Dark matter halos, simulated in CDM N-body experiments, exhibit universal density profiles (e.g., NFW or Einasto), but when interpreted through the lens of fractal geometry, we propose that:
Halo substructure follows multifractal distributions,
The intermediate-scale correlation length is layer-dependent,
Self-similar halo merging generates scale-free power-law spectra observable in weak lensing.
This fractal interpretation yields a generalized power spectrum for matter density fluctuations:
P(k)k,=3DHP(k) \propto k^{-\beta}, \quad \beta = 3 - D_H
Allowing us to relate power spectrum slope directly to observed fractal dimensionality.
4. Cosmic Microwave Background (CMB) Implications
Residual anisotropies in the CMB at large angular scales (e.g., quadrupole suppression, axis of evil) may be reframed as shadow projections of a fractal universe---where large-scale voids and over-densities across layers scatter and lens the CMB photons non-uniformly.
This introduces:
Directional variance in lensing convergence maps,
Non-Gaussian kurtosis excesses due to fractal-induced fluctuations,
Spectral distortions from light traversing nested voids (layer junctions) with variable effective dimensionality.
5. Observational Signatures and Predictions
If the universe is fractal-layered rather than strictly homogeneous:
The transition to homogeneity (where DH3D_H \to 3) may never be reached globally.
Galaxy surveys (e.g., SDSS, Euclid) should detect persistent power-law clustering at scales >200 Mpc in certain directions.
CMB angular correlation function will show hemispheric variance consistent with large-scale fractal void alignment.
Weak lensing maps should reveal anisotropic convergence patterns aligned with large-scale fractal sheets.
6. Theoretical Implications and Fractal-Topology Synthesis
Combining this fractal structure with our Multilayer Topology (II.2):
