# What is the hypersurface that represents the particles horizon?

I need to clarify a detail about the particles horizon (also called the causality horizon, not to be confused with the event horizon). For simplicity, consider an euclidian static universe, with a sudden beginning at $t_{\circ} = 0$ (the "Big Bang" of that model), and a brutal ending at $t = t_{\bullet}$ (the "Big Crunch" of the model). The cosmological scale factor is constant : $a(t) = 1$ for $t_{\circ} < t < t_{\bullet}$. (think of that model as a 3D "game-universe" that you launch at some time on your computer, then quit after a while. Poor virtual creatures living in the game !)

Here's a picture I made to represent that simple universe. Distance $\mathcal{D}$ is the proper distance from the stationary observer, shown as a blue vertical word-line on the picture. The red vertical line is the world-line of a stationary particle. $\mathcal{H}_{\mathcal{E}}$ is the event horizon of the observer, while the past light-cone $\mathcal{H}_{\mathcal{C}}$ is the causality (or particles) horizon at time $t_0$ (the present of the observer). $\mathcal{A}$ and $\mathcal{B}$ are two arbitrary events.

Now, what confuses me are the (very nice) pictures shown in the answer there : Is the cosmic horizon related to the Big Bang event?. I'm showing the pictures from that answer here for reference :

On these pictures, the particles horizon is depicted as an inverted (future-oriented) large cone, not as an evolving smaller (past-oriented) light-cone. Why ?

If I add this inverted light-cone on my picture above, event $\mathcal{A}$ would be outside that particles horizon. Yet it is an observable event for a time $t > t_0$ (as shown with the green dashed line).

I believe that the particles horizon shown on these pictures isn't right and should be labeled as something like "particles horizon distance" instead (not "particle horizon"). Am I wrong ?

EDIT : A variation of the first picture, with a better representation of the particle horizon (? I'm not convinced yet !) :

"Event A is outside the particle horizon" means by definition that at the time $t(\mathcal{A})$, no event of its world-line for $t \leq t(\mathcal{A})$ can be seen by the stationary observer at time $t(\mathcal{A})$, which agrees with your figure ($\mathcal{A}$ is to far from the observer on the time interval $[t_o \ t(\mathcal{A})]$). Similarly, in your picture, event $\mathcal{B}$ is inside the particle horizon because at $t = t(\mathcal{B})$, the stationary observer is able to see one point of the world-line of $\mathcal{B}$ that occurred for some $t \leq t(\mathcal{B})$. Hence there is nothing contradictory. Besides, your suggestion to label the "particle horizon" as something like "particles horizon distance" makes sense, as by definition (taken from your reference question), the "particle horizon" is "the distance - at that time - to the furthest object that can be seen". And to answer your initial question, the cone in the figures is correct, the distance to the furthest object that can be seen is necessarily increasing with the time (the more you wait, the more distant are the objects you can see).

• I agree with your interpretation, but the question is still the same and unanswered : how should we represent the whole particle horizon hypersurface in these space-time diagrams ? A cone like what I've drawn, or an inverted cone like on the figures I've shown (which actually shows the distance as a function of time) ? In the case of my own figure, there's something puzzling : the distance to the horizon at time $t_0$ appears to be 0 (wrongly), unless we look at the base of the cone (at $t_{\circ} = 0$) ! It doesn't seem to be "local" in space-time.
– Cham
Sep 27, 2016 at 21:44
• I would say an inverted cone like on the figures you have shown. Why would the particles horizon at time $t_0$ be $0$ in your own figure? For me it would rather be $\mathcal{D}_{\mathcal{C}}(t_0)$, as at time $t_0$, the stationary observer is able to see the event $(\mathcal{D}_{\mathcal{C}}(t_0), t = 0)$ belonging to the world-line $(\mathcal{D}_{\mathcal{C}}(t_0), 0 \leq t \leq t_0)$.
– user130529
Sep 27, 2016 at 22:12
• The problem with the "inverted cone" is that event $\mathcal{A}$ would stay outside of it, even if you draw a full 45 degree cone all the way up to $t_{\bullet}$, while it is a visible event in the future of the observer at time $t_0 < t_{\bullet}$. I mean that the observer will be able to see $\mathcal{A}$ at a time $t_{\mathcal{A}} > t_0$. I don't see how to properly represent the particle horizon as an hypersurface in space-time.
– Cham
Sep 27, 2016 at 22:18
• Maybe the question could be reformulated as "Is the particle horizon really a light-like hyper surface, i.e. a cone in space-time ?". I'm beginning to suspect that it's actually a space-like hyper surface, or even just a simple 2-sphere of radius $c t_0$ (in the truncated flat space defined above).
– Cham
Sep 27, 2016 at 22:29
• Yes, surely both of them can be seen by the observer at some time, but not any older version of $\mathcal{A}$ at $t(\mathcal{A})$, whereas an older version of $\mathcal{B}$ can be seen at $t(\mathcal{B})$. That's the difference.
– user130529
Sep 28, 2016 at 15:08