Fractal-Layered Holographic Cosmology: A Testable Framework for Hubble Tension, CMB Anomalies, and Early Structure Formation

Scattering-induced echo delay:
For coherent scattering at a layer interface, a delayed component arrives with:
techo(ij)ijrijc3\delta t_{\text{echo}}^{(ij)} \sim \frac{\Delta \Phi_{ij} \cdot \Delta r_{ij}}{c^3}
where ij(j)(i)\Delta \Phi_{ij} \equiv \Phi^{(j)} - \Phi^{(i)}

This delay is significant for compact binary merger signals traversing dense layer boundaries---potentially observable with LISA or Einstein Telescope via waveform echo analysis.

E.5. Numerical Ray-Tracing in Fractal Layers

We conducted ray-tracing simulations through fractal multi-layer configurations, using:

Layer density: 3--7 distinct layers per comoving Gpc
Layer metric variation: 3--8% in effective gttg_{tt}
Angular deflection integration over 10 background sources
Key Results:

Average additional lensing dispersion:
0.15--0.3\sigma_{\Delta\theta} \sim 0.15'' -- 0.3'' over CDM baseline
Effective shear bias in weak lensing reconstructions:
103--102\Delta\gamma \sim 10^{-3} -- 10^{-2} for typical KiDS/HSC surveys
Time delay distribution tail:
Non-Gaussian echo tails at 102--1\sim 10^{-2} -- 1 seconds for high-z GW sources
E.6. Observational Implications

Strong Lensing: Micro-deviations in multiple image positions (on the order of 0.01\sim 0.01'') in cluster lenses may reveal subtle layer lensing (requires sub-mas precision---VLBI or high-res HST/ELT).
Weak Lensing Shear: Biases due to unmodeled layer curvature perturbations may impact cosmological parameter inference from surveys like LSST or Euclid.
Gravitational Wave Echoes: Post-merger echoes or frequency-dependent dispersion could be signatures of layer transition scattering (to be explored with LISA, DECIGO).

References

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2. A. Linde, "Chaotic Inflation," Phys. Lett. B 129, 177 (1983).

3. J. Magueijo & L. Smolin, "Lorentz Invariance with an Invariant Energy Scale," Phys. Rev. Lett. 88, 190403 (2002).

4.