Minimizing S\mathcal{S} under constraints yields self-similar solutions, indicating that fractal geometry is a statistical attractor of cosmic evolution.
We have modeled the cosmic matter distribution using non-integer power laws derived from Hausdorff fractal dimensions, allowing for scale-invariant structure, inhomogeneity, and self-similarity. This approach aligns with galaxy clustering observations, explains large-scale anisotropies, and integrates naturally within our multilayer cosmological framework. The resulting modifications to the stress-energy tensor and gravitational dynamics offer testable predictions distinguishable from CDM expectations.
III.4. Topological Interference Fields
To rigorously describe the cross-layer coupling and nonlocal correlations in our proposed Multilayered Cosmological Topology, we introduce the concept of topological interference fields. These fields emerge from phase correlations across distinct cosmological layers and capture the imprint of nontrivial global topologies on local dynamics.
1. Motivation: Correlated Structures Across Cosmological Domains
In our multilayer universe framework, each spacetime layer i\Sigma_i evolves with its own local metric g(i)g_{\mu\nu}^{(i)}, expansion history, and fractal matter content. However, observed cosmic anisotropies and alignment phenomena (e.g., CMB quadrupole-octopole alignments, galaxy spin correlations) suggest the existence of coherent correlations across otherwise causally disconnected regions.
These correlations motivate the need for a field-theoretic description of phase entanglement or interference among topologically distinct cosmological layers.
2. Defining the Topological Interference Field
We define a complex interference scalar field interf(x)\Phi_{\text{interf}}(x), encoding the interference pattern of multiple spacetime layers, as follows:
interf(x)exp(i(x))d4x\Phi_{\text{interf}}(x) \sim \int \exp\left(i \theta(x)\right) \, d^4x
where:
