Cloud–Bulk Interaction Field Equations and the Geometric Cosmological Parameter Λ

Cosmological observations indicate a non-zero cosmological constant, \( \Lambda \), commonly interpreted as a constant vacuum-energy density. When General Relativity is combined with quantum field theory, vacuum contributions enter the field equations as terms proportional to \( g_{\mu\nu} \), producing the well-known discrepancy between the observed and predicted vacuum-energy densities: the cosmological constant problem. In addition, recent analyses combining SNe Ia, H (z), BAO, LSS, BBN and CMB data prefer a time-dependent cosmological parameter at up to \( 4.2\sigma \), or \( 3.2\sigma \) with revised supernova calibrations. Together, these challenge the identification of \( \Lambda \) with a fixed vacuum-energy density. The present framework instead models the quantum vacuum as a dynamical background (Bulk) with its own metric \( \tilde g_{\mu\nu} \), in which astrophysical systems are embedded as localized manifolds (Clouds) described by the metric \( g_{\mu\nu} \). The resulting interaction field equations reduce locally to the Einstein field equations, with a constant Newtonian coupling, supplemented by an explicit rank-two Geometric Slip tensor that carries the environmental tidal influence at the Cloud--Bulk interface, a role played in General Relativity only implicitly by the Weyl curvature. The Slip closes in the fully synchronized limit (\( \beta\to\infty \)), just as General Relativity reduces to Newtonian gravity for weak fields and slow motion (\( c\to\infty \)). In the unlocked cosmological regime, the effective cosmological term, \( \Lambda_g \), emerges from the incomplete Cloud–Bulk synchronization, with the Bulk curvature and expansion rate setting its geometric scale. Complementary to other alternatives to the \( \Lambda \)CDM model, the framework thus provides an effective geometric description in which the small observed value of \( \Lambda_g \) reflects the weak curvature of the present Universe, while remaining compatible with current observational constraints.