Cosmology's sole "rate of expansion" is declining

[marcus]

In a quantitative science verbal phrases like "rate of expansion" have to correspond to mathematically well-defined quantities.

As far as I know, in Cosmology the only quantitative handle in common use that corresponds to "rate of cosmic expansion" is the Hubble constant, and as best we can tell it has been declining over time since around the start of expansion.

According to the standard cosmic model it is expected to continue declining but not to zero.

The declining rate of expansion H(t) is expected to level out at a positive value H_∞.
That is the effect of the cosmological constant, in fact H_∞² = Λc²/3.

You could say that the cosmo constant Λ is just an alternative form of the longterm expansion rate H_∞---or vice versa the longterm expansion rate is a concrete practical expression for "dark energy" alias the cosmo constant.

I was struck by the seeming confusion that arose in another thread, that began with this clear straightforward question by one of our members. The discussion seemed to get lost amid ill-defined or purely verbal expressions.

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?...

I'd like to make a fresh start on that and deal with it in well-defined definite quantitative terms from the very start. Maybe others would like to help, or take over at this point, for that matter.

 

[marcus]

Andrew Kirk had some really clear words about this in the other thread--conveying good understanding of the actual quantities. I hope to quote them---but first let's look at the conventional form of the present-day Hubble constant, 68 km/s per Mpc.

The conventional units make that rather hard to picture but we can easily convert to to percentage increase per million years. It turns out that cosmic scale distances distances are currently growing by about 0.007 percent per million years.
More exactly according to most recent Planck measurements, the expansion rate is 1/144 % per million years.

To make the conversion you simply type or paste this into google and press return:
68 km/s per Mpc in (percent per million years)

It comes out very close to 0.007, more exactly 0.00695...or to 1/144.

The longterm expansion rate H_∞ works out to about 1/173 percent per million years.

Quantitatively, that's the story about the cosmic distance growth rate H(t). In a nutshell, H(now) = 1/144% per million years and H_∞ = 1/173% per million years.

AFAIK, there is no other commonly recognized mathematical quantity that corresponds to the universe's expansion rate.

In other words there is no standard mathematical symbol standing for a recognized quantity that you could call cosmic expansion rate that is increasing. The moment you start talking about a universal "expansion rate" that is increasing you are injecting confusion into the discussion, potentially misleading beginners.

An individual distance between two particular galaxies can of course have a growth speed which increases as the size of the distance grows. That's down at the level of particulars. The universe as a whole has no well-defined expansion speed. A particular distance growing at increasing speed could simply be showing the ordinary exponential growth that one expects from a constant or slowly declining (near constant) expansion rate.

 

[PeterDonis]

there is no standard mathematical symbol standing for a recognized quantity that you could call cosmic expansion rate that is increasing.

Isn't the quantity $\dot{a}$ (the Hubble constant times the scale factor) increasing? Or, to put it another way (if you object that $\dot{a}$ has units of distance/time instead of just 1/time), isn't the quantity $\ddot{a} / a = \dot{H} + H^2$, which can be interpreted as the "acceleration" of the expansion (and which has units of 1/time^2, no distance), currently positive?

 

[marcus]

Good! I like that. a-dot is not as widely recognized as the Hubble constant but if someone wants to go into how it is defined and use it clearly in discussion as a kind of expansion speed, that would be fine. What bothers me is the vagueness

 

[timmdeeg]

To my understanding $\ddot{a}$ is positive if the universe expands accelerated and negative if it expands decelerated.

As to $\dot{a}$, I think in a matter-dominated universe $\dot{a}$ should decrease (because the matter density goes with $1/a^3$) and should increase if the universe is dominated by vacuum energy (because of its constant density).

I doubt that one can interpret $\dot{a}$ as a speed (at least not in the sense of special relativity), as in curved space-time relative velocities aren't well defined.

 

[marcus]

BTW a simple, but possibly fun, thing to notice is this:
Consider people living in year 12 billion, year 13 billion, 14 billion and so on. They can all agree about what the Hubble constant is at any given time.
H(year 12B) = 74 ppm per million years, or if you like, 1/135% per million years.
H(year 13B) = 71 ppm per million years, or 1/140%
H(year 14B) = 69 ppm per million years, or 1/145%
H(year 15B) = 67 ppm per million years, or 1/149%
And $\dot{a}/a$ = H does have a definite meaning the same for all of them.
Also $\ddot{a} / a = H^2 + \dot{H}$ has a universally unambiguous meaning.These things have universal meaning because based on H, the consistently well-defined expansion rate.

