Trying to understand Dark Energy

[rede96]

Hi, I wonder if anyone can help me understand the concept of dark energy better.

I understand the very basic concept that the universe is expanding and that the rate of expansion may be speeding up (I've read somewhere that some don't think the rate is actually increasing) but how does that lead to there being some sort of dark energy?

From my very elementary understanding of this, I imagined that dark energy was actually part of space, where as dark matter and visible matter are things that exist in space. Is that correct?

Also, as the universe is expanding, then does this mean the amount of dark energy per cubic meter of space becomes more dilute or is the rate of dark energy per cubic meter of space a constant? (Or increasing?) If so how is that possible? Doesn't that contradict conservation of energy?

Another thing that confuses me is how are things like the Higgs field effected by the expanding universe? Does the Higgs field expand with the universe, so for any particles effected by the Higgs field, their mass remains constant? If so then how is that possible? Also does dark mater affect the Higgs field?

Or is there any line of thought that says as the universe expands, then the Higgs field becomes more dilute and thus has less of an effect on matter meaning matter doesn't attract as much over larger distances and therefore that could contribute to expansion speeding up?

Sorry for the multiple questions! And please excuse my terminology, I have no background in physics at all it just fascinates me :)

 

[andrewkirk]

Lots of questions there. I'll answer a couple.

From my very elementary understanding of this, I imagined that dark energy was actually part of space, where as dark matter and visible matter are things that exist in space. Is that correct?

'A part of space' is not a well-defined term. I find it more helpful to just think of it as a feature of spacetime. If we start thinking of it as some sort of 'stuff' we can end up getting quite confused.

Also, as the universe is expanding, then does this mean the amount of dark energy per cubic meter of space becomes more dilute or is the rate of dark energy per cubic meter of space a constant? (Or increasing?) If so how is that possible? Doesn't that contradict conservation of energy?

It remains constant. It doesn't contradict conservation of energy because

(1) it's not really energy in the same way that heat and work are ('Dark Energy' is just a fancy name to make Einstein's Cosmological Constant (which is what it really is) sound cool); and

(2) Conservation of energy is a local phenomenon that does not apply in General Relativity, because energy is a frame-dependent measure.

 

[rede96]

Lots of questions there. I'll answer a couple.

Thanks, very much appreciated!

'A part of space' is not a well-defined term. I find it more helpful to just think of it as a feature of spacetime. If we start thinking of it as some sort of 'stuff' we can end up getting quite confused.

Ah ok, that make sense. I guess my logic was that galaxies can recede from each other at speeds greater than c, as they are not moving through spacetime. So it is the fabric of space that is actually expanding. So as it isn't a 'force' that is pushing the galaxies apart through space, it must be a part of whatever 'space' is. If that makes sense.

I just had another question pop into my head, sorry! Is it correct to say all matter can warp spacetime, but Dark Energy doesn't warp spacetime or counteract matter from warping spacetime?

It remains constant. It doesn't contradict conservation of energy because

(1) it's not really energy in the same way that heat and work are ('Dark Energy' is just a fancy name to make Einstein's Cosmological Constant (which is what it really is) sound cool); and

(2) Conservation of energy is a local phenomenon that does not apply in General Relativity, because energy is a frame-dependent measure.

Ah ok, I see thanks. But as I understand it, the Cosmological Constant was a fudge factor to keep a static universe. So there must be some work being done to counteract gravity? Also if space is expanding, there must be some sort energy or work being done somewhere to generate the extra space? So I am finding it difficult to understand where all this extra 'Dark Energy' is coming from.

I also assumed that as the amount of matter isn't growing, the percentage that dark energy makes up of the total universe, which is 68% ish now I think, would be increasing with time.

 

[phinds]

I understand the very basic concept that the universe is expanding and that the rate of expansion may be speeding up ...

There is no "may" involved. It is expanding at an accelerating rate. That's what dark energy DOES. If it were not expanding at an accelerating rate, there would be no need to posit dark energy.

