To account for gigaparsec-scale underdensities, void-Hubble structures, and asymmetric clustering, we propose a Multilayer FRW extension, where spacetime is composed of discrete, dynamically coupled regions (or layers), each with its own local scale factor ai(t)a_i(t), matter content, and curvature.
The modified metric in this scenario is a piecewise-continuous warped product, written for layer Li\mathcal{L}_i as:
dsi2=i2(t)dt2+ai2(t)[dr21kir2+r2d2]ds_i^2 = -\alpha_i^2(t) dt^2 + a_i^2(t)\left[\frac{dr^2}{1 - k_i r^2} + r^2 d\Omega^2\right]
Where:
i(t)\alpha_i(t) is a lapse function encoding clock-rate variations across layers.
kik_i is the effective curvature of layer Li\mathcal{L}_i.
Cross-layer matching is enforced through boundary field conditions discussed below.
This framework naturally supports locally varying Hubble constants Hi=ai/(iai)H_i = \dot{a}_i / (\alpha_i a_i), which in turn provides a theoretical basis for the observed Hubble tension---i.e., different observational paths (e.g., CMB vs. local ladder) probe different layers or transition zones.
2. Inter-Layer Topological Connectivity
While each layer is locally described by an FRW-like metric, their interconnections are topological rather than metrical---mediated by a shared pre-geometric substrate. This leads us to introduce a field-theoretic formulation of connectivity between layers Li\mathcal{L}_i and Lj\mathcal{L}_j using boundary matching conditions.
Let \phi be a scalar mediator field propagating in the underlying topological bulk. The action of the full system is:
S=iLigi(Ri16G+Limatter)+ijLint[i,j,ij]S = \sum_i \int_{\mathcal{L}_i} \sqrt{-g_i} \left( \frac{R_i}{16\pi G} + \mathcal{L}_i^{\text{matter}} \right) + \int_{\Sigma_{ij}} \mathcal{L}_{\text{int}}[\phi_i, \phi_j, \gamma_{ij}]
Where:
ij\Sigma_{ij} denotes the interface between two adjacent layers.
ij\gamma_{ij} is the induced metric on the interface.
Lint\mathcal{L}_{\text{int}} encodes coupling between fields in Li\mathcal{L}_i and Lj\mathcal{L}_j, such as junction conditions or interference terms.
In analogy with brane-world cosmology and AdS/CFT-type dualities, this structure permits nonlocal correlations and phase interference effects across layers---potentially explaining large-angle CMB anomalies or parity asymmetries.
