The conformal diagram of de Sitter spacetime looks like this
I think I understand the causal properties of this diagram. Someone who is static in the south pole can send messages only to the upper right triangle, while they can only receive messages from the lower right triangle.The union of both is the right diamond, which plays a similar role to the exterior of a black hole in a Schwarzschild diagram.
My question is: Where in this diagram are the usual horizons one discusses in cosmology? For example, where in this diagram can I see the particle horizon? Or the Hubble horizon? It seems to me that at future infinity, an observer at thesouth pole can access all space (Not all spacetime!). That is, a signal from the north pole will reach the south pole at future infinity. That clashes with my understanding of cosmology where there are places in space whose light will never reach us, no matter how much we wait.
For the particle horizon, that should be defined by the time light had to travel to you since the beginning. I understand that the beginning of our universe was not de Sitter and de Sitter spacetime doesn't even have a singularity, but still, I think that we can see the particle horizon as the subregion of a space slice that intersects our past lightcone
Similarly, in our universe there exists a Hubble horizon. This is the boundary after which we will no longer see things. I believe this would just be the diagnal going from the north pole in the past to the south pole in the future. Even if things were initially in our past lightcone, as time goes by, as soon as they cross this boundary, they are gone from our sight. Something like this?
Does this make any sense?