We define the fractal torque field (r)\vec{\tau}(\vec{r}) arising from asymmetric gravitational potentials i(r)\Phi_i(\vec{r}) in nested layers as:
(r)=abcIbddcfractal(r)\vec{\tau}(\vec{r}) = \epsilon_{abc} \, I_{bd} \, \partial_d \partial_c \Phi_{\text{fractal}}(\vec{r})
Where:
IbdI_{bd} is the inertia tensor of the forming halo,
fractal=ii(r)cos(ij)\Phi_{\text{fractal}} = \sum_i \Phi_i(\vec{r}) \cos(\theta_i - \theta_j) includes phase interference terms from topological interactions,
Anisotropic patterns are amplified in regions of high inter-layer gradient.
As a result, galaxies in these zones acquire spins preferentially aligned with either:
The major axis of their local filament (parallel),
Or the normal to the void surface they border (perpendicular),
depending on the local layer curvature and torque vector field.
3. Predictions and Expected Signal
The model predicts:
Quadrupolar or hexadecapolar spin-filament alignment patterns on scales >50h1Mpc> 50 \, h^{-1} \, \text{Mpc},
Redshift evolution in alignment strength A(z)A(z), peaking around z1.5z \sim 1.5,
Directional asymmetries in alignment angle distributions across the sky (dipole or octupole modulation),
Stronger alignment in fractal interlayer zones than in homogenous regions.
4. Detection Strategies with SKA
The Square Kilometre Array (SKA), particularly SKA Phase 2, is ideally suited for testing these predictions:
Key Capabilities:
Full-sky HI surveys of ~10 galaxies up to z2z \sim 2,
Resolved rotation curves and spin vectors via 21cm emission,
High-precision mapping of cosmic web filaments using galaxy distributions.
Methods:
Construct 3D filament skeletons using DisPerSE or T-ReX,
Compute galaxy spin vectors L\vec{L} from HI disk orientations,
Measure alignment angle =arccos(L^F^)\theta = \arccos(\hat{L} \cdot \hat{F}) relative to filament axis,
Analyze P(cos)P(\cos \theta) for excesses over random alignment,
Perform sky hemisphere comparisons to detect large-scale asymmetries.
5. Forecasted Signals
