# What is the evidence for the existence of a Hubble sphere beyond which galaxies recede faster than light

The title has been changed from the original title, because that is what I meant, but I misused the terminology, and probably some physics, asking about evidence for the existence of a horizon for the observable universe?

Though my question was a mess, Christoph put some work into giving it a cogent answer, clarifying some points, pointing out errors, and adapting to what I really wanted, which I explained in a comment. As I was asking him, he suggested however that I do not change much the question, which contains probably common misconceptions.

With my thanks to him, here is the question as originally asked:

From what I understand, this observable universe horizon results from the expansion, so that when galaxies are sufficiently far away, the expansion speed relative to us is faster than the speed of light. That is clean and simple reasonning.

But, isn't it possible to imagine a geometric structure of the universe such that the speed of light is never exceeded despite the expansion. Then there would not be an observation horizon.

After all, we found with relativty that speeds add in a more complex way that the simple addition we were used to. Could it be that the simple expansion analysis that we use to justify the existence of a horizon for the observable universe could be refined and give a different result?

What we observe is not very far (since the universe is only 13,7 Glyr old). More precisely, we observe it from a time when it was not so far (sorry for the over-simplified statement), even though what we may observe is today much further away. But there is no way to observe things that approach the suggested distance of the horizon (around 80 to 90 Glyr, from various posts).

From what I understand, this observable universe horizon results from the expansion, so that when galaxies are sufficiently far away, the expansion speed relative to us is faster than the speed of light. That is clean and simple reasonning.

Maybe that's clean and simple, but it's also wrong. If you mean recession velocities by "expansion speed relative to us", those reach $$c$$ at the Hubble sphere instead of the event horizon, and we do see objects more distant than that. A better fit would be relative velocities as computed by parallel transport along the light trajectory - except that sources outside the event horizon are causally disconnected from us, ie there is no null geodesic connecting us with them and hence no way to compute relative velocities.

What is the evidence for the existence of a horizon for the observable universe

It's a prediction of our model, sensitive to the time evolution of the scale factor $$a$$. We think the end of the universe will happen at a finite conformal time $$\eta_\text{max} = \int_0^\infty\frac{dt}{a(t)} \lt\infty$$ The cosmic event horizon is the past light cone at that time. In terms of conformal time $$\eta$$ and (appropriately defined) comoving radial distance $$\chi$$ it's given by $$\chi(\eta) = \eta_\text{max} - \eta$$ Light emitted by sources beyond that boundary would take a 'longer than infinite' cosmological time to reach us.

If the time evolution of the scale factor were such that the integral above wasn't finite, there wouldn't be a horizon, and the observable universe would continue to grow unconstrained instead of asymptotically.

Regarding your clarification, the Hubble sphere doesn't have that that much physical significance. It's the place where recession velocities reach $$c$$, but those aren't true relative velocities, but apparent velocities based on a somewhat arbitrary space-time-decomposition at a distance.

Take the galaxy GN-z11, the most distant astronomical object with spectroscopically determined redshift, with $$z\approx11$$. When the light we receive today was emitted, the galaxy had a recession velocity on the order of $$4c$$, and is still receding 'today' (ie at current cosmological time) with a velocity of more than $$2c$$. Nevertheless, we're able to see it - though we'll never be able to see it as it exists 'today' as it has already crossed the event horizon a couple of billion years ago.

How can that be, you may wonder? For an analogy that isn't really exact, but might get across some of the features involved, consider the following situation in special relativity:

You're sitting on a space station. Two ships launch in oppostite directions, with velocities of $$0.6c$$ each. From your perspective, the distance between the ships grows at a rate of $$1.2c$$. And yet, the ships are able to send light signals between each other due to the invariance of the speed of light: From your perspective, the signal starts out with a velocity of $$c$$ despite the emitting ship's motion, and hence can catch up to the other ship; there also isnt really anything special about ships moving with $$0.5c$$ each so their apparent recession velocity adds to $$c$$.

A better analogy could propably be created using accelerating ships (as this will involve Rindler horizons), but I'd have to work out the details first.

• Sorry, I goofed, and misused the terminology. As you can obviously see, I do not use it too often. My question was only about "recession velocity" and the existence of the Hubble sphere, and I did not mean to involve what actually is called "event horizon", as I read it from your answer. I guess we might see more distant objects, but from a time they were inside the sphere (How else, unless expansion slows down?). Now, do I ask my question again, or do I change my existing question, trying to account for your existing answer? Or do I still misunderstand? Feb 26 '20 at 9:35
• I amended my answer. I'd leave your question largely as-is, and just add some clarifying details Feb 26 '20 at 10:42
• Thanks for the pedagogical effort. I did learn some ... though layman intuition is often nonsense in this context. Feb 26 '20 at 11:05
• I just thought that the nuance should be added, that one of those two ships does not see the other as travelling at super luminal speeds, but rather sub luminal - which is why they can communicate, when taken from their point of view. Feb 26 '20 at 23:59
• And we don't actually see things travelling away from us at faster than the speed of light, we see things that in a certain frame are now travelling away faster than the speed of light - but we see them as they were earlier in that frame, and travelling slower than light. As the cross the boundary, their light becomes infinitely red shifted and takes infinitely long to get to us. So, our vision of them freezes as the information about them reaching light speed reaches us. Feb 27 '20 at 0:12