How exactly inflation solves the Horizon Problem

[Arman777]

I am reading an article, Inflation and CMBR by Charles H. Lineweaver.

He explains the inflation period as the shrinking of the event horizon in the comoving coordinate system. Which it makes sense since the inflation was a period of $\Lambda$. And In this period of time event horizon shrinks down to $0$ as time goes to infinity (in future). And in the solution part of the horizon problem, the author defines a new surface last scattering due to the inflation.

I am having trouble to understand how can shrinking event horizon can lead to a new surface of the last scattering and solve the horizon problem.

 

[Arman777]

I guess no one knows the answer :/ I ll add the link of the article. https://www.mso.anu.edu.au/~charley/papers/canberra.pdf (Page 5/13) maybe it helps more ?

 

[PeterDonis]

I guess no one knows the answer :/

No, I think it's just that you didn't provide a link to the article before so people had no way of reading the actual reference, and without that we don't know if the issue is you misinterpreting something the reference said, or an actual problem with the reference itself.

I am having trouble to understand how can shrinking event horizon can lead to a new surface of the last scattering and solve the horizon problem.

Lineweaver is not saying that shrinking the event horizon leads to a new surface of last scattering. He is saying that adding a period of inflation in the very early universe leads to a new surface of last scattering (as compared with a model in which there is no inflation). Shrinking the event horizon (in terms of comoving distance) is another side effect of inflation (and of our current dark energy dominated expansion).

That said, I am not sure I agree with the way Lineweaver describes the solution to the horizon problem in this paper, because his description implies that adding inflation to the model can change the conformal time and scale factor at the surface of last scattering (just look at the diagram: the new surface has a scale factor of about 0.8, whereas the original surface has a scale factor of about 0.001, with a corresponding difference in conformal time). But those things are fixed by observations: the ratio of the current CMB temperature to its temperature at last scattering (which is calculated from the properties of atoms), and also the redshift of the CMB (which can be observed directly, though it's difficult), give the ratio of scale factor now to scale factor at last scattering, which also gives the conformal time at last scattering. So those things have to be as they are in the Fig. 1 diagram in the paper (i.e., down at the bottom of the diagram as Lineweaver draws it).

I think the proper description of the effect of inflation, in terms of the conformal time diagram, is that it hugely extends the diagram downward, i.e., it adds a lot more conformal time prior to the surface of last scattering (by greatly increasing the ratio of the scale factor at last scattering to the scale factor at the start of the model, i.e., at the bottom of the diagram). With the diagram extended a lot more downward, there is sufficient room for the light cones of A and B, at the correct surface of last scattering (i.e., scale factor about 0.001), to overlap before the bottom of the diagram is reached. This solves the horizon problem, but without changing any scale factors at or after the surface of last scattering, which, as noted, can't be changed because they are fixed by observations.

 

[Arman777]

Lineweaver is not saying that shrinking the event horizon leads to a new surface of last scattering. He is saying that adding a period of inflation in the very early universe leads to a new surface of last scattering (as compared with a model in which there is no inflation).

Yes well. You are right. I thought that those two are related in core. Since the article explians the inflation as shrinking of the ccomoving coordinates and then asa we know infliaton "solves" the horizon problem.

I think the proper description of the effect of inflation, in terms of the conformal time diagram, is that it hugely extends the diagram downward, i.e., it adds a lot more conformal time prior to the surface of last scattering (by greatly increasing the ratio of the scale factor at last scattering to the scale factor at the start of the model, i.e., at the bottom of the diagram). With the diagram extended a lot more downward, there is sufficient room for the light cones of A and B, at the correct surface of last scattering (i.e., scale factor about 0.001), to overlap before the bottom of the diagram is reached. This solves the horizon problem, but without changing any scale factors at or after the surface of last scattering, which, as noted, can't be changed because they are fixed by observations.

I did not quite understand. They are not overlaping at the a(t)=0.001 ? How can they overlap if the conformal time moves downward...? Whats the meaning of adding more conformal time..? Are you saying that without the inflation comformal time would be less then 45.6 Gy ?

