Thus, the local energy density behaves as a non-integer power law:
(r)rDH3\rho(r) \sim r^{D_H - 3}
This scaling naturally captures the dense cores and diluted peripheries observed in large-scale structures.
3. Generalized Energy-Momentum Tensor
To incorporate fractal scaling into the field equations, we define a fractal-modified energy-momentum tensor:
Tfractal=(r)uu=0(rr0)DH3uuT^{\mu\nu}_{\text{fractal}} = \rho(r) \, u^\mu u^\nu = \rho_0 \left( \frac{r}{r_0} \right)^{D_H - 3} u^\mu u^\nu
where:
0\rho_0 is a reference density at scale r0r_0,
uu^\mu is the fluid 4-velocity,
This form reduces to standard perfect fluid energy-momentum tensor in the limit DH3D_H \to 3.
4. Embedding in the Multilayer Spacetime
In our multilayered topological universe (Section II.2), the fractal matter distribution applies within each nested layer, while inter-layer transitions may show abrupt shifts in DHD_H. Each layer i\Sigma_i has:
i(r)rDH(i)3,withDH(i)(2,3]\rho_i(r) \sim r^{D_H^{(i)} - 3}, \quad \text{with} \quad D_H^{(i)} \in (2, 3]
This structure creates naturally evolving voids and attractors without requiring dark energy to fine-tune the expansion.
