Lensing degeneracy limit	$C_\ell^{\kappa\kappa} \propto \sigma_8^2\,\Omega_{\mathrm{m}}^{2}$ (approx.)	Shear power scales roughly as $\sigma_8^2\Omega_{\mathrm{m}}^2$ , so only the $S_8$ combination is well measured; $\sigma_8$ alone stays degenerate.

Common trapConfusing sigma-8 with S-8, or treating either as a directly observed rms

Do not conflate $\sigma_8$ with $S_8$ . Weak lensing constrains $S_8$ tightly but $\sigma_8$ and $\Omega_{\mathrm{m}}$ only weakly and degenerately, so comparing raw $\sigma_8$ values from surveys with different $\Omega_{\mathrm{m}}$ priors is close to meaningless — compare along the measured direction, $S_8$ . A second trap: $\sigma_8$ as reported is the linear-theory extrapolation, an idealised amplitude the field would have if growth stayed linear, not a directly observable rms — the actual matter on $8\,h^{-1}\,\mathrm{Mpc}$ is already mildly non-linear, so the "8" names a scale, not a map reading. Finally, a low lensing $S_8$ does not by itself indict the CMB: $S_8^{\mathrm{CMB}}$ is a $\Lambda$ CDM forward-extrapolation, and the mismatch could live in late-time physics or in shear systematics.

Worked examples

1 · From Planck's chain to the tension significance (fully worked)

Planck 2018 reports $\sigma_8 = 0.811$ and $\Omega_{\mathrm{m}} = 0.315$ for base $\Lambda$ CDM. Compute the implied $S_8$ , then quantify its tension with KiDS-1000, which measures $S_8 = 0.766^{+0.020}_{-0.014}$ .

Step 1. Apply the definition with the Planck central values: $S_8 = \sigma_8(\Omega_{\mathrm{m}}/0.3)^{0.5} = 0.811\times(0.315/0.3)^{0.5} = 0.811\times(1.05)^{0.5} = 0.811\times1.0247$ . This gives $S_8 = \mathbf{0.831}$ , consistent with Planck's directly reported $0.834\pm0.016$ (the small difference reflects full-chain marginalisation rather than plugging in central values).

Step 2. Form the tension metric of equation (7). Take $\sigma_{\mathrm{CMB}}=0.016$ and, for the asymmetric KiDS error, $\sigma_{\mathrm{lens}}\approx0.017$ . The offset is $\lvert 0.834-0.766\rvert = 0.068$ .

Step 3. Combine the errors in quadrature: $\sqrt{0.016^2+0.017^2} = \sqrt{0.000256+0.000289} = \sqrt{0.000545} = 0.02334$ . Then $T = 0.068/0.02334 = \mathbf{2.9}$ . So the KiDS-1000 versus Planck tension is $\approx2.9\sigma$ — a genuine pull, but comfortably below the $5\sigma$ threshold at which cosmologists would call it a discovery rather than a hint.

2 · How much must late-time growth slow to close the gap? (fully worked)

Suppose the tension is entirely physical, caused by a mode-independent suppression of the linear growth factor $D(a)$ at low redshift. By what fraction must $D(1)$ fall below its $\Lambda$ CDM value, and roughly when would that suppression have to act?

Step 1. Because linear growth is scale-independent, $\sigma_8\propto D(1)$ and therefore $S_8\propto D(1)$ at fixed primordial amplitude. The required rescaling is just the ratio of the measured to the predicted value: $D(1)_{\mathrm{obs}}/D(1)_{\Lambda\mathrm{CDM}} = S_8^{\mathrm{lens}}/S_8^{\mathrm{CMB}} = 0.766/0.834$ .

Step 2. Evaluate: $0.766/0.834 = 0.918$ . So $D(1)$ must sit about $\mathbf{8\%}$ below the $\Lambda$ CDM growth the CMB predicts. This is a small number — new physics need not overturn the growth history, only trim its final stretch by a few percent.

Step 3. The suppression must switch on after the CMB is imprinted and largely after matter- $\Lambda$ equality ( $z\sim0.3$ ), since BAO distances and the CMB peaks already agree with $\Lambda$ CDM and must not be disturbed. A late-onset $\sim8\%$ growth suppression below $z\sim0.5$ is the target any physical resolution must hit — enough to lower $S_8$ , little enough to leave the well-measured geometry alone. This is exactly the window dynamical dark energy, weak fifth forces, and decaying-dark-matter models try to exploit.

