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Spatial flatness constrains the total density, not matter alone. A flat model can havewith a substantial dark energy component. A positive cosmological constant has pressure $p_{\Lambda }=-{\rho }_{\Lambda }{c}^2$ and does not dilute like matter; it can produce the observed accelerated expansion.

If positive $\Lambda$ persists, $a(t)$ becomes exponential and the future conformal time intervalis finite. For two comoving civilisations separated by $\chi$, each one-way signal consumes $\chi /c$. Thus a message completing $n$ legs at finite time requiresOnly finitely many legs, and hence finitely many full exchanges, can then occur. A reply to even the first new message requires $2\chi /c<\Delta {\eta }_{\mathrm{future}}$, the cosmological round-trip communication horizon condition. Depending on the density parameters, the initial outward message might arrive while its reply can never reach us, as in the right-hand panel of part d.

The source gives no matter and dark-energy fractions, so it does not specify an exact count for the $z=1$ pair in the real Universe. Flatness and lack of matter dominance alone do not prove a finite horizon: a radiation-dominated flat universe, for example, has divergent future conformal time if that power-law expansion persists. The finite-message conclusion uses persistent accelerated expansion such as positive constant $\Lambda$.

Set $a(t)=a_0(t/t_0{)}^2$. The remaining future conformal time isA light ray sent to a comoving reflector at separation $\chi$ and immediately returned consumes $2\chi$ of conformal time. The cosmological round-trip communication horizon therefore has supremum physical distanceFinite-time reception requires $d_0<d_{\max }$. Indeed the total travel equation is $2d_0/a_0=(t_0/a_0)(1-t_0/t_r)$, giving $t_r=t_0/(1-2d_0/t_0)$. At the limiting distance the return time tends to infinity. This round-trip radius is half the current physical cosmological event horizon radius.