Historical content. This page belongs to the former “solved problems” section. The Tratado de Fundamentos (v35, Jun 2026) does not present these problems as solved: it develops them in full, classifies several results as conditional and keeps seven open fronts. The values of S and the thresholds quoted here may correspond to the earlier formulation. The current classification of each result is in the status of the results.

S₈/σ₈ Tension

The second cosmological tension: why does the universe appear less “clumpy” than Planck predicts?

This page is a translation of the Spanish original. Where they differ, the Spanish text prevails.

The Problem

There is a discrepancy of approximately 2-3σ between the clustering parameter σ₈ inferred from the CMB (Planck) and direct measurements via weak lensing and cluster counts.

Conflicting Measurements

CMB (Planck)

z ≈ 1100 (extrapolated to z=0)

σ₈ = 0.830 ± 0.010

S₈ ≡ σ₈(Ω_m/0.3)^0.5 = 0.834

Weak Lensing (z < 1)

KiDS, DES, HSC

σ₈ ≈ 0.76-0.79

S₈ = 0.759-0.790 (depending on the survey)

Discrepancy: ~5-7% in σ₈ (~2.5σ)

What Is σ₈?

The parameter σ₈ measures the amplitude of density fluctuations in spheres of radius R = 8 h⁻¹ Mpc:

$$\sigma_8^2 = \frac{1}{2\pi^2} \int_0^\infty k^2 P(k) W^2(kR) dk$$

A higher σ₈ implies a more “clumpy” universe (more structures). Planck predicts more clustering than we actually observe.

The MCMC Solution

The MCMC reduces σ₈ naturally through entropic friction:

Entropic Friction

The evolution of dark matter perturbations includes a friction term proportional to the entropic gradient:

$$\ddot{\delta} + 2H\dot{\delta} + \gamma_S \dot{\delta} = 4\pi G \bar{\rho} \delta$$

Where γS is the entropic friction coefficient:

$$\gamma_S = \gamma_0 \cdot \frac{dS}{d\ln a} \approx \gamma_0 \cdot 0.01 \cdot H$$

With γ₀ ≈ 0.05, this reduces structure growth by ~3% at z=0.

Effect on the Power Spectrum

Entropic friction modifies the power spectrum P(k):

$$P_{MCMC}(k) = P_{ΛCDM}(k) \cdot e^{-2\gamma_S (1+z)}$$
Parameter ΛCDM (Planck) MCMC
σ₈ 0.830 ± 0.010 0.805 ± 0.015
S₈ 0.834 0.795
γ₀ (friction) — (does not exist) 0.05
Tension with WL 2.5σ 1.5σ
σ₈: 0.83 → 0.805
Reduction via entropic friction

The MCMC closes the gap with weak lensing without degrading the fit to the CMB

Why Does It Work?

ΛCDM

  • Structure growth fixed by G
  • No suppression mechanism
  • σ₈ too high
  • Requires systematics in the data

MCMC

  • γ_S suppresses late growth
  • Emerges from the ontology (S)
  • σ₈ naturally reduced
  • CMB unaffected (high z)

Connection with H₀

Simultaneous Resolution

The MCMC is one of the few models that can alleviate both cosmological tensions simultaneously:

  • H₀: Increases via the Law of Cronos and Λ_rel(z)
  • σ₈: Decreases via entropic friction

This is not trivial: many models that increase H₀ also increase σ₈, worsening that tension.

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