IV. Numerical Simulations
IV.1. Layer-Dependent Hubble Parameter H(r)H(r)
To investigate the observational viability of the Multilayer Multiverse topology introduced in Section II.2 and its mathematical formalism in Section III.1, we perform numerical simulations modeling the radial dependence of the local Hubble parameter H(r)H(r). This simulation probes how the inhomogeneous and topologically layered geometry produces variations in cosmological expansion, offering an avenue to address persistent observational anomalies such as the Hubble tension and cosmic anisotropies.
1. Parametrization of Layered Structure
In our model, the observable universe is partitioned into quasi-spherical concentric domains Di\mathcal{D}_i, each characterized by:
A local expansion rate HiH(ri)H_i \equiv H(r_i),
A density profile i(r)\rho_i(r),
A local curvature parameter kik_i,
A Hausdorff dimension DH,iD_{H,i}, representing fractal inhomogeneity.
The effective Hubble parameter as a function of radius rr is then described as a layer-weighted sum:
H(r)=iWi(r)HiH(r) = \sum_i W_i(r) \cdot H_i
where:
Wi(r)W_i(r) is a normalized spatial weighting function, e.g., a smooth bump or Gaussian centered at radius rir_i, satisfying iWi(r)=1\sum_i W_i(r) = 1.
This formalism allows continuity of expansion across domains while maintaining sharp gradient transitions in density and curvature between layers.
2. Density Inputs and Initial Conditions
To constrain simulation inputs, we use observationally informed density profiles derived from:
