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Chapter 21: Dark Sector (Relics)

21.4 Cosmic Coincidence​

A central mystery of standard cosmology is the timing of cosmic acceleration: why the energy densities of matter and the vacuum are comparable today. This section resolves the Cosmic Coincidence problem by showing that this equivalence is a dynamic necessity of the Master Equation's logistic approach to the stable attractor.

21.4.1 Lemma: Saturation Epoch Convergence​

Coincidence of Matter and Vacuum Densities as Natural Feature of Logistic Growth Approach to Attractor Saturation
  • Coincidence Problem: Standard cosmology struggles to explain why the matter density Ωm\Omega_m and vacuum energy density ΩΛ\Omega_\Lambda are of comparable orders of magnitude today, given that they dilute at different rates during cosmic expansion.
  • Attractor Saturation: In QBD, the evolution of the graph towards the stable attractor ρ∗≈0.037\rho^* \approx 0.037 (§5.2.2) follows a logistic growth curve.
  • Crossover Epoch: The comparable magnitudes of Ωm\Omega_m and ΩDE\Omega_{DE} is not a coincidence, but a natural, extended epoch corresponding to the transition era where the logistic growth curve approaches saturation at the stable fixed point ρ∗\rho^*, matching the observed crossover.

21.4.2 Proof: Saturation Epoch Convergence​

Verification of Saturation Epoch Convergence by Phase Portrait Analysis of Cosmic Evolution
  • Phase Portrait Construction: The proof maps the phase portrait of the Master Equation coupled to the cosmic fluid expansion equations.
  • Extended Coincidence Window: It solves for the timeline of the attractor convergence, demonstrating that the ratio Ωm/ΩDE\Omega_m / \Omega_{DE} remains within a single order of magnitude for a substantial fraction of the active lifetime of the 4D manifold, resolving the coincidence problem dynamically without fine-tuned initial parameters.