C.3. Modified Tidal Tensor Statistics
Define the tidal tensor smoothed on scale RR:
Tij(R)=ij(x)WR(x)T_{ij}(R) = \partial_i \partial_j \Phi(\mathbf{x}) * W_R(\mathbf{x})
In Fourier space, using Poisson's equation /k2\Phi \sim \delta / k^2, the variance becomes:
TijTklkikjkkklk4P(k)W~2(kR)d3k\langle T_{ij} T_{kl} \rangle \sim \int \frac{k_i k_j k_k k_l}{k^4} P(k) \tilde{W}^2(kR) \, d^3k
Plugging in P(k)k(3DH)P(k) \sim k^{-(3 - D_H)} yields a scale dependence:
TijTklR(1+DH)\langle T_{ij} T_{kl} \rangle \sim R^{-(1 + D_H)}
Thus, in a fractal field, the tidal tensor varies more strongly with scale than in the scale-invariant case.
C.4. Fractal Inertia Tensor and Mass Distribution
Assuming that protohalos form in overdensities following the same fractal scaling, the mass within radius RR scales as:
M(R)RDHM(R) \sim R^{D_H}
