Emergent Einstein–Friedmann Dynamics from Universal Wavefunction Geometry

In this work, we demonstrate how an effective spacetime description and an Einstein--Friedmann structure can emerge naturally from the geometry of a universal wavefunction, without postulating gravitational field equations or introducing matter fields explicitly. By treating the flux hypersurface associated with a conserved wavefunction current as an embedded Lorentzian manifold, we show that its induced geometry is necessarily of Friedmann--Robertson--Walker type under minimal assumptions of homogeneity and isotropy. We show that the intrinsic curvature of the induced metric is fully determined by the embedding geometry itself. In particular, de Sitter spacetime arises as a special, maximally symmetric case corresponding to a constant-curvature hyperboloid in the ambient space. More generally, for physically relevant and normalisable classes of wavefunction envelopes, the embedding geometry remains hyperboloid in character but exhibits a time-dependent curvature scale. In this regime, the effective vacuum curvature term is approximately constant at early times, giving rise to de Sitter behaviour, and subsequently decays as the hypersurface evolves, leading asymptotically to a linear expansion law. By identifying a conserved, potential-like geometric invariant inherited from the universal wavefunction, we recover an effective Einstein--Friedmann structure on the hypersurface without invoking gravitational dynamics. This invariant fixes the scaling of the dominant contribution to the effective energy density and determines the value of the effective gravitational coupling. For closed slicing, this contribution cancels identically against the spatial curvature term in the Friedmann equation, leaving the late-time expansion governed solely by the residual vacuum-like sector. These results position general relativity as an emergent effective theory arising from a deeper wavefunction-based geometric structure.