# A layman's guide to the accelerating expansion of space

1. Jun 19, 2015

### valenumr

I am writing this thread because I have too often had to explain this and I basically wanted to have a resource available to direct folks to, and, frankly, the Wikipedia topics on this are pretty vague on the explanation. Any questions, comments, or corrections are greatly appreciated. So, here we go...

Often, the argument is made that the expansion of the universe must be decelerating which is incorrect. The first argument is that more distant galaxies are receding more quickly than nearer galaxies, while the second is that more distant galaxies are older than nearer galaxies. While both statements are scientifically accepted, the conclusion that galaxies were recessing faster in the past and therefore the expansion of the universe is decelerating is completely incorrect.

Another incorrect assumption often made is that this expansion, which leads to the recession of distant galaxies, is equated with proper relative motion, and that the galaxies have a relative velocity component due to this recession. This also is not correct. It is more appropriate to consider that galaxies remain fixed, with the empty space between expanding over time. Certainly, there can be true relative motion among galaxies, and it would be quite unphysical if there were not.

Before you can convince yourself or anyone else that the expansion of the universe is, in fact, accelerating, you first must understand how the universe would appear to a fixed observer if the expansion rate were constant. I will attempt to explain this as clearly as possible.

The above diagram illustrates how this constant rate expansion would occur in a single spatial dimension for seven uniformly spaced galaxies over two time intervals, with time progressing from bottom to top. The green dots represent a fixed observer galaxy, while the blue, purple, and red dots represent progressively more distant galaxies. So initially the blue galaxies are 1 unit away, the purple galaxies are 2 units away, and the red galaxies are 3 units away.

After one time interval, the configuration of the galaxies is now such that the blue galaxies are 2 units away, the purple galaxies are 4 units away, and the red galaxies are 6 units away, so space has expanded uniformly to twice it's original size. But notice that the change in distance from the fixed, green observer is progressive: the blue galaxies have recessed by 1 unit, the purple by 2, and the red by 3! Now it should be clear that this change in distance is directly proportional to the observed distance, which in this case, the ratio is 1/2.

As mentioned above, this change of distance is entirely different from physical motion, and it should be clear from the diagram that it is happening very uniformly at any instant in time. Space is not expanding faster anywhere regardless of distance. The fact that distant galaxies appear to be receding faster is entirely an observer effect due to the fact that there is more space between an observer and a more distant galaxy.

Finally, after a second interval of time, we can see in the diagram that the space between all galaxies has again doubled uniformly, so this is how constant expansion would appear to our fixed observer over time. The blue galaxy is now 4 units away, the purple 8, and the red 12. But again notice that the change in distance is progressive, directly proportional to the observed distance, and that the ratio is still 1/2, which demonstrates that, at least in this scenario, the rate of expansion is constant.

So now that we understand the expansion of the universe more clearly, how can we determine if the rate of expansion is decreasing, constant, or, increasing? Well, we can take the top line of the diagram and either slightly stretch it (accelerating expansion) or slightly compress it (decelerating expansion). Now I'll leave it to you to deduce the observational effects of either case, but remember while doing so, that we can only observe the present (galaxies in the top line of the diagram). Hint: red-shift is cumulative over time.

As a final note, scientists used this same process to search for evidence that this expansion was slowing down due to gravitational effects. They made specific predictions so they would know where to search, and their observations actually demonstrated the opposite of what was expected. Granted there is still a lot of debate over the accuracy of the measurements and variety of other factors involved in these experiments, but the tide seems to be in favor of accelerating expansion. Sorry Dark Energy Haters.

2. Jun 19, 2015

### phinds

I think putting it that way makes the case MUCH less strongly than it should be made. It's not a "tide in favor of" it's an "overwhelming tidal wave" of evidence. Accuracy can be debated, but that's quantitative detail. The qualitative results are, I believe, indisputable at this point.

3. Jun 19, 2015

### valenumr

Just trying to keep it objective. I would say there is at least a minority effort to refute the claim, and there is an ongoing effort to strengthen its support as well. I do agree that general consensus accepts accelerating expansion as indisputable given our present understanding.

4. Jun 19, 2015

### Bandersnatch

It's not completely incorrect. For a good part of the history of the universe the expansion was indeed decelerating. Look up evolution of the scale factor.

5. Jun 19, 2015

### marcus

It's a good suggestion, to look up the evolution of the scale factor. When you solve the equation of the standard cosmic model (which basically everybody uses) you get a scale factor curve that looks like the righthand side of this plot

The time scale is 1 = 17.3 billion years.
The present age is 0.8 = 13.8 billion years.
You can see that the deceleration lasted until around age 0.44.
It is very subtle but after 0.45 it the size of distances begins to "accelerate" in the sense that the scale factor slope increases.

