Cosmology's sole "rate of expansion" is declining

[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.

 

[Jorrie]

You are both right, because we are approaching both an constant H and exponential expansion - only to be realized in the long term...

 

[Bandersnatch]

I was objecting to the specific wording of that quote. Accelerated expansion does not mean that the rate of expansion (H) is 'dropping, however less fast than in the case of decelerated expansion', since H was always dropping more slowly with every year, even during the period of decelerated expansion. It's not a sufficient statement to describe accelerated expansion.

 

[Jorrie]

It may not be a sufficient statement to describe accelerated expansion, but it is not fundamentally wrong. A comparative curve shows that for zero Lambda (with everything else the same), the expansion rate H is indeed 'dropping faster' than is the case with the present Lambda. Such wording may cause confusion though.

 

[Bandersnatch]

It may not be a sufficient statement to describe accelerated expansion, but it is not fundamentally wrong. A comparative curve shows that for zero Lambda (with everything else the same), the expansion rate H is indeed 'dropping faster' than is the case with the present Lambda. Such wording may cause confusion though.

Since this thread is about clarifying confusing language, let me hammer this issue a bit more. I know it's not that big of a problem, but hopefully it's not on the level of pointless semantics.
The problem with wording here is that the presence of lambda does not automatically imply accelerated expansion - not throughout the whole history.
As you said above, that H falls 'less fast' is an indication of the presence of $\Lambda$. But not of whether the expansion is accelerating or not. H falls slower with lambda regardless of whether the universe undergoes a period of decelerated or accelerated expansion.
So while your statement is true, it is not the same statement as timmdeeg's.

If somebody asks 'what does accelerated expansion of the universe mean'? Answering that it means H falls slower with time is not correct. That is the (partial) answer to the effect of having lambda.

 

[Jorrie]

I agree. It is good to point out the potential semantic pitfalls.
The comparison is nevertheless interesting. It also shows one of the other consequences of a zero Lambda: other things remaining equal, a much smaller present cosmological time - around 9.5 Gy for the observed H₀.

 

[nikkkom]

If you are searching for a quantity which is "increasing" when lambda is non-zero, it's simply the velocity of distant galaxies (as measured by redshift, for example).

In Λ=0 case and flat curvature, each individual galaxy slows down (as observed by us). As matter density falls, this "slowing down" also decreases, asymptotically to zero.

In Λ>0 case and flat curvature, galaxies also do slow down *in the early Universe*, but then "slowing down" goes to zero, and then velocity of any given galaxy starts to *increase*.

For laymen, this explanation would be easier than explaining what Huble constant is, why it is not really a constant, and so on.

 

[PeterDonis]

If you are searching for a quantity which is "increasing" when lambda is non-zero, it's simply the velocity of distant galaxies (as measured by redshift, for example).

But this "velocity" is not a physical velocity; for example, it can be faster than light.

For laymen, this explanation would be easier than explaining what Huble constant is, why it is not really a constant, and so on.

I'm not so sure. We get just as many (if not more) questions about how distant galaxies can be moving away from us faster than light as we do about why the Hubble "constant" isn't really constant.

 

[Jorrie]

In Marcus' post #13 above, we have converged on the following:

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

Do you guys have a problem with these statements?

 

[PeterDonis]

Do you guys have a problem with these statements?

Not for the purpose intended, which is to reduce people talking past each other by picking well-defined terms.

But you're still going to have to explain to some people why it's ok that the "recession speed" can be faster than the speed of light. There's no terminology we can pick that will forestall all questions, because the physics itself is counterintuitive in some respects.

 

[marcus]

There's no terminology we can pick that will forestall all questions, because the physics itself is counterintuitive in some respects.

Very true! We can only make small gradual improvements. No terminology is perfect. There will always be questions (Maybe that is part of the fun : ^)

We get just as many (if not more) questions about how distant galaxies can be moving away from us faster than light as we do about why the Hubble "constant" isn't really constant.

We have to repeatedly make the point that distance growth (in the expansion process) is not like ordinary motion that we are used to---nobody gets anywhere by it, nobody approaches any goal. It's not relative motion such as is governed by the speed limit.
But distance growth can have a well-defined speed---Δx/Δt. We can try telling them them not to say "galaxies moving away from us" because its distance increase, not relative motion. Actually each of us probably has their own way of responding to that question about fasterthanlight recession. Whatever works.

Peter and Jorrie, glad you are both on hand and taking part in this thread. I've had to be off line a fair amount today and it's also a real advantage to have several different voices.

 

[marcus]

The easiest explanation for laymen is not necessarily always the best---it can be a dead end. I tend to think a good explanation of expansion is one that puts a beginner on a path where he or she can eventually get to the Hubble expansion rate H(t) and the Friedman equation---the simple differential equation that governs how H(t) evolves.

The beginner may or may not eventually get to the next level of understanding---but I think we shouldn't block progress with a barrier of non-quantitative verbiage. An explanation shouldn't try to be final---and it shouldn't lead to confusion if the person starts to examine it. Ideally (if possible) should be a bridge to the right next question to ask. Nikkom's post got me thinking along these lines (may not always be practical to attempt).

 

[nikkkom]

But this "velocity" is not a physical velocity; for example, it can be faster than light.

The redshift is physical. And according to it, distant galaxies "accelerate" - if you would repeatedly measure redshift of the very same distant galaxy, it is increasing.

 

[nikkkom]

In Marcus' post #13 above, we have converged on the following:

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

Do you guys have a problem with these statements?

The second statement needs qualification what is meant by "expansion rate".

 

[Pizza]

Hi marcus,
very interesting thread. I'm a beginner too and new here, trying to learn some equations and more. I wondered what H_∞ means?

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 thought, ∞ (infinite) is undefined. Isn't that a problem?

(sorry for my nickname, couldn't find something else, hope you get hungry ;^)

 

[PeterDonis]

The second statement needs qualification what is meant by "expansion rate".

Which is explained in other posts in this thread.

 

[PeterDonis]

I thought, ∞ (infinite) is undefined. Isn't that a problem?

$\infty$ is just being used as a handy label in this case; the physical meaning is "the value that the Hubble constant approaches in the very far future".

 

[timmdeeg]

I'd rather say that accelerated expansion means the universe is approaching exponential expansion.
Here's the plot:
View attachment 91581

Thanks. The inflection point of the a-curve separates deceleration and acceleration. In case inflation is included the H-curve also should show an inflection point. But how would you describe its physical meaning? I don't see any.

I think that accelerated expansion means that the second derivative of the scale factor is positive, which includes exponential expansion.

 

[Pizza]

Thank you, Peter!