But each group defines a(t) differently, because they normalize differently. So they all disagree as to what a, and $\dot{a}$, and $\ddot{a}$ are.
a(t) has no definite meaning which everybody accepts.

The people in year 12B say that a(year 12B) = 1
The people in year 13B say that a(year 13B) = 1
The people in year 14B say that a(year 14B) = 1
The people in year 15B say that a(year 15B) = 1

We, in year 13.8B say that a(year 13.8B) = 1
Everybody says something different, there is universal disagreement about a(t)

 

[marcus]

It may help to try some concrete numbers in the Friedman equation.
H² - H_∞² = [const] ρ
where rho is the combined ordinary and dark matter density. Matter density is declining so H is converging towards its longterm value H_∞.
Present-day matter density expressed as energy equivalent per volume is 0.24 nanojoule per cubic meter. That's the same as 0.24 joule per cubic kilometer, which might be easier to imagine. A joule is enough energy to lift a 1 kilogram book up 10 cm. Or it's the thud when you drop the book back down on the desk.

If matter density is expressed in nanojoule per cubic meter, and H is in percent growth per million years, then the numerical value of the constant in the equation is 6.2×10^-5 sometimes written as 6.2e-5
Here as a numerical check is the Friedman equation applied to presentday expansion rate (1/144 percent per million years) and presentday matter density.
1/144² - 1/173² = 6.2e-5 × 0.24
Here are some more values of the expansion rate, showing its decline leveling off to longterm value of 1/173 percent per million years (determined by the cosmo constant).

year        fraction of percent expansion per million years
1 billion        1/15
...
...
11 billion      1/128
12 billion      1/135
13 billion      1/140
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.)

 

[Jorrie]

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.

I think we must completely avoid the word 'speed' in this context and use distance growth rate between two galaxies, which is a-dot, not H (the latter being the fractional distance growth rate).
The problem is that most beginners would have read that the present expansion rate is increasing, or that expansion is presently accelerating and now we tell them that the "expansion rate (H)" is presently decreasing. I can see some potential confusion caused by the semantics. I recommend always adding the qualifier "fractional" to verbal descriptions of H.

 

[marcus]

I think we must completely avoid the word 'speed' in this context and use distance growth rate between two galaxies, which is a-dot, ...

Why avoid the word speed? When you look at a particular distance between two specified galaxies the way you quantify the growth is Δx/Δt
which is a speed.
How fast the distance is growing depends very much on the size of the distance, so it is NOT a-dot. a-dot does not tell you how fast the distance is growing because it does not contain that information.

I think it's better when talking to beginners to call a speed a speed.

And reserve the word 'rate' for fractional rates---percentage expansion rates.

 

[marcus]

The expansion speed (of a particular designated distance) is what is increasing.

The expansion rate (namely the Hubble constant H(t)) is what is decreasing.

Let's keep the language simple for beginners, use words of one syllable and have them refer to definite quantities.

 

[Jorrie]

The expansion speed (of a particular designated distance) is what is increasing.

The expansion rate (namely the Hubble constant H(t)) is what is decreasing.

I can live with this if you would replace "expansion speed" with "recession speed" (v = Hd).
In Lineweaver&Davis' article and papers (e.g http://arxiv.org/abs/astro-ph/0310808), "recession velocity" is the operative word throughout, but I have no problem with using speed instead.

I do have a problem with the idea of 'expansion having a speed'. And I think it is more than just semantics or philosophy - the scale factor 'a' is the underlying parameter that defines expansion, but as you have pointed out, it does not have a speed. It does have a rate of change and also an acceleration derivative, which is, AFAIK, where the "accelerating expansion" term comes from.

