Thermodynamics and Relic Density of Topological Defects in Quantum Information Geometry

Motivated by the continued null results in direct detection, we explore a purely geometric dark matter paradigm. By modeling the quantum information vacuum as a CP 1 manifold, we demonstrate that localized topological defects dynamically generate a conserved energy-momentum tensor. Identifying the defect core radius with the physical symmetry-breaking scale ( r c ≃ 1/ T c ), these macroscopic entities acquire a topologically protected mass ( M g e o m ≃ π κ k 2 2 T c ) and behave dynamically as pressureless dust ( 〈 w 〉 = 0 ). Applying the Kibble-Zurek mechanism to their primordial formation, we derive a Master Geometric Constraint which, remarkably, transforms into a predictive cosmological framework. Assuming a fundamental winding number k = 1 and topological stiffness κ ≈ 1, the universe dictates a symmetry-breaking temperature of T c ≈ 2.6 × 10 11 GeV, rigidly fixing the geometric dark matter mass at M geom ≈ 4.1 × 10 11 GeV. This purely geometric derivation perfectly reproduces the observed relic abundance (Ω CDM h 2 ≈ 0.120). Furthermore, this GUT-scale regime naturally explains the absence of direct detection signals and yields an exceptionally diffuse gravitational compactness ( C ∼ 10 − 14 ). Consequently, these geometric defects possess a non-zero tidal deformability, offering a distinct astrophysical signature for future gravitational wave observatories, effectively distinguishing them from primordial black holes.