Expansion of the universe, acceleration limit

[tim9000]

So if the universe expansion is accelerating due to dark energy, does that mean that (assuming there is) one end of the universe relative to the other end of the universe will see it moving away at speeds greater than the speed of light? Or is the expansion capped by relativity?
Or does the expansion of space not count as 'movement' in relativity?...I mean it must be though, because if we can no longer see past a point, it means that point is moving away from us faster than light from that point can get to us? Why is this allowed by special relativity?

 

[alw34]

There is no 'far end' nor 'near' end of the universe as far as is currently known.

Or does the expansion of space not count as 'movement' in relativity?

That's the general idea.

Galaxies far enough away from each other, solar systems and planets as well, are RECEDING from us at 'faster than light' but that is not the same as MOVING. Most of the galaxies that can be seen from Earth are receding from us faster than the speed of light, but this is a result of increasing distances, and doesn't have a lot to do with motion through space. It's still impossible for a massive particle near such a galaxy or planet to move faster than c relative to that galaxy or planet.

It IS difficult to understand when one first comes across it. Our everyday notion of distance doesn’t work on cosmological scales, so everyday intuition is useless.

At the 'Hubble radius', the recession velocity is c; beyond the Hubble radius galaxies move away from us at greater than c. We also move away from such observers out there at greater than c. This is no way invalidates relativity; in fact it is based on relativity.

The 'distance' measure used in the best cosmology model of our universe is most commonly the 'proper distance', and is based on a particular metric [distance measure]. You even have to define 'stationary' observers at great distances, in cosmology called 'co moving observers'. [[Roughly analogous to separated observers on Earth standing still: if one observer is moving, the measures will be constantly changing.] One needs some standard against which to make calculations.

The ‘superluminal’ velocity [greater than the speed of light beyond the Hubble radius] at great distances tells us ONE of many different possible definitions of distance changes; Other metrics[distance measures] that use different co-ordinates may not contain any apparent superluminal recession.

For more, checkout the Balloon Analogy by pHinds from these

forums: http://www.phinds.com/balloonanalogy/

 

[tim9000]

There is no 'far end' nor 'near' end of the universe as far as is currently known.

What I'm saying is that, assuming the universe is finite, it has to have a skin of matter, like a galaxy that is the last one before nothing. (nothing being either objective, or this Hubble radius you mentioned...which I suppose would be a time dependant definition) So whatever shape the universe is, maybe an egg or something, that's kind of like arbitrary points on the skin which bisect the universe, psudo-ends? Sort of what I had in mind.

I'll have to read your comment again tomorrow, it's late where I am, but when you say 'receeds' do you mean like they're stationary in space, in that as far as relativity is concerned the units of space between them are fixed, but the size of the units themself are growing...wait that wouldn't work, because the speed of light would be in the same units, so wouldn't it seemingly speed up due to the in crease? Sorry, I'm still so confused...

 

[alw34]

Almost everyone is confused at first. If not, you should definitely be a cosmologist!

The universe is most likely infinite in 'size'...but that's another whole discussion.

 

[phinds]

What I'm saying is that, assuming the universe is finite, it has to have a skin of matter, like a galaxy that is the last one before nothing. (nothing being either objective, or this Hubble radius you mentioned...which I suppose would be a time dependant definition) So whatever shape the universe is, maybe an egg or something, that's kind of like arbitrary points on the skin which bisect the universe, psudo-ends? Sort of what I had in mind.

I'll have to read your comment again tomorrow, it's late where I am, but when you say 'receeds' do you mean like they're stationary in space, in that as far as relativity is concerned the units of space between them are fixed, but the size of the units themself are growing...wait that wouldn't work, because the speed of light would be in the same units, so wouldn't it seemingly speed up due to the in crease? Sorry, I'm still so confused...

You are still very confused, but some reading will likely clear it up for you. I will note only that there IS no "skin" / "edge", even if the universe is finite. If it is finite it is unbounded so there is still no center and no edge. Did you read my balloon analogy article that you were given a link to? It should answer some of your questions.

EDIT: and by they way, I agree w/ alw34 ... EVERYBODY is confused about this when they first hear about it because it is so counter-intuitive.

 

[tim9000]

You are still very confused, but some reading will likely clear it up for you. I will note only that there IS no "skin" / "edge", even if the universe is finite. If it is finite it is unbounded so there is still no center and no edge. Did you read my balloon analogy article that you were given a link to? It should answer some of your questions.

EDIT: and by they way, I agree w/ alw34 ... EVERYBODY is confused about this when they first hear about it because it is so counter-intuitive.

Yes, thanks for that, although I already had a pretty good idea. I had pictured it more as like rubber jelly/'jello' blob with coins sprinkled through it, and the big mass is being stretched out from all sides. Rather than a balloon. So the distance between my fingers isn't getting bigger, because its an electrictrostatically and gravitation-ally bounded system.
What I was saying is, say the universe isn't an infinite amount of galaxies, say it's limited, then the skin of matter would be like a sponge porous surface, where there is matter and radiation, and where there is just empty space.
Anyway back on topic, the part where they said "galaxies at the edge of our observable universe are receding from us at something like 3 times the speed of light." This is what I mean, so is there a frame of reference where it is 'getting further away' faster than light or not? It seemed to contradict itself as it says it's not moving, but getting further away. Like saying the derivative of distance isn't speed, if they're 'receding' 3 times faster than light, how can we see them?

If they can see it's accelerating, than surely they can calculate when the edge of 'our' observable universe will be right around Earth?

 

[phinds]

What I was saying is, say the universe isn't an infinite amount of galaxies, say it's limited, then the skin of matter would be like a sponge porous surface, where there is matter and radiation, and where there is just empty space.

No, it would not. You need to study topology. There is no "skin"

Anyway back on topic, the part where they said "galaxies at the edge of our observable universe are receding from us at something like 3 times the speed of light."

Yes, that is what I said, that is what I meant, and that is what is true.

This is what I mean, so is there a frame of reference where it is 'getting further away' faster than light or not?

yes, there is, but keep in mind that it is a recession velocity, not "moving" in the sense that you are accustomed to think of "moving", which is proper motion.

It seemed to contradict itself as it says it's not moving, but getting further away.

Yes, that's what I meant when I said this is counter-intuitive.

Like saying the derivative of distance isn't speed, if they're 'receding' 3 times faster than light, how can we see them?

The derivative of PROPER MOTION is speed. There is no proper motion involved here.

If they can see it's accelerating, than surely they can calculate when the edge of 'our' observable universe will be right around Earth?

That is no a meaningful concept since when the size of the observable universe was the size that the Earth is now, the milky way hadn't even been formed, much less our solar system, and even less still the Earth.

 

[tim9000]

Hi, thanks for the reply.

That is no a meaningful concept since when the size of the observable universe was the size that the Earth is now, the milky way hadn't even been formed, much less our solar system, and even less still the Earth.

But I thought the size of the observable universe was shrinking not growing?
Because things are receding further...ok, I just did some looking around on youtube, and I found a video, at 10:40 he says how the horizon is shrinking:


Is this a different concept than what you were talking about? I understand that this is either a hypothesis or a theory, but surly they can calculate when our cosmological horizon will be just around Earth (even though the Earth will be destroyed long before).

I'm going to pursue this line of inquiry just in case I learn something: Say there is objectively a finite amount of galaxies in our universe, but space is infinite, than surely what we call the universe resembles the surface of a sponge, with contours and holes around the edge where there is no matter or radiation yet. If not, than why not? I mean, I'm making the assumption that surrounding and in between all the galaxies there is at least some sort of radiation, no matter how minuscule, if you didn't agree with that than I can see why you're saying there is no skin (just pockets of matter and energy in infinitely empty space).Anyway, back to the main event: So there is no actual motion involved, in the electromagnetic field or whatever of the universe they're static. But to us it seems like they're moving away, how do we know, if they're receding three times the speed of light, can we still see them, and if so, how?
Cheers
P.S. I looked up the definition of 'topology' but I'm not sure what you mean.