 

[PeterDonis]

the article explians the inflation as shrinking of the ccomoving coordinates

Where are you getting that from?

They are not overlaping at the a(t)=0.001 ?

Not as the diagram is drawn, because the diagram shows almost no conformal time below that point.

How can they overlap if the conformal time moves downward...?

Imagine extending the diagram downward a lot. The past light cones of A and B at a(t) = 0.001 would then have room to expand a lot (since the past light cones expand as you go downward in the diagram), to the point where they overlapped.

Whats the meaning of adding more conformal time..?

Conformal time is the integral of proper time divided by the scale factor. So how much conformal time there is before a(t) = 0.001 depends on how much the scale factor changed before that. With inflation, the scale factor changes by a much, much larger ratio (if there are at least 60 e-foldings during inflation, that means roughly an extra ratio of $10^{26}$) before a(t) = 0.001; that means much, much more conformal time.

Are you saying that without the inflation comformal time would be less then 45.6 Gy ?

No, I'm saying that with inflation, there should be a lot more conformal time than 45.6 Gy before "now".

 

[Arman777]

Where are you getting that from?

Actually, It should have been "shrinking of the event horizon with respect to the comoving coordinates" / Page 5

So you are saying that without inflation, the conformal time would be less and there would be not enough "time" to points to reach a heat equilibrium. With inflation, the conformal time would be higher and there's enough "time" to reach thermal equilibrium ? And the reason is that the scale factor increases a lot with inflation.

If above explanation is true then why the author defined a new scattering surface ? You said that its wrong description due to the redshift and that's ofc true. But then why he defines such thing... I don't get it.

 

[PeterDonis]

It should have been "shrinking of the event horizon with respect to the comoving coordinates"

Ah, ok. Then you are misinterpreting what the article says: the "shrinking of the event horizon" in comoving coordinates shown in the diagram is not due to inflation; it's due to dark energy, i.e., the acceleration of the expansion of the universe that is happening now, not the inflation that happened in the very early universe.

It's true that inflation also caused a shrinking of the (then) event horizon in comoving coordinates; but that's not shown in the diagram (because the diagram compresses the period of inflation into such a short increment of "height" that the effects of inflation on the event horizon are not visible). Also, that "shrinking" of the event horizon is not what solves the horizon problem, as I said before.

So you are saying that without inflation, the conformal time would be less and there would be not enough "time" to points to reach a heat equilibrium. With inflation, the conformal time would be higher and there's enough "time" to reach thermal equilibrium ?

Basically, yes; but also, the conformal time that's shown in the diagrams is the conformal time without inflation--even though some of the diagrams claim to show what happens with inflation.

And the reason is that the scale factor increases a lot with inflation.

Increases a lot in a very short interval of proper time, yes. In other words, during inflation you have a very short period of proper time that corresponds to a very large period of conformal time. But the diagrams don't show that.

If above explanation is true then why the author defined a new scattering surface ?

I don't know. No explanation for that claim is given in the paper; it's just asserted.

 

[Arman777]

h, ok. Then you are misinterpreting what the article says: the "shrinking of the event horizon" in comoving coordinates shown in the diagram is not due to inflation; it's due to dark energy,

Yes, of course, I know that. I am not misinterpreting.

I was just trying to say that when the $\Lambda$ becomes dominant the universe goes into the inflationary period. Of course, the graphs show the future. But this process is the same in the inflationary period, since the universe was dark energy dominated in that period of time.

Basically, yes; but also, the conformal time that's shown in the diagrams is the conformal time without inflation--even though some of the diagrams claim to show what happens with inflation.

Increases a lot in a very short interval of proper time, yes. In other words, during inflation, you have a very short period of proper time that corresponds to a very large period of conformal time. But the diagrams don't show that.

I don't know. No explanation for that claim is given in the paper; it's just asserted.

Its interesting...I understand it now. Thanks a lot.

