A.3. Modified Einstein Equation
Varying the total action SiS_i with respect to the metric g(i)g_{\mu\nu}^{(i)} yields the Einstein equation for layer ii:
G(i)=8G(T(i)+T(int,i))G_{\mu\nu}^{(i)} = 8\pi G \left( T_{\mu\nu}^{(i)} + T_{\mu\nu}^{(\text{int},i)} \right)
Assuming isotropy and homogeneity, the 00-component gives:
(aiai)2=8G3ikiai2+8G3i(int)\left( \frac{\dot{a}_i}{a_i} \right)^2 = \frac{8\pi G}{3} \rho_i - \frac{k_i}{a_i^2} + \frac{8\pi G}{3} \rho_i^{(\text{int})}
A.4. Interference Energy Density
We now derive i(int)\rho_i^{(\text{int})} from the interaction Lagrangian:
i(int)=1ai3Sint(i)tjiijcos(ij)\rho_i^{(\text{int})} = \frac{1}{a_i^3} \frac{\delta S_{\text{int}}^{(i)}}{\delta t} \sim \epsilon \sum_{j \neq i} \alpha_{ij} \cos(\theta_i - \theta_j)
Here, the cosine term arises from vacuum phase overlap, motivated by analogy to Josephson junctions in condensed matter and phase entanglement in quantum field theory across topological domains.
Thus, we arrive at the interference-modified Friedmann equation:
Hi2=(aiai)2=8G3ikiai2+jiijcos(ij)H_i^2 = \left( \frac{\dot{a}_i}{a_i} \right)^2 = \frac{8\pi G}{3} \rho_i - \frac{k_i}{a_i^2} + \epsilon \sum_{j \neq i} \alpha_{ij} \cos(\theta_i - \theta_j)