 

[Bandersnatch]

at 10:40 he says how the horizon is shrinking

This is a bit more complicated than what Phil Plait says in that video. In fact, his explanation borders on misleading.
If by event horizon we understand the proper distance* from beyond which no signal can ever reach us, even after infinite time, then that distance is not shrinking - it is and always will be growing, albeit at an ever slower pace, asymptotically approaching 17.3 billion light years. At least that's what the best-fit models tell us.
Here's the evolution of the event horizon in the currently favoured model plotted on a graph (the outer thick solid line is marked 'event horizon'):

However, galaxies (the dotted lines on the graph) are constantly carried away beyond this horizon by the expansion. This means that every second there is less and less matter that we will be able to observe in the far future as it is now. I.e., today a galaxy on the edge of the horizon might emit light that will eventually be seen far in the future, but the light that it emits tomorrow might never reach us, so it will never be observable in its state as it will be from tomorrow onward.

In the far future, you'd end up with a universe where all but the closest of galaxies will have left the observable universe. So it's not that the horizon is shrinking, rather, galaxies vanish from sight one by one.

Only they don't really, at least not in a straightforward way - due to the fact that the closer an emitter is to the horizon, the more stretched its light becomes, and the longer it takes for it to reach us (with light emitted at the horizon requiring infinite travel time, but still getting there), it will be possible to observe, if not now, then some time in the future, any galaxy that was inside the event horizon at least once (but the farther away a galaxy is, the less 'up to date' the picture will be). This means that everything up to the current(!) proper distance of 63-ish billion light years will be eventually observable, and everything already observable will always remain so, but in increasingly outdated and redshifted state.

Only not even that is entirely correct, as the increasingly more and more stretched wavelengths of light emitted by objects near the event horizon will eventually get stretched so much that not even a receiver the size of the observable universe would be able to detect them, so even objects in principle observable by the virtue of having been once inside the event horizon will eventually, and in a finite time, disappear from sight for all practical purposes.

In the video, around the indicated time stamp, Phil talks about the possibility of non-constant Dark Energy component - if it were to turn out to be increasing (no indication so far, but it's still a possibility), then the horizon would indeed shrink, and the universe would eventually end up in a big rip. However, most models, and most people you might hear talking about it, will likely use the constant dark energy component (this class of models is known as Lambda-CDM, where lambda indicates constant dark energy, i.e. the 'cosmological constant') - and here the horizon is always growing as shown on the graph above.

* i.e. a distance you'd get if you froze the expansion of the universe and went out with a measuring stick to see how far something is - it's necessary to keep in mind what kind of distances we're talking about in cosmology, since there's a few 'types' (and in fact, if you choose the right 'type', you can make the horizon 'shrink' even in LCDM, but it's unlikely the video had this one in mind, and besides - that'd qualify as what you might call a 'mathematical trick').As for your line of inquiry regarding limited amount of matter and a 'skin', or 'edge' - remember that we must always base our models on observations, and there is no indication that there is any edge or skin, or very large voids of space. The observations point to a remarkably homogeneous and isotropic universe - meaning, pretty much the same stuff and the same distribution everywhere you look.

 

[tim9000]

Wonderful reply. Although I'm still not sure how we can tell something is there if it is receding at 3 times the speed of light...

Why/how is the event horizon asymptotic at 17.3 b yrs?

So let me see if I've got this straight: so the amount of galaxies we can see is and always has become more and more numerous, as time goes on we can see further away, however everytime a new galaxy comes into view it is of poorer and poorer quality because it it red-shifted more and more as it they recede faster and faster? But at some point they'll be receding so fast the wavelength of the light will pretty much just be a flat-line? How much all depends on the impact of dark energy...

