Effort to get us all on the same page (balloon analogy)

Mark M

One thing to note is that the cosmological constant in the EFE is the same thing as a constant negative pressure vacuum energy. If you solve the EFE for the stress energy tensor, you'll notice that the cosmological constant contributes to the total energy.

It still doesn't change the fact that QFT predicts this vacuum energy should be much, much larger than the observed value.

marcus

But as it actually appears in the Einstein GR equation or "EFE" λ is not an energy and not a pressure. It is the reciprocal of length squared (a measure of curvature).
One can talk about the amount of energy you would need to create that curvature if it were not already there. As far as we know that is a fictitious energy.

My point is that thinking about λ in terms of pressure or energy is a bad idea. It involves dragging a curavture term on the lefthand side of the equation over onto the righthand side and multiplying it by stuff to turn it into a fictitious energy and a negative pressure.

This is explained more clearly and at greater length in the article we were just quoting. You might want to take a look at it!

http://arxiv.org/abs/1002.3966/
Why all these prejudices against a constant?
__

For the more math-inclined, Cosmology is a mathy science so explanation/understanding involves fitting data to equation-models. But the models are relatively simple! I'm sure lots of layfolk diffident about their math can nevertheless understand cosmology basics just fine!

The basic equation of cosmology is called the Friedmann equation, and is derived from Einstein GR equation after making simplifying assumptions.
Ever since very early times, matter has vastly outweighed radiation. We can handle other conditions but it makes a later equation slightly more complicated. For simplicity let's assume matter dominates and also that space is nearly flat. We can put the Friedmann equation in a particularly simple form:

H² - H_∞² = (8πG/3)ρ

Here rho is the current density of matter (dark+ordinary) and the small contribution from radiation. We could write ρ(t) to show the time dependence. H is a number-per-unit-time which is the current fractional growth rate of distance. And H_∞ is a constant number-per-unit-time which is the fractional growth rate of distance in the far future, which the H(t) is heading towards. Its square is the same as the cosmo constant Lambda except for a factor of c²/3. So we can treat it as a look-alike or stand-in for λ.

Measuring H_∞² is really the same as measuring λ. Whatever you get for the former, you just multiply by 3 and divide by c² and presto that is your value of λ.

That's because by definition H_∞² = λc²/3.
So measuring one is equivalent to measuring the other.

Let's take an imaginary example--forgetting realistic numbers. We can see that the equation is of the form x² - C = y, where C is some constant to be determined by plotting lots of (x,y) datapoints. You look back in the past and estimate density (y) was and also what the expansion rate (x) was at particular times in the past. You do that with lots of cases (in reality wth supernovae in lots of galaxies). So you get a curve. If C, the constant, equals zero, then the curve is simply x² = y.
If C = 1, then the curve will look different: x² - 1 = y, so you can tell the difference and in this way decide what the constant C is.

Mark M

Thanks for the link!

I still don't understand how the cosmological constant is a 'natural' part of the EFE. They show the modified Einstein-Hilbert action (Eq. 5) to prove their point, but the original Einstein-Hilbert action for the gravitational field doesn't contain a [itex] \Lambda [/itex]. You have to put it in by hand (That's just how I've understood it, correct me if I'm wrong).

The point I was making is that the cosmological constant is physically equivalent to a negative pressure vacuum energy. That's why the cosmological constant is measured in units of energy. However, QFT does predict - it necessitates - some kind of vacuum energy. Even if the number is vastly incorrect, the concept flows directly from quantum mechanics. So, if there was no vacuum energy, that would be extremely odd. I just assume there is a cutoff at which current particle physics no longer apply, where whatever higher energy physics that exist there fix the problem. I'm aware that that's a cop out.

Their argument against a QFT vacuum energy is that even assuming a finite cutoff, the value is still too high. To which I don't have the expertise to respond to.

marcus

Let's take another look at our basic expansion-rate equation
H² - H_∞² = (8πG/3)ρ

Just using Freshman calculus we can differentiate it, and some nice things happen. The constant term drops out and we just have.