Physics of the portrait

$S_8$ has no single discoverer, but the statistic it is built from — the variance of galaxy counts in $8\,h^{-1}\,\mathrm{Mpc}$ spheres — and the whole machinery of comparing clustering amplitudes to a growth prediction descend directly from one person, who turned cosmic structure from a qualitative story into a quantitative science.

Origins · 1 minJames Peebles and how this was discovered

James Peebles

b. 1935, Nobel Prize in Physics 2019 · physical cosmology and the statistics of large-scale structure

Phillip James Edwin Peebles built the statistical language in which the $S_8$ tension is even stated. Beginning in the 1960s at Princeton, he treated the galaxy distribution not as a catalogue of objects but as a random field, introducing the two-point correlation function $\xi(r)\approx(r/r_0)^{-1.8}$ with $r_0\approx5\,h^{-1}\,\mathrm{Mpc}$ and its Fourier partner, the power spectrum $P(k)$ . His 1980 monograph The Large-Scale Structure of the Universe codified the variance-in-spheres statistic, and the 1983 analysis with Marc Davis established that the rms galaxy-count fluctuation is of order unity averaged over spheres of radius $8\,h^{-1}\,\mathrm{Mpc}$ — the very reason that peculiar scale, and the symbol $\sigma_8$ , became the field's standard clustering normalisation. Peebles also derived the linear growth equation in an expanding universe, computed early transfer functions with Yu, and was among the architects of cold dark matter, the framework whose late-time growth prediction the tension now tests. His 2019 Nobel Prize recognised precisely this transformation of cosmology into a precision, statistics-driven science; the modern comparison of a CMB-anchored $\sigma_8$ against a directly measured one is his program carried to its natural, and now slightly uncomfortable, conclusion.

History

1980

— Peebles publishes The Large-Scale Structure of the Universe, formalising the correlation function and the variance-in-spheres statistic

1983

— Davis and Peebles find the galaxy count variance is of order unity in 8 h⁻¹ Mpc spheres, fixing the fiducial scale of σ₈

2013

— the first Planck cosmology release pins Aₛ and Ωₘ, yielding a relatively high predicted σ₈

2017

— KiDS-450 and DES Y1 cosmic-shear analyses report a low S₈, opening the tension

2021

— KiDS-1000 and DES Y3 sharpen S₈ ≈ 0.77, a 2 to 3σ offset from Planck

Going further

Texts. Ryden, Introduction to Cosmology, 2nd ed., Ch. 12 (structure formation, $\sigma_8$ ) · Dodelson, Modern Cosmology, 2nd ed., Ch. 7–9 (power spectrum, growth, lensing) · Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., Ch. 30 (cosmic structure) · Mo, van den Bosch & White, Galaxy Formation and Evolution, Ch. 4 & 6 (perturbation growth, statistics) · Binney & Tremaine, Galactic Dynamics, 2nd ed., Ch. 9 (perturbation growth) · Planck 2018 results VI, A&A 641, A6 (2020) · Heymans et al., KiDS-1000, A&A 646, A140 (2021) · DES Y3, Phys. Rev. D 105, 023520 (2022). The key numerical anchors are: Planck 2018 $S_8=0.834\pm0.016$ , $\sigma_8=0.811$ , $\Omega_{\mathrm{m}}=0.315$ ; KiDS-1000 $S_8=0.766^{+0.020}_{-0.014}$ ; DES Y3 $S_8\approx0.776$ . The tension arithmetic gives $(0.834-0.766)/\sqrt{0.016^2+0.017^2}\approx2.9$ , and the implied growth ratio is $0.766/0.834\approx0.918$ .

Prerequisites. Density Perturbations and the Density Contrast · Linear Growth of Structure · Transfer Function and the Matter Power Spectrum · The Primordial Power Spectrum · The $\Lambda$ CDM Concordance Model. Leads to. Dark Energy and the Cosmological Constant · Cold Dark Matter · The CMB Anisotropy Power Spectrum.

Ask the tutor: Ask me to walk through why cosmic shear can only pin down $S_8$ and not $\sigma_8$ and $\Omega_{\mathrm{m}}$ separately, or to work a growth-suppression model that lowers $S_8$ without disturbing the CMB.

Is the S-8 tension the same kind of problem as the Hubble tension?