You can calculate this curve, the universe's distance growth curve, yourself using google calculator. Anyone can.
It is simply the hypersine "sinh" raised to the 2/3 power

the usual notation for the scale factor is a(t) so the equation is $$a(t) = \sinh^{2/3}(\frac{3}{2}t)$$ for convenience it is customary to divide this by its present value to "normalize" it so that it equals ONE at present age. So you divide that by 1.3 or more precisely 1.3115. and then a(now) = 1
What I have plotted here is $$a(t) = \frac{\sinh^{2/3}(\frac{3}{2}t)}{1.3115}$$ You can see that at the present age, t = 0.8, you have a(.8) = 1

6. Jun 19, 2015

### eltodesukane

I don't understand.
"We can only observe the present", but looking far away we see into the past.
So if we are the green dot on the top line, we can see the blue dot on the 2nd line, the purple dot on the 3rd line, and the red dot on the missing 4th line.
In both case we observe the same thing, no acceleration or deceleration.

7. Jun 19, 2015

### valenumr

Fair enough, but that is out of context. Perhaps I should have worded that better. My point is, the conclusion that the expansion is decelerating now based on the two stated premises is completely incorrect.

My point is really to illustrate how constant expansion leads to Hubble's law without getting into too much math.

8. Jun 19, 2015

### valenumr

Both statements there are true in their own way. We can only observe the light arriving at the present from distant sources, but we know it was emitted in the past. As a gross simplification, think of the top line as what we observe at present, and the two bottom lines as two different alternate histories as to how the universe evolved to the "present", with each history leading two different observations.

This is actually one aspect of cosmology that blows my mind though. When we look out to the edge of the observable universe in opposite directions, we can say that those two regions are very far from each other. But when light from those regions was emitted, they were much closer together. Looking outward into space is like looking inward into the past.

9. Jun 19, 2015

### phinds

Well, actually, it's not "like" looking into the past, it IS looking into the past.

10. Jun 19, 2015

### marcus

My comment would be that the detection of a cosmological constant was based, as we know, on redshift-distance data on standard candle supernovae. For a given redshift (1+z is the wavelength enlargement factor, which is also the distance enlargement while the light was in transit) the standard supernovae proved to be DIMMER.
SO MORE DISTANT than was expected, given the amount of expansion that had occurred while the light was traveling.

turn that around and it says for standard sources observed at a given distance, there has been LESS DISTANCE EXPANSION RECENTLY while the light was coming to us. Less redshift, for a given distance. than would have been expected with zero cosmo. const.

Because more distance (more dimness) for a given redshift than you'd get with zero cosmological constant.

that seems contradictory, paradoxical. It does not seem to compat in a simple way with our verbal description and our naive mental image of "acceleration"
====================

I don't know of a simple verbal explanation that makes this intuitive. Maybe someone else here can explain. I think you have to address how we actually SEE the positive Lambda in the plot that compares redshift to luminosity (dimness----i.e. distance).

you have to tackle the fact of what is observed

after all, we don't actually SEE things "accelerating". we can't even watch things "move away" because distances expand so slowly (in human terms)
currently 1/144 of one percent in a million years.

and redshift is not a DOPPLER effect of some "speed" at a particular moment like when the light was emitted. It is so important to realize that!
It is not related to any particular "speed" at any particular moment. Cosmological redshift measures the entire cumulative effect of all the expansion over the whole time the light was in transit. This is not related in any simple way to the recession speed at any given moment.
====================

So my feeling is that it is not trivial to try to explain the effect on observations of a positive cosmo. const. Lambda.

Probably the first fact to grasp is what the Hubble expansion rate H(t) represents and to realize that it has been DECLINING since very early times and according to standard cosmology it is expected to CONTINUE declining.

this is the key fact, how H(t) is declining. People often get the idea from all the "expansion" and "dark energy" talk that it must be increasing, but this is a confusion.

11. Jun 19, 2015

### eltodesukane

Yes, and if we could look further to the time of the Big Bang (which we can not cause the universe becomes opaque to light at a certain point), those two regions would actually be at the same location. (which is why I don't understand why inflation is said to be necessary to causally link these two regions).

12. Jun 19, 2015

### valenumr

Is it possible that there is a greater effect of gravitation nearby, or has this already been accounted for or otherwise considered and rejected?

13. Jun 19, 2015

### valenumr

For the first part of your statement, that would have to assume the universe is finite in extent, which is unknown.

Regarding inflation, think of it this way. We are barely causally connected to these two regions at present. But if you think about it, without inflation, they never have been, but perhaps would be sometime in the future.

