Ratio between the sizes of the Observed and the Entire Universe

[Cerenkov]

Hello.
I have some questions regarding Alan Guth's book on the Inflationary Universe.
The following, from pages 185 & 186, has piqued my interest. I have reproduced Figure 10.6, above. Please read on. My questions follow. Please also take note that I can only understand these matters at a basic level and would appreciate any answers at that level. Thank you.Implicit in Figure 10.6 is a remarkable prediction of the inflationary theory. Due to the enormous expansion during the inflationary period, the size of the observed universe before inflation was absurdly small. There is no reason, however, to suppose that the size of the entire universe was this small. While inflationary theory allows a wide variety of assumptions concerning the state of the universe before inflation, it seems very plausible that the size of the universe was about equal to the speed of light times its age, or perhaps even larger. If the universe were smaller than this, then it almost certainly would have already collapsed into a crunch.

Applying this reasoning to the sample numbers shown on Figure 10.6, we find that the entire universe is expected to be at least 10²³ times larger than the observed universe!

These numbers are highly uncertain, since they depend sensitively on the duration of the period of inflation, which in turn depends on the decay rate of the false vacuum. Without knowing the correct grand unified theory and the values of all its parameters, the decay rate of the false vacuum cannot be even approximated. Nonetheless, the qualitative behavior shown in Figure 10.6 seems to be typical of all inflationary universe calculations. If the inflationary theory is correct, then the observed universe is only a minute speck in a universe that is many orders of magnitude larger.In a footnote Guth writes that the ratio between the size of the observed universe and the size of the entire universe would persist after the inflationary phase. So today the entire universe would still be at least 10²³ times larger than the observed universe.

Now to my questions.

1.
Given that all of the above dates from 1998, has any subsequent data called it into question? I ask because I’ve read that the WMAP and Planck CMB data did indeed rule out some early inflationary models.2.
Given the vintage of Guth’s statements, is his claim about the qualitative behavior of Figure 10.6 seeming to be typical for all inflationary universe calculations, still valid?3.
Is the ratio cited above still an accepted part of current inflationary theory or has it been superseded or ruled out in some way?4.
Is the persistence of this ratio after the inflationary period still part of current inflationary theory?5.
Guth posits that the early universe would have collapsed into a crunch if it were any smaller than the value he uses in his calculation. Is this line of reasoning still part of current inflationary theory?6.
Given the highly uncertain nature of the values used in Guth’s calculations, in your opinion, is his claim that the entire universe would still be at least 10²³ times larger than the observed universe?

A. Wishful thinking.
B. Informed speculation.
C. Something else (please specify).

Thanks in advance,

Cerenkov.

 

[mfb]

The number depends on your favorite model but it is huge in all of them, and it is just a lower limit - the universe could be much larger (or even of infinite size).

4.
Is the persistence of this ratio after the inflationary period still part of current inflationary theory?

As far as I understand it is a statement about the observable universe today, but certainly a statement about the universe after inflation. As the expansion rate is the same everywhere in space that ratio doesn't change with time at all.

 

[kimbyd]

The number depends on your favorite model but it is huge in all of them, and it is just a lower limit - the universe could be much larger (or even of infinite size).As far as I understand it is a statement about the observable universe today, but certainly a statement about the universe after inflation. As the expansion rate is the same everywhere in space that ratio doesn't change with time at all.

I'm not sure this is strictly true, though it probably depends a bit upon what you mean by "observable universe". If you mean the current proper distance to the particle horizon (the particle horizon is the location from which particles could have reached us in the age of the universe), then that distance increases a bit faster than the rate of expansion. The reason is simple: as time passes, we see objects further away as light has had a longer time to travel to reach us.

There is a hard limit to how far the particle horizon will grow, however. And the current particle horizon is quite close to this limit (roughly 98% by my calculations). So I think what we can say is that at the current time, the ratio between the observable universe and the entire universe is evolving very slowly.

That said, eventually the particle horizon will cease to be a meaningful measure of "observable": by about 2 trillion years nothing beyond the local supercluster will be visible because the light from everything else will have redshifted so much that their wavelengths will be longer than the cosmological horizon. After that time, the ratio of the effectively-observable universe to the entire universe will be shrinking exponentially (because the entire universe grows exponentially while the effectively-observable universe will be close to constant in size).

 

[mfb]

though it probably depends a bit upon what you mean by "observable universe"

As far as I understand the source means the part of the universe observable today. If we look at the size evolution of that part it is identical to the size evolution of the overall universe. If we look at "the observable universe at an earlier point in time" then the ratio changes, of course.

 

[kimbyd]

As far as I understand the source means the part of the universe observable today. If we look at the size evolution of that part it is identical to the size evolution of the overall universe. If we look at "the observable universe at an earlier point in time" then the ratio changes, of course.

I don't think the ratio of observable/total changing over time makes sense unless we consider the part of the universe observable at different points in time.

But yes, I agree that if we're only considering the universe we can observe today and how that universe expands over time, the ratio should be largely constant.

However, this makes me think of a huge caveat: this entire argument, both yours and mine, makes a big assumption: that the cosmological principle applies on very large scales. This doesn't apply, for example, in the context of eternal inflation, where the "total universe" tends to grow exponentially faster than the observable universe.

If inflation isn't eternal, then this might not be a huge issue. Inflation always ends when the density reaches a certain point. And if inflation isn't eternal, all regions, no matter their original density, will tend to have inflation stop within a fraction of a second of one another. So non-eternal inflation should typically obey the cosmological principle on large scales as well.

Still, even in the case of eternal inflation, you can say that you're only restricting to the patch which stopped inflating at about the same time as us when considering the ratio above. The size of the observable universe relative to that patch will obey the constraints we've discussed here. I believe most other alternatives to inflation will either obey the cosmological principle or there will be a concept of a local patch that obeys the cosmological principle.

 

[Bandersnatch]

There is a hard limit to how far the particle horizon will grow, however. And the current particle horizon is quite close to this limit (roughly 98% by my calculations)

I thought this would be the ratio between the comoving extent of the bases of the current and far-future light cones. I.e., how much of comoving space can be seen. So, whatever that comes to, 46/61 ish.
Since that's hardly 98%, what are you thinking of instead?

 

[kimbyd]

I thought this would be the ratio between the comoving extent of the bases of the current and far-future light cones. I.e., how much of comoving space can be seen. So, whatever that comes to, 46/61 ish.
Since that's hardly 98%, what are you thinking of instead?

Oops, you're right. My idea of estimating it was entirely wrong-headed.

 

[Cerenkov]

My thanks for the replies.

Cerenkov.