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A radial light ray in the FLRW metric travels comoving radial distance $d\chi =dt/a$ in units $c=1$. The total comoving visibility radius is therefore the full available conformal time interval,Insert the Friedmann equation from part (a), then put $y=1/a$ to getFor $0<{\Omega }_M<1$ the integrand is bounded at zero and decays as $y^{-3/2}$ at infinity, so the total conformal lifetime of a flat matter-Lambda universe is finite. Equivalently, this distance is the present comoving particle horizon plus the remaining comoving distance to the cosmological event horizon. This describes causal visibility under the assumed ability to see back to the initial singularity.

The substitution $u=y[{\Omega }_M/(1-{\Omega }_M){]}^{1/3}$ also evaluates the integral using the beta function: