Size of the Universe: Estimate Conflicts & Layman's Guide

[Hasu]

I've found several conflicting estimates of how many times larger, in volume, the entire universe is than the observable universe.

http://cseligman.com/text/galaxies/expansion.htm" says "hundreds or thousands"

http://www.physics.ucsd.edu/~tmurphy/phys10/universe.pdf" claims "125, 000 times larger"

Wikipedia puts it at 10^23 times both http://en.wikipedia.org/wiki/Alan_Guth" .

And http://www.newscientist.com/article/dn14098?DCMP=ILC-hmts&nsref=news1_head_dn14098" New Scientist article claims 10^100

All of these numbers are wildly different. I'm having trouble finding anything that I would consider a "good source," and am unsure how recently there have been revised estimates of cosmic inflation. I'm very much a layman on this subject and would appreciate being pointed in the right direction.

 

[Chalnoth]

Nobody knows exactly. All we have are lower bounds. But it could easily be vastly, vastly larger, as due to the properties of inflation, we expect it to be vastly larger than the observable region.

 

[SpaceExplorer]

I've found several conflicting estimates of how many times larger, in volume, the entire universe is than the observable universe.

http://cseligman.com/text/galaxies/expansion.htm" says "hundreds or thousands"

http://www.physics.ucsd.edu/~tmurphy/phys10/universe.pdf" claims "125, 000 times larger"

Wikipedia puts it at 10^23 times both http://en.wikipedia.org/wiki/Alan_Guth" .

And http://www.newscientist.com/article/dn14098?DCMP=ILC-hmts&nsref=news1_head_dn14098" New Scientist article claims 10^100

They all may be wrong.Because all we humans can do is to predict since the whole universe is beyond our rich.The problem gets harder as the 'Cosmic inflation' is accelerating i.e the farther the galaxies are apart, the faster they move away from each other.So, at each point of time the universe is getting bigger and bigger which also means that our prediction at one point of time must not agree with our predictions at another point of time.

 

[marcus]

I'm having trouble finding anything that I would consider a "good source," ...

A good source, which gives a conservative lower bound, is the official NASA publication of 5th year WMAP data in January 2009. A number of the world's top cosmologists signed off on this (David Spergel, Joanna Dunkley, Ned Wright, Eiichiro Komatsu... to name just a few). WMAP is a premier project, and they also factored in the other two main bodies of data (galaxy count surveys and supernova data) to give a "WMAP+BAO+SN" result in a separate column of their table.

So it doesn't get much better. Look for Komatsu et al.

The present radius of the observable, if you could freeze expansion so as to measure it in today's distance terms, is 45 billion LY.
Komatsu et al give a lower bound on the "radius of curvature" assuming the entire universe is finite and using the conventional model. Their figure translates to a present-day circumference of 600 billion lightyears (lower bound). Again, this is if you could freeze expansion and measure distance as it is today.

This means that the most distant matter from us, today, is 300 billion lightyears (at least, it is a lower bound).

This is a 95% confidence figure, based on their 95% confidence estimate of the overall curvature.

So a kind of authoritative current conservative lower bound estimate of the ratio would be 300/45. The whole thing is somehow at least 6 or 7 times bigger. But notice that in the finite case of the standard model the whole thing is a hypersphere. The 3D analog of the 2D surface of a sphere. So I am telling you size as a circumference or half circumference.
But we think of the observable part as a solid 3D ball (a small region contained in that hypersphere) so the measure of size we use is the radius of the ball.

It is like having a small circular 2D patch on the curved 2D surface of a balloon. We compare the radius of the patch with the circumference or half-circumference of the balloon.

Komatsu et al is hard to read, but I will give the link. Their figure for this is in Table 2 on page 4.
http://arxiv.org/abs/0803.0547
This is a 2008 preprint, the thing was published in 2009. It is also online at an official NASA WMAP website, along with other reports.

Actually with their numbers the 95% confidence lower bound for the circumference would be closer to 630 billion lightyears, but I preferred to round off so as not to make it look too precise. I'd rather say about 600. We don't know what it is, we only have rough lower bounds, it could of course be much much bigger, or infinite. But the extent of knowledge today is this lower bound and we cannot reasonably say more until we have more precise data.
In fact a new spacecraft Planck intended to get more precise data on this and other features was launched this year and is already making observations, so this lower bound estimate will certainly be improved sometime in the next few years.

 

[Hasu]

This is awesome and exactly what I was looking for. Thanks a lot.

