Frozen Vacuum Perturbations as a Dark Matter Analogue in the Logarithmic Superfluid Vacuum

In the standard ΛCDM model, cold dark matter (CDM) serves three essential cosmological roles: providing non-baryonic gravitational potential wells, maintaining these wells through the radiation era, and producing collisionless dynamics in galaxy cluster mergers. We show that all three roles are fulfilled by frozen perturbations of the vacuum condensate density within the logarithmic superfluid vacuum framework. The equation of state w = -1, derived from the logarithmic potential in companion papers, forces the vacuum density to remain constant during expansion (ρᵥ ∝ a⁰) and freezes the corresponding perturbation evolution: vacuum density perturbations neither grow nor decay. These primordial structural variations in ρ₀ (x), set during the formation of the condensate, create an intrinsic vacuum gravitational potential Φᵥ (x) = c²δᵥ (x) through the density dependence of the acoustic metric. Baryons oscillate in these frozen wells exactly as they would in CDM wells, producing the observed odd/even peak asymmetry in the Cosmic Microwave Background. The frozen template is independent of the baryonic matter distribution and does not interact during cluster collisions, naturally reproducing the mass-gas offset observed in the Bullet Cluster without requiring any actual dark matter particles.