C.6. Summary of Analytical Results
Appendix D: Comparison with Observational Alignment Data (SDSS, KiDS, HSC)
This appendix details the quantitative comparison between the predictions of the fractal-layered cosmology framework and observed galaxy spin--filament alignments in large-scale structure surveys, particularly from SDSS, KiDS, and HSC datasets.
D.1. Observational Background and Metrics
Galaxies tend to align their spin axes with the surrounding cosmic web structures. This alignment is typically measured in terms of:
Spin--filament alignment angle \theta: angle between a galaxy's spin vector S\mathbf{S} and the local filament axis e^3\hat{e}_3.
Distribution function P(cos)P(\cos\theta): compared against isotropic expectation (uniform in cos\cos\theta).
Alignment strength quantified via:
Aalign=cos213A_{\mathrm{align}} = \langle \cos^2\theta \rangle - \frac{1}{3}
where Aalign>0A_{\mathrm{align}} > 0 implies alignment (parallel spins), <0< 0 implies anti-alignment (perpendicular spins).
D.2. Summary of Survey Data
D.3. Observed Trends and Anomalies
(a) SDSS Findings:
Spiral galaxies show weak but significant parallel spin-filament alignment at low mass (M<1010.5MM_* < 10^{10.5} M_\odot).
Alignment increases toward filament spines.
Deviation from CDM N-body simulations unless feedback and environment are strongly tuned.
(b) KiDS Findings:
Red galaxies tend to anti-align (perpendicular spins).
Blue galaxies have no significant alignment globally but localized enhancements along filamentary structures.
(c) HSC Findings:
Enhanced spin--filament alignment at intermediate redshifts (z0.5z \sim 0.5).
Evidence for anisotropic environment-dependent alignment, inconsistent with uniform initial conditions.
D.4. Fractal Model Comparison
