Cosmological horizon crossing

A mode of comoving wavenumber $k$ and physical wavelength $2\pi a/k$ crosses a conformal horizon of physical size $ac\tau$ when

During a decelerating expansion the comoving Hubble radius grows. A fixed comoving wavenumber then passes from a superhorizon scale to a shorter scale when $k\simeq aH$. In radiation domination, $aH=1/\tau$, so this occurs at $k\tau \simeq 1$. Here horizon entry refers to Hubble radius crossing; a true causal horizon instead depends on a conformal-time integral.

With $a(t_0)=1$, $k_0=2\pi /(c{t}_0)$, and $1+z_{\mathrm{eq}}=(t_0/t_{\mathrm{eq}}{)}^{2/3}$,

A mode crossing during matter domination acquires the growth factor $({\tau }_0/{\tau }_H{)}^2$. An initial amplitude proportional to ${\tau }_H^2{k}^{1/2}$ therefore becomes proportional to $k^{1/2}$ today.