2HH' = (8πG/3)ρ'

But density ρ is essentially just some mass M divided by an expanding volume proportional to the cube of the scalefactor: a³
(M/a³)' = -3(M/a4)a' = -3ρ(a'/a) = -3ρH
Because by definition H = a'/a

2HH' = (8πG/3)(-3ρH) = - 8πGρH, and we can cancel 2H to get:

H' = - 4πGρ

I've highlighted that because it comes in a few lines later. Again by definition H = a'/a so we can approach H' from another direction:
H' = (a'/a)' = a"/a - (a'/a)² = a"/a - H²
It's great how much of the first 2 or 3 weeks of a beginning calculus course comes into play: chain rule, product rule, (1/xⁿ)'...

Now the Friedman equation tells us we can replace H² by H_∞² + (8πG/3)ρ. So we have
H' = a"/a - H² = a"/a - H_∞² - (8πG/3)ρ = - 4πGρ

Now we group geometry on the left and matter on the right, as usual, and get:
a"/a - H_∞² = (8πG/3)ρ - 4πGρ = - (4πG/3)ρ
Here we used the arithmetic that 8/3 - 4 = - 4/3

This is the socalled "second Friedmann equation" in the matter-dominated case where pressure is neglected.
a"/a - H_∞² = - (4πG/3)ρ

We can make another application of our basic Friedmann equation to replace
(4πG/3)ρ by (H² - H_∞²)/2
a"/a = H_∞² - (4πG/3)ρ = H_∞² - (H² - H_∞²)/2
= (3H_∞² - H²)/2

This will tell us the time in history when the INFLECTION occurred. When the distance growth curve slope stopped declining and began to increase. This is the moment when a" = 0. It marks when actual acceleration of distance growth began---i.e. when a" became positive.
To find that time all we need to do is find when
H² = 3H_∞²
since then their difference will be zero, making a" = 0. That means
H = √3 H_∞ = √3/163 percent per million years = 1/94 percent per million years.
That happened a little less than 7 billion years ago. In other words when expansion was a bit less than 7 billion years old. You can see that from the table. The 1/94 fits right in between 6 billion years ago and 7 billion years ago. In between 1/100 and 1/90.

The present Hubble rate is put at 70.4 km/s per Mpc which means distances between stationary observers increase 1/139 percent per million years. And the Hubble radius (a kind of threshhold within which distances are expanding slower than c) is currently 13.9 billion LY.
So by analogy you can see how the Hubble rate has been declining, while the Hubble radius (reciprocally) has extended out farther and farther.

marcus

I've seen Steven Weinberg reason "natural" this way in another situation. It's highbrow physics, but very common. When you write down a theory you are supposed to include ALL THE TERMS ALLOWED BY THE SYMMETRIES of the theory. It is almost a ritual mantra.
The symmetries (whatever they are for that particular theory) determine what terms "belong" and which do not.

In the case of GR the symmetries are the DIFFEOMORPHISMS (all the invertible smooth maps of the manifold). It is a large powerful group and it shows its power by excluding all terms except G and λ terms. Einstein called diffeo-invariance by the name "General Covariance". He had decided to make a theory that was "general covariant" and so the action had to be what you see and the EFE arising from it had to be what you see. There isn't anything put in by hand.

So the λ was always LATENT in the theory even though it may have been omitted in the very first publications. My guess is that it was seen to belong even at the start but not much talked about. Somebody else who knows the history better should clarify this. Presumably DeSitter needed λ to get his DeSitter space (1917) solution, which has no matter but does have a positive λ. And Levi-Civita came up with the same solution at just the same time. My guess is they both must have known the constant was there and that they were not putting anything in "by hand". Just my guess, could be wrong.

I think Bianchi and Rovelli discuss the λ naturalness, on the basis of diffeo-invariance. Maybe we should check back and see exactly what they say.

http://arxiv.org/abs/1002.3966/
Why all these prejudices against a constant?

