The Probability of an Infinite Universe

[Athanasius]

For every infinite value, there are an infinity of values less than it that are finite (since infinity minus one equals infinity). So wouldn't a huge but finite universe with very slight, undetectable curvature be infinitely more probable than an infinite flat universe?

If that is so, let's say you have an infinite number of universe-causing quantum fluctuations. Then the probability of an infinite universe is exactly one. But what are the odds that infinite universe would be a low entropy universe like ours? 1/10^10^123, according to Roger Penrose (The Emperor's New Mind, Oxford University Press, 2002, page 445.)

So wouldn't the odds of an infinite universe with low entropy such as ours has be extremely low (1/10^10^123)?

 

[Chalnoth]

For every infinite value, there are an infinity of values less than it that are finite (since infinity minus one equals infinity). So wouldn't a huge but finite universe with very slight, undetectable curvature be infinitely more probable than an infinite flat universe?

If that is so, let's say you have an infinite number of universe-causing quantum fluctuations. Then the probability of an infinite universe is exactly one. But what are the odds that infinite universe would be a low entropy universe like ours? 1/10^10^123, according to Roger Penrose (The Emperor's New Mind, Oxford University Press, 2002, page 445.)

So wouldn't the odds of an infinite universe with low entropy such as ours has be extremely low (1/10^10^123)?

Yes, I think it's generally expected that some tiny amount of curvature exists, even if we can't detect it. Simple quantum fluctuations during inflation would cause some net curvature within our observable universe, for example.

So when you hear a cosmologist say, "The universe is flat," you should hear, "The observable universe is flat enough that we can't detect any overall curvature, yet."

 

[PeterDonis]

For every infinite value, there are an infinity of values less than it that are finite (since infinity minus one equals infinity). So wouldn't a huge but finite universe with very slight, undetectable curvature be infinitely more probable than an infinite flat universe?

This reasoning is not correct. You can always take the power set of an infinite set, and the power set of any set has strictly greater cardinality than the set itself; so for any infinite set, there are an infinity of values *greater* than it, as well as an infinity of values less than it.

wouldn't the odds of an infinite universe with low entropy such as ours has be extremely low (1/10^10^123)?

Yes. AFAIK the only way we currently have of explaining this is anthropic reasoning: *all* of the possible universes "exist" (perhaps as the result of quantum fluctuations in some vast underlying medium--the "landscape" of string theory is something like this as well), and we just happen to be in one that started in a very low entropy state because observers like us can only exist in such a universe.

 

[PeterDonis]

Simple quantum fluctuations during inflation would cause some net curvature within our observable universe, for example.

But the curvature perturbations due to quantum fluctuations during inflation would average out to zero on a large enough scale, wouldn't they?

 

[Chalnoth]

But the curvature perturbations due to quantum fluctuations during inflation would average out to zero on a large enough scale, wouldn't they?

I'm not so sure that's true. Inflation is finite into the past, meaning that either the universe is finite and simply-connected, or there are regions which have never been in causal contact with one another.

In the first case, there's no reason to believe that the quantum fluctuations would average out to be exactly zero, because you're only combining a finite number of random variables. Even if their distribution has zero-mean, they still won't add up to be identically zero.

In the second case it just isn't likely at all that any reasonable averaging is possible, as the lack of causal connection makes it difficult to say anything at all about the overall curvature.

 

[Athanasius]

This reasoning is not correct. You can always take the power set of an infinite set, and the power set of any set has strictly greater cardinality than the set itself; so for any infinite set, there are an infinity of values *greater* than it, as well as an infinity of values less than it.

I see where you are coming from, but because I said "for every infinite value," I was speaking of all higher cardinality infinite values, too. This would be an aleph-null infinite anyway, so I don't think the bigger infinities would be relevant. Even if they were, each one of that infinity of greater infinite values is a single value vastly outnumbered by the numbers less than it. And so even if those odds are overcome by an infinity of fluctuations, making the odds of an infinite fluctuation occurring to be one in one, the odds of a low entropy infinite universe like ours still would work out to one times (1/10^10^123).

 

[PeterDonis]

I see where you are coming from, but because I said "for every infinite value," I was speaking of all higher cardinality infinite values, too.