I recommend the post linked to in my signature

 

[rede96]

There is no "may" involved. It is expanding at an accelerating rate. That's what dark energy DOES. If it were not expanding at an accelerating rate, there would be no need to posit dark energy.

Ah, ok. I thought dark energy was responsible for expansion and the accelerating rate of expansion came from the reduction in gravitational forces as things get further apart.

I recommend the post linked to in my signature

Thanks for that.

 

[phinds]

Ah, ok. I thought dark energy was responsible for expansion and the accelerating rate of expansion came from the reduction in gravitational forces as things get further apart.
Thanks for that.

There's a good discussion in the link I gave you about the three aspects of expansioy in cosmology (inflation, expansion, and acceleration of expansion).

 

[andrewkirk]

the Cosmological Constant was a fudge factor to keep a static universe.

It can be used for that but, from what I have read, that was not the main reason for its original inclusion. My understanding is that it's more like a constant of integration. In developing his gravitation equation, Einstein was reasoning that the curvature of spacetime, which is a 4 x 4 tensor (the Einstein Tensor) must vary according to the density of mass-energy, which is represented by the 4 x 4 stress-energy tensor. But you can add a constant (wrt mass-energy) term to that and it will still keep that dependency on mass-energy. That constant term is $\Lambda$, the cosmological constant multiplied by $g^{\alpha\beta}$ the metric tensor (another 4 x 4 tensor).

I have read that Einstein later regretted including it, because there was no evidence for its being nonzero, and it made the equation 50% messier (three terms instead of two). He even called it a 'mistake'. He didn't live long enough to see himself vindicated by the work of Brian Schmidt et al that demonstrated it was nonzero.

By the way, the term 'accelerating expansion' is ambiguous and can be misleading. As I recall, what that means is that the rate at which any given object recedes from us will increase over time. However, I think it is believed that the recession rate of objects at a fixed distance is actually decreasing. That is, the recession rate of a (co-moving) object currently 100 parsecs away is greater than the expected recession rate, a billion years from now, of another co-moving object that will be 100 parsecs away at that time. So the recession rate is increasing if we are fixing on an object, and decreasing if we are fixing on a distance.

 

[rede96]

It can be used for that but, from what I have read, that was not the main reason for its original inclusion. My understanding is that it's more like a constant of integration. In developing his gravitation equation, Einstein was reasoning that the curvature of spacetime, which is a 4 x 4 tensor (the Einstein Tensor) must vary according to the density of mass-energy, which is represented by the 4 x 4 stress-energy tensor. But you can add a constant (wrt mass-energy) term to that and it will still keep that dependency on mass-energy. That constant term is Λ \Lambda, the cosmological constant multiplied by g αβ g^{\alpha\beta} the metric tensor (another 4 x 4 tensor).

Thanks for the information, but to be honest that's a bit too advanced for me to understand. I am guessing it has something to do with gravity and creating a flat universe?

By the way, the term 'accelerating expansion' is ambiguous and can be misleading. As I recall, what that means is that the rate at which any given object recedes from us will increase over time.

Yes, that's pretty much what I understood too.

However, I think it is believed that the recession rate of objects at a fixed distance is actually decreasing. That is, the recession rate of a (co-moving) object currently 100 parsecs away is greater than the expected recession rate, a billion years from now, of another co-moving object that will be 100 parsecs away at that time. So the recession rate is increasing if we are fixing on an object, and decreasing if we are fixing on a distance.

That confused me a little. I was reading the Balloon Analogy that is posted in phinds signature. The section on the acceleration of expansion says

it was found that not only is the rate of expansion NOT slowing down, it is ACCELERATING.

Which suggests that the rate at which an object of a fixed distance (eg 100 parsecs) is moving away is increasing not decreasing. Or have I misinterpreted that?

