Particle and Event Horizons in the ΛCDM Model

[JimJCW]

TL;DR Summary

In the ΛCDM (Big Bang) model, the expansion of space generates a particle horizon and an event horizon that have complicated and interesting properties. Let’s plot the radii of these horizons as functions of time and discuss the results.

According to Wikipedia,

The particle horizon is the maximum distance from which light from particles could have traveled to the observer in the age of the universe. It represents the boundary between the observable and the unobservable regions of the universe, so its distance at the present epoch defines the size of the observable universe.​

In astrophysics, an event horizon is a boundary beyond which events cannot affect an observer.​

These horizons can be calculated with calculators provided by Jorrie and Gnedin. They use different names for them:

Using Jorrie’s calculator and PLANCK Data (2015), one can get the following plots:

In the above figure, the particle horizon indicates the size of the observable universe as a function of time. The radius of the event horizon was small during the early times and has a value of 16.5 Gly now.

An interesting point to note is that light emitted outside the event horizon at various times will never reach our location. It can be demonstrated that lights emitted inside of and close to the event horizon have peculiar behaviors such as prolonged journeys to reach our location.

Please share with us your understandings and/or questions about these horizons.

 

[PeterDonis]

According to Wikipedia,

The particle horizon is the maximum distance from which light from particles could have traveled to the observer in the age of the universe. It represents the boundary between the observable and the unobservable regions of the universe, so its distance at the present epoch defines the size of the observable universe.​

The Wikipedia definition gets this backwards. The particle horizon is the maximum distance light can have traveled away from our location in the age of the universe. Since the particle horizon has for quite some time been further away than the event horizon, light emitted from the particle horizon during that time (since about t = 4 billion years) can never reach us. Therefore, the particle horizon is not the boundary of the observable universe; that boundary is always inside the event horizon.

The best reference I know of for straightening out these things is Davis & Lineweaver:

https://arxiv.org/abs/astro-ph/0310808

 

[Bandersnatch]

The Wikipedia definition gets this backwards. The particle horizon is the maximum distance light can have traveled away from our location in the age of the universe. Since the particle horizon has for quite some time been further away than the event horizon, light emitted from the particle horizon during that time (since about t = 4 billion years) can never reach us. Therefore, the particle horizon is not the boundary of the observable universe; that boundary is always inside the event horizon.

I don't think this is quite right. Whether we think of the outbound signal, or the inbound one, doesn't really matter, since those two are symmetrical - the farthest comoving distance the outbound signal has reached by now is the same as the comoving distance to the farthest observable emitter. I.e. the comoving extent of the base of the past light cone - the observable universe - is the mirror image of the particle horizon (as the comoving coordinate graph in the L&D paper shows).

 

[PeterDonis]

Whether we think of the outbound signal, or the inbound one, doesn't really matter

It does if we infer, incorrectly, that the particle horizon is the boundary of the observable universe, which is what the Wikipedia definition appears to be saying.

the comoving extent of the base of the past light cone - the observable universe - is the mirror image of the particle horizon (as the comoving coordinate graph in the L&D paper shows).

This is true in conformal coordinates, yes (the bottom of the three graphs in Figure 1 of the paper). But, as you note, they are mirror images of each other, which is not the same as being identical.

 

[Bandersnatch]

I don't know. They both have the same extent at any given time slice. So in what way is the particle horizon not the boundary of the OU?

 

[PeterDonis]

in what way is the particle horizon not the boundary of the OU?

Because the "observable universe" is a region of spacetime, not space. The particle horizon is not the boundary of that region of spacetime; our past light cone is.

Here's another way of looking at it: objects at the particle horizon now are far outside the event horizon and we will never see them as they are now. We are just now receiving the first light rays from those "points in space" as they were just after the Big Bang; but what was there then (hot, dense, rapidly expanding plasma) is very, very different from what is likely to be there now (galaxies and galaxy clusters). So saying that some galaxy 47 Glyr away now is "in our observable universe" seems like a very misleading way to put it. Such a galaxy happens to be at the same comoving coordinate as a particular region of hot, dense plasma just after the Big Bang that we are just receiving light rays from now (although we actually aren't because any light rays emitted from that plasma were scattered for a few hundred thousand years before the universe became transparent to radiation). But that in no way means we can see the galaxy.

 

[Bandersnatch]

Because the "observable universe" is a region of spacetime, not space.

Yes, I understand that. But it seems pretty clear to me that what's being talked about in that Wikipedia article, and here by extension, is the boundary between regions in space at a moment in cosmic time. It's not about what's inside vs what's outside the entire light cone.

 

[PeterDonis]

it seems pretty clear to me that what's being talked about in that Wikipedia article, and here by extension, is the boundary between regions in space at a moment in cosmic time.

Which, as I noted, is misleading because the observable universe is not a region in space, it's a region of spacetime.

It's not about what's inside vs what's outside the entire light cone.

But it should be. Certainly we should not be saying something else just because Wikipedia says so. The light cone is the boundary in spacetime of what we can see. That's how the concept of "observable universe" should be explained.