But you can't do that; whatever the "cardinality of the universe" is, if it's infinite, it corresponds to some particular infinite cardinality, not all of them at once.

This would be an aleph-null infinite anyway

Not if the universe is a continuum, which is what it's currently modeled as in all of our physical theories (except for some speculative versions of quantum gravity).

each one of that infinity of greater infinite values is a single value vastly outnumbered by the numbers less than it.

And by the infinite cardinalities *greater* than it, which was the point of my argument. In fact, the number of infinite cardinalities greater than any infinite cardinality is uncountable, whereas the set of finite values is countable; so the set of infinite cardinalities greater than any given infinite cardinality is *larger* than the set of finite values.

the odds of a low entropy infinite universe like ours still would work out to one times (1/10^10^123).

As I said in my first post in this thread, this is true, and I'm not disputing it. (It's worth noting, though, that AFAIK the odds of a low entropy universe like ours are similarly low whether the universe is spatially finite or infinite.) I'm only disputing the particular argument you are making involving infinite cardinalities.

 

[Athanasius]

But you can't do that; whatever the "cardinality of the universe" is, if it's infinite, it corresponds to some particular infinite cardinality, not all of them at once.

I agree, and that is not what I was trying to communicate. What I meant was whatever form of infinity it may be, that is one value - the greatest possible given the conditions that produced that fluctuation. If so, there are no possible magnitude values under those conditions that would be greater than it, and the values less than it will vastly outnumber it. That would make the odds of an infinite fluctuation occurring very low, unless an infinity of fluctuations occur. Even if an infinity of fluctuations occur , that makes the odds of an infinite fluctuation to be one in one. However, the odds of that infinite universe being a low entropy one like ours would be extremely low.

Are you saying that the magnitude of a quantum fluctuation can be anyone of an infinity of infinite values, even if they all occur in the same environment, and are all measured by the same metric? My reasoning is that if the fluctuations are measured by the same metric, only one kind of infinity is possible. If you believe that more than one kind would be possible in the same field environment when measured by the same metric, please let me know how. In that case, my argument fails.

Not if the universe is a continuum, which is what it's currently modeled as in all of our physical theories (except for some speculative versions of quantum gravity).

OK, agreed.

And by the infinite cardinalities *greater* than it, which was the point of my argument. In fact, the number of infinite cardinalities greater than any infinite cardinality is uncountable, whereas the set of finite values is countable; so the set of infinite cardinalities greater than any given infinite cardinality is *larger* than the set of finite values.

No matter how great a finite value, you can always add one to it. So from that perspective, there is an infinite set of possible finite values. However, I agree that there are infinitites of greater cardinality than that infinite. The question is, would those greater cardinality infinites apply here?

As I said in my first post in this thread, this is true, and I'm not disputing it. (It's worth noting, though, that AFAIK the odds of a low entropy universe like ours are similarly low whether the universe is spatially finite or infinite.) I'm only disputing the particular argument you are making involving infinite cardinalities.

What is the argument you would use to arrive at the same conclusion, given an infinity of fluctuations?

 

[Athanasius]

PeterDonis, it just occurred to me that an infinite fluctuation producing an infinite universe would have to involve an entire infinite field, so only one infinite fluctuation would be possible at a time. Is that how you arrived at the same conclusion?

 

[PeterDonis]

Are you saying that the magnitude of a quantum fluctuation can be anyone of an infinity of infinite values, even if they all occur in the same environment, and are all measured by the same metric?

It depends on what theoretical model you are using; all of this is pure speculation, with no evidence either way that I'm aware of. But all of the theoretical models I'm aware of model the underlying "medium" of quantum fluctuations as a continuum.

My reasoning is that if the fluctuations are measured by the same metric, only one kind of infinity is possible.

The assumptions you are making about things like "probability of a fluctuation" do not even make sense for a continuum; you have to use different mathematical tools.

No matter how great a finite value, you can always add one to it. So from that perspective, there is an infinite set of possible finite values. However, I agree that there are infinitites of greater cardinality than that infinite. The question is, would those greater cardinality infinites apply here?

Yes, since a continuum is one such infinity.

