# Event horizon and particle horizon

The furthest distance that we can see is defined by the radius of the particle horizon, which is nearly 46 Gly. However, the event horizon is nearly 16 Gly. Does this mean the galaxies that are further than 16 Gly will stay the same in the sky? Since their light can never reach us, in other words, their images on the sky will never change?

And after the event horizon becomes stable at 17.6 Gly, every galaxy that crosses that distance will stay on that horizon and we will see them as getting redshifted to infinity?

• Commented Oct 16, 2020 at 19:07

Here's a crude diagram from a similar answer of mine that illustrates what's going on.

                   Z                      <- future infinity
/ \
/   \
/     \
D   C    B   A   B    C   D         <- now
.   .   /   / \   \   .   .
.   .  /   /   \   \  .   .
.   . /   /     \   \ .   .
.   ./   /       \   \.   .
.   /   /         \   \   .
.  /.  /           \  .\  .
. / . /             \ . \ .
./  ./               \.  \.            last scattering
~~~~~~d~~~c~~~~~~~~a~~~~~~~~c~~~d~~~~~~   <- (universe opaque below this)


The horizontal axis on this diagram is "comoving distance", with respect to which objects that move with the Hubble flow are at rest. The vertical axis is "conformal time", with respect to which light travels along diagonal lines of a constant slope (in combination with the comoving distance). A is our current location and Z is the ultimate location in the far future of the same matter, supposing it doesn't deviate much from the Hubble flow. The diagonal lines are the past light cones of A and Z, which are what we can see at those times.

The distance from A to B is 16 Gly and the distance from A to C is 62 Gly. If matter is between B and C then we can (in principle) see it, but we will never see light that it emits at the time labeled "now". That doesn't mean it's frozen in the sky, because we aren't currently seeing light from "now", but from earlier times. We will see this matter continue to evolve along a vertical line from the inner triangle to the outer triangle. The light from that finite proper time will be stretched over the infinite proper time from A to Z.

From any point in the history of matter than ends up at Z, the horizonal metric distance to the past light cone of Z is the distance to the cosmic event horizon. While it looks like this goes to zero, it actually approaches a limit of around 18 Gly because the metric scale factor increases in rough inverse proportion to the apparent distance to Z on the diagram. The point at which any galaxy crosses that horizon is the last point in its evolution that you'll ever see, if you end up at Z.

The difference between comoving and proper distances in defining the observable universe

Does this mean the galaxies that are further than 16 Gly will stay the same in the sky? Since their light can never reach us, in other words, their images on the sky will never change?

No, galaxies further than 16 Gly will NOT stay the same in the sky.

Look at the above image. Consider, for example, a galaxy at 20 Gly from us. This galaxy is currently outside our EH (red line) but we can see it now thanks to the light it emitted in the past, about 2.5 Gyr after the Big Bang (the point where its worldline at 20 Gly intersects our current past light cone, the orange line).

In the future, we will continue to receive light from this galaxy, but light emitted later in time. In fact, from today $$t_0$$ to $$t=\infty$$ in our rest frame we (or better, our decendents) will be able to witness the evolution of the galaxy from $$t_1=2.5$$ Gyr to $$t_2=11$$ Gyr in the rest frame of the galaxy ($$t_2=11$$ Gyr corresponds to the event where the worldline of the galaxy intersects our EH). We will never be able to see anything that occurs in the galaxy after $$t_2=11$$ Gyr in the rest frame of the galaxy, because after that time the galaxy is located beyond our EH and therefore its light can never reach us.

As you can see, as times goes on, the image of the galaxy in the sky will always be changing.

One more thing. As $$t\to \infty$$ in our rest frame, the light from the galaxy emitted near $$t_2=11$$ Gyr will be redshifted out of detectability. So in the far future, we will see a vanishing image of the galaxy "frozen" in the sky at the time frame $$t_2=11$$ Gyr. The mark 17.6 Gly (the proper distance to the EH when $$t\to \infty$$) is irrelevant in this case since our galaxy is already beyond 17.6 Gly.

"Particle Horizon" is similar in meaning to the radius of the spherical boundary of the observable universe. That is,

... the maximum distance from which light from particles could have traveled to the observer in the age of the universe.

Although this was completely unfamiliar to me, I found a definition that I think is what you are referring to when you use the term for "Event Horizon". (Previously the only definition I knew was regarding black holes.)

... the event horizon is the largest comoving distance from which light emitted now can ever reach the observer in the future.

Although your descriptions of the phenomena related to this concept seem to be accurate, the values you give (16 Gly and 17.6 Gly) seem odd to me. I would appreciate your posting a citation to a reference of how these values were calculated. The equation in Wikipedia,

$$d_e(t) = a(t)\int_t^{t_{max}} \frac{cdt'}{a(t')},$$

assumes that $$t_{max}$$ is some future time. What value of $$t_{max}$$ is used to have

$$d_e(t) = 16$$ Gly?

• @PM 2Ring Thank you for your edits.
– Buzz
Commented Nov 3, 2020 at 19:33