 

[phinds]

By the way, the term 'accelerating expansion' is ambiguous and can be misleading. As I recall, what that means is that the rate at which any given object recedes from us will increase over time. However, I think it is believed that the recession rate of objects at a fixed distance is actually decreasing. That is, the recession rate of a (co-moving) object currently 100 parsecs away is greater than the expected recession rate, a billion years from now, of another co-moving object that will be 100 parsecs away at that time. So the recession rate is increasing if we are fixing on an object, and decreasing if we are fixing on a distance.

Yes, that's correct, but it's very confusing to bring the concept of a changing Hubble Constant into the conversation this way. You can see the confusion you've caused in @rede96.

 

[phinds]

T

Which suggests that the rate at which an object of a fixed distance (eg 100 parsecs) is moving away is increasing not decreasing. Or have I misinterpreted that?

Your understanding is correct. Andrew is bringing in a whole new concept, which is based on the fact that the rate of acceleration is very slowly decreasing over time. It IS still accelerating just not quite as fast. He is talking about an object that is a fixed distance from us now and an object that is the same distance away from us far in the future. The are both receding away from us at an ever increasing rate but the rate of that acceleration is just slightly less in the future.

 

[rede96]

Your understanding is correct. Andrew is bringing in a whole new concept, which is based on the fact that the rate of acceleration is very slowly decreasing over time. It IS still accelerating just not quite as fast. He is talking about an object that is a fixed distance from us now and an object that is the same distance away from us far in the future. The are both receding away from us at an ever increasing rate but the rate of that acceleration is just slightly less in the future.

Ah ok, I think I understand now. So when we talk about the ACCELERATION of expansion, then what that really means is that objects further away are moving away at a faster and faster rate. But that 'rate' of that acceleration is decreasing over time as measured for objects of the same distance now, and objects of the same distance in the future.

Is that right?

 

[phinds]

Ah ok, I think I understand now. So when we talk about the ACCELERATION of expansion, then what that really means is that objects further away are moving away at a faster and faster rate. But that 'rate' of that acceleration is decreasing over time as measured for objects of the same distance now, and objects of the same distance in the future.

Is that right?

you got it.

 

[rede96]

you got it.

Ok thanks for that.

There's a good discussion in the link I gave you about the three aspects of expansioy in cosmology (inflation, expansion, and acceleration of expansion).

I've read the links you mentions but I'm still a little confused about how that leads to some sort of dark energy being present. Couldn't the acceleration observed simply be caused by the energy produced from the Big Bang / inflation?

 

[phinds]

I've read the links you mentions but I'm still a little confused about how that leads to some sort of dark energy being present. Couldn't the acceleration observed simply be caused by the energy produced from the Big Bang / inflation?

No, that energy caused the expansion we see, but it requires an ongoing process, not something happened billions of years ago, to cause the acceleration of the expansion. Cosmologists 30 years ago firmly believed that the expansion would be found to be slowing down and would keep slowing down or would actually reverse (the "big crunch") but to everyone's surprise, when the first measurements were made, it was found to be accelerating. Thus dark energy was posited, and "dark energy" is really just a place-holder name that means "we know what's happening but we don't know why, so we're going to call the reason for it dark energy until we figure it out".

 

[rede96]

No, that energy caused the expansion we see, but it requires an ongoing process, not something happened billions of years ago, to cause the acceleration of the expansion. Cosmologists 30 years ago firmly believed that the expansion would be found to be slowing down and would keep slowing down or would actually reverse (the "big crunch") but to everyone's surprise, when the first measurements were made, it was found to be accelerating.

As I understand it, 'space' itself is expanding, which leads to more distant objects moving away faster (accelerating). So as I am understanding what is observed was simply that. The more distant the object the faster it moved away. This situation would be the same for any rate of expansion (Hubble constant)

So I was wondering why it needed to be an ongoing process? I suppose I imagined it like motion, e.g. Every object in a state of uniform motion tends to remain in that state of motion unless an external force is applied to it. So I saw the initial inflation or the expansion of space as a sort of motion. There was in initial energy to cause space to expand and without any other forces acting on it space would continue to expand. This causes more distance objects to move away faster. And it would make sense (to me anyway!) that gravity would be responsible for the gradual decrease of the Hubble constant.

