Universe Is Infinite: Mathematical Proofs

[marcus]

Inflation scenarios work equally well starting from an infinite universe or a finite universe. All inflation scenario needs to do is expand space by a factor of exp(60) or so, which it can do starting from an infinite space or a finite one.

Cosmologists do not know that inflation occurred. It is so speculative that the word "scenario" is used instead of "theory".

Flatness or near-flatness is observed.
Inflation is a speculative scenario proposed to explain some observed facts, like flatness.

Q:When cosmologists say there are theoretical reasons from inflation theory why the universe has to be exactly flat, do they mean exactly flat and truly infinite, or just very close to exactly flat and enormously large (compared to the Hubble scale)?

The observations can't distinguish between infinite (which is certainly simpler!) and finite with slight positive curvature but very very big.

All the observations can do is say Omega = 1.02 +/- 0.02, the current best estimate. If in fact Omega is exactly one (which would be nice and simple) then the U is infinite and flat. If instead Omega is 1.01 or 1.03 then the U is finite.

For now all we can do is measure Omega and narrow down the uncertainty about it as instruments get better.
This is all IMO and AFAIK.



Q:I know that inflation is supposed to blow everything up by a factor of 10^oodles, but that doesn't make anything infinite, just huge.

That is right. the expansion factor that people talk about is exp(60) or "60 e-foldings". That is only 10 to the 26 by my calculator so it is not THAT big. Infinite expands to be infinite and finite expands to be finite.

Q:So does it have to do with eternal inflation being eternal...

Does what have to do with inflation being eternal? As far as I know "eternal inflation" scenarios are really speculative. They are fantasies about how things could be without empirical substance to them. In reply I would say no. Nothing has to do with inflation being eternal. However the scenario of eternal inflation can be used to generate gorgeous computer graphics, as Andrei Linde and others have done.

there is a trick to this however. Lee Smolin has proposed a Darwinian explanation of why the physical constants like 1/137 are so nice. this explanation depends on their having been many generations of
spacetimes with slightly different physical constants which means slightly different physical laws (the constants are the coefficients in the laws of physics so different constants means differnt laws).
the values of the constants have evolved so that they confer "reproductive success" to the universe by encouraging the condensation of gas into stars and stars into black holes, which bud out the backside to form offspring spacetimes. those worlds with the most conducive constants reproduce more. so the ensemble of all worlds consists largely of those with physical constants which encourage black hole reproduction. Smolin uses this to explain why numbers like 1/137 are what they are. They are incidentally also supportive of our kind of life but that is just a tangential byproduct and not part of Smolin's story.

this is an intelligent speculation but it has nothing to do with finitude or infinitude of individual spacetimes, so it is not relevant to your question.

the best introductory cosmology paper I know is Lineweaver "Inflation and the Cosmic Microwave Background"
http://arxiv.org/astro-ph/0305179

and his Figures 4 and 5 are about inflation. On pages 10-13 he discusses it as a way of addressing the flatness problem and the horizon problem

 

[Ontoplankton]

Marcus, thanks for your reply. If I'm understanding you correctly, you're claiming that the only theoretical reason why an inflationary universe should be exactly flat is that this is simpler than an almost exactly flat universe. Wouldn't this mean that inflation theory makes an infinite universe less likely, rather than more likely? If omega has a value close to but not equal to one, wouldn't that be a bigger coincidence in a non-inflationary universe than in an inflationary universe, considering that inflation brings omega to near one from wherever it started out?

Also, are you saying that eternal inflation is (much) more speculative than inflation itself? They do claim in the paper I linked above something like "a generic feature of inflationary models is that inflation goes on forever". They also claim this means that the universe is infinite "seen from the inside" (for one choice of coordinates).

 

[marcus]

Originally posted by Ontoplankton
Marcus, thanks for your reply. If I'm understanding you correctly, you're claiming that the only theoretical reason why an inflationary universe should be exactly flat is that this is simpler than an almost exactly flat universe.

I should not have given that impression. Sorry if I was careless or overly simple! People do give theoretical reasons and plausibility arguments why the universe should be exactly flat (since it is approximately). I tend to discount what seems to me primarily theoretical reasoning and put more faith in observation but that is a personal bias perhaps. Simplicity is not the only reason. People who want to find theoretical reasons can find others. All I believe is that omega is tantalizingly close, if not exactly, equal to one.
I believe the 1.02 +/- 0.02 and even that involves a leap of faith since one has to trust the people using the instruments.

Originally posted by Ontoplankton
... Wouldn't this mean that inflation theory makes an infinite universe less likely, rather than more likely? If omega has a value close to but not equal to one, wouldn't that be a bigger coincidence in a non-inflationary universe than in an inflationary universe, considering that inflation brings omega to near one from wherever it started out?

YES! You are absolutely right about that. Inflation scenarios are attempts to solve among other things "flatness problem" and the "horizon problem". Without imagining inflation it would be extremely difficult to see how the U could be so flat without having started out 100 percent flat. Because ordinary expansion (in a matter or radiation dominated U) exaggerates curvature. While, as you say, dark energy dominated expansion flattens out curvature.

You know all about this. If you didnt already know, that Lineweaver article explains the nuts and bolts of it with a few simple equations
on page 11 and 12. It is such a good pedagogical article! Makes the mechanical details of it simple and understandable.

Originally posted by Ontoplankton

Also, are you saying that eternal inflation is (much) more speculative than inflation itself?