 

[ogg]

In his Theoretical Minimum videos, Susskind states that the increase in distance is "indistinguishable" from velocity. (I think I've correctly remembered the term he used), that it isn't meaningful to claim it is somehow different from velocity. I'm usually in Feynman's camp when it comes to discriminating between what we know from what we guess to be true; claims of what H was 13 billion years ago or what it will be in 13 billion years (or whether it even has a convergent value) is model dependent, and not something we "know", not something we've measured. Or are we at the point where the Standard Model is something we know to be true? I don't think so...? It would be nice if we could use a different font face or color to indicate whether what we are saying is derived from model or from data, we usually just mention the model being used once and then make all sorts of more or less absolute claims about the nature of our Universe, mixing what we know as facts and our best guesses. I've yet to see (I'm not well versed in Cosmology) a development of this from a historical point of view which includes the fact that Hubble & Humason's original determination that recessional speed (or velocity, but note it is speed|velocity not something else) is proportional to distance requires that any two non-bound objects are (currently) experiencing increasing (exponentially) velocity apart. The question of whether this increase in velocity is constant, increasing or decreasing is model dependent, as I understand it. Of course, there's always the problem that explaining the model becomes tedious if we constantly qualify our language, its poor pedagogy while at the same time it's poor pedagogy to give a single interpretation of the facts when others are seriously being considered (even if they aren't "consensus"). I guess my musings lead me to the conclusion that even in "quantitative science", phrases like "rate of expansion" do NOT necessarily have a single "well-defined" meaning, that meaning is context dependent. I'm doubting that enforcing some sort of "consensus" prescription of how words are to be defined/used is the right way to go...(Although I do agree that an author shouldn't mix semantics during development of the math, and when context changes any change in definition should be made explicit...but, really, how many actually do that?)

 

[marcus]

I can live with this if you would replace "expansion speed" with "recession speed" (v = Hd).
In Lineweaver&Davis' article and papers (e.g http://arxiv.org/abs/astro-ph/0310808), "recession velocity" is the operative word throughout, but I have no problem with using speed instead.
...

Good! Fine-tuning language to minimize the chance of confusion. Thanks, Jorrie.

The recession speed is what is increasing.
The expansion rate is is what is decreasing.

 

[Hellmut1956]

I am kind of intimidated by you experts. I thought that my knowledge about the expansion rate of the universe since about 5 billion years the dark energy seems to be accelerating the expansion of the universe. This force that is responsible for the expansion is only effective over very large distances. To my understanding this dark energy stands for a still unknown kind of force or energy, analog to the dark matter. But i do repeat, I am just an aficionado or in the terms of you experts a "beginner"!

 

[marcus]

Heh, heh, no need to feel intimidated (or maybe you are just kidding about that : ^)
Your words sound very good to me in the right context---like a casual conversation, say at a family gathering. Where the nieces and nephews, as I'm imagining, don't have any reason to want to push their understanding further, to a quantitative level. Your informal style sounds congenial, Hellmut, and just right for a certain audience.

But since we are in PF context let's see if we can rephrase things in a potentially more quantitative way so it can be preparing listeners to get up to the next "level" of understanding. I hope this can be a bit helpful to you as well.

"... this dark energy stands for a still unknown kind of force or energy, analog to the dark matter."

There is no compelling evidence, so far, that it stands for a force or anything we would normally call an energy.
THE OBSERVATIONS SO FAR ARE BEAUTIFULLY CONSISTENT WITH SOMETHING VERY SIMPLE which has always been in our equation of gravity, the Einstein equation, and is one of two natural physical constants which belong in the equation. So the observations are consistent with there simply being a non-zero value of one of those two constants, which until recently was thought to be zero because measurements had not revealed its effects.

So unless future measurements show up something unexpected (like this constant Lambda is not constant but is changing in its effects in some strange way) we don;t have any "still unknown kind" of anything. We just have something that is an unavoidable constant in the equation of gravity and involves no "force" or "energy". Just a slight intrinsic constant curvature...
A spacetime curvature, to be sure. That is what our equation of gravity is primarily about...

Oops more to say but must go to supper, back later.

Back now.

A small constant spacetime curvature would eventually reveal itself in the growth of distances at a constant percentage rate, after matter and energy had thinned out enough so they no longer dominated.
But for the first part of cosmic history (in our case the first 8 billion years or so) the effect of matter and radiation energy overwhelm this very slight curvature and there is no exponential growth to be seen.

So it explains the observations so far in a very simple way, and it could possibly be the right way. There is so far no solid evidence to the contrary. So we should not forget the simple explanation. No need to make it all seem mysterious, or hype it up to get public attention.

"..., analog to the dark matter"

Also the cosmo constant effect is in no way analogous to dark matter AFAICS.

People are misled by similarity of words, dark this and dark that. It is partly hype and should have been avoided. We see dark matter by its lensing effect and can map its clouds. It gathers into understandable bunches of higher density whose contours we can map. This is very different from a constant residual curvature extending throughout spacetime.