What is the argument you would use to arrive at the same conclusion, given an infinity of fluctuations?

The usual argument looks at how precisely the initial conditions of the universe would have to be specified in order for it to not have either collapsed into a Big Crunch by now, or expanded so much that we could not see any other galaxies at all (i.e., every other galaxy except ours would be beyond our cosmological horizon). That turns out to require a precision in the initial conditions that equates to the low entropy figure you give.

 

[marcus]

...
If that is so, let's say you have an infinite number of universe-causing quantum fluctuations...

Are you assuming that our universe was caused by a quantum fluctuation? I follow the quantum cosmology literature and in what I see by way of new research papers there's a lot of bounce cosmology these days, and very little talk of "quantum fluctuation" starting expansion.
There's "matter bounce", and modified gravity f(M) or f(S) bounce, and loop gravity bounce, and several other versions---also now and then a REVIEW paper covering recent developments in the different bounce approaches.

So it seems to me that it may be too facile an assumption to assume that our universe's expansion was triggered by some unspecified "quantum fluctuation". I'd say that has a kind of Nineties air to it, or even 1980s. Not all contemporary quantum cosmology researchers find the idea any longer especially interesting.

So wouldn't the odds of an infinite universe with low entropy such as ours has be extremely low (1/10^10^123)?

It's been pointed out that just because the most common MODEL cosmologists work with is INFINITE one does not assume the U actually is spatially infinite. It simply means that infinite gives a simple excellent fit to the data. The U is spatially big enough so there is no detectable curvature. So flat infinite is easy to use and fits. But there is no EVIDENCE that it is actually infinite, rather than merely very large. So let me get rid of the word "infinite" and take another look:

So wouldn't the odds of a universe [starting] with low entropy such as ours [had] be extremely low (1/10^10^123)?

That's an intriguing question! Our U apparently began expansion with very low entropy. How did that happen? Wouldn't it be unlikely?

According to these people (Aurelian Barrau and Linda Linsefors) low entropy was inevitable. This is a new paper, we'll have to see how it is received.
http://arxiv.org/abs/1406.3706
Our Universe from the cosmological constant
Aurelien Barrau, Linda Linsefors
(Submitted on 14 Jun 2014)
In this article, we consider a bouncing Universe, as described for example by Loop Quantum Cosmology. If the current acceleration is due to a true cosmological constant, this constant is naturally conserved through the bounce and the Universe should also be in a (contracting) de Sitter phase in the remote past. We investigate here the possibility that the de Sitter temperature in the contracting branch fills the Universe with radiation and causes the bounce and the subsequent inflation and reheating. We also consider the possibility that this gives rise to a cyclic model of the Universe and suggest some possible tests.
5 pages

You know the standard LambdaCDM cosmic model assumes a positive cosmological constant Lambda. What they say is if that is true (which it seems very likely to be) then putting that together with other widely accepted things, and the cosmological bounce assumption (that quantum corrections akin to the Heisenberg uncertainty principle make gravity repel at extreme density and resist further compression), you get that a low entropy beginning is NOT improbable.
On the contrary it eventually must occur---is in fact inevitable.

 

[Chalnoth]

Are you assuming that our universe was caused by a quantum fluctuation? I follow the quantum cosmology literature and in what I see by way of new research papers there's a lot of bounce cosmology these days, and very little talk of "quantum fluctuation" starting expansion.

Perhaps that's because quantum cosmology is synonymous with bounce cosmology, and generally isn't a term used to refer to quantum fluctuations birthing new regions?

 

[marcus]

Perhaps that's because quantum cosmology is synonymous with bounce cosmology, and generally isn't a term used to refer to quantum fluctuations birthing new regions?

There's been an interesting trend. For example during the five years 1995-1999 there were a much larger fraction of quantum fluctuation based cosmology papers. This is in what the Stanford-SLAC Inspire data base classifies as quantum cosmology

"quantum cosmology" 1995-1999, Inspire search:
http://inspirehep.net/search?ln=en&...search=Search&sf=&so=d&rm=citation&rg=25&sc=0 (395 found as of 28 June 2014)

"quantum cosmology" and not "loop" 1995-1999, Inspire search:
http://inspirehep.net/search?ln=en&...search=Search&sf=&so=d&rm=citation&rg=25&sc=0 (368 as of 28 June 2014)

As you can see there are almost no "Loop" papers i.e. based on loop quantum gravity (which yields a cosmological bounce).
But in a comparable period, 2009-present, the same category (which of course still contains "quantum fluctuation" type papers, as before) now contains a large fraction of Loop papers. That is the primary place you get bounce cosmologies although I have been seeing a growing number of alternative bounce ones lately.

