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The conformal diagram of de Sitter spacetime looks like this

enter image description here

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.

Attempted answer

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

Attempt at finding the particle horizon in the dS diagram

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?

enter image description here

Does this make any sense?

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Everything that you said is correct, and the reason it doesn't match ΛCDM cosmology is just that the ΛCDM manifold doesn't have that shape. Without inflation, it has this causal structure:

...____________...  t = ∞    η
        /\                   ^
       /  \                  |
...___/____\___...  t ≈ 0    +--> x

This is a plot of $(η,x,y,z)$ conformal/comoving coordinates with $y$ and $z$ suppressed. Note that $η$ on this diagram is not the same as $η$ on yours. It's also not a Penrose diagram, since it extends indefinitely to the left and right, but it illustrates the causal structure well enough. "$t\approx 0$" is the traditional radiation-dominated big bang singularity, or a very early time in the radiation-dominated epoch ("end of inflation"), or the CMBR emission time—it makes little difference. "$t=\infty$" is of course the de Sitter future infinity. $η(t=\infty) - η(t\approx 0)$ is finite and not very large (about 16 Gyr), while the comoving size of the universe is apparently much larger, if not actually infinite.

The idea of adding an inflationary epoch is to move the beginning of time to an earlier $η$ so that the past light cone encompasses a larger area. De Sitter space in these coordinates is infinite in three directions, as past infinity is at $η=-\infty$ (but these coordinates only cover half of the space, the upper right triangle of your diagram for instance).

I would like to show a Penrose diagram for ΛCDM that looks like your diagram at late times, to tie the pictures together, but I'm not sure how to do it, and I can't remember ever seeing one. Your diagram can be obtained by embedding de Sitter space in $\mathbb R^{4,1}$, compactifying the embedding space (which turns the de Sitter space into the hypercylinder $S^3\times [-π/2,π/2]$), and suppressing two of the angular coordinates. You could (I think) do the same thing with a $k=+1$ approximation to ΛCDM with a large radius of curvature, but the resulting diagram wouldn't be flat: it would be a cylinder with an oddly shaped end cap for the radiation and matter (and optional inflationary) epochs. I may be missing something, though.

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