In the standard CDM model, primordial fluctuations seeded by inflation are modeled as a nearly scale-invariant Gaussian field:
P(k)knsP(k) \sim k^{n_s}
where:
ns0.96n_s \approx 0.96 from Planck CMB measurements (indicating slight red tilt),
This spectrum is imposed by inflationary dynamics under slow-roll conditions,
It is modulated by the transfer function, incorporating baryon acoustic oscillations and radiation damping at small scales.
However, CDM assumes a statistically homogeneous and isotropic universe, neglecting scale-dependent clustering bias and internal structural complexity---an assumption challenged by observations of cosmic voids, filaments, and walls at scales >100 Mpc.
2. Fractal Cosmology Prediction
In contrast, the fractal-layered model derives the matter power spectrum from first principles of coarse-grained spatial structure, characterized by a Hausdorff (fractal) dimension DHD_H. The density field scales as:
(r)rDH3\rho(r) \sim r^{D_H - 3}
Using Fourier transform relations, this leads to a power-law power spectrum:
