5. Implications for Cosmic Anisotropy and Acceleration
Fractal scaling affects both geometry and dynamics:
Geodesic deviation and Ricci focusing behave differently in fractal backgrounds.
Observers located within lower-DHD_H layers perceive apparent acceleration due to inhomogeneous expansion.
This provides a natural explanation for observed cosmic anisotropies and late-time acceleration without invoking a cosmological constant.
6. Fractal Power Spectrum and Observables
From the density scaling, we derive a fractal power spectrum P(k)P(k) for large-scale structures:
P(k)k,=3DHP(k) \sim k^{-\gamma}, \quad \gamma = 3 - D_H
Observations suggesting 1.2\gamma \sim 1.2 correspond to DH1.8D_H \sim 1.8, matching filamentary structures in cosmic web simulations.
This aligns with:
Two-point correlation functions (r)r\xi(r) \sim r^{-\gamma},
Galaxy number counts in magnitude-redshift space.
7. Fractal Distribution as an Entropic Attractor
The emergence of fractality may be rooted in information-theoretic entropy minimization, leading to criticality in early universe dynamics.
We define an information-encoded entropy functional:
S=ipilogpi,pi=riDH3jrjDH3\mathcal{S} = - \sum_i p_i \log p_i, \quad p_i = \frac{r_i^{D_H - 3}}{\sum_j r_j^{D_H - 3}}
