CMB anisotropy maps (e.g., Planck 2018, WMAP),
Large-scale structure surveys (e.g., SDSS, DES, Euclid forecast),
Type Ia supernova distance moduli,
BAO data, used to calibrate the radial dependence of matter and dark energy content.
We model the mass-energy density (r)\rho(r) in each layer using the fractal distribution function:
(r)=0(rr0)DH3\rho(r) = \rho_0 \left( \frac{r}{r_0} \right)^{D_H - 3}
where:
DH(2.0,2.9)D_H \in (2.0, 2.9), varies across simulations to probe different self-similarity regimes,
r0r_0 is a scaling constant fixed at the typical scale of local voids (150Mpc\sim 150\, \text{Mpc}).
3. Modified Friedmann Equation per Layer
Within each domain Di\mathcal{D}_i, the evolution of the local scale factor ai(t)a_i(t) follows a modified Friedmann equation accounting for both local curvature and fractal corrections:
Hi2=(aiai)2=8G3i(r)kiai2+i(r)H_i^2 = \left( \frac{\dot{a}_i}{a_i} \right)^2 = \frac{8\pi G}{3} \rho_i(r) - \frac{k_i}{a_i^2} + \delta_i(r)
where:
i(r)\delta_i(r) represents fractal correction terms or inter-layer coupling contributions (modeled as perturbative deviations from standard CDM).
4. Simulation Implementation
We implement the simulations using:
Finite-difference methods for radial evolution of H(r)H(r),
Adaptive mesh refinement near layer transitions to resolve sharp gradient regions,
CosmoMC-modified solvers to incorporate fractal density and non-FRW curvature structure.
Boundary conditions:
At r=0r = 0, normalization to locally observed H0local=731.5km/s/MpcH_0^{\text{local}} = 73 \pm 1.5 \, \text{km/s/Mpc},
At large rr, asymptotic matching to Planck-inferred global value H0CMB=67.40.5km/s/MpcH_0^{\text{CMB}} = 67.4 \pm 0.5 \, \text{km/s/Mpc},
Smooth junction conditions across layer boundaries enforced via Birkhoff-like conditions generalized for non-homogeneous settings.
5. Results and Key Features