 

[Arman777]

I understand it verbally but how can show that increasing the scale factor increases the conformal time, by using math

Conformal time is,
$$\eta= \int dt/a(t)$$ so we have

$$d\eta= dt/a(t)$$
$$a(t)=dt/d\eta$$

In other words, during inflation you have a very short period of proper time that corresponds to a very large period of conformal time.

From here I see that $a(t)$ should get smaller...?

 

[PeterDonis]

when the $\Lambda$ becomes dominant the universe goes into the inflationary period

No. The inflaton field--the thing that drove the expansion of the universe during inflation--is not the same as $\Lambda$, the dark energy that is currently driving the universe's accelerated expansion. The dynamics happen to be similar, but that doesn't mean they're the same thing.

how can show that increasing the scale factor increases the conformal time

That's not what I said.

You actually wrote down an equation that helps: $d\eta = dt / a(t)$. Write this as:

$$
\frac{d\eta}{dt} = \frac{1}{a(t)}
$$

In other words, the smaller $a(t)$ is, the more conformal time elapses for a given increment of proper time.

Now consider the interval of proper time from, say, $10^{-35}$ seconds to the proper time of last scattering. This is a fixed interval of proper time. Without inflation, the scale factor would change by some ratio during this interval of proper time; so at the starting proper time, $10^{-35}$ seconds, the scale factor would be smaller than the scale factor at the ending proper time by that ratio--and we know that ending scale factor compared to the scale factor now, that's fixed by observations. Integrating that gives some interval of conformal time.

But with inflation, the scale factor changes by a much larger ratio from $10^{-35}$ seconds proper time to the proper time of last scattering. So integrating that gives a much larger interval of conformal time, because the scale factor at the start is much smaller (by at least 26 orders of magnitude, or 60 e-foldings).

One thing to bear in mind: the scale factors in the diagram are normalized so the scale factor now is 1. But that means that the actual proper distance now is model-dependent: or, to put it another way, how much proper distance a unit of scale factor corresponds to is model-dependent. It's much, much larger with inflation than without inflation. So a better way to describe the effect of inflation might be to say that it hugely increases the "proper scale factor" (the scale factor in actual proper distance units instead of normalized so the scale factor now = 1) for all times after the end of inflation. And that hugely increases the amount of conformal time that is in the past of all times after the end of inflation. But it doesn't change any conformal times after the end of inflation; so it doesn't move the surface of last scattering relative to now (which is what the diagram in the paper is showing, and why that diagram is wrong); instead, it moves the bottom of the diagram way, way down relative to the surface of last scattering.

 

[Arman777]

I guess I understand it .. thanks

 

[PeterDonis]

And that hugely increases the amount of conformal time that is in the past of all times after the end of inflation.

One other thing to bear in mind: when I say "hugely increases" here, I mean comparatively, not necessarily in absolute values. Or, to put it another way, I don't know which diagram actually has the conformal times on the left labeled wrong: Fig. 1 or Fig. 4. I have been talking as if Fig. 1 has the conformal times on the left labeled correctly (as well as the scale factors on the right). But it could be that Fig. 1 has the conformal times on the left labeled wrong, and the scale factors on the right labeled correctly; in which case, Fig. 4 would have the conformal times on the left labeled correctly, and the scale factors on the right labeled wrong. That would have the same effect as I described, just in a different way (by "compressing" the scale of the diagram). But either way, something has to be wrong about the way the diagrams are labeled, unless I'm completely missing something.

 

[PeterDonis]

I have found a reference that describes the solution to the horizon problem the way I have been describing it. See pp. 32-33 here:

http://www.damtp.cam.ac.uk/user/db275/Cosmology/Lectures.pdf

Note how Fig. 2.2, on p. 32, is similar to Fig. 1 in Lineweaver's paper; but Fig. 2.3 is very different from Lineweaver's Fig. 4 (and is basically what I've been describing--adding a lot more conformal time on the bottom of the diagram).

 

[Arman777]

I had that paper..but never read it fully. Now by looking at the picture the idea is more clear to me now.