= Gaussian horizon-crossing variance for a Harrison-Zeldovich spectrum
{c}
{title2=$\sigma_R^2=A(\tau/\tau_{\rm eq})^4/(4\pi^2r^4),\quad r=R/a$}
For the growing matter-era spectrum $P(k,\tau)=Ak(\tau/\tau_{\rm eq})^4$ 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 $k^3P/(2\pi^2)$ 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
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