Mark M

Thanks for that correction. I hadn't known that the cosmological constant emerged from a symmetry of GR.

Still, what of the QFT vacuum energy? If it exists, it certainly makes a contribution to the acceleration of the universe. So, if it does exist, then it seems simpler to say that it is the cause of the acceleration. The only way I see a way around that is to say that QFT either doesn't predict vacuum energy, or it's very much misunderstood. Since the first scenario (as far as I know) can't be true, we would need to go with the second. And I don't think that's very desirable, considering the success of QFT.

marcus

I'll expand the earlier table and recap the easy calculus derivations from before.
The present Hubble rate is put at 70.4 km/s per Mpc which means distances between stationary observers increase 1/139 percent per million years. And the Hubble radius (a kind of threshhold within which distances are expanding slower than c) is currently 13.9 billion LY.
So by analogy you can see how the Hubble rate has been declining, while the Hubble radius (reciprocally) has extended out farther and farther.

By definition H = a'/a, the fractional rate of increase of the scalefactor.

We'll use ρ to stand for the combined mass density of dark matter, ordinary matter and radiation. In the early universe radiation played a dominant role but for most of expansion history the density has been matter-dominated with radiation making only a very small contribution to the total. Because of this, ρ goes as the reciprocal of volume. It's equal to some constant M divided by the cube of the scalefactor: M/a³.
Differentiating, we get an important formula for the change in density, namely ρ'.
ρ' = (M/a³)' = -3(M/a⁴)a' = -3ρ(a'/a) = -3ρH
The last step is by definition of H, which equals a'/a

Next comes the Friedmann equation conditioned on spatial flatness.
H² - H_∞² = (8πG/3)ρ
Differentiating, the constant term drops out.
2HH' = (8πG/3)ρ'
Then we use our formula for the density change:
2HH' = (8πG/3)(-3ρH) = - 8πGρH, and we can cancel 2H to get the change in H, namely H':

H' = - 4πGρ

I've highlighted that because it gets used a few lines later. Again by definition H = a'/a so we can differentiate that by the quotient rule and find the change in H by another route:
H' = (a'/a)' = a"/a - (a'/a)² = a"/a - H²

Now the Friedman equation tells us we can replace H² by H_∞² + (8πG/3)ρ. So we have
H' = a"/a - H² = a"/a - H_∞² - (8πG/3)ρ = - 4πGρ

We group geometry on the left and matter on the right, as usual, and get:
a"/a - H_∞² = (8πG/3)ρ - 4πGρ = - (4πG/3)ρ
Here we used the arithmetic that 8/3 - 4 = - 4/3

This is the socalled "second Friedmann equation" in the matter-dominated case where radiation pressure is neglected.
a"/a - H_∞² = - (4πG/3)ρ
In the early universe where light contributes largely to the overall density a radiation pressure term would be included and, instead of just ρ in the second Friedmann equation, we would have ρ+3p.

Now using the second Friedmann equation we would like to discover the time in history when the INFLECTION occurred. When the distance growth curve slope stopped declining and began to increase. This is the moment when a" = 0. It marks when actual acceleration of distance growth began---i.e. when a" became positive.

We can use the MAIN Friedmann equation to replace
(4πG/3)ρ by (H² - H_∞²)/2 in the second equation.
a"/a = H_∞² - (4πG/3)ρ = H_∞² - (H² - H_∞²)/2
= (3H_∞² - H²)/2

Now to find the inflection time, all we need to do is find when it was that
H² = 3H_∞²
since then their difference will be zero, making a" = 0. That means H = √3 H_∞ = √3/163 percent per million years = 1/94 percent per million years. As one sees from the table, that happened a little less than 7 billion years ago. In other words when expansion was a bit less than 7 billion years old. You can see that from the table. The 1/94 fits right in between 6 billion years ago and 7 billion years ago. In between 1/100 and 1/90.

RayYates

Is there a non mathimatical definition of λ ?

I know ∏ is the relationship between a circles radius and its circumference; c is the speed of light. Those are definitions I can get my head around.

I googled Definition: cosmological constant and got an arbitrary constant in the equations of general relativity theory. --- worse than useless! Other definitions were contradictory with many referencing dark energy.

Anyone care to give it a try?

marcus

Heh heh, you won't believe me, will you? One of the main points in my recent posts has been to try to give you just exactly that. A definition of λ which is intuitively meaningful---something you can visualize and hang some concrete meaning on.

FACT: our universe has this pattern of expanding distances, by a small fraction of a percent per million years.

Get to know your universe: the rate is 1/139 of one percent expansion per million years.

That's a fact of life, your life, my life and the life of the Aliens on Planet Gizmo.

There is another important quantity that is a basic feature of the universe, which everybody should know (Aliens included )

FACT: The 1/139% rate is not steady but is tending towards 1/163 of a percent per million years.

What is λ? Probably the most immediate handle on it is what you get when you multiply it by c² and divide by 3.

You get the SQUARE of that 1/163 percent rate.

On Planet Gizmo they probably don't use exactly the same λ that our Albert wrote they probably use a constant χ which is the same as λc²/3.

Because the relativity equation their Albert wrote had a c² and a 3 in it. So to compensate, to make their equation give the same answers, they have to adjust the constant and make it χ.

They also don't use ∏ = 3.14 they use a constant called something else which is 6.28. It is the ratio of the RADIUS to the circumference. That's what they use as a constant, instead of the DIAMETER to the circumference like we do. And they probably have different sexual practices as well. Or barbecue differently.

A constant is just as good if you multiply it by 2, or divide it by 3, you just have to adapt the formulas so as to compensate.

Morally the cosmological constant Λ is the same as that 1/163 of a percent growth rate which our universe will eventually get to. A kind of trend inherent in its 4d geometry. A rate of distance increase that our geometry thinks is "just right", no more and no less. (And on this planet we do not yet know WHY it is this particular 1/163 size and not some other size. QG (4d quantum geometry) may eventually explain the size of that 1/163 rate since it is a feature built in to the geometry and QG looks at the quantum foundations of geometry.

marcus

Ray, you claim to be non-mathy but this you can get for sure, Einstein in 1915 was driving at a relation between geometry (which he put on the left side of the equation) and matter (which he put on the right). describing their influence on each other.
I'm saying that Lambda is a feature of geometry and belongs on the LEFT. If you see an equation that puts Lambda on the right, then all one can say is it shows a deplorable lack of judgment and good taste. Something fishy about it: A fictitious energy, a spurious term--something that doesn't belong, pretending to be part of the matter side.

Lambda is an inherent minimal growth rate that nature's geometry has a built-in tendency towards. For most of history because it got such a terrific kickoff at the start, the growth rate has been much bigger. but now it is settling down. As the density thins out, H² is getting closer and closer to H_∞²
That's what the Friedmann equation says. The amount it has left to go is proportional to the density
H² - H_∞² = (8πG/3)ρ
and as distances (and volumes) enlarge, the density gets less and less.

Sorry if you think the proportionality constant 8πG/3 is a bit clunky and elaborate. On Planet Gizmo they probably write it with a single symbol K. We humans, by a series of historical accidents, just happen to write it 8πG/3. It's a proportionality between density and the square of growth rates. If you have a density (mass per unit volume) and you multiply by K what you get is the square of some percent per unit time growth rate. So the proportionality looks a bit clunky but please don't be put off by that! It's really very nice that there is such a clean simple relation between matter conditions and the changing geometry features.

RayYates

I'm pretty mathy but your a couple orders of magnitude ahead of me.

If I asked, "What is ∏?" One could say = approximately 3927/1250. That would be a more precise answer than "the ratio between a diameter and circumference" but doesn't say what it is.

Not to beat the dead horse.
I've seen it defined as "represents the energy of empty space", but I know you agree with that so since its geometry and not energy... it would be.... called..... a relationship between.......

marcus

Comment welcome. I'm working on improving this presentation of basic cosmology for newcomers. It assumes elementary differential calculus: chain rule, product rule...for derivatives. Otherwise very basic.

For definiteness I use the key model parameters from the 2010 WMAP7 report by Komatsu et al, namely 0.272, 0.728, and 70.4 km/s per Mpc.
The current Hubble growth rate of 70.4 km/s per Mpc means distances between stationary observers are currently increasing by 1/139 of a percent per million years. By the same token, the Hubble radius (a kind of threshhold for incoming photons, within which distances are expanding slower than c) is currently 13.9 billion lightyears.
The table shows how the Hubble rate has been declining, while the Hubble radius (reciprocally) has extended out farther and farther.

You can see that, according to the standard cosmic model, 13 billion years ago the Hubble rate of distance expansion was 1/11 of a percent per million years. The Hubble radius then was only 1.1 billion lightyears. Expansion had, by then, been in progress for 0.7 billion years and light emitted at that time would be received by us redshifted by about z = 7.19. This means that distances and wavelengths from that time have increased by a factor of 8.19.
The factor is always 1+z, one more than the stated redshift.
A key tool here is the scalefactor curve a(t) which tracks proportional increase in distances over time. It is a rising curve which is arbitrarily set equal to one at the present time: a(now) = 1. The Hubble rate is the fractional rate of increase of the scalefactor:
By definition H = a'/a.

You can see that the Hubble rate has been declining more and more slowly over the millennia. This is the most important point: according to our best understanding (especially since the supernova studies reported in 1998) the decline is leveling off towards a constant limiting rate of about 1/163 of one percent per million years.

This is the expected longterm expansion rate which is here denoted H_∞.
The Hubble expansion rate H, which we can measure and infer past values for, is declining ever more slowly and in the distant future will approach H_∞ as its limit.

This was implicit in the GR equation early on but it was assumed by most students of cosmology that this limit was ZERO. It was only in 1998 that it became generally accepted that the limit is non-zero. H_∞ is one guise of a constant Λ that appears naturally in the GR equation and is called the "cosmological constant"*.

The simple model of the universe which generates these numbers, producing a remarkably good fit with observation, is called the Friedmann equation.
H² - H_∞² = (8πG/3)ρ
It's a simplification of the 1915 equation of General Relativity obtained by assuming overall uniformity of the universe--an assumption that so far has proven to be quite reasonable and makes the model a lot easier to use.
I want to explain the terms in this equation and help see how it works. (To start with we're focusing on the spatially "flat" or k=0 version. It's widely used because at large scale space does seem to have little or no overall curvature.)

We'll use ρ to stand for the combined mass density of dark matter, ordinary matter and radiation. In the early universe radiation played a dominant role but for most of expansion history the density has been matter-dominated with radiation making only a very small contribution to the total. Because of this, ρ goes as the reciprocal of volume. It's equal to some constant M divided by the cube of the scalefactor: M/a³.
Differentiating, we get an important formula for the change in density, namely ρ'.
ρ' = (M/a³)' = -3(M/a⁴)a' = -3ρ(a'/a) = -3ρH
The last step is by definition of H, which equals a'/a

The idea now is to work with the Friedmann equation and get it to tell us some things. First let's differentiate it--the constant term will drop out:
H² - H_∞² = (8πG/3)ρ
2HH' = (8πG/3)ρ'
Then we can use our formula for the density change:
2HH' = (8πG/3)(-3ρH) = - 8πGρH, and we can cancel 2H to get the change in H, namely H':

H' = - 4πGρ

Again by definition H = a'/a, so we can differentiate that by the quotient rule and find the change in H by another route:
H' = (a'/a)' = a"/a - (a'/a)²
H' = a"/a - H²
Now we have two different expressions for H', this one and the highlighted one, so we can write:

a"/a - H² = - 4πGρ

The Friedman equation tells us we can replace H² by H_∞² + (8πG/3)ρ. So we have
a"/a - H² = a"/a - H_∞² - (8πG/3)ρ = - 4πGρ

We group geometry on the left and matter on the right--then, noticing that 8/3 - 4 = - 4/3, we get:
a"/a - H_∞² = (8πG/3)ρ - 4πGρ
a"/a - H_∞² = - (4πG/3)ρ

This is the socalled "second Friedmann equation" in the matter-dominated case where radiation pressure is neglected. In modeling the early universe, when light contributed largely to the overall density, a radiation pressure term would be included and, instead of just ρ in the second Friedmann equation, we would have ρ+3p.

The second Friedmann equation is also called the "acceleration Friedmann equation" because it yields information about a". We would like to use it to discover at what moment in history an INFLECTION occurred in the scalefactor a(t) distance growth curve. When did the slope of the scalefactor curve stop leveling out and begin to get steeper? This is the moment when a" = 0. It marks when a" changed from negative to positive and actual acceleration of distance growth began.

The way we do this is to rearrange the "second Friedmann" slightly:
a"/a - H_∞² = - (4πG/3)ρ
a"/a = H_∞² - (4πG/3)ρ
and use the main Friedmann equation to replace (4πG/3)ρ by (H² - H_∞²)/2.

a"/a = H_∞² - (H² - H_∞²)/2
a"/a = (3H_∞² - H²)/2

Now to find the inflection time, all we need to do is discover when it was that
H² = 3H_∞²
since then their difference will be zero, making a" = 0.
That means H = √3 H_∞ = √3/163 percent per million years
H = 1/94 percent per million years.
As one sees from the table, that happened a little less than 7 billion years ago. In other words when expansion was a bit less than 7 billion years old. You can see that from the table. The 1/94 fits right in between 6 billion years ago and 7 billion years ago. In between 1/100 and 1/90.

*The relation is H_∞² = Λc²/3

marcus

One thing that you might wish to try is actually CALCULATING something using the current Hubble expansion rate of 1/139 percent per million years. Often that means pasting or typing something into the google window (which doubles as a scientific calculator able to convert units and supply physical constants.)

As a reminder, here's the Friedmann equation.
H² - H_∞² = (8πG/3)ρ

Here's a sample calculation: try pasting this into the google window. It will give the CRITICAL MATTER DENSITY required for spatial flatness at this time in history, expressed as an energy density (joules per cubic meter).

c^2*(1/139^2 - 1/163^2)*(percent per million years)^2/(8pi*G/3)

Basically what you are doing is solving for ρ, to get the mass density, and then multiplying by c² to turn that into the equivalent energy density.

If you look closely at what is to be pasted into the calculator you will see that
(1/139^2 - 1/163^2)*(percent per million years)^2
is just a version of the familiar H² - H_∞², the lefthand side of the Friedmann equation.
To solve for the density ρ all we need to do is divide by (8pi*G/3).

So what I'm suggesting you paste into the window should make sense in terms of the preceding discussion. You should get 0.23 nanopascals. That is 0.23 nanojoules per cubic meter. We know that the matter density of our universe is pretty close to that, because spatially it's pretty close to flat.

One way to think of it is to translate the energy density into 0.23 joules per cubic kilometer. It's easy to get an idea of a joule of energy: just drop a conventional physics textbook (one-kilogram) from a height of 10 centimeters. For 0.23 joules, drop it from 2.3 centimeters. It makes a little thud. That thud is the energy equivalent of how much mass a cubic kilometer of today's universe, on average, contains.

marcus

I've added to the table in post #318. The first few columns show how the Hubble rate has been declining, while the Hubble radius (reciprocally) has extended out farther and farther. The new columns show the proper distance (in Gly) of an object seen at given redshift z both now and back when it emitted the light we are currently receiving.

You can see that, according to the standard cosmic model, 13 billion years ago the Hubble rate of distance expansion was 1/11 of a percent per million years. The Hubble radius then was only 1.1 billion lightyears. Expansion had, by then, been in progress for 0.7 billion years and light emitted at that time would be received by us redshifted by about z = 7.19.

