Codex @codex  0

Conservation of the comoving galaxy population gives $n(z)=n_0{a}^{-3}=n_0(1+z{)}^3$. Holding the proper galaxy radius fixed, integrate a:With $r_G=0.015\mathrm{Mpc}$, $n_0=0.01\mathrm{Mp}{\mathrm{c}}^{-3}$, and $c/H_0=4282.75\mathrm{Mpc}$, set $C=c\pi r_G^2{n}_0/H_0=0.030273$. ConsequentlyFor random uncorrelated intersections, the zero-intersection probability is $e^{-\tau }$ from the Poisson distribution, so the fraction of the sky covered by foreground galaxies out to $z=1$ isIgnoring overlaps gives $f\simeq \tau \simeq 3.7\%$. A literal shell of zero thickness exactly at $z=1$ has zero coverage; the integrated foreground interpretation is needed. Clustering changes the exact overlap probability, although the mean optical depth above still follows from the mean number density.

Finally,At this optical depth, the Poisson sky coverage is $1-e^{-1}\simeq 63\%$, not $100\%$. The large inferred cosmological redshift extrapolates conserved galaxy counts and fixed radii far beyond their plausible physical range; it is the answer within the prescribed toy model.