 

[yogi]

The cseligman article cited in post 3 - even though the site refernces events 2008-9paragraph 6 would seem to be outdated - that would also affect conclusions as to the present size of the observable universe - anyone have any comments

 

[apap68]

Try this one http://arxiv.org/abs/hep-th/0610199

 

[Jeff Root]

Hey there! This is my very first post on Physics Forums!

Komatsu et al give a lower bound on the "radius of curvature"
assuming the entire universe is finite and using the conventional
model.
...
But notice that in the finite case of the standard model the
whole thing is a hypersphere. The 3D analog of the 2D surface
of a sphere.

So the figures are all for the maximum curvature / minimum size possible
given the inability to detect any overall curvature?

What if there actually is no overall curvature? What can be said about the
minimum possible size in that case?

-- Jeff, in Minneapolis

 

[Ich]

Then the universe is infinite, except for the possibility that curvature does not necessarily dictate the shape of the universe. It could also have so-called non-trivial topolgy (with holes in it) and still be finite.

 

[Chronos]

I prefer the here and now universe - 13.7 billion light years - the only portion accessible to our instruments. The rest is speculation. Inferences beyond that are coffee table talk, imo. I agree curvature is a very important measurement, but less sure how accurate our measurements are, or what meaning it has. We know it is close to dead flat, suggesting a possibly infinite universe. But, without observational access, is that really meaningful?

 

[Jeff Root]

Then the universe is infinite, except for the possibility that
curvature does not necessarily dictate the shape of the universe.
It could also have so-called non-trivial topolgy (with holes in it)
and still be finite.

I would have no problem with an infinite "multiverse" of some kind,
but it is clear to me that the matter/energy that is participating in
the cosmic expansion must be finite. An infinite amount of matter
spread throughout an infinite volume of space could not all acquire
the same properties in finite time. Clearly, everything participating
in the expansion must have been in causal contact at some time,
or it would not be participating in the expansion.

If "Universe" means "everything participating in the expansion", and
curvature does not necessarily dictate the shape of the Universe,
is the minimum possible size just the size of the visible Universe?

-- Jeff, in Minneapolis

 

[Ich]

it is clear to me that the matter/energy that is participating in
the cosmic expansion must be finite.

So let's hope that it's clear to the universe, too.

is the minimum possible size just the size of the visible Universe?

It could be smaller. There is not a boundary somewhere, structures would just be repeated over and over again. See http://arxiv.org/abs/0802.2236" for example.

 

[marcus]

Hey there! This is my very first post on Physics Forums!


So the figures are all for the maximum curvature / minimum size possible
given the inability to detect any overall curvature?

That's right. There's a 95% confidence interval or error bar that includes zero. Komatsu et al take the top end, max curvature, of that interval and convert it to a "radius of curvature" for the hypersphere case. That is the most common model of finite spatial volume. (There is also a toroidal model analogous to the 2D surface of a donut, but not much discussed.)

They get a radius of curvature of about 100 billion LY. This is a lower bound.

What if there actually is no overall curvature? What can be said about the
minimum possible size in that case?

If average largescale curvature is zero then finite volume is still not ruled out, for a curious reason. The toroidal or 'pac man' case can actually be spatially flat according to the definitions. People don't consider this much or discuss it. They usually assume if curvature is zero then space extends indefinitely (like conventional 3D Euclidean). I'm certainly of that mind. If the errorbar shrinks tenfold and still contains zero, then I would tend to think of the volume as infinite.

But you have to also consider the 'pac man' case. Our space models are not embedded in any higher dimension surround space. So take a 2D analog. A 2D torus surface can be construced just by taking a 2D flat square and 'identifying' opposite edges. So if pac man runs off the east edge of the screen he reappears coming in from the west edges. A cylinder is flat. As curvature is calculated, it has zero curvature. And when you see him run off the north edge and reappear coming from the south edge then you know he lives in a flat 2D torus.

The same construction can be made with a 3D cube by identifying (joining) opposite faces.
So mathematically you can describe a zero-curvature spatial flat finite volume universe.

So then you ask what can be said about minimum size? Well then we have papers by Neil Cornish, David Spergel and Glenn Starkman, where they looked for periodic patterns in the stars. They got some lower bound. They did not find any periodicity so they said, in effect, "if we are in a Pac Man type situation then it must be at least so and so big". Because if it weren't that big we would have noticed some repetitions. I'm probably misrepresenting somewhat here. I don't take these periodic models as seriously as my betters do. Cornish Spergel Starkman are all very major cosmology scholars. You can look up their papers on arxiv.

The short answer to your question is that I can't give you a lower bound size estimate in the flat case, but some people people have cranked one out.

