where:
TjljlT_{jl} \equiv \partial_j \partial_l \Phi is the local tidal tensor (second derivative of the gravitational potential),
IlkI_{lk} is the inertia tensor of the mass distribution,
ijk\epsilon_{ijk} is the Levi-Civita symbol.
In Gaussian random fields with scale-invariant power spectra, the statistical alignment between L\mathbf{L} and the principal axes of the tidal tensor is well-characterized.
However, for fractal density fields with long-range correlations and nontrivial scaling, the statistical properties of both TjlT_{jl} and IlkI_{lk} are modified.
C.2. Fractal Density Field Properties
We assume the matter density perturbation (x)\delta(\mathbf{x}) satisfies:
(x)(x+r)r(3DH)\langle \delta(\mathbf{x}) \delta(\mathbf{x} + \mathbf{r}) \rangle \propto r^{-(3 - D_H)}
This autocorrelation implies a power spectrum:
P(k)k(3DH)P(k) \sim k^{-(3 - D_H)}
Using this, the variance of density perturbations smoothed on scale RR behaves as:
2(R)R(3DH)\sigma^2(R) \sim R^{-(3 - D_H)}
This modified scaling directly affects both the tidal and inertia tensors.
