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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:

Use the stipulated common normalization $A=H_0{\Omega }_{m,0}$ and plot $R=A{r}_h/c$. The Einstein-de Sitter universe has $H_{0,E}=A$, whereas the $({\Omega }_{m,0},{\Omega }_{\Lambda ,0})=(0.3,0.7)$ model has $H_{0,\Lambda }=A/0.3$. Their comoving particle horizons are thereforeBoth decrease with increasing $z$, or increase with cosmic time. The Einstein-de Sitter universe curve diverges at $z=-1$ and has $R_E(0)=2$, $R_E(2)=2/\sqrt{3}$. The matter-plus-vacuum curve remains finite at $z=-1$: the integrand approaches a constant there, and its high-redshift tail is integrable. Moreover it lies below the Einstein-de Sitter curve for this particular normalization, since, with $u=1+z^{\prime }>0$,The high-redshift ratio of the two curves tends to $\sqrt{0.3}$, not one. Holding $H_0\sqrt{{\Omega }_{m,0}}$ fixed would give a different comparison; it is $H_0{\Omega }_{m,0}$ that the original question specifies.

A positive cosmological constant eventually makes the scale factor grow exponentially. The future integral ${\int }_t^{\infty }cd{t}^{\prime }/a(t^{\prime })$ then converges, so the remaining conformal time is finite. The comoving particle horizon approaches a finite limiting visibility radius; it does not shrink. The remaining distance to that limit is the comoving radius of the cosmological event horizon. Thus a signal emitted sufficiently far away today can never reach us, even though the proper particle-horizon radius continues to grow. This is the total conformal lifetime of a flat matter-Lambda universe.