Gaussian horizon-crossing variance for a Harrison-Zeldovich spectrum (${\sigma }_R^2=A(\tau /{\tau }_{\mathrm{eq}}{)}^4/(4{\pi }^2{r}^4),r=R/a$)

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