Harrison-Peebles-Zeldovich spectrum ($P(k,\tau )\propto {\tau }^{4}k$)

The Harrison-Peebles-Zeldovich spectrum is the scale-invariant primordial growing-mode spectrum: in a fixed conformal time slice before cosmological horizon crossing, the matter power spectrum is proportional to ${\tau }^{4}k$ in the usual growing-mode density convention. At horizon entry ${\tau }_H\sim k^{-1}$, its dimensionless mass variance $k^{3}P(k,{\tau }_H)$ is independent of $k$. This statement fixes a spectral shape, not its amplitude. Radiation domination subsequently suppresses small-scale density growth through the Mészáros effect, producing the asymptotic matter power spectrum $P\propto k^{-3}{\ln }^2(k/k_{\mathrm{eq}})$ after matter-radiation equality.

For the growing matter-era spectrum $P(k,\tau )=Ak(\tau /{\tau }_{\mathrm{eq}}{)}^4$ and Gaussian Fourier window $W(kr)=e^{-(kr{)}^2/2}$, integration gives the displayed smoothed matter density variance. A physical Hubble-radius window has $r=1/(aH)=\tau /2$, making its variance independent of crossing time. The dimensionless mode variance $k^{3}P/(2{\pi }^2)$ is likewise constant at $k\tau =2$. A pure $P\propto k$ spectrum needs ultraviolet-convergent smoothing: a real-space top-hat alone leaves a logarithmic divergence.

// Target: cosmology.bigb