Historical content.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.

Block 3: N-body

Halo formation simulations with entropic friction, generating Burkert profiles.

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

3

N-body Simulations

This block introduces entropic friction into the simulations of cosmic structure formation, with the aim of addressing the “cusp-core” problem of dark matter.

The Cusp-Core Problem

Problem in ΛCDM

The standard model (ΛCDM) predicts dark matter halos with NFW profiles that have a “cusp” at the center: the density diverges as ρ ∝ 1/r.

However, observations show flat cores in dwarf and LSB galaxies.

MCMC proposal (earlier corpus)

In the proposal of the earlier corpus, entropic friction slowed particles in regions of high density, which prevented the formation of cusps and produced flat cores in those simulations (historical record, not reproduced today: FE-08-2026, E7).

Entropic Friction

The friction coefficient depends on the local density:

$$\eta(\rho) = \alpha \times \left(\frac{\rho}{\rho_c}\right)^{1.5}$$

Where α = 0.1 is the friction parameter and ρc is the critical density.

Burkert Density Profile

Entropic friction generates Burkert-type profiles:

$$\rho(r) = \frac{\rho_0}{(1 + r/r_c)(1 + (r/r_c)^2)}$$

Unlike the NFW profile, this one has a flat core (ρ → ρ₀ as r → 0).

Core-Mass Relation

The core radius scales with the halo mass:

$$r_{core} = 1.8 \times \left(\frac{M}{10^{11} M_\odot}\right)^{0.35} \text{ kpc}$$

Parameters

Parameter Value Description
α (friction) 0.1 Entropic friction coefficient
exponent 1.5 Density exponent
r_s (NFW) ~20 kpc Typical NFW scale radius

Comparison: NFW vs Burkert

NFW (cusp)
Burkert (flat core)
r_core
1.80 kpc
r_s (NFW)
20.0 kpc
Mass
10¹¹ M☉

Physical Interpretation

Why Does It Work?

Entropic friction acts as a “brake” that disperses the kinetic energy of dark matter particles in dense regions:

  • At the center of the halo, where ρ is high, the friction is maximal
  • The particles lose speed and cannot concentrate any further
  • A core of constant density forms

Connection with Philosophy

Entropic friction is the mechanism by which the “resistance of space” limits the excessive concentration of mass, maintaining cosmic equilibrium.

Implementation in Python

import numpy as np def friccion_entropica(rho, rho_c, alpha=0.1, exp=1.5): """Entropic friction coefficient.""" return alpha * (rho / rho_c) ** exp def perfil_nfw(r, rho_s, r_s): """NFW profile (central cusp).""" x = r / r_s return rho_s / (x * (1 + x)**2) def perfil_burkert(r, rho_0, r_c): """Burkert profile (flat core).""" x = r / r_c return rho_0 / ((1 + x) * (1 + x**2)) def radio_core(M, M_ref=1e11): """Core radius as a function of mass (kpc).""" return 1.8 * (M / M_ref) ** 0.35 # Example M_halo = 1e11 # Solar mass r_c = radio_core(M_halo) print(f"Radio del nucleo: {r_c:.2f} kpc")

📁 File: simulations/bloque3_nbody.py

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