The table also shows that when we observe a galaxy as it was 10 billion years in the past we know that the light we are getting was emitted while the galaxy was receding faster than c. This is revealed by the fact that the then-distance (5.79 Gly) exceeded the then-Hubbleradius (5.4 Gly). This would be true for any galaxy observed to have redshift z > 1.64 or thereabouts. This means most of the galaxies we can see---the distance to any such galaxy was increasing faster than c when it emitted the light we're getting and has continued to increase faster than c all the while the light has been traveling to us.

The redshift z = 1.64 is also interesting because it marks the angular size minimum. Objects with the same physical size will look bigger (take up a larger angle in the sky) if their redshift is greater than about 1.64 and if it is less. Then-distance peaks right around z = 1.64

Incidental information: According to Peeble's Cosmic Inventory ( http://arxiv.org/abs/astro-ph/0406095 ) ordinary matter makes up about 16% of the total matter of the universe. Roughly a sixth, in other words.

marcus

In another thread someone had a question concerning the PRESENT rate of distance expansion in real terms--namely how much was it currently accelerating. So I wanted to show an easy way to address the question of acceleration and come up with a definite number.
Of course to get a definite speed one has to specify some particular distance between two stationary observers and see how fast that is growing and how much the growth is speeding up.

What distance one chooses to look at is somewhat arbitrary---I picked 13.9 billion lightyears because it makes the numbers simple. It is a bit less than a third of the current radius of the observable region.

For definiteness I use the key model parameters from the 2010 WMAP7 report by Komatsu et al, namely 0.272, 0.728, and 70.4 km/s per Mpc.
The current Hubble growth rate of 70.4 km/s per Mpc means distances between stationary observers are currently increasing by H = 1/139 of a percent per million years. And according to the standard model the limit that H is tending to is H_∞ = 1/163 of a percent per million years*

The Friedmann equation model in the spatially flat case is
H² - H_∞² = (8πG/3)ρ
where ρ is the density of all kinds of matter and radiation (excluding the cosmological constant, which I'm taking to be simply that: the cosmological constant.)
In the case where the contribution of radiation to ρ is small compared with that of dark and ordinary matter, the acceleration equation takes this form:
a"/a - H_∞² = - (4πG/3)ρ

So then we have:
a"/a = H_∞² - (4πG/3)ρ
and using the main Friedmann equation to replace (4πG/3)ρ by (H² - H_∞²)/2, we have:
a"/a = H_∞² - (H² - H_∞²)/2
a"/a = (3H_∞² - H²)/2, and factoring out H² we get:
a"/a = [(3(H_∞/H)² - 1)/2]H²
a"/a = [(3(139/163)² - 1)/2]H²
a"/a = 0.59 H²

Since we are asking about acceleration at the present time and by convention the scalefactor a(now) = 1 we can just write a" = 0.59 H²
and if we choose, as mentioned earlier, the distance R = 13.9 billion lightyears to be the present separation between the pair of stationary observers or objects then the acceleration is just gotten by multiplying on both sides by R:
a"R = 0.59 H²R

Now HR = c, because that's how R was chosen, and so
a"R = 0.59 Hc
This means that the current acceleration is 0.59/139 = 1/236 of a percent of the speed of light per million years.

I like this example because it gives an idea of how slow the acceleration is. The distance itself is currently increasing at the speed of light. And that rate is scarcely changing at all! Indeed after a million years it will still only be just slightly (a small fraction of a percent) larger than the speed of light.


*The relation to the cosmological constant is H_∞² = Λc²/3

marcus

Since we've turned a page I'll bring forward the table from post #318, to which more columns have been added. The Hubble rate is shown both in conventional units (km/s per Mpc) and as a fractional growth rate per d=10⁸y. The first few columns show how the Hubble rate has been declining, while the Hubble radius (reciprocally) has extended out farther and farther. The columns on the right show the proper distance (in Gly) of an object seen at given redshift z both now and back when it emitted the light we are currently receiving.

To illustrate: according to the standard cosmic model with final WMAP parameters, 13 billion years ago the Hubble rate of distance expansion was 1/11 of a percent per million years. For brevity this can also be written as a fractional growth rate of 1/11 per d (= 10⁸y). The Hubble radius then was 1.1 billion lightyears. Expansion had, by then, been in progress an estimated 0.757 billion years and light emitted at that time is now being received by us redshifted by about z = 7.19.

The redshift z = 1.64 marks the angular size minimum. Objects with the same physical size will look bigger (take up a larger angle in the sky) if their redshift is greater than about 1.64 and also if it is less. "Distance-then" peaks right around z = 1.64

Incidental information: According to Peeble's Cosmic Inventory ( http://arxiv.org/abs/astro-ph/0406095 ) ordinary matter makes up about 16% of the total matter of the universe. Roughly a sixth, in other words.

The timescale d=10⁸ years turns out to be convenient to work with so to get an intuitive feel for it as an interval of time here's a geological timeline:
http://www.ucmp.berkeley.edu/help/timeform.php/
For example the Paleozoic Era = about 3 of this unit
and is divided into 6 roughly equal Ages (Cambrian, Ordovician, Silurian, Devonian, Carboniferous, Permian ) each of these lasting approximately 1/2 unit.
In this context the names don't matter, only the idea that geological ages tend to be on the order of one of these d=10⁸y. It is a length of time during which something can happen which is distinctive enough in terms of geology or biological evolution so that the relevant professionals decide to give it a name.

It is also a period of time during which distances between pairs of observers at rest wrt background can grow by some fractional amount, such as 1/139 at present, or 1/11 much earlier.

marcus

Key quantities in cosmology are fractional distance growth rates, and average density. The basic equation of cosmology, the Friedmann equation, relates (the square of) the fractional growth rate to overall matter density ρ.

H² - H_∞² = (8πG/3)ρ

Here ρ is the mass-equivalent density of all kinds of matter and radiation (not the cosmological constant, which I'm taking to be simply that: the cosmological constant.)
Using the convenient time unit d=10⁸ years, the present and eventual values of the distance growth rate can be written 1/139 per d, and 1/163 per d.
It's convenient to solve for the energy-equivalent form of the density: ρc² which will come out in nanopascals, that is nanojoules per cubic meter.

[3c²/(8πG)](H² - H_∞²) = ρc²

It turns out that the coefficient 3c²/(8πG) = 16144 nanopascal d².

So it's a straightforward calculation to find the density (in energy-equivalent form). The d² cancels and we have:

16144 nanopascal(1/139² - 1/163²) = 0.228 nanopascal

As basic arithmetic, this works in the google calculator. Pasting in
16144(1/139^2 - 1/163^2)
gives 0.228

Expressed in energy-equivalent form, the average matter density in the universe today is presumably close to 0.228 nanojoule per m³. Or in other words 0.228 joule per cubic kilometer. About 16% of this is ordinary matter and most of the rest is dark matter.

What we calculated there is actually the critical matter density---that necessary for overall spatial flatness. Since it continues to be found that the cosmos is nearly flat---at large scales the overall spatial curvature is at least very close to zero---the current critical matter density is a good estimate for the actual one.
For simplicity the version of the Friedmann equation used here assumes spatial flatness.

The essential takeaway message here is that if you know two fractional growth rates (the present and the future target rate), namely 1/139 and 1/163 per d then this simple arithmetic:
16144(1/139² - 1/163²) = 0.228
gives you the estimated current matter density (expressed as energy equivalent per unit volume.)
A joule of energy (dropping a kilo textbook from about 10 cm) is easy to imagine. Or think 2.28 centimeters to get the 0.228 joule figure. And that amount of work has to be contained in a cubic kilometer.
The 3c²/(8πG) = 16144 nanopascal d² thing can be thought of as a constant of nature---relating fractional expansion rate to density. You can get the 16144 nanopascals for yourself, from google calculator, just by pasting in
3c^2/(8pi*G)/(10^8 year)^2