As for your line of inquiry regarding limited amount of matter and a 'skin', or 'edge' - remember that we must always base our models on observations, and there is no indication that there is any edge or skin, or very large voids of space. The observations point to a remarkably homogeneous and isotropic universe - meaning, pretty much the same stuff and the same distribution everywhere you look.

I understand that there is absolutely no value in unfounded speculation, I was just saying, if there was a finite amount of matter in the universe, that would be how you describe it. I can totally accept that it is an infinitely large isotropic universe.Thanks

 

[alw34]

Bandersnatch's description is as nice a brief one as you'll get. Suggest you save it for future reference.

Why/how is the event horizon asymptotic at 17.3 b yrs?

It's results from a defined parameter in conjunction with the FLRW cosmological model, based on Einstein's general relativity tuned with 'best fit' cosmological parameters. It's what the best model tells us.

FYI: Just so you are aware, in your original post you referenced 'expansion' and in post #8 have migrated to 'size'. In cosmology especially, those are rather different measures and concepts relative to everyday special relativity.

Here are some posts I saved when I first studied these ideas to help explain to myself what is going on:

[There is a LOT here descriptive of some underlying mathematics. Not necessarily easy to visualize. Nothing here should be assumed to conflict with what Bandersnatch had to say already.]

{Just a reminder} Locally,nothing 'moves' faster than light. Even the most distant observers measure light at ‘c’ locally. No galaxy can 'outrun' the speed of light.

Marcus: Yet the vast majority of the galaxies which we can see today emitted the light which we are now receiving when they were already receding FTL. [details below]

Wallace: All this ‘superluminal’ velocity at great distances tells us is how one of many different possible definitions of distance changes. If you accept the FLRW metric then you have to live with that. Other metrics that use different co-ordinates but make the same physical predictions do not contain any apparent superluminal recession.

Marcus: The properties such as “cosmological time”, , “spatial distance”, “time”, “preferred coordinate systems”, “preferred metrics (even with cross terms)”, “expansion”, “Hubble flows” etc are by and large properties of a particular solution to the Einstein equation…..

Comoving and proper distances are not the same concept ...as in special relativity. It is important to the definition of both comoving distance and proper distance in the cosmological sense (as opposed to proper length in special relativity) that all observers have the same cosmological age. For instance, if one measured the {cosmological} distance along a straight line or spacelike geodesic between two points, observers situated between the two {distant} points would have different cosmological ages when the geodesic path crossed their own world lines, so in calculating the distance along this geodesic one would not be correctly measuring comoving distance or cosmological proper distance.

In introductory physics, “expansion”, say of a heated rod, is measured relative to a practical invariant standard, say an invariant. measuring tape in an instant of time.. In cosmology, “expansion”, refers to something altogether more strange and unfamiliar to practical ordinary life: a change in a metric coefficient a(t) in the expression...

An implication of this is that the way ‘increasing distances’ change is NOT well represented in the balloon analogy. In a contracting universe a distant particle could move away, or in an expanding universe {like ours} a distant particle could come toward you. You don’t intuitively expect this behavior if you think of the universe as the model loaf of rising bread filled with raisins! In the balloon analogy, you might [naively] assume ...since space is expanding the {distant} galaxy will start moving away from you, joining the Hubble flow eventually. However in a decelerating (but still expanding) universe {like ours} the particle actually comes towards you!

A shared criterion of rest* is what allows a shared idea of time, and allows us to talk about distances between observers at a particular moment in time. These are called proper distances [Hubble Law type distance] and they are what you would measure by any conventional means if you could freeze expansion at that moment (to give yourself time to measure without the distance changing.) *as with respect to the CMBR

The Hubble rate [a’[t]/a[t] is decreasing and will continue to decrease {to a constant value}. The current Hubble rate says that large scale distances (like between widely separated galaxies) increase 1/140 of a percent every million years. This percentage is expected to decline towards around 1/160 of a percent every million years.

