3. Derivation from RG-Driven Effective Action
Let SeffS_\text{eff} be the effective action describing matter distributions on a coarse-grained spacetime manifold M\mathcal{M}. Under scale transformations rrr \rightarrow \lambda r, we expect:
(r)=DH3(r)\rho(\lambda r) = \lambda^{D_H - 3} \rho(r)
This scaling behavior emerges when the RG beta functions for the gravitational and matter sectors flow toward non-Gaussian fixed points, producing anomalous dimensions for energy density operators. The resulting effective stress-energy tensor TeffT_{\mu\nu}^{\text{eff}} exhibits non-integer scaling under dilation symmetry.
In particular, the energy-momentum tensor contributes to the Einstein field equations with an effective matter profile:
T00(r)=(r)rDH3T^0_0(r) = -\rho(r) \sim r^{D_H - 3}
which leads to modified dynamics in the Friedmann or LTB-like equations, without the need for exotic dark energy components.
4. Physical Interpretation and Observational Implications
The fractal geometry leads to:
Large-scale anisotropies in the cosmic microwave background due to inhomogeneous gravitational potentials,
Scale-dependent gravitational lensing and redshift drift,