Ich says it very concisely. He refers to the Toroidal model, and other periodic stuff, as having "non-trivial topology". A Pac Man universe does not literally "have holes in it" unless you embed it in some higher dimensional surround, which there is no evidence for.
But there are loops which cannot be shrunk to a point. (Loops going all the way around the donut.) This can be interpreted as "having holes" that prevent the loop from being shrunk.

Then the universe is infinite, except for the possibility that curvature does not necessarily dictate the shape of the universe. It could also have so-called non-trivial topolgy (with holes in it) and still be finite.

...it is clear to me that the matter/energy that is participating in the cosmic expansion must be finite. An infinite amount of matter spread throughout an infinite volume of space could not all acquire the same properties in finite time. Clearly, everything participating in the expansion must have been in causal contact at some time,
or it would not be participating in the expansion.

I don't see how to rebut. There are philosophical problems with infinity, I guess. The fact is observational cosmologists are practical people and they like to work with the spatially flat infinite volume model. It is mathematically simple.
They don't ask "where did it ultimately come from?" They just want to fit the data.

If "Universe" means "everything participating in the expansion", and curvature does not necessarily dictate the shape of the Universe, is the minimum possible size just the size of the visible Universe?

Personally I can't answer with confidence. There are competing Occam Razor demands. Say we want a model that obeys either General Relativity or the simplest most straightforward quantization of it. We don't want some weird Baroque modification of our theory of how geometry behaves.

That means no boundaries, because GR doesn't describe what the behavior would be. If there was no existence beyond some distance...well it just doesn't make sense. The dynamics would have to be cooked up.

The simplest thing is assume uniformity, space and matter more or less the same everywhere.

Then you get a mathematical model with the fewest variables, basically very simple, with a good fit to the data. Occam is happy.

The model cosmologists use is the Friedman model---formulated around 1923 by Alex Friedman. And recently also a quantized version of the Friedman model, which when put on a computer and run, predicts a bounce. But otherwise looks the same after a small interval of time. The Friedman model comes in the space-infinite version or a positive curve space-finite version (depending on a curvature parameter, which we are still trying to determine.) In all cases the Friedman model indicates a size which is beyond what we can actually see. It is the simplest way to get a satisfactory dynamic geometry based on GR, which is well tested at accessible scales like in the solar system.

So what do you do? You either accept that the universe extends beyond what we can see or you need a more complicated, and untested, theory of gravity.

The most distant matter which we can see, according to the standard picture, is now about 45 billion lightyears from us. That means, if we could freeze expansion, and send a flash of light, the flash would take 45 billion years to reach that matter.

That matter is what the CMB is the glow from, and when it emitted the light we now receive as CMB it was over 1000 times closer, estimated 41 million lightyears.

If you want to model some universe that is radically curtailed, no existence either of space or matter beyond 41 million LY, back then...somehow expanding...it seems very hard. How would it work? Why are we at the very center? What is physics like out there at the boundary? What do "they" see, the hypothetical observers out there. To me, it seems to get thorny and unsimple.

 

[Jeff Root]

marcus,

Thanks for your extensive and very understandable reply!
I'm going to ask about just one of the areas you touched on.

I can understand how a cylinder is geometrically "flat". Shapes
drawn on a flat sheet are not distorted when the sheet is rolled.
I'm not sure I can comprehend how 3-D space can be analogously
curved without being curved.

It seems to me that curving a cylinder into a torus would distort
the sheet in much the same way that the sheet would have to
distort to form a sphere. Wouldn't it? How can a torus be said
to be geometrically "flat"?

My problem with comprehending how 3-D space could be rolled
into a cylinder without distortion goes doubly for a 3-D torus.

Assuming (without comprehension) that a torus is geometrically
"flat", still, how could anything, including the Universe, have the
characteristics of a torus without being "curved"? My impression
is that the 2-D surface of a torus is closed in the same way that
the surface of a sphere is closed. A torus looks to me very much
like a sphere, but more complicated.

You can see from my argument about causality that I would not
find plausible any model in which the "stuff" participating in the
cosmic expansion is infinite in quantity or extent. So I'm looking
at models of the Universe in which that stuff can be finite.

The torus is certainly such a model, but it strikes me as absurdly
contrived. An ad hoc construction that a mathematician would
suggest to fit a set of given parameters, rather than the natural
result of a physical circumstance. It seems very artificial. Maybe
I just don't understand what it is.

-- Jeff, in Minneapolis

 

[Chalnoth]

It seems to me that curving a cylinder into a torus would distort
the sheet in much the same way that the sheet would have to
distort to form a sphere. Wouldn't it? How can a torus be said
to be geometrically "flat"?