"quantum cosmology" since 2009, Inspire search:
http://inspirehep.net/search?ln=en&...search=Search&sf=&so=d&rm=citation&rg=25&sc=0 (703 found as of 28 June 2014)

"quantum cosmology" and not "loop" since 2009, Inspire search:
http://inspirehep.net/search?ln=en&...search=Search&sf=&so=d&rm=citation&rg=25&sc=0 (347 as of 28 June)

It is interesting to look back at the papers from the earlier 5 year period. Among those leading in the number of citations one sees authors like Andrei Linde, Alexander Vilenkin, Raphael Bousso, Gabriele Veneziano, Stephen Hawking… Many associated with some kind of universe from "nothing" except some kind of "fluctuation".
Now if you do the same search, and look for the highly cited papers, it's much more authors who are interested in the expansion starting from a rebound from an earlier collapsing phase.
There has definitely been a shift in researcher interest and research emphasis within the area of research that Inspire data base categorizes as "quantum cosmology".

It is almost, as you suggest, as if the very MEANING of the name of this professional research category has changed! Thanks for calling attention to the trend---it has been a remarkable shift in focus!

 

[Haelfix]

GR-QC is a place for lqg + the older approaches like the Wheeler-DeWitt canonical superspace formalism. As the inspire data suggests, the amount of papers on the old approach is roughly constant and there are now in addition about 100 or so papers a year that are decidedly loopy. This is more a bookkeeping change than anything else.

In the old days, LQG was often grouped into hep-th, but that was changed about 10 years ago or so and its rather rare nowdays to actually see a crossover.

In any event, approaches where quantum fluctuations start cosmologies is more or less the accepted method and is essential to how inflation starts. Inflation papers incidentally are often somewhat confusingly posted in hep-th. That is done on purpose to separate methods that rely on semiclassical treatments vs those that rely on much more speculative and ill understood theories of quantum gravity.

 

[Athanasius]

In any event, approaches where quantum fluctuations start cosmologies is more or less the accepted method and is essential to how inflation starts.

Wouldn't a quantum fluctuation also be needed to initially get a quantum bounce-crunch cycle started?

 

[Athanasius]

That's an intriguing question! Our U apparently began expansion with very low entropy. How did that happen? Wouldn't it be unlikely?

According to these people (Aurelian Barrau and Linda Linsefors) low entropy was inevitable. This is a new paper, we'll have to see how it is received.
http://arxiv.org/abs/1406.3706
Our Universe from the cosmological constant
Aurelien Barrau, Linda Linsefors
(Submitted on 14 Jun 2014)
In this article, we consider a bouncing Universe, as described for example by Loop Quantum Cosmology. If the current acceleration is due to a true cosmological constant, this constant is naturally conserved through the bounce and the Universe should also be in a (contracting) de Sitter phase in the remote past. We investigate here the possibility that the de Sitter temperature in the contracting branch fills the Universe with radiation and causes the bounce and the subsequent inflation and reheating. We also consider the possibility that this gives rise to a cyclic model of the Universe and suggest some possible tests.
5 pages

You know the standard LambdaCDM cosmic model assumes a positive cosmological constant Lambda. What they say is if that is true (which it seems very likely to be) then putting that together with other widely accepted things, and the cosmological bounce assumption (that quantum corrections akin to the Heisenberg uncertainty principle make gravity repel at extreme density and resist further compression), you get that a low entropy beginning is NOT improbable.
On the contrary it eventually must occur---is in fact inevitable.

Last night I listened to an interview of some LQG bounce cosmologists who stated that the thermodynamics could be "reset" during the quantum bridge phase. That would certainly solve the thermodynamics problem with the old oscillating cosmology thought to exist under GR. Is this derived entirely from the maths, or is it simply an assumption?