So could you help me understand why it needs to be an ongoing process?

 

[andrewkirk]

There was in initial energy to cause space to expand and without any other forces acting on it space would continue to expand. This causes more distance objects to move away faster. And it would make sense (to me anyway!) that gravity would be responsible for the gradual decrease of the Hubble constant.

So could you help me understand why it needs to be an ongoing process?

If the expansion were simply an inertial phenomenon, whereby things keep moving away from one another because of an impetus given them long ago, an object that is currently receding at 10 m/s would continue to recede at that rate if space were flat and hence infinite. One way to visualise that is that the gravitational pull of objects nearer to us than the object pull it towards us and retard its recession, but they are balanced by gravitational forces of objects farther from us pulling it away.

Alternatively, if space were elliptical (bounded), the recession would slow because of gravitational forces pulling things back together.

The real answer is in the gravitational equation and the Friedman equation. But that simplified visualisation works for me.

 

[rede96]

If the expansion were simply an inertial phenomenon, whereby things keep moving away from one another because of an impetus given them long ago, an object that is currently receding at 10 m/s would continue to recede at that rate

Sure, but I was referring to space expanding not objects moving through space. Again, as I understand it, the reason galaxies recede from each other is because of the space between them expanding, not the actual galaxies moving through space. Hence why we can have recession velocities greater than c.

So I was looking at space expanding having inertia (if that is the right term) if expanding space had inertia and expanding space is responsible for galaxies receding, then it makes sense that the further a galaxy was away from us, the faster it would be moving. So in that case we wouldn't necessarily need dark energy.

If galaxies moving away from each other is not due to expanding space, but rather some unknown 'energy' that permeates all space which is pushing against all matter in all directions, then I would understand that to be dark energy.

Does that make sense?

 

[phinds]

There is no inertia involved in metric expansion. If there were nothing but expansion, then as andrew said, the rate of recession of a distant galaxy would be constant. It is not constant. When it is X light years from us it is receding at R. When it is 2X LY from us, it is receding at 2.1R. When it is 3X LY from us, it is receding at 2.4R (I'm making these numbers up but they illustrate the point). This cannot be simple ongoing expansion of space. Something is causing the expansion to accelerate.

 

[marcus]

The real answer is in the gravitational equation and the Friedman equation...

Andrew I liked very much your posts 2 and 7 in this thread. I liked your ability to look at things in different ways.
One form of the Friedman equation for the spatially flat case is
H² - H_∞² = const ρ

where rho ρ is the combined density of ordinary and dark matter. (In energy density terms 0.24 nanojoule per m³) There is no contribution from the cosmological constant since that is on the left side.
It's just a fact that the cosmological constant is 3H_∞² so it appears in the equation that way, similarly to how it does in the GR equation ("gravitational equation") you referred to in your post just now.

One way to look at this is that the square of the expansion rate is a kind of spacetime curvature and the constant gives a measure of the flexibility of spacetime geometry (or conversely it let's you know how stiff geometry is--how much matter density you have to put to get a given amount of curvature.)

The bigger rho is, the bigger the squared expansion rate.
Then as space expands the combined matter density goes to zero, so the lefthand side goes to zero so that H converges to H_∞. Matter density DOES cause the expansion rate to decline, but it doesn't have to decline all the way to zero! There is a residual expansion rate given by the cosmo constant.
Some refer to it as "vacuum curvature"---it just complicates things to imagine it caused by some imagined "dark energy"---it behaves like a constant intrinsic feature of geometry.