 

[JimJCW]

The Wikipedia definition gets this backwards. The particle horizon is the maximum distance light can have traveled away from our location in the age of the universe. Since the particle horizon has for quite some time been further away than the event horizon, light emitted from the particle horizon during that time (since about t = 4 billion years) can never reach us. Therefore, the particle horizon is not the boundary of the observable universe; that boundary is always inside the event horizon.

The best reference I know of for straightening out these things is Davis & Lineweaver:
https://arxiv.org/abs/astro-ph/0310808

According to Davis & Lineweaver,

The particle horizon marks the size of our observable universe because we cannot have received light from, or sent light to, anything beyond the particle horizon.​

Jorrie’s calculator is based on the ΛCDM model. In his Tutorial Part III – How Things are Computed, he gives Davis’s Ph.D. Thesis as a reference, in which it is stated,

The particle horizon at any particular time is a sphere around us whose radius equals the distance to the most distant object we can see.​

According to Wikipedia,

The particle horizon is the maximum distance from which light from particles could have traveled to the observer in the age of the universe. It represents the boundary between the observable and the unobservable regions of the universe, so its distance at the present epoch defines the size of the observable universe.​

It seems the definitions given above are all similar.

The Big Bang model depicts a complicated picture of the universe and the consequences of space expansion can be very confusing. Let’s use some calculation results to illustrate this. GN-z11 is currently the oldest and most distant known galaxy in the observable universe, having a redshift of z = 11.09. Using Jorrie’s calculator and PLANCK Data (2015), one can get the following information about this galaxy:

The figure below shows that at the emission time, the photons and GN-z11 were 2.66 Gly away from our location. Starting from there, GN-z11 continues to move away from our location due to space expansion according to the equation,

The observed photons have a journey coincident with part of the Journey of an observed CMB Photon.

Adding the particle and event horizons to the figure, one gets the following:

At emission, GN-z11 was inside the event horizon and, therefore, the emitted photons can reach our location. These photons, however, arrived 13.8 - 0.41 = 13.4 Gyr later. Around t = 3.84 Gyr, GN-z11 moved out of the event horizon and light emitted by it since can never reach our location.

To examine the calculation results further, let us draw an imaginary line from our location to the current GN-z11 location and beyond, and use D to represent various proper distances from our location at the present time. Jorrie’s calculator can give the following information:

The present-time proper distances of the starting location of the CMB photons and GN-z11 are both inside the observable universe (particle horizon). That’s why we can observe them, but what we observe are the photons coming from them long time ago; the lights emitted by them now will never reach our location.

To generalize, we can observe, at the present time, all objects between the event horizon and the particle horizon, but only their past history reaching us just now. Any lights emitted by them now, however, will never reach our location.

 

[PeterDonis]

According to Davis & Lineweaver,

The particle horizon marks the size of our observable universe because we cannot have received light from, or sent light to, anything beyond the particle horizon.​

Yes, but it's important to put this sentence in context:

The particle horizon marks the size of our observable universe. It is the distance to the most distant object we can see at any particular time. The particle horizon can be larger than the event horizon because, although we cannot see events that occur beyond our event horizon, we can still see many galaxies that are beyond our current event horizon by light they emitted long ago.​

The bolded part is the crucial part. Your discussion later on in your post is consistent with it.

 

[JimJCW]

The particle horizon can be larger than the event horizon because, although we cannot see events that occur beyond our event horizon, we can still see many galaxies that are beyond our current event horizon by light they emitted long ago.

What do you think about changing ". . ., we can still see many galaxies that are beyond our current event horizon by light they emitted long ago." to

. . ., we can still see all galaxies that are beyond our current event horizon, but inside the particle horizon, by light they emitted long ago.​

 

[PeterDonis]

What do you think about changing ". . ., we can still see many galaxies that are beyond our current event horizon by light they emitted long ago." to

. . ., we can still see all galaxies that are beyond our current event horizon, but inside the particle horizon, by light they emitted long ago.​

That change is correct and does help to clarify the meaning, yes.

 

[mbond]

". . ., we can still see all galaxies that are beyond our current event horizon, but inside the particle horizon, by light they emitted long ago."

Is this beyond our current event horizon or is it beyond our Hubble sphere (radius c/H0)?

 

[Bandersnatch]

Is this beyond our current event horizon or is it beyond our Hubble sphere (radius c/H0)?

In terms of distance - both. At present, the particle horizon is the farthest, then the event horizon, then the Hubble sphere. Anything beyond the EH is necessarily beyond the Hubble sphere.
In this discussion, though, only the horizons matter. The Hubble sphere isn't one (i.e. it's not any sort of a barrier to incoming signals, that could limit observation).

 

[mbond]

According to Rindler (1956) the particle horizon divides "particles" into 2 classes: those observable at t0, those that are not;
the event horizon divides "events" into 2 classes: those that are forever inobservable, those that are not.