 

[jerromyjon]

constant residual curvature extending throughout spacetime.

I've been pondering something which started with thinking about entanglement across an event horizon and somehow lead to the thought about particles being drawn into the singularity and thinking about how particles would naturally converge as they are drawn to a singularity and I was wondering if that makes any sense at all in relation to the RATE at which the increase in SPEED is decreasing. I think I said what I mean.

 

[Hellmut1956]

UUps...! I am really a beginner! I confess that I do not understand your response! I have just a clue that you refer to the fact that gravitation is not really a force, but a curvature of the spacetime and that in consequence what is often called dark matter can be indirectly detected by its effect onto the deformation of the spacetime! I have read about this "lense" effect detected.

I have read about the dark energy as something which effect is opposed to that of "attraction" attributed to the gravity. Its existence, so I thought to have understood it, is the result of trying to identify if the universe will either collapse or expand forever! What I had understood was, that by studying the "speed" at which galaxies retreat form us, by identifying a certain type of solar explosion with exactly specified energy and as a consequence intensity and studying the speed by using the effect of doppler, since about 5 billion years the universe is not slowing down as a result of the gravitational influence but accelerating and this is attributed to a still not understood "dark energy"!

I confess that my mathematical skills eroded in the last 4 decades since I studied at a german university limit my ability to look and analyze equations! I use to define myself as a mathematical analphabet of mathematical legastheniker.

 

[Chronos]

The problem can be reduced to a differential equation if you can properly identify and quantify the relevant variables. IIRC that is exactly what they did with the orginial supernova survey data and how they the dark energy hypothesis was derived. Reducing the data to its differential form provides a residual that is mathematically equivalent to an acceleration. In physics, an acceleration requires force, hence a form of energy. Since the nature of this energy was unknown, it was called 'dark enegy', applying the same logic used to coin the term 'dark matter' - an unknown form of matter whose presence is evident by virtue of gravitational effects. You can achieve the same effect by adding an integration constant to the Einstein field equation, hence the term 'cosmological constant' is simply another way of saying 'dark energy'.

 

[marcus]

Note: original papers do not talk about using supernovae to measure "dark energy". they talk about measuring the cosmological constant, or measuring Lambda.
http://arxiv.org/abs/astro-ph/9505022
http://arxiv.org/abs/astro-ph/9805201
http://arxiv.org/abs/astro-ph/9812133
Lambda has been around for around 100 years.
"dark energy" is probably best regarded as hype that does not even necessarily refer to an energy. It could be the curvature constant Lambda, or it could be ? [speculative] {you name it} the power of darkness? "quintessence"
IMHO Simplest for now just to treat it as the constant Lambda in Einstein equation, until there is real evidence that that does not fill the bill.

 

[marcus]

Hello Hellmut!
At an informal verbal level everything you say sounds good and is I think well expressed. Good overview and summary.
You might like to read the article "Why all these prejudices against a constant?" by Carlo Rovelli
google some words of the title to get it:
http://arxiv.org/abs/1002.3966
it could be a partial antidote to all the "dark energy" talk
"since about 5 billion years the universe is not slowing down as a result of the gravitational influence but accelerating and this is attributed to a still not understood "dark energy"!"
we do have a pretty good understanding in terms of Lambda, but will further finer measurements show that understanding is not adequate?
Wording is partly a matter of taste.

 

[Hellmut1956]

Hi Marcus, tough reading of your article mentioned. I have stored it on my PC to reread it and to digest it over the future. I am working, within the limits given to me by my health problems, to refresh my mathematical skills and to look into what has been developed in mathematics in this last 4 decades. Thank you for this link!

 

[Chronos]

Agreed, marcus. Dark energy was a deduction based on the supernova data. DE does not invalidate the cosmological constant explanation, merely places it in a different light. Paul Lutus has a nice discussion here http://vps.arachnoid.com/dark_energy/index.html.

 

[Hellmut1956]

For others that might not have heard about lambda before, I I had not. Due to the data in Marcus reply to me I went to the course offered from MIT for free in the Internet, where Alan Guth gives lectures about the early universe in general and his inflation theory. See here! I have seen that this lectures and the reading material that goes with that course should help me to understand the response from Marcus.

 

[marcus]

For others that might not have heard about lambda before, ...

