Invariant manifold structure and conditional obstruction to phantom divide crossing in single-clock cosmology

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Abstract
In this work the dynamical role of the phantom divide is investigated within minimally coupled single-clock cosmological effective descriptions. Under general conditions including Einstein gravity, covariant energy–momentum conservation, autonomous closure of the background dynamics, and perturbative stability, the hypersurface Σ = ρ + P = 0 ( w = − 1 ) is shown to define an invariant manifold of the cosmological flow provided the effective clock is non-transversal at the phantom divide . In this case, the phase space vector field is tangent to the hypersurface, implying that the sign of Σ is preserved along dynamical trajectories and that continuous evolution between quintessence and phantom regimes is dynamically excluded in finite cosmic time within this subclass. The phantom divide is thereby identified as a dynamical separatrix of the background flow. The invariant manifold structure is conditional and not universal. In transversal realizations, for which the effective clock does not degenerate at the phantom divide ( c ˙ ≠ 0 at c = 0 ), the tangency condition is violated and continuous phantom crossing may occur within a regular effective field theory description. The results therefore provide a geometric characterization of phantom crossing in terms of phase space structure, yield a dynamical classification of single-clock cosmologies based on transversality, and offer a theoretical framework for interpreting observational constraints on the dark energy equation of state.

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