In quantum field theory, renormalization group (RG) flows describe how physical parameters evolve with scale. Analogously, in our cosmological framework:
Each layer of the universe corresponds to a different effective scale,
The cosmic metric and energy density evolve as a function of "scale-layer depth" ll, akin to RG flow in statistical physics.
The energy density at scale rr evolves as:
(r)rDH3,\rho(r) \sim r^{D_H - 3},
where DHD_H is the Hausdorff fractal dimension, resulting from coarse-graining across layers.
This is not merely geometric --- the metric evolution and coupling constants also change under the influence of:
Inter-layer phase correlations (akin to coupling constants),
Holographic energy exchange (akin to anomalous dimensions),
Curvature feedback mechanisms (renormalizing gravitational coupling).
3. Embedding RG Flow into Cosmological Dynamics
We propose a generalization of the Friedmann equation under RG-flow-inspired evolution:
Hi2=8G3ikiai2+jiijcos(ij),H_i^2 = \frac{8\pi G}{3} \rho_i - \frac{k_i}{a_i^2} + \epsilon \sum_{j \neq i} \alpha_{ij} \cos(\theta_i - \theta_j),
where:
ii indexes the coarse-grained layer,
ij\alpha_{ij} encode cross-scale coupling strengths (analogous to beta functions),
i\theta_i are phase angles that modulate entanglement,
\epsilon controls overall interlayer interference strength.
Such a formulation allows self-similarity and feedback between structure formation at different cosmic epochs, providing a formal mechanism to connect early- and late-time anomalies.
4. Comparison with Traditional Hierarchical Models