The scale factor a[t] [= 1/[1+z] ] is increasing as defined by it’s time derivative and that is what most people mean when they say “expansion” is accelerating. It’s really the expansion RATE that is accelerating.

I'm pretty sure this comes from Marcus:
The rate of expansion is currently slowing, and appears to be approaching a constant value. This is referred to as an accelerated expansion, velocity per distance, because the distances between objects are accelerating. Currently the value is close to 70km/s/Mpc. Galaxies that are 100 megaparsecs away average 7,000 km/sec recession velocity. Galaxies 200 megaparsecs away average 14,000 km/sec recession velocity, and so on. So as galaxies get further away, they speed up relative to each other. While the expansion rate is dropping slowly, galaxies speed up as they move further from us. Distances are increasing at a percentage rate of 1/144 % per million years, and are epected to eventually level out at around 1/173% percent per million years. Reference https://www.physicsforums.com/threads/at-what-speed-is-our-universe-expanding.836825/#post-5254818

Please excuse the slight differences as "1/44" versus "1/140"...As better cosmological parameters are measured, calculations change over time.

 

[alw34]

Although I'm still not sure how we can tell something is there if it is receding at 3 times the speed of light...

Forgot to answer your question.

You cannot detect anything today, right now.
But you might be able to 'tomorrow'.
[That is, in the future.]
If you use Jorrie's cosmo calculator you can figure out just what will and what will not be in,say, your likely lifetime as an example.

 

[tim9000]

Forgot to answer your question.

You cannot detect anything today, right now.
But you might be able to 'tomorrow'.
[That is, in the future.]
If you use Jorrie's cosmo calculator you can figure out just what will and what will not be in,say, your likely lifetime as an example.

Thanks another couple fantastic replies in addition to the already outstanding responses.
So '3' times the speed of light is an approximation, and we can't actually see them...

Preface: I'm going to need to re-read this thread again when I can focus more. But just on my first read through of you reply, am I right to take it that:
So all the galaxies are 'moving' away from us, or co-moving away from us, like relatively, but they're slowing down, I.e. the second derivative is negative. However the space between them is also getting bigger, at an accelerated rate, hence the recession.

I know this is off topic, but many almost 10 yrs ago when I did HS physics, I remember the teacher telling us that there was no relative stationary point in the universe. As it happens we did astronomy or something as the HSC topic, (because that's what our teacher got his physics doctorate in), rather than sub-atomic particles, as some other schools did. So I never learned the standard model, but I was wondering if there are various fields through the universe, and ripples in them create photos, and particles, then surely these fields themselves are sort of like a non-physical imaginary reference of being stationary? Surely should we be fortunate enough to have a civilisation that lasts thousands or millions of years, and technological breakthroughs beyond our dreams are humanly possible (warp speed etc.) than we could exploit the fields and calculate how to make a particle be stationary relative to the field and watch it fall out of phase with observability?
just a thought...sorry about the mad tangent

 

[alw34]

"The evolution of the universe and its horizons in proper distances."

There is a nice dynamic evolution chart of the same type illustration Bandersnatch posted, here:

https://en.wikipedia.org/wiki/Comoving_distance#Uses_of_the_proper_distance

If you click on the illustration, it not only expands but becomes a dynamic illustration as an observer [on the vertical center line] moves through time [vertically up] with the evolution of the universe. It is interesting to see
how light arriving at the observer asymptotically approaches the event horizon in the future which is itself expanding asymptotically but much less rapidly.

Earlier in the same article is a similar plot with coordinate distances.

Is anything like this in FAQ's? Seems like an explanation accompanying such a dynamic illustration would be a nice learning tool. Either that, or somebody might explain the plots and put them in Wikipedia; shame Wikipedia did not explain this.

