Historical content.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.

Cusp-Core Problem

Why do the centers of dwarf galaxies have flat density when ΛCDM predicts pronounced peaks (cusps)?

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

The Problem

N-body simulations with cold dark matter (CDM) predict density profiles with central cusps (NFW profile: ρ ∝ r⁻¹), but the observed rotation curves show flat cores.

Conflicting Profiles

NFW Profile (ΛCDM)

$$\rho_{NFW}(r) = \frac{\rho_s}{(r/r_s)(1+r/r_s)^2}$$
  • Cusp: ρ → ∞ as r → 0
  • Slope: γ = -1 at the center
  • Universal in CDM simulations

Burkert Profile (Observed)

$$\rho_B(r) = \frac{\rho_0 r_c^3}{(r+r_c)(r^2+r_c^2)}$$
  • Core: finite ρ(0) = ρ₀
  • Slope: γ = 0 at the center
  • Observed in dwarf galaxies

Visual Comparison

Density Profiles

NFW (ΛCDM)
Burkert (MCMC)

The MCMC Solution

The MCMC introduces two ingredients that naturally transform cusps into cores:

1. Lattice Density (ρ_lat)

In regions of high baryonic density, the entropic lattice introduces an effective pressure that counteracts gravitational collapse:

$$P_{lat} = \frac{\hbar c}{a^4} \cdot f(S)$$

Where a is the lattice spacing and f(S) is a function of the entropic state. This pressure acts as a “floor” that prevents the formation of cusps.

2. Adrián Field (Φ_Ad)

The scalar field Φ_Ad mediates the mass → space transition, creating an effective “friction” at the center of the halos:

$$\nabla^2 \Phi_{Ad} = 4\pi G \rho - \lambda_{Ad} \Phi_{Ad}$$

The term λ_Ad acts as a regulator that smooths density gradients, transforming cusps into cores naturally.

Cusp → Core Transformation

The effective MCMC profile interpolates between NFW and Burkert:

$$\rho_{MCMC}(r) = \frac{\rho_s}{(r/r_s + r_c/r_s)(1+r/r_s)^2}$$

Where rc (core radius) emerges from the lattice scale:

$$r_c \approx \left(\frac{M_{halo}}{10^{10} M_\odot}\right)^{0.3} \text{ kpc}$$
Galaxy Type M_halo [M☉] Predicted r_c Observed r_c
Ultra-faint dwarf ~10⁸ 0.3 kpc 0.2-0.5 kpc
Classical dwarf ~10⁹ 0.6 kpc 0.4-0.8 kpc
Irregular ~10¹⁰ 1.2 kpc 0.8-1.5 kpc
LSB (low brightness) ~10¹¹ 2.5 kpc 2-4 kpc
γ = -1 → γ = 0
Cusp-to-core transformation

The MCMC reproduces the observed Burkert profiles without ad hoc baryonic feedback

Why Does It Work?

ΛCDM + Feedback

  • Requires supernovae to “heat” DM
  • Efficiency tuned ad hoc
  • Difficult in galaxies without SF
  • Does not work in ultra-faints

MCMC

  • ρ_lat is intrinsic to the model
  • Requires no external adjustments
  • Works at all masses
  • Scales with M^0.3 naturally

Observational Evidence

  • THINGS Survey: 34 galaxies with flat cores
  • LITTLE THINGS: Irregular dwarfs with γ ≈ 0
  • Fornax/Sculptor: MW dwarfs with measured r_c
  • SPARC: Universal correlation r_c vs M_halo

The MCMC quantitatively reproduces the observed r_c-M_halo relation. The profile is tabulated with a calibration against SPARC, and the result is pending reproduction with the corrected scheme (Cronos v3).

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