I realize this is extremely vague and handwavy but maybe I can illustrate with some real quantities given by actual numbers

 

[marcus]

Here are some actual expansion rate numbers

year        fraction of percent expansion per million years
1 billion        1/15
...
...
12 billion      1/135
13 billion      1/140
13.8 billion    1/144  (present rate)
14 billion      1/145
15 billion      1/149
...
...
50 billion      1/173  (approx. equal to H∞ the longterm rate)

The point about the rate tailing off to a constant is that to the extent that we have a constant expansion rate we have exponential distance growth at a constant percentage growth rate. So if you look at a specific distance between two essentially stationary galaxies, that distance is growing exponentially. The speed of distance growth, for that particular distance you are tracking, is of course increasing and naturally it is not limited by c because this is geometry change, not ordinary motion. Nobody gets anywhere by it, everybody just becomes farther apart. Not good to think of it as ordinary relative motion (which is limited by c.)

 

[marcus]

... So in that case we wouldn't necessarily need dark energy...

Right. The simplest explanation for the way H(t) is changing over time, for the distance-redshift data we observe, is a non-zero cosmological curvature constant. There is no credible evidence that this arises from anything we'd normally call an "energy".

Some people like to fantasize a "dark energy" that underlies the observed constant residual spacetime curvature. It sounds good and it may improve the research funding situation to refer to "dark energy" a lot.

But it is not a necessary part of the picture. The Lambda curvature constant belongs naturally in the Einstein GR equation because allowed by the symmetries of the theory. [For an explanation, google "why all these prejudices against a constant" ]
For a long time Lambda was thought to be zero and then in 1998 careful measurements showed it was not zero.

 

[rede96]

If there were nothing but expansion, then as andrew said, the rate of recession of a distant galaxy would be constant. It is not constant. When it is X light years from us it is receding at R. When it is 2X LY from us, it is receding at 2.1R. When it is 3X LY from us, it is receding at 2.4R (I'm making these numbers up but they illustrate the point).

Right, I think this might be were I was getting confused. Galaxies further away are moving away faster than near by galaxies, but the rate at which those further away galaxies are moving away is accelerating exponentially.

So could I look at this way, using the balloon analogy...

If I take a balloon and pump air into it at a fixed rate, (assuming the balloon can expand infinitely) then the expansion of the balloon will slow down because the constant rate of air will have less and less of an effect on the size of the balloon as the volume of the balloon gets bigger. Thus the prediction is that the balloon (thus the universe) would move closer and closer to a fixed size (after an infinite amount of time.) This would be expansion without dark energy.

But what has been observed is like someone has been turning up the flow rate of the air into the balloon causing the expansion of the balloon to accelerate. And this extra pressure is like dark energy. (I know that isn't quite correct but it helped me to picture what was going on)

If that is correct then the only thing I am still struggling with is how the Hubble constant can be getting smaller with time?

 

[marcus]

If that is correct then the
only thing I am still struggling with is how the Hubble constant can be getting smaller with time?

It has to be getting smaller with time because the density of ordinary and dark matter is positive.
Einstein GR equation describes how matter/energy curves spacetime and cosmology uses a simplified version of the GR equation called the Friedman equation.
Distance expansion can be seen as a kind of spacetime curvature that spreads geometry out.
The spatially flat case of Friedman is
H² - H_∞² = [const] ρ where rho is the density. The density is declining so H is converging towards its longterm value H_∞.
Check out these numbers.

Here are some actual expansion rate numbers

year        fraction of percent expansion per million years
1 billion        1/15
...
...
11 billion      1/128
12 billion      1/135
13 billion      1/140
13.8 billion    1/144  (present rate)
14 billion      1/145
15 billion      1/149
...
...
50 billion      1/173  (approx. equal to H∞ the longterm rate)

The point about the rate tailing off to a constant is that to the extent that we have a constant expansion rate we have exponential distance growth at a constant percentage growth rate. So if you look at a specific distance between two essentially stationary galaxies, that distance is growing exponentially. The speed of distance growth, for that particular distance you are tracking, is of course increasing and naturally it is not limited by c because this is geometry change, not ordinary motion. Nobody gets anywhere by it, everybody just becomes farther apart. Not good to think of it as ordinary relative motion (which is limited by c.)