 

[tim9000]

In the far future, you'd end up with a universe where all but the closest of galaxies will have left the observable universe. So it's not that the horizon is shrinking, rather, galaxies vanish from sight one by one.

Only they don't really, at least not in a straightforward way - due to the fact that the closer an emitter is to the horizon, the more stretched its light becomes, and the longer it takes for it to reach us (with light emitted at the horizon requiring infinite travel time, but still getting there), it will be possible to observe, if not now, then some time in the future, any galaxy that was inside the event horizon at least once (but the farther away a galaxy is, the less 'up to date' the picture will be). This means that everything up to the current(!) proper distance of 63-ish billion light years will be eventually observable, and everything already observable will always remain so, but in increasingly outdated and redshifted state.

Only not even that is entirely correct, as the increasingly more and more stretched wavelengths of light emitted by objects near the event horizon will eventually get stretched so much that not even a receiver the size of the observable universe would be able to detect them, so even objects in principle observable by the virtue of having been once inside the event horizon will eventually, and in a finite time, disappear from sight for all practical purposes.

It was an excellent summary.
So what's the difference between the 'light cone' and the Hubble sphere?
So we can see superluminal recession because the Hubble sphere is expanding toward the light, but I'm still having trouble understanding the diagram, why is the event horizon a region and not completely to the right and left outside the Hubble Sphere?

There is a nice dynamic evolution chart of the same type illustration Bandersnatch posted, here:

Thanks very much for pointing me to that too, I need to meditate on it and read it again.

I'm still having trouble understanding proper space changing, and light. I mean and I suppose time. So light sees no time because it's traveling at C, however if a metre is defined using C and time, then as 'space' expands, light has to travel further and thus takes longer to get there, does this mean that the Planck length and metre change, or are by definition of the units adjusted too (automatically)? I can't tell.

Thanks

 

[Kenygreen]

I have been reading a lot of clever stuff and ti has made my head feel like an analogy of the expanding Universe ?

I began my study career as a propective medic in the course of which I studied Botany. There I learned that the linear scale "belonga Man" and all natural scales are logarithmic:
i.e.they do not have a beginning (minimum) nor an end (maximum). Hence I get confused when confronted by expessionns such as "infinite, infinity,"
Out in Space you cannot fix a starting point and so absolute measurements must be replaced by relative measurements - relative to some declared point that has an existence such as the Sun ?

Afraid that I am the last person to dip an oar here.

KG

 

[Drakkith]

Out in Space you cannot fix a starting point and so absolute measurements must be replaced by relative measurements - relative to some declared point that has an existence such as the Sun ?

All measurements are relative, even the ones not in space.

 

[tim9000]

lol, hence why I'm asking relative to light.

Cheers

 

[pbuk]

Most of the galaxies that can be seen from Earth are receding from us faster than the speed of light

This is misleading. No galaxy that can be seen from Earth was receding faster than the speed of light when it emitted the light that enables us to see it. The statement you made is based on the concept of proper motion, which is not an easy one to grasp.

The rest of your post shows a reasonable awareness of the concepts of cosmology but these are difficult to communicate accurately in a forum post. Perhaps you should stick to the link to phinds site and the excellent Insights article on the balloon analogy.

 

[Bandersnatch]

This is misleading. No galaxy that can be seen from Earth was receding faster than the speed of light when it emitted the light that enables us to see it. The statement you made is based on the concept of proper motion, which is not an easy one to grasp.

I don't understand why you say that. The graph provided in post #9 clearly shows that current light cone includes sections tracing through regions of space lying outside Hubble sphere.
And here's a snapshot of outputs from Jorrie's LightCone calculator:

 

[Bandersnatch]

So what's the difference between the 'light cone' and the Hubble sphere?

