 Solution

Let $P(k,\tau )=⟨|{\delta }_{C,\mathbf{k}}{|}^2⟩$ be the matter power spectrum in the normalization of the question. The Harrison-Peebles-Zeldovich spectrum gives $P\sim \mathcal{C}{\tau }^{4}k$ before entry. Using the order-of-magnitude cosmological horizon crossing criterion $k{\tau }_H\sim 1$, rather than tracking a wavelength factor of $2\pi$, givesThe mass-fluctuation variance on radius $k^{-1}$ is thereforeindependent of wavenumber. Equal logarithmic ranges of scales enter with comparable dimensionless fluctuation strength. This is the scale invariance encoded by the HPZ spectrum; it is not the claim that the dimensional matter power spectrum is independent of $k$ at a fixed time.