14. Jun 19, 2015

### valenumr

Yes, I think this is a common misconception, which I tried to articulate:

15. Jun 19, 2015

### marcus

All this stuff works out with standard 1915 Einstein GR gravity when you put it in equation terms. Nothing exotic like modified gravity. The seeming unintuitive contradictions come when you try to explain in words. Or when you picture distance expansion as things "moving". Like you said, it is not things moving thru space
good point.

16. Jun 19, 2015

### eltodesukane

No, but we still can not see beyond that Big Bang distance because the light beyond it had no time to reach us. So the universe beyond that is inaccessible to us, whether it is finite or infinite.

17. Jun 19, 2015

### valenumr

Actually, we can see the CMB, which is essentially the radiation from the surface of last scattering. There is nothing visible before this, because according to theory, the universe was opaque.

18. Jun 19, 2015

### marcus

How the redshift-distance observation data turns into positive Lambda is still a bit unintuitive at least I think so. Maybe someone else can help intuitize it.
As I see it, at the moment, the key thing is a CHANGE OF VARIABLE OF INTEGRATION (just a Freshman college calculus thing, but still a mental obstacle at the most basic level). Distance now (i.e.dimness) is after all an INTEGRAL of the expansion while the light was traveling.

You know the scale factor a(t) that I graphed earlier.

$$D_{now} = \int_{then}^{now} \frac{cdt}{a(t)}$$

this is the integral you first have to understand and then do the change of variable on.

to understand is easy. a(t) is normalized to be 1 at present, so 1/a(t) is the factor by which the little bit of distance cdt is expanded by the time it gets to us.
then is when emitted
so you add up all those little steps taken at all different times, and each is multiplied by how much it got expanded during transit.

19. Jun 19, 2015

### valenumr

I think I understand... I guess on the scale that the effects of expansion can be observed, gravity can be essentially ignored due to homogeneity. Would that be accurate?

20. Jun 19, 2015

### marcus

I think so.
I want to tell you what I think is the key step in understanding the dependence of the redshift-distance curve on the positive cosmo curvature constant Lambda.
Are you all right with:

$$D_{now} = \int_{then}^{now} \frac{cdt}{a(t)}$$?

A nice thing is you can define a variable S = 1/a = z+1 called the stretch factor, the factor by which wavelengths and distances got stretched between then and now, while the light was in transit.
And then you can change variable:
$$D_{now} = \int_1^S \frac{ds}{H(s)}$$

where H(s) is the Hubble expansion rate, which we can write as a function of s.

Last edited: Jun 19, 2015
21. Jun 19, 2015

### valenumr

Absolutely. If a(t) is a constant, then everything would look the same regardless of distance (integrating a constant). This doesn't match observation.

22. Jun 19, 2015

### valenumr

I think part of the challenge is what we mean by "now" and "then". I mean, in a sense there is a universal now everywhere, but we only can see "here and now", and everything else is "then and there". But if we could magically teleport 20 billion light years away to "there and now" everything would still pretty much look the same.

23. Jun 19, 2015

### marcus

the secret is that positive Lambda makes H(t) decline more slowly with increasing time, and since S increases as you go BACK in time a positive Lambda makes 1/H(s) decline more slowly with increasing s.
So with positive Lambda you get a bigger integral ds, a bigger distance, a dimmer supernova. It fits the data better!
I don't have time to explain better right now. Here are some H(s) curves for different cosmo const. Lambda. the black one is right, it gives bigger distances (areas under the curve) than the orange one, which is essentially for zero LAMBDA.

24. Jun 19, 2015

### marcus

25. Jun 19, 2015

### marcus

If you want to know what Lambda is, really, in terms of actual observations and fitting model parameters to data,
it is given by the longterm value of H(t)
As Einstein introduced Lambda in the GR equation in 1917 it is a RECIPROCAL AREA (a curvature constant in the GR equation)
So if you multiply by c2 you get reciprocal time-squared. the square of a growth rate
Λc2 is a Time-2 quantity, per second2 or per year2,
whatever time unit you are using.

As t →∞, H(t) → H, the longterm expansion rate and
H2 = Λc2/3
So it's simple. If Λ is zero (the way most people thought before 1998) then H declines to zero.
If Λ is some positive curvature constant, then H does not decline all the way to zero.

It levels out at a positive rate H

If Λ is positive the H(t) curve is not so steep

and if you run it backwards in time it is not so steep increasing, so 1/H is not so steep declining.
And increasing S is going backwards in time. So you can see in the plot of 5 sample curves.
The black one (which is right) does not decline so steeply with increasing S.

So integrating, taking area under it, gives larger distances, and that is what they found in 1998.
They found dimmer supernovas than they would have expected using the ORANGE curve, which is for a tiny Lambda almost zero.

But the black curve is just right. If you go to the green one above it, labeled 16.3, you get distances that are too big. the green curve declines TOO slowly.

Last edited: Jun 19, 2015