Light cones trace the path of signals (light) traveling through spacetime. A past light cone with the tip at the observer (i.e. here and now) shows the path of all signals that you're seeing now. If you e.g. look at where the line of a lightcone crosses with the dotted line representing redshift z=1:

you can see where in the spacetime the emitter was. That is, at what time the presently observed signal of a z=1 galaxy was emitted, and how far it was at the time of emission. At the time of observation (apex of the light cone) you can't see anything outside the light cone shown, and you have seen everything inside the light cone at some time in the past.

In a static, non-expanding space light cones would look like straight lines forming a triangle, i.e. like this:

(or a cone, if you use 2+1 dimensions rather than 1+1, hence light cone). If the space is expanding, those nice triangular light cones get distorted into what you see on the first graph. But notice that as the lines converge on the observer, where expansion is less and less significant, it looks more and more triangular in shape.

A triangular light cone in a static universe would expand to cover arbitrary large distance - as you draw triangles with vertices at successively further times (higher on the t axis), the base gets progressively and indefinitely larger. This means that in a static universe you could see signals from arbitrarily far away, as long as you waited long enough for the signal to reach you.

In expanding universes, the distorted light cones also grow indefinitely in time, but, not being triangular, their bases don't cover arbitrarily large distances.
Here's an animation of how light cones evolve in our expanding universe:
http://yukterez.net/lcdm/lcdm-flrw-animation.gif
(taken from: http://yukterez.net/)
(Hmmm, something's wrong with the gif. I'll see if I can fix it - in the meantime, the animation can be seen here: http://yukterez.net/lcdm/i.html#plot)
The one on the right is the exact same graph, only drawn in different coordinates that allow us to better see what's going on at the base of the graph. The comoving distance scale can be understood as proper ('regular') distance at the present time, and to translate it to proper distance at any other time, it has to be multiplied by the scale factor 'a' shown on the right (growing from 0 to infinity), so that the base of the graph, despite being so splayed, actually represents very short proper distances (and the top shows very far ones).
The Hubble sphere, on the other hand, is just a sphere with a radius marking recession velocity equal to the speed of light. It is quite unrelated to a light cone.

So we can see superluminal recession because the Hubble sphere is expanding toward the light, but I'm still having trouble understanding the diagram, why is the event horizon a region and not completely to the right and left outside the Hubble Sphere?

I don't think the dashed area is meant to convey anything special, apart from 'colour'-coding the graph for ease of use. The horizon is definitely just the outer line, not the whole area.

Why/how is the event horizon asymptotic at 17.3 b yrs?

This has to do with the dark energy content in the universe. There is a certain (constant as per the LCDM) density of dark energy per unit volume of space in our universe, and that determines how far away the horizon will be.
If not for all the matter and radiation in the universe, i.e., if we had an universe with dark energy only, there would be no asymptotic approach of the horizon to that value, but it'd be exactly that, forever. It would also be equal to where the Hubble radius (aka sphere) would be, since without any matter or radiation to 'pull' everything together and slow the expansion down, anything (a test point of sorts, since there's no galaxies in this universe) finding itself receding more rapidly than c, would never be able to send a signal to the observer.
As the universe expands and radiation and matter get diluted, their meddling influence on what dark matter 'wants' the universe to expand like gets diminished. That's why in the far future (infinity), when matter and radiation densities approach zero, the event horizon, as well as Hubble radius approaches the value they'd have without any matter or radiation.

 

[alw34]

(t=0, x=0, i.e. here and now)

It's not 'here an now' but 'way back then', right??

 

[Bandersnatch]

It's not 'here an now' but 'way back then', right??

Oh, sh*t. I was thinking about the second graph when I wrote it. Yeah, the first graph has the t=0 at the beginning of expansion.
Let me fix that.

 

[pbuk]

I don't understand why you say that. The graph provided in post #9 clearly shows that current light cone includes sections tracing through regions of space lying outside Hubble sphere.
And here's a snapshot of outputs from Jorrie's LightCone calculator:
View attachment 95814

Ah yes, thank you for correcting my schoolboy error - and my apologies to @alw34