Historical content.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.

H₀ Tension

The most important discrepancy in modern cosmology: why do local and cosmic measurements give different values for H₀?

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

The Problem

There is a discrepancy of approximately 5σ between measurements of the Hubble parameter using the cosmic microwave background (CMB) and local measurements using Type Ia supernovae.

Conflicting Measurements

CMB (Planck)

z ≈ 1100 (early universe)

H₀ = 67.4 ± 0.5 km/s/Mpc

Indirect measurement through the pattern of CMB fluctuations

SH0ES (Local)

z < 0.1 (current universe)

H₀ = 73.0 ± 1.0 km/s/Mpc

Direct measurement using Cepheids and Type Ia supernovae

Discrepancy: ~6 km/s/Mpc (~5σ)

The MCMC Solution

The earlier corpus presented the MCMC as resolving this tension through two complementary mechanisms that emerge naturally from its ontology:

1. Law of Cronos

The temporal metric depends on the entropic state S and the local density ρ:

$$ds^2 = -[1 + \alpha(\rho_{local}/\rho_c)^\gamma] c^2 dt^2 + a^2(t) d\mathbf{x}^2$$

This implies that the temporal flow is different in dense versus empty regions:

$$dt_{local} = dt_{cosmo} / \sqrt{\Theta(S,\rho)}$$
  • In dense regions (galaxies): Θ > 1 → slower time
  • In empty regions (CMB): Θ ≈ 1 → standard time

2. Dynamic Dark Energy Λ_rel(z)

A small relaxation of the cosmological constant:

$$\Lambda_{rel}(z) = 1 + \delta_\Lambda \cdot e^{-z/2} \cdot (1+z)^{-0.5}$$

With δΛ ≈ 0.012 (~1.2% variation), this allows a higher H₀ without degrading the fit to the CMB or the BAO.

Fit Parameters

Parameter ΛCDM (Planck) MCMC
H₀ [km/s/Mpc] 67.7 ± 0.7 69.8 ± 1.1
Ωm 0.315 ± 0.007 0.300 ± 0.015
ε (relaxation) — (fixed at 0) 0.012 ± 0.003
ztrans — (not applicable) 8.9 ± 0.4
σ₈ 0.830 ± 0.010 ≈ 0.805
χ²min/Ndof 1.09 1.03
4σ → 1.5σ
Reduction of the tension with SH0ES

The MCMC reduces the discrepancy from ~4σ to ~1.5σ without introducing ad hoc parameters

Diagram: Solution Flow

         ┌─────────────────────────────────────────────────┐
         │            H₀ TENSION IN COSMOLOGY              │
         └─────────────────────────────────────────────────┘
                              │
                              ▼
    ┌───────────────────────────────────────────────────────────┐
    │   CMB (Planck)                  Local Measurements        │
    │   z ≈ 1100                      (SH0ES, z < 0.1)          │
    │   ────────────                  ────────────────          │
    │   H₀ = 67.4                     H₀ = 73.0                 │
    │   (uniform vacuum,              (galactic structures,     │
    │    Θ ≈ 1)                        Θ > 1)                   │
    └───────────────────────────────────────────────────────────┘
                              │
                              ▼
    ┌───────────────────────────────────────────────────────────┐
    │                    MCMC RESOLUTION                        │
    └───────────────────────────────────────────────────────────┘
                              │
          ┌───────────────────┼───────────────────┐
          ▼                   ▼                   ▼
    ┌───────────┐       ┌───────────┐       ┌───────────┐
    │ Law of    │       │ Λ_rel(z)  │       │ Result    │
    │ Cronos    │       │ dynamic   │       │           │
    ├───────────┤       ├───────────┤       ├───────────┤
    │ dt_local  │       │ ε = 1.2%  │       │ H₀ = 69.8 │
    │ ≠ dt_cosmo│  +    │ z_trans   │  →    │ ± 1.1     │
    │           │       │ = 8.9     │       │           │
    │ Θ(S,ρ)    │       │           │       │ Tension   │
    │ variable  │       │ Early     │       │ reduced   │
    │           │       │ transition│       │ 4σ → 1.5σ │
    └───────────┘       └───────────┘       └───────────┘

Why Does It Work?

ΛCDM

  • Assumes constant Λ for all z
  • CMB (z=1100) fixes H₀ = 67.4
  • When extrapolated to z=0, it does not fit
  • 5σ discrepancy

MCMC

  • Λ_rel(z) evolves with z
  • CMB sees Λ₀(1+ε) → consistent
  • z < z_trans: Λ “relaxes”
  • Closes the gap to 1-1.5σ

Ontological Connection

It Is Not an Ad Hoc Patch

Unlike other models (wCDM, Early Dark Energy), the MCMC does not introduce arbitrary parameters. The resolution emerges from the ontology of the model:

Ontological Element Physical Manifestation
Mp/Ep Tension Generates ρ_id(z), dynamic dark energy
Law of Cronos Temporal flow depends on density/entropy
Entropic Collapses Create “transitions” such as z_trans
Adrián Field Φ_Ad Mediates the mass→space conversion

Empirical Validation

The MCMC fit uses the following datasets:

  • Pantheon+: 1701 Type Ia supernovae
  • BAO: eBOSS + DESI (baryon acoustic oscillations)
  • Planck prior: Only Ωb and ns (NOT H₀)

The result (Δχ² = -12 relative to ΛCDM) indicates that the MCMC provides a better fit to the data without significantly increasing the degrees of freedom.

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