 

[rede96]

Right. The simplest explanation for the way H(t) is changing over time, for the distance-redshift data we observe, is a non-zero cosmological curvature constant. There is no credible evidence that this arises from anything we'd normally call an "energy".

Thanks for reply, and sorry to be a pain but in layman's terms does that mean a zero cosmological constant leads to a flat universe and dark energy to explain the red-shift data observed, but a non zero (negative?) cosmological constant explains the red shift observations without dark matter?

 

[marcus]

Too many questions to answer in one post. For starters, make a sharp distinction between dark matter and dark energy. DM has been mapped by the way it clusters and lenses background features. It has been known about for a long time and has played an important role in formation of structure.

DE was made up in the excitement that followed the 1998 discovery of positive cosmo constant. It has aspects of hype, myth, and unnecessary baggage.
If evidence is discovered that the longterm evolution of H(t) is NOT behaving according to simple Friedman model with constant Lambda then that will be exciting evidence that "dark energy" is REAL! But so far people keep looking for anomalous behavior that can't be simply explained by constant Lambda, and not finding it. This has been going on since 1998 with all kinds of exotic fantasies that don't pan out. So the "dark energy" idea is getting tired.
So any way keep the DM and DE ideas separate. DM is a solid part of the standard cosmic model, called LCDM for "lambda cold dark matter" and contour maps of DM cloud density have been made.

 

[marcus]

Rede, you asked what if Lambda were zero. In cosmology the basic equation is
H² - H_∞² = [const] ρ

And Lambda (up to a factor of 3) is just the longterm Hubble expansion rate squared H_∞²

So if Lambda were zero, H(t) would just tail off to zero, instead of tailing off to a positive percentage distance growth rate.

The basic equation would just look like this:
H² = [const] ρ
so as distances expand, density goes to zero, and so H must go to zero.

(*psst* so no exponential growth! you only get exponential growth of distances if the decline of H stabilizes at a constant percentage growth rate)

Think: at the start of expansion, which might have been a bounce (quantum effects making gravity repellent at extremely high density), expansion got a big KICK. H(t) started off ENORMOUS. And the ordinary and dark matter immediately started causing H(t) to decline (courtesy the Einstein GR equation or its simple cousin the Friedman) and H(t) has been declining ever since. But we realized in 1998 that when you track the decline over time (using distance redshift data) it looks like it is not declining to zero! It looks like it is on track to level out at a positive percentage distance growth rate!
Namely about 1/173 of a percent growth per million years, as best as we can estimate. That's one way to measure the cosmological curvature constant Lambda.

Which can then be mythologized if you like by attributing it to a mysterious "dark energy"

 

[rede96]

I've been doing a bit more reading, but just so I can be confident I have understood the basics could someone just clarify if the following statements, which I know are in layman's terms, are true.

1) The universe has been expanding since the big bang (but not at the same rate due to factors like radiation vs matter dominated universe)

2) Observations taken show the rate at which stars in distant galaxies have been measured to be receding from us is accelerating, meaning a star at a given distance of x today will be moving away slower than a star at the same distance x in the future.

3) The rate at which a star at a fixed distance of x is accelerating ( as mentioned above) is actually decreasing slightly over time. Meaning that if I measure the recession velocity of an object at a distance x today, then an object at distance x in 100 years and then an object at distance x in 200 years, the rate of acceleration is decreasing.

 

[phinds]

Yep, you got it.

Be aware, the measurements are not with individual stars ... WAY to faint to be seen ... but with galaxies. The principle is the same of course but let's stay practical here.

I would also add to #1 the emerging dominance of dark energy, which has been around all along but only about 7 billion years ago began to overtake the effects of gravity.

 

[rede96]

Yep, you got it.

Great, thanks to all!

Be aware, the measurements are not with individual stars ... WAY to faint to be seen ... but with galaxies. The principle is the same of course but let's stay practical here.

