These are just some rather handwaving comments related to why the spectrum of the fluctuations in the CMB are of expected to be scale invariant.

The variations of the CMB we observe today correspond to the expanded initial over-dense and under-dense regions created by vacuum fluctuations.

The theoretical expectation of scale invariance of the spectrum of these fluxtuations can for example be explained by looking at a scalar field with the Lagrangian density

\(L = \frac{1}{2}\dot{\phi^2} + \frac{1}{2} (\nabla\phi)^2 - V(\phi)\)

Multiplying by the volume of space $a^3(t)$, assuming that any spatial gradients are stretched out by inflation, and making use of the Euler-Lagrange equations it can be shown that the equation of motion corresponds to damped harmonic oscillator

\(\ddot{\phi} + 3 H \dot{\phi } = -\frac{\partial V}{\partial \phi}\)

where the second term in the L.H.S. is the so-called Hubble friction. Together with the quantum fluctuations, a scale invariant CMB spectrum can then theoretically be expected, as quantum fluctuations are excited at small scales, subsequently enlarged by the expansion of the universe, and finally dissipated (frozen out) by the Hubble friction at large scales as soon as their wave length gets larger than the light horizon (separation over which information can be exchanged).

I am not sure what processes there could be that lead to deviations of the CMB spectrum from scale invariance.