Intuition Behind Scale Invariance Power Spectrum

[center o bass]

In the book "Statistical physics for cosmic structures" at p. 171 a read a definition of scale invariance (leading to the so called scale invariant power spectrum) given as the requirement that $\sigma^2_M(R=R_H(t)) = constant$, where $R_H(t)$ is the horizon, i.e. the maximal distance that light could have traveled in cosmological time $t$.

In other words, the normalised mass variance over a sphere of radius the horizon distance should be independent of time. So if we computed the mass variance at some time $t_0$ when the horizon was $R_H(t_0)$ this should be the same as if we computed it at any other $t_1$ when the horizon was $R_H(t_1)$ even though $R_H(t_0)$ might be much smaller than $R_H(t_1)$.

I am trying to get intuition for why one would believe such a requirement to be true. Does anyone have some enlightening explanations/insights?

 

[bapowell]

Scale invariance is only true for de Sitter expansion in which \dot{\rho} = 0. The background is steady state, so there are no time-dependent dynamics that would impart a scale dependence on the perturbations.