No, though they are cousins. Both compare an early-Universe $\Lambda$ CDM extrapolation from the CMB against a direct low-redshift measurement, and both find the late-Universe value on the "quieter" side — a lower expansion rate eases $H_0$ , and lower growth eases $S_8$ . But the Hubble tension is sharp, roughly $5\sigma$ , and resistant to any single systematic, which is why it dominates attention. The $S_8$ tension is milder, $2$ – $3\sigma$ , and genuinely systematics-limited: baryonic feedback and photometric redshifts can each move it by the size of the effect. So $S_8$ is treated as suggestive corroboration rather than an independent smoking gun, and a model resolving both at once would be far more compelling than one touching only $S_8$ .

Why is the pivot 0.3 and the exponent 0.5 in the definition of S-8?

Both are conventions, not physics. The exponent $0.5$ approximates the slope of the weak-lensing degeneracy: shear power scales roughly as $\sigma_8^2\Omega_{\mathrm{m}}^2$ for a single scale and source plane, which suggests an exponent of $1$ , but averaging over a real survey's multipoles and redshift bins flattens the best-constrained direction to about $\sigma_8\,\Omega_{\mathrm{m}}^{0.5}$ — and different surveys sometimes fit slightly different exponents. The pivot $\Omega_{\mathrm{m}}=0.3$ is cosmetic: it centres $S_8$ near $\sigma_8$ for the concordance cosmology so the number reads near unity. Neither choice is fundamental; they exist so one clean number captures what lensing actually measures.

Could the tension just be baryonic feedback rather than new physics?

Very possibly — this is the leading mundane explanation. The $\sigma_8$ integral draws most of its weight from $k\sim0.1$ – $0.4\,h\,\mathrm{Mpc}^{-1}$ , where feedback from active galactic nuclei and supernovae blows gas out of halos and suppresses the small-scale matter power spectrum by up to $\sim10\%$ . Since $\sigma_8\propto\sqrt{\int P}$ , mismodelling that suppression can bias the inferred $S_8$ by several percent — the whole size of the tension. Analyses that marginalise more aggressively over feedback, or discard the most non-linear scales, tend to nudge shear $S_8$ upward toward Planck. Disentangling feedback from genuine growth suppression is exactly why Stage-IV surveys pair shear with X-ray, kinetic Sunyaev-Zel'dovich, and CMB-lensing cross-correlations.

Does a low S-8 mean the CMB is wrong?

No. The CMB directly measures the early-Universe amplitude $A_{\mathrm{s}}$ and the matter density $\Omega_{\mathrm{m}}$ from the acoustic peaks, and those measurements are extraordinarily robust. What it reports as $S_8^{\mathrm{CMB}}$ is not an early-Universe observable but a $\Lambda$ CDM forward extrapolation: it takes the CMB-fixed initial conditions and integrates $D(a)$ to today. A low lensing $S_8$ therefore points to one of two things — either late-time growth genuinely ran slow, so the extrapolation and not the CMB is at fault, or the lensing measurement is biased low by systematics. Tellingly, CMB-lensing amplitudes from Planck, ACT, and SPT sit high, consistent with the extrapolation, sharpening suspicion on the low-redshift shear regime.

What is the difference between sigma-8 and f-sigma-8, and why measure both?

$\sigma_8$ is the amplitude of clustering — how tall the density fluctuations are today. The combination $f\sigma_8$ multiplies it by the linear growth rate $f=\mathrm{d}\ln D/\mathrm{d}\ln a$ , and it is what redshift-space distortions measure: peculiar velocities, which distort the observed clustering in redshift space, are sourced by $f$ while their scale is set by $\sigma_8$ . Measuring both matters because they probe growth differently — $\sigma_8$ from lensing integrates growth up to now, whereas $f\sigma_8$ from spectroscopic surveys is a snapshot of its instantaneous rate at each redshift. New physics that suppresses late-time growth should leave correlated fingerprints in both, but they carry independent systematics, so agreement between them strengthens any claim considerably.

Why not just quote sigma-8 and Omega-m separately and be done with it?

Because weak lensing does not measure them separately — it measures a blend. The shear signal depends on the combination $\sigma_8\,\Omega_{\mathrm{m}}^{0.5}$ , so any survey's likelihood is a long banana in the $\sigma_8$ – $\Omega_{\mathrm{m}}$ plane rather than a tight blob. Quoting $\sigma_8$ alone would force you to fix $\Omega_{\mathrm{m}}$ by a prior, and two surveys with different priors would report different $\sigma_8$ even if their data agreed perfectly. $S_8$ is the projection onto the short axis of the banana — the one direction the data actually constrain — so it is reproducible across surveys and cleanly comparable to the CMB: the honest summary of what lensing knows and does not know.