We quantify this structure via the Hausdorff dimension DHD_H, a measure of spatial complexity that generalizes the notion of Euclidean dimension. For a point distribution embedded in 3D space, the mass--radius relation for a fractal structure is given by:
M(R)RDHM(R) \propto R^{D_H}
Where:
M(R)M(R) is the total mass (or number of galaxies) enclosed within a radius RR,
DH<3D_H < 3 implies fractal, scale-invariant clustering.
Empirical studies suggest:
On scales 1 Mpc<R<100 Mpc1 \text{ Mpc} < R < 100 \text{ Mpc}, galaxy distributions follow DH2.0D_H \approx 2.0 to 2.32.3,
On larger scales, a transition toward DH3D_H \to 3 is expected but not conclusively observed.
We hypothesize that this transition is not universal, but layer-dependent as outlined in II.2, meaning the fractal scaling may persist asymmetrically across layers, imprinting observable anisotropies.
2. Fractality in Galaxy Clustering and Voids
The two-point correlation function (r)\xi(r) of galaxy clustering behaves as:
(r)(rr0),1.8\xi(r) \sim \left(\frac{r}{r_0}\right)^{-\gamma}, \quad \gamma \approx 1.8
This exponent corresponds to a correlation dimension D2=31.2D_2 = 3 - \gamma \approx 1.2, supporting a fractal-like structure in galaxy clustering up to tens of Mpc. Similarly, void size distributions, cluster mass functions, and filamentary structures display hierarchical nesting, suggesting the Universe is organized according to recursive self-similar dynamics.
In our framework, these structures emerge from:
Nonlinear self-gravitating dynamics on a fractal metric background,
Inter-layer phase interference modulating structure growth,
Topological constraints on expansion within each layer (see II.2).
3. Fractal Dark Matter Distribution
