We derive the modified Friedmann equation that governs the evolution of each cosmological layer in a universe with multi-layered, phase-entangled topology. The formalism generalizes standard FLRW dynamics by incorporating interference terms arising from cross-layer couplings due to phase differences in vacuum configurations.
A.1. Setup: Layered Mini-Universe Patch
Assume the universe is composed of NN distinct but gravitationally coupled cosmological layers Mi\mathcal{M}_i, each characterized by its own:
Scale factor ai(t)a_i(t),
Energy density i(t)\rho_i(t),
Spatial curvature kik_i,
Phase field i(t)\theta_i(t), representing vacuum configuration or tunneling coordinate,
Effective coupling ij\alpha_{ij} to other layers jij \neq i.
We begin with the Einstein field equations applied to each layer's metric:
dsi2=dt2+ai2(t)(dr21kir2+r2d2)ds_i^2 = -dt^2 + a_i^2(t) \left( \frac{dr^2}{1 - k_i r^2} + r^2 d\Omega^2 \right)
A.2. Effective Action and Interlayer Coupling
Define a generalized action for the ii-th layer:
Si=d4xgi[Ri16GLm(i)]+Sint(i)S_i = \int d^4x \sqrt{-g_i} \left[ \frac{R_i}{16\pi G} - \mathcal{L}_m^{(i)} \right] + S_{\text{int}}^{(i)}
where the interaction term is postulated as:
Sint(i)=jiijd4xgicos(ij)S_{\text{int}}^{(i)} = \epsilon \sum_{j \neq i} \alpha_{ij} \int d^4x \sqrt{-g_i} \cos(\theta_i - \theta_j)
with 1\epsilon \ll 1 as a dimensionless interference amplitude encoding quantum tunneling between vacuum layers.