I am not certain what you mean by "eternal inflation". I will post what I have so far and think about it.

 

[marcus]

two notions of eternal inflation

inflation scenarios normally have some kind of "graceful exit"
by which they imagine that inflation stops of its own accord after some fraction of a second during which the 60 e-foldings happen.

Lineweaver discribes this.
--------------

It seems to me that "eternal inflation" could mean two things.

One is Andrei Linde idea (adopted by others such as Lee Smolin, who puts it to Darwinian evolutionary use).
Andrei Linde likes to imagine universes budding off of universes, like cactus. Well that is not the only image of it. I am not a good one to explain this. But perhaps it should be called "endlessly REPEATED" inflations. or endlessly repeated big bangs. Universe after universe comes into existence as an expanding bubble or expanding bud from some prior entity.

this is actually a poetical idea. some ideas are so appealing to the psyche that they develop a power that transcends scientific scrutiny and evaluation. I will not be such a curmudgeon as to say that Linde's idea is bad because it is speculative. I like it because it is excellent and visual poetry.

But (because it contemplates a large number of individual universes, connected by impenetrable bigbang budding events or like oases separated by untraversable deserts) this IS more speculative than simply picturing an inflation scenario lasting some fraction of a second at the beginning of the universe which we witness and belong to.

The customary inflation scenario that people invoke is a brief one-shot episode that is assumed in order to explain two or three things about the world we see.

Endlessly reiterated inflation is a different story.

THE SECOND MEANING of "eternal inflation" might be just inflation that goes on for ever.
that is not a feature of the usual bigbang inflation scenario but
one could give meaning to the idea

One could say that "inflation" by definition is any expansion which accelerates due to the neg pressure of a vacuum energy term or a cosmological constant.

For most of the history of the U, according to the mainstream view, expansion has been slowing down.
Lineweaver has a picture of this. Figure 14 "the size and destiny of the universe"

But starting a few billion years ago the dark energy term began to dominate and expansion began to speed up. You will see in Figure 14 an inflection point where that happened.

Figure 3 shows how a matter dominated U (decelerating expansion) can give way (as matter thins out) to a dark energy dominated U (accel. expan.)

So you can say that as of a couple of billion years ago we are in a new inflationary stage and we see no end to it
(the only "graceful exit" would be if the dark energy term would decrease and people play with models in which the cosmological constant does change---but there is no observational evidence that it changes so that is so-far just model-play)
Thus you could say we are not in an "eternal inflation" era that began a couple of billion ago.

But we can not really say that it is going to be eternal because we don't know if Lambda changes or not. And anyway what is going on now is not what people usually mean by an inflation scenario.

I get the idea this is familiar ground to you---but am spelling things out like this partly to provide you an opportunity to disagree if you have different ideas.

 

[marcus]

Onto, I had some comment on the paper by Garriga and Vilenkin
that you linked to, but withdrew it as too argumentative.

 

[Ontoplankton]

It was some time ago when I read the paper, but IIRC their main claim is that you get an infinite universe with infinitely many Hubble-sized regions within every one (no longer inflating) bubble. Whether there are many such bubbles shouldn't matter. I encountered this through Tegmark's paper "Parallel Universes" (Tegmark seems to agree with Garriga and Vilenkin's claim that inflation generically leads to an infinite universe).

You've mentioned a counterexample, so it can't be true that this sort of eternal inflation happens in all kinds of inflation. But do you think there are many such counterexamples? Would it be right to say that most, but not all, types of inflation that are being seriously considered are like Garriga and Vilenkin descibe (with inflation that never ends, but not necessarily with more than one infinite bubble-universe)? Or would it be more correct to say that it applies only in one specific example of inflation?

Edit: I had already read your previous post; one thing you said was that of the two types of eternal inflation you mentioned, the one they mean is the first one. I suppose that is what I mean by eternal inflation, too, though I don't understand enough to mean anything exact. (This just for general clarification.)

 

[marcus]

Originally posted by Ontoplankton
... I encountered this through Tegmark's paper "Parallel Universes" (Tegmark seems to agree with Garriga and Vilenkin's claim that inflation generically leads to an infinite universe)...

was that the SciAm article on Multiverses? I haven't seen his paper "Parallel Universes" do you have a link or some such handle on it.

Are you comfortable with the idea that in our own universe (the one we see) inflation stopped.

that is, the inflation people usually mean when they talk about the "inflationary scenario" around the time of the big bang, was not eternal but was very brief

 

[Ontoplankton]

Originally posted by marcus
was that the SciAm article on Multiverses? I haven't seen his paper "Parallel Universes" do you have a link or some such handle on it.

http://arxiv.org/abs/astro-ph/0302131

It's a watered-up version of the SciAm article.

Are you comfortable with the idea that in our own universe (the one we see) inflation stopped.

that is, the inflation people usually mean when they talk about the "inflationary scenario" around the time of the big bang, was not eternal but was very brief

Right, but, as I understand it, in "eternal inflation" the rest of the universe outside of our thermalized bubble is supposed to still be inflating, even though our bubble itself is not. IIRC Tegmark/Garriga/Vilenkin claim that in nicer coordinates, such a thermalized bubble is actually infinite instead of always growing.

(By the way, I haven't looked at Lineweaver's tutorial just now, but I think I did back when you first posted it here; I agree it's very good and remember especially liking the explanations of event/particle horizons, etc.)