Deviations from Poissonian distributions in galaxy number counts.
These empirical features are better characterized by fractal dimension DH<3D_H < 3, rather than a homogeneous 3D Euclidean distribution. This suggests the need for a theoretical framework that naturally predicts:
(r)rDH3\rho(r) \sim r^{D_H - 3}
where (r)\rho(r) is the average mass-energy density enclosed within a sphere of radius rr, and DHD_H is the Hausdorff (fractal) dimension of the matter distribution.
2. Coarse-Graining and Emergent Dimensionality
We posit that fractal behavior arises not as a fundamental property at the Planck scale, but as an emergent feature of cosmic evolution, driven by:
Coarse-graining of quantum fluctuations during the Universe's early expansion,
Nonlinear gravitational clustering under general relativity,
The renormalization group flow of effective field degrees of freedom as resolution scale increases.
In this view, the early Universe exhibits local quantum fluctuations with full spacetime symmetry (D = 4), but as the Universe expands and structures form hierarchically, gravitational interactions dominate the RG flow, breaking scale-invariance and producing a lower effective dimension.
This process is analogous to dimensional reduction in condensed matter systems, where critical behavior near phase transitions exhibits fractal scaling due to nontrivial fixed points of RG trajectories.