Hi Hellmut! Good idea to say something for people who like yourself hadn't heard of this constant.
If you google "general relativity" and look at the Wikipedia article you see, in a box, this equation in large type as a kind of "icon"

This is the GR equation and if you go to many other subsidiary related Wikipedia articles they show the same iconic equation near the top over to the right always. This ties all the articles together as a kind of symbol. It is our law of gravity and it has two gravitation constants. Newton G and the curvature constant Lambda that Einstein first discovered belonged in the equation.

Cosmologists name for their standard cosmic model that they all are using is "Lambda Cold Dark Matter" or ΛCDM.

The main equation in cosmology is DERIVED from the GR equation by making simplifying assumptions (uniformity: homogeneity isotropy) so that the GR equation simplifies to

H² - Λ/3 = (8πG/3c²)ρ

where H(t) is the main thing they want to know how evolves over time, the expansion rate, and rho ρ is the energy-equivalent density of ordinary and dark matter. H(t) is basic to cosmology and drives everything else.

Or if you like mass density instead of energy density you can write:
H² - Λ/3 = (8πG/3)ρ
with rho being mass density.

And Λ/3 is the longterm limit that H² must go to as space expands making ρ go to zero.

So we can define the longterm limit of H as H_∞ and say H→H_∞ and
H_∞² = Λ/3

This is just renaming. I haven't explained very much, maybe not anything at all, I've just given a background or cultural perspective. Gravity is the interaction of matter with geometry. There is room for just two physical constants in our law of gravity (aside from c which is everywhere). The two constants of Nature that are specifically related to gravity are Newton G and this slight curvature constant Λ that Einstein discovered in 1917.

(There was confusion about it at first and most people thought it was zero for most of the time until 1998. Some people suspected it was small and positive and observations in 1998 appeared to vindicate them. But it has a rightful place in the equation and has always been lurking there since 1917.)

For most of the time Λ has been around it has not been associated in people's minds with an "energy". That is a symptom of recent excitement and makes things more complicated. It would be if it were eventually discovered that Λ was varying over time, and interchanging with known forms of energy somehow. But it doesn't seem to be doing that. We've been watching since 1998 and looking for evidence of that, to no avail.

 

[117]

Hey everyone,
I'm new here.
Upon reading that the Hubble rate of expansion is decreasing I have just one question, namely. Some have said that DE could be caused by a cosmological constant - found in the equations of General Relativity. How can a cosmological constant be reconciled with a decreasing rate of expansion?

Or maybe I have just misunderstood a Cosmological Constant

 

[Jorrie]

Welcome here 117!

Upon reading that the Hubble rate of expansion is decreasing I have just one question, namely. Some have said that DE could be caused by a cosmological constant - found in the equations of General Relativity. How can a cosmological constant be reconciled with a decreasing rate of expansion?

The best way to look at it is that for a spatially flat universe, if the cosmological constant was zero, the expansion rate would have dropped to zero in the long term. With a positive cosmological constant, it eventually settles at a constant expansion rate (1/173 % per million years). Now a constant expansion rate actually accelerates the rate at which the distance to a specific distant galaxy increases.

Just like a constant (compound) interest rate on your investment will increase the amount of money you gain every year. Accelerated expansion! We see this in the way the cosmological scale factor (a) is 'flexing' upwards over time.

 

[Chronos]

Nice analogy, Jorrie

 

[marcus]

...
Or maybe I have just misunderstood a Cosmological Constant

I like Jorrie's explanation. I'll amplify by basically saying the same thing in different words.

The role of the Cosmological Constant is to determine the longterm expansion rate H_∞ to which H(t) declines.

The Λ does not prevent H from steadily declining, but if Λ=0 then H will decline to zero, whereas if Λ is positive H will not decline to zero but will decline and level out towards a small positive expansion rate determined by Λ.

The relation between that longterm rate H_∞ and Lambda is very simple:
H_∞² = Λ/3

That is how we get the value for H_∞ = 1/173% per million years.
We take the value for Λ and divide by 3, and take the square root.

At any given time in history, all distances are expanding at the the same rate---the same declining and now approximately constant percentage rate.
And the distances are expanding at all different SPEEDS. Because the speed depends on the distance's size. Bigger grow at faster speed.
And as any given distance gets bigger, it grows at a faster speed, of course. Because the expansion rate, although declining, is now changing very slowly, it is nearly constant.

