To explain large-scale matter clustering, early galaxy formation, and cosmic web filamentation within our proposed framework, we adopt a fractal geometry perspective. This section formalizes the use of non-integer dimensional analysis to characterize the spatial distribution of mass-energy and its observational consequences.
1. Motivation: Observational Clues to Fractality
Recent surveys---including SDSS, 2dF, and DESI---have revealed evidence of scale-invariant clustering patterns and power-law correlations in the spatial distribution of galaxies up to scales of several hundred megaparsecs.
Standard CDM cosmology assumes homogeneity beyond the so-called homogeneity scale RH100MpcR_H \sim 100 \, \text{Mpc}, yet increasing data show:
Void self-similarity,
Filamentary cosmic web structure,
Inhomogeneities persisting across scales.
These features motivate modeling the universe's matter distribution with fractal geometry.
2. Fractal Scaling and Hausdorff Dimension
In a fractal mass distribution, the mass contained within a sphere of radius rr centered at any point scales as:
M(r)rDHM(r) \sim r^{D_H}
where:
DHD_H is the Hausdorff (fractal) dimension,
0<DH30 < D_H \leq 3; for DH=3D_H = 3, the distribution is uniform.
Differentiating with respect to volume yields the mass-energy density:
(r)dMdVddr(rDH)r2rDH3\rho(r) \sim \frac{dM}{dV} \sim \frac{d}{dr}\left( r^{D_H} \right) \cdot r^{-2} \sim r^{D_H - 3}