Oh ok, but what about type 1a supernova? I thought they used these as a standard candle as they were as bright as galaxies at some point during the collapsing process? Also I thought the intrinsic brightness of a 'galaxy' is different for each one and would be unknown, so distance couldn't be measured?

 

[phinds]

Oh ok, but what about type 1a supernova? I thought they used these as a standard candle as they were as bright as galaxies at some point during the collapsing process? Also I thought the intrinsic brightness of a 'galaxy' is different for each one and would be unknown, so distance couldn't be measured?

True. I equate "star" with "regular star" but you are right. We DO measure red-shift of galaxies, but standard candles are, I believe, what gave the first solid information.

 

[marcus]

...
2) Observations taken show the rate at which stars in distant galaxies have been measured to be receding from us is accelerating, meaning a star at a given distance of x today will be moving away slower than a star at the same distance x in the future.
...

Yep, you got it.

No, Phinds. It looks to me from Rede's 2) that the (overly verbal, insufficiently quantitative) discussion in this thread has left Rede with some confusion.

 

[phinds]

No, Phinds. It looks to me from Rede's 2) that the (overly verbal, insufficiently quantitative) discussion in this thread has left Rede with some confusion.

OOPS. Good catch. I didn't see that he got that backwards; I glossed over it.

 

[marcus]

Rede, it might help you get over the confusion if you would practice doing something:
when you mean speed, say speed, not "rate"
use the word "accelerate" only when talking about a speed. That's the root meaning of "accelerate".

Reserve the word "rate" for the percentage rate of increase.

Like the current rate of distance growth is 1/144 percent per million years.

Don't talk about the rate accelerating or decelerating because the expansion RATE is not a SPEED.

You can talk about the expansion rate INCREASING or DECREASING over time. Actually it has always been decreasing since very early times near start of expansion.

I think it would help to focus on consistent use of words and practice making those distinctions.

 

[marcus]

...
2) Observations taken show the rate at which stars in distant galaxies have been measured to be receding from us is accelerating, meaning a star at a given distance of x today will be moving away slower than a star at the same distance x in the future.
...

No actually. A galaxy that today is 14.4 billion LY from us is receding at speed c.
In the future a galaxy at that distance will be receding at speed < c.

Rede and Phinds both, I would suggest this, as a way out of confusion and confusing other people--if someone says to you "the universe's rate of expansion is accelerating" then I would say that person is feeding you sloppy verbiage and you should pay no attention to them from that point on.

To say "accelerating" suggests that the person is talking about speed. But the universe has no well-defined expansion speed in the usual sense of speed. The speed a distance is growing is proportional to its size (a certain percentage of its size, per unit time). So there is no one speed.

There is one universal percentage RATE, called H(t). And that rate has always been declining (don't say "decelerating", it is not a speed ). Moreover according to the standard cosmic model everybody uses it is slated to continue declining.

 

[marcus]

Rede and Phinds, I think maybe some concrete numbers might help instead of only words. Words alone are inherently vague---one has to keep in mind what math quantities they are tied to, or they drift.
Maybe this table, or maybe you can work out something neater by way of quantitative examples. The table just happens to be handy.

year        fraction of percent expansion per million years
1 billion        1/15
...
...
11 billion      1/128
12 billion      1/135
13 billion      1/140
13.8 billion    1/144  (present rate)
14 billion      1/145
15 billion      1/149
...
...
50 billion      1/173  (approx. equal to H∞ the longterm rate)

according to the table a distance whose size is 14.4 billion LY now is growing at speed c. Now is year 13.8 billion
but in year 15 billion a distance of size 14.4 billion LY will only be growing at speed 144/149 c. That is, a speed less than c.

In the long run a distance of size 14.4 billion LY will be growing at speed 144/173 c.

And if you pick two widely separated galaxies the size of the distance between them will be changing, of course, over time. In the long run it will be growing exponentially at 1/173 % per million years.

Oops have to get ready to go out. Back later...