Let's model the universe with the FLRW metric
$$ ds^2 =-dt^2 +a(t)^2\big(d\chi +R_k(\chi)^2 d\Omega^2\big)$$
where $a(t)$ is the scale factor and $R_k(\chi)$ is $\chi$ for a spatially flat ($k=0$) spacetime, equal to $\sin(\chi)$ for a spacetime with positive spatial curvature ($k=1$) and equal to $\sinh(\chi)$ for a spacectime with negative spatial curvature ($k=-1$).
If there's only one component in the universe (matter, radiation or dark energy), we get
$$ a(t)=\begin{cases}t^{2/3}\ \ \text{ for matter}\\ t^{1/2}\ \ \text{ for radiation}\\ e^{Ht}\ \ \ \text{for dark energy}\end{cases}$$
Now, let's consider the conformal Penrose diagram of each universe. I understand that both matter and radiation dominated universes start with a Big bang and have a future null infinity. On the other hand, the dark energy dominated universe is eternal and has no null boundary; its conformal diagram is that of a de Sitter spacetime (i.e. a box with only asymtptotic future and past spacelike boundaries but no boundaries "on the sides").
I understand that the universe started being radiation dominated, then switched to matter dominated and currently is (and forever after will be) dark energy dominated.
My question is: How are we supposed to glue the different conformal diagrams? Is there a way to make an actual, realistic conformal diagram of the universe as it switches through each era? Does it even make sense to talk about an asymptotic boundary for the first eras of the universe? if the asymptotic region is supposed to be the place where things go after an infinite time, it doesn't matter if radiation shot to infinity during the radiation dominated era, by the time it "gets there" the universe will have become dark energy dominated so there will be no null boundary.
My guess is that the conformal diagram of our universe should look like the dark energy dominated diagram, but with a Big bang in the beginning, which should be the only noticeable difference. It doesn't matter that there would be a null future infinity during matter/radiation dominace, because in the far future the universe is dark energy dominated and there is no null future infinity anyways.
P.S.: Side question, does the value of $k$ affect the conformal diagram at all?