 

[marcus]

BTW Jorrie and Chronos since Λ is such a small curvature it's often awkward to express it quantitatively (as a minuscule number per unit area, or per unit time squared.) I just now tried Λ ≈ 1.0025e-20 per square lightyear

That seems to work all right in the Google calculator. We can divide this version of Λ by 3 and take the square root and the reciprocal will be the longterm Hubble radius 17.3 billion lightyears, that we use in the Lightcone calculator.

At least when I paste this in
(1.0025e-20/3 per square lightyear)^(-1/2) in lightyears
I get 17.3 billion LY.

And google calculator also gives me the longterm Hubble time if I paste in using the "per square time" version of Lambda. Dividing that by 3, taking the reciprocal square root,
(1.0025e-20/3 per square year)^(-1/2) in years
I get 17.3 billion years.

 

[Jorrie]

BTW Jorrie and Chronos since Λ is such a small curvature it's often awkward to express it quantitatively (as a minuscule number per unit area, or per unit time squared.) I just now tried Λ ≈ 1.0025e-20 per square lightyear

Number-wise, it is actually a little easier if we use time units of Gy; it becomes Λ ≈ 0.01 Gy^-2, so that Λ/3 ≈ 0.00333 Gy^-2 and taking the inverse square root gives the long term Hubble time as T_H_∞ ≈ 17.3 Gy.

Something that is bugging me a little is that in an accelerating expansion universe, spacetime curvature should be negative and hence the radius of curvature should be viewed as an imaginary number. Is the fact that we view the cosmological constant as positive just a convention, or do I have the negative spacetime curvature idea wrong?

 

[marcus]

That bugs me too, something I've not satisfactorily resolved. In a Euclidean signature (Riemannian) manifold you'd expect timelike lines spreading apart would correspond to negative curvature. Company coming, I can't think about this right now, will try later. George Jones might be able to clarify this.

 

[PeterDonis]

In a Euclidean signature (Riemannian) manifold you'd expect timelike lines spreading apart would correspond to negative curvature.

In a Riemannian manifold there is no such thing as "timelike" lines (or null lines). The metric is positive definite, so all curves are spacelike.

 

[marcus]

Ah Peter, good you are there, maybe you can help clarify this. What I had in mind were lines in a 1+3D Riemannian case that would be timelike in the actual Pseudo-Riemannian---very rough idea trying to get some intuition.
It seems like the lines spreading apart, if you go over to Lorentzian signature, might correspond to positive curvature. Could that be right?

Or take another approach. Notice the + sign in the GR version, contrasting with the minus sign in the Friedman equation. As the universe expands the T tensor on RHS goes to zero. So the G_μν tensor must go to -ΛgSUB]μν[/SUB].

Can you shed some light on how this corresponds to what happens with the Friedman equation model universe? It should be in some sense analogous since one is a simplification of the other

$$H^2 - H_\infty^2 = [const] ρ$$

I suspect this is the kind of thing that Jorrie says is bugging him (although he can doubtless be more precise about it). We normally associate lines converging (diverging) with positive (negative) curvature---so why is the cosmological curvature constant positive when its effect is a residual rate of expansion.

The company has arrived. I'm taking time away from our visit to type but really would like to get some better understanding of this.

 

[timmdeeg]

Nice plot.

The best way to look at it is that for a spatially flat universe, if the cosmological constant was zero, the expansion rate would have dropped to zero in the long term. With a positive cosmological constant, it eventually settles at a constant expansion rate (1/173 % per million years). Now a constant expansion rate actually accelerates the rate at which the distance to a specific distant galaxy increases.

As constant expansion rate $H$ means that the universe is expanding exponentially, accelerated expansion just means that the expansion rate is dropping, however less fast than in the case of decelerated expansion.
It would be interesting (perhaps also for didactical purposes) to show $H$ and $a$ as a function of time in one plot. I have been searching for that unsuccessfully, but perhaps someone around here can help?

 

[Bandersnatch]

As constant expansion rate HH means that the universe is expanding exponentially, accelerated expansion just means that the expansion rate is dropping, however less fast than in the case of decelerated expansion.

I'd rather say that accelerated expansion means the universe is approaching exponential expansion.
Here's the plot:

For plot bonanza go straight to Jorrie's calc: http://www.einsteins-theory-of-relativity-4engineers.com/LightCone7s/LightCone.html and play with the chart display functionality.