I understand that "particle" may means a galaxy, I don't understand what means "event" ... Why this distinction between particle and event? why not saying that the EH divide the galaxies, etc into 2 classes: those forever unobservable, those already observable or that will be observable if we wait long enough?

 

[Ibix]

An event is exactly what it sounds like - a thing that happened. More precisely, it is a place at a particular time. We use "point" to describe a location in space and "event" to describe a location in spacetime.

So the history of a particle can be described as a sequence of events: now it is here, a second ago it was there, two seconds ago it was there, etcetera. For a particle that crosses an event horizon, the events of its history that are this side of the horizon are visible (eventually) and those that are the other side never are.

 

[JimJCW]

In terms of distance - both. At present, the particle horizon is the farthest, then the event horizon, then the Hubble sphere. Anything beyond the EH is necessarily beyond the Hubble sphere.
In this discussion, though, only the horizons matter. The Hubble sphere isn't one (i.e. it's not any sort of a barrier to incoming signals, that could limit observation).

Jorrie's calculator and PLANCK Data (2015) give the following result:

 

[Bandersnatch]

I understand that "particle" may means a galaxy, I don't understand what means "event" ... Why this distinction between particle and event? why not saying that the EH divide the galaxies, etc into 2 classes: those forever unobservable, those already observable or that will be observable if we wait long enough?

Just adding a bit to Ibix's comment. Think of a galaxy as a string of events, together forming its history.
I.e. there's some gas early on; as time advances the gas coalesces and the galaxy forms. Over time, star formation ceases and the stars fade until it's just some inert mass drifting in the void forever. On cosmological scales, all these events happen pretty much in the same place (same spatial coordinate).
For an observer located within the galaxy, the string of events will form a straight line (or, a 'world line'), where all the events happen 'here', just at different times, like so:

(In this post, a 'galaxy' will refer to the entire world line, from formless plasma from which it formed, to a collection of stellar husks in the far future.)

The world line of a similar, but distant galaxy, receding with the expansion of space, will be a curve:

I.e. as time advances, the observer at the 'here' spot will see each consecutive event at ever farther distances.

The event horizon cuts off a part of the world line of a distant galaxy, so that only some of the events can be ever observed:

In this example, everything from birth to mergers could be observed at some point by the 'here' observer, but the stages of evolution including final star formation and stars going out is forever out of reach.

Particle horizon marks the reach of the signal emitted from 'here' at t=0.

As the signal passes consecutive galaxies on its way, the t=0 signals from those same galaxies reach the 'here' spot at the same moment.
The particle horizon is the farthest we can see at any given moment in the cosmic time. It's the farthest world line from which light could reach us. But it says nothing about how much of the world line - i.e. how much of the history of a galaxy at that spot - will be observable. That's what the event horizon determines:

To reiterate: the particle horizon and the event horizon provide answers to two different questions. PH tells us what is the farthest spatial coordinate (in the comoving sense - heuristically, the farthest galaxy) from which light signals are reaching us now. EH tells us what range of space-time coordinates can ever be observed - or, in other words, what is the last thing (if any) that we could see from any given place.

(Note: these are eyeballed drawings, not graphs - while they were made to somewhat resemble the real thing, the scaling and the curves will be off. For properly drawn graphs refer to the calculators mentioned in the OP, or to fig.1 in the paper linked in post #2)

 

[JimJCW]

An interesting point to note is that light emitted outside the event horizon at various times will never reach our location. It can be demonstrated that lights emitted inside of and close to the event horizon have peculiar behaviors such as prolonged journeys to reach our location.

To illustrate this point, we continue to use GN-z11 as an example. Let’s draw an imaginary line from our location to GN-z11 and use D to represent its proper distances at various times. Starting from t = 0.410 Gyr and D = 2.66 Gly, GN-z11 moved away from our location along the line. Around t = 3.89 Gyr, it reached the event horizon at D = 12.0 Gly, as shown in the figure below:

Any light emitted by GN-z11 outside of the event horizon will never reach our location. We are, therefore, interested in knowing when will the lights emitted during the portion of the journey with D < 12.0 Gly and t < 3.89 Gyr arrive at our location. The calculated result using the two calculators mentioned in this thread from t = 0.410 to t ≲ 3.89 Gyr is shown in the following figure (the calculations were somewhat tedious):

Note that while the light emitted at t = 0.41 Gly reaches our location at t = 13.8 Gyr, light emitted at t = 2 Gly will reach our location around t = 30 Gyr. As GN-z11 approaching the event horizon, the increase of arrival time becomes even more rapid, indicating that the journey of the photons is further prolonged. For example, light emitted at t = 3.8 Gly will reach our location around t = 90 Gyr. This trend is consistent with the property that lights emitted after GN-z11 moved out of the event horizon (around t = 3.89 Gyr) will never reach our location.

 

[JimJCW]

The figure below shows that at the emission time, the photons and GN-z11 were 2.66 Gly away from our location. Starting from there, GN-z11 continues to move away from our location due to space expansion according to the equation,