 

[marcus]

Last night I listened to an interview of some LQG bounce cosmologists who stated that the thermodynamics could be "reset" during the quantum bridge phase. That would certainly solve the thermodynamics problem with the old oscillating cosmology thought to exist under GR. Is this derived entirely from the maths, or is it simply an assumption?

People have various arguments, derived from the math, to the effect that, coming out of the bounce, the entropy is low. Or alternatively that it is low in the observable region after a bounce followed by inflation. Low entropy is supported by reasons in bounce cosmology, and is not simply an assumption.

On the other hand I have not seen any paper that purports to give the one "official" explanation of why low entropy is not a problem for bounce cosmology. Maybe someone else has. I may simply have missed it or not read widely enough.

This bothers me. I would like to hear one official explanation that finally settles the question for me, rather than several lines of argument. If you can remember who it was you heard being interviewed and would be kind enough to send me their names (by private message if you prefer that) I would consider writing them.

 

[Athanasius]

People have various arguments, derived from the math, to the effect that, coming out of the bounce, the entropy is low. Or alternatively that it is low in the observable region after a bounce followed by inflation. Low entropy is supported by reasons in bounce cosmology, and is not simply an assumption.

On the other hand I have not seen any paper that purports to give the one "official" explanation of why low entropy is not a problem for bounce cosmology. Maybe someone else has. I may simply have missed it or not read widely enough.

This bothers me. I would like to hear one official explanation that finally settles the question for me, rather than several lines of argument. If you can remember who it was you heard being interviewed and would be kind enough to send me their names (by private message if you prefer that) I would consider writing them.

Hi Marcus,

The Interviewees were Abhay Ashtekar (Penn State University), and Ivan Aguillo (Cambridge University).

The Interview is here: .

You will find the question and comment at 17:36. Note that Dr. Aguillo uses the word "may" several times, so he does not seem to be speaking with certainty.

By the way, I noted that the producers of this interview credited you for your advice on the forum!

 

[marcus]

Hi Marcus,

The Interviewees were Abhay Ashtekar (Penn State University), and Ivan Aguillo (Cambridge University).

The Interview is here: .

You will find the question and comment at 17:36. Note that Dr. Aguillo uses the word "may" several times, so he does not seem to be speaking with certainty.

By the way, I noted that the producers of this interview credited you for your advice on the forum!

Hi Athanasius,
Thanks for posting that link! I had watched the video some months earlier and I found it even more interesting and informative the second time. It's really well done! Plus they were answering questions that had become sharply delineated for me in the meantime.
Agullo and Ashteker do a good job on the entropy question. I think this may turn out to be the "official" or best explanation I was looking for.

The repellent effect of gravity, at extreme density, effective at a short range of 10 Planck lengths, irons out the region, during the bounce, which will become the observable region after expansion.
Also Ashtekar gave an interesting description of how the entropy associated with the horizon is annihilated when the horizon itself disappears---this is a concrete way of accounting for the way entropy depends on the perspective of the observer and there is a radical change between the observer looking forward towards the collapse and the observer in the expanding phase looking back at the start of expansion.

 

[Athanasius]

This bothers me. I would like to hear one official explanation that finally settles the question for me, rather than several lines of argument. If you can remember who it was you heard being interviewed and would be kind enough to send me their names (by private message if you prefer that) I would consider writing them.

If entropy is a state of disorder, and low entropy is a state of order, then we have to ask ourselves, "What specific order or information state are we talking about?" There are many conceivable low entropy universes that would not produce life. In fact, most of them would not. In our universe, our particular low entropy state includes the physical laws of our universe and their fine-tuning parameters. So a simple "reset" of entropy as Aguillo was speaking of would hardly explain the odds of our particular low entropy state of order arising. We would need a lot of bounces. But since our bounce seems to be the last one, since Omega equals one, there can't be an infinity of them. Since time is directional, like a ray, there could not be an infinity past of bounces, either.

We also have to talk about a reduction in expansion energy after each bounce. If ours is the last bounce, there had to be a reduction in it from the previous bounce to this one. Otherwise this bounce could not have occurred. If you can get an answer to these questions when you write, I would love to hear them!