Block 2: Cosmology

Historical content. Part of the technical content of this block comes from the corpus prior to the Tratado de Fundamentos (v35) and may use threshold values from that formulation. The current classification of each result is in the status of the results.

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

ΛCDM vs MCMC

ΛCDM Model

Λ = constant

$$E(z) = \sqrt{\Omega_m(1+z)^3 + \Omega_\Lambda}$$

MCMC Model

Λ = Λ(z) dynamic

$$E(z) = \sqrt{\Omega_m(1+z)^3 + \Omega_\Lambda \cdot \Lambda_{rel}(z)}$$

Indeterminate Channel ρ_id(z)

The density of the indeterminate channel (emergent dark energy) has a transitional behavior:

$$\rho_{id}(z) = \begin{cases} \rho_0 \cdot (1+z)^3, & z > z_{trans} \\ \rho_0 \cdot [1 + \varepsilon(z_{trans} - z)], & z \leq z_{trans} \end{cases}$$

z > z_trans (High redshift)

Matter-like behavior: ρ ∝ (1+z)³

The indeterminate channel follows the dynamics of dark matter.

z ≤ z_trans (Low redshift)

Quasi-constant behavior

Transition to effective dark energy, modulated by ε.

Transition Parameters

Parameter	Value	Description
z_trans	8.9 ± 0.4	Transition redshift
ε	0.012 ± 0.003	Transition coefficient
ρ₀	0.305	Normalized base density

Relative Lambda (Simplified Approximation)

For quick calculations, the MCMC correction can be approximated as:

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

This approximation with δΛ ≈ 0.02 captures the main behavior in the observable regime.

Correspondence with ΛCDM

The three primary components of the MCMC are mapped onto the parameters of standard cosmology:

Determinate Mass

Ω_b = 4.93%

Baryonic matter

MCV (Latent)

Ω_DM = 26.6%

Dark matter

ECV (Indeterminate)

Ω_Λ = 68.5%

Dark energy

MCMC Cosmological Parameters

Parameter	Value	Prior Range	Description
H₀	69.6 ± 1.1 km/s/Mpc	[50, 100]	Hubble constant of the earlier corpus (production fits: 68.19 ± 0.40, with ΛCDM favored)
ρ_b0	0.0493	[0.01, 1.0]	Normalized baryonic density
ρ₀	0.305	[0.01, 2.0]	Base indeterminate density
z_trans	8.9 ± 0.4	[0.1, 3.0]	Transition redshift
ε	0.012 ± 0.003	[-0.5, 0.5]	Transition coefficient
r_d	147 Mpc	[130, 160]	Sound horizon (BAO)
M	-19.4 mag	[-21, -18]	SNe Ia absolute magnitude

H(z) Comparison: MCMC vs ΛCDM

δΛ correction: 0.020

ΛCDM

MCMC

H(z=0) ΛCDM

67.40

H(z=0) MCMC

68.08

Difference z=0

+1.0%

Age of the Universe

13.7 Gyr

Physical Interpretation

Why a Dynamic Λ?

In the MCMC, the cosmological constant is not really “constant.” It reflects the continuous process of conversion of mass into space:

At high z (distant past): Λ_rel → 1 (ΛCDM recovered)
At low z (present): Λ_rel slightly greater than 1
The difference is small but measurable

Hubble Tension

The production fits run in August 2026 do not reproduce an alleviation of the Hubble tension: they give H₀ = 68.19 ± 0.40 km/s/Mpc and favor ΛCDM (ΔBIC +14.5). The current result is in the comparison.

Implementation in Python

import numpy as np from scipy.integrate import quad class CosmologiaMCMC: """Block 2: Cosmology of the MCMC.""" def __init__(self, H0=67.4, Om=0.315, OL=0.685, dL=0.02): self.H0 = H0 self.Om = Om self.OL = OL self.dL = dL def Lambda_rel(self, z): """MCMC correction to Lambda.""" return 1 + self.dL * np.exp(-z/2) * (1+z)**(-0.5) def E_LCDM(self, z): """E(z) parameter for LCDM.""" return np.sqrt(self.Om * (1+z)**3 + self.OL) def E_MCMC(self, z): """E(z) parameter for MCMC.""" return np.sqrt(self.Om * (1+z)**3 + self.OL * self.Lambda_rel(z)) def H(self, z, modelo='mcmc'): """Hubble parameter H(z).""" E = self.E_MCMC(z) if modelo == 'mcmc' else self.E_LCDM(z) return self.H0 * E def edad(self, modelo='mcmc'): """Age of the universe in Gyr.""" E_func = self.E_MCMC if modelo == 'mcmc' else self.E_LCDM integrand = lambda z: 1 / ((1+z) * E_func(z)) integral, _ = quad(integrand, 0, 1000) return integral * (977.8 / self.H0) # Gyr

📁 File: mcmc_core/bloque2_cosmologia.py

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