Well, it's true that a real torus isn't flat everywhere. There's a bit of positive curvature on the outer side, and a bit of negative curvature on the inner. Overall, though, there is no net curvature.

Typically when we talk about a torus being "flat" we're actually not talking about a real torus, though: we're talking about an idealized torus that is actually identically flat everywhere. You can't actually physically make one of these, but it's easy enough to describe mathematically: you just take a flat rectangular surface, and define it such that when an object moves past one edge, it reappears on the opposite edge. Just like in the game Asteroids. This is a toroidal topology (to physically make it so that going off one edge makes an object appear on the other, you'd wrap the surface back on itself and connect the edges twice, making a torus), but it is also perfectly flat everywhere.

My problem with comprehending how 3-D space could be rolled
into a cylinder without distortion goes doubly for a 3-D torus.

Well, the main difficulty here is that you're assuming that for space to wrap back on itself, it must curve through some higher dimension. There is no fundamental reason why this needs to be the case. It is not mathematical nonsense for our universe to be rather like the game Asteroids in its topology. It would be weird, to be sure, but just being weird isn't a valid reason to rule out a possibility.

You can see from my argument about causality that I would not
find plausible any model in which the "stuff" participating in the
cosmic expansion is infinite in quantity or extent. So I'm looking
at models of the Universe in which that stuff can be finite.

The torus is certainly such a model, but it strikes me as absurdly
contrived. An ad hoc construction that a mathematician would
suggest to fit a set of given parameters, rather than the natural
result of a physical circumstance. It seems very artificial. Maybe
I just don't understand what it is.

Well, you could have a real universe that does indeed wrap back on itself through higher dimensions, and thus have a fair amount of curvature, and we'd still be unable to measure that curvature because of inflation. I don't think most theorists believe for an instant that our universe is genuinely, perfectly flat. Rather the general expectation is that at one time there was quite a lot of curvature, but that inflation diluted that curvature away so much that it became very, very small, perhaps unmeasurable.

 

[Jeff Root]

Thanks for the excellent reply, Chalnoth!

Right now I just want to check that what seems obvious is correct:

All else being equal, is the overall curvature of the Universe directly
proportional to its density? So that the denser the Universe, the
more strongly it curves?

If so, what about a universe with high density but small mass?
Like a universe consisting of a single, large elliptical galaxy?

-- Jeff, in Minneapolis

 

[Chalnoth]

Right now I just want to check that what seems obvious is correct:

All else being equal, is the overall curvature of the Universe directly
proportional to its density? So that the denser the Universe, the
more strongly it curves?

No, that's not at all the case. In particular, here we're not talking about total space-time curvature, but just the spatial component. The spatial component of curvature in an expanding universe depends upon how the energy density relates to the rate of expansion. If your expansion is too slow (or your energy density too high), then you have a "closed" universe. If, on the other hand, your expansion is too fast (or your energy density too low), then you have an "open" universe.

Inflation drives the universe to flatness because as time goes forward, it pushes the expansion rate to almost exactly equal that required for flatness.

If so, what about a universe with high density but small mass?
Like a universe consisting of a single, large elliptical galaxy?

When we talk about the overall curvature of the universe, we are only talking about universes that are, on large scales, the same from place to place. If you have a universe that is exceedingly different from place to place, such as this example with only one galaxy, then it just doesn't even make sense to use that language.

 

[hamster143]

Well, the main difficulty here is that you're assuming that for space to wrap back on itself, it must curve through some higher dimension. There is no fundamental reason why this needs to be the case. It is not mathematical nonsense for our universe to be rather like the game Asteroids in its topology. It would be weird, to be sure, but just being weird isn't a valid reason to rule out a possibility.

Theoretically speaking, there is a theorem (Whitney embedding theorem) that basically says that you can embed any n-dimensional smooth manifold into a Euclidean (flat) space with dimension m >= 2n. So, you could curve the 3-d space through a 6-d space, forming a 3-d torus / "pac-man field", maintaining zero curvature everywhere.

But, like you said, there's nothing in the theory that requires us to assume any kind of embedding.

 

[Chalnoth]

Theoretically speaking, there is a theorem (Whitney embedding theorem) that basically says that you can embed any n-dimensional smooth manifold into a Euclidean (flat) space with dimension m >= 2n. So, you could curve the 3-d space through a 6-d space, forming a 3-d torus / "pac-man field", maintaining zero curvature everywhere.

But, like you said, there's nothing in the theory that requires us to assume any kind of embedding.

Ah, that's interesting. I seem to remember hearing about that. But yeah, some of my differential geometry is pretty fuzzy, and my topology is very slim.