Is Our Universe Infinite or Spherical?

[Nasher]

Lets get a bit of brain-storming going...

What shape is the universe?


Is it infinite in all three dimensions of x, y, z ?

Is it spherical?

 

[smallphi]

All astronomical information comes from our past light cone - a sphere of light of decreasing radius created at big bang and sweeping through space, and focusing on us right now. Everything outside that is unknown. The light cone has finite extent since the universe existed for finite time after the big bang.

We have no idea what is outside of our light cone, so questions like weather the universe is finite and what shape it has cannot be answered based on the current observational data. The simplest is to assume it is infinite which we currently do.

 

[setAI]

the universe is likey a shapeless infinite dimensional phase space of all possible mathematical/formal/computational structures in superposition- sub-sets of these states correspond to all the states of all histories of every possible causal rule-system- including all the possible world-states of the hilbert space of our universe-

now the spatial shape of causal space-times with locality like ours seen by observers should all be n-spheres with a surface-area b where n is the number of large spatial dimensions of the observer's environment and b is the n-1-surface area of observability determined by the Beckenstein Bound of the observer's local environment-

in our case with an expanding universe the surface area is probably that of the Hubble volume within our cosmological event horizon [although I would say that ALL causal structures have a positive cosmological constant because I think it is related directly to Entropy- and all causal structures are subject to Information Entropy]

 

[Chris Hillman]

Suggest a good book

What shape is the universe?
Is it infinite in all three dimensions of x, y, z ?
Is it spherical?

Jeffrey Weeks, The Shape of Space, is a well-illustrated nontechnical book which explores this question very carefully. It should be just what you want.

 

[xantox]

Lets get a bit of brain-storming going...

What shape is the universe?

Is it infinite in all three dimensions of x, y, z ?

Is it spherical?

Current observational constraints:

• suggest that our universe is not smaller than 24 gigaparsec, but are unable to decide whether it is finite or infinite;
• find the usual 3 spatial dimensions, but are unable to decide whether there are small extra dimensions like eg proposed by string theory;
• indicate constant curvature at large scale, but are unable to decide its sign;
• are unable to rule out non-trivial topologies (eg positive curvature does not necessarily imply a 3-sphere as there are infinite topologies sharing the same property).

[EDIT: I second the suggestion by Chris Hillman and recommend this excellent software also by J. Weeks, to get some intuitive understanding of non trivial topologies: http://www.geometrygames.org/CurvedSpaces/ ]

 

[Nasher]

Thank you all for your feedback...

When I think of the big bang, I naively think of these galaxies moving away from some big bang centre point in a sort of spherical space and that the universe expending is like the space getting stretched in the three usual dimension of space.

If it was like that in this sort of 3D classical space then I would presume that some of the galaxies would be on the peripheral of the universe and from looking out one side of such a peripheral galaxy one would see other galaxies on one side but looking out the other side there would be just darkness.

Of course, that is a bit naive as I should be thinking more in terms of a higher dimenional spacetime.

From any typical planet going around any typical sun like star in any typical galaxy would one expect the view to be more or less a typical as the view we have?

Would it be better to think of 3D space being to spacetime like what
a 2D surface is to a regular sphere?

 

[marcus]

...
Would it be better to think of 3D space being to spacetime like what
a 2D surface is to a regular sphere?

that's a good picture to have in mind as one possibility. Keep in mind that we don't know the overall shape of space (that you asked about) or even whether it is finite or infinite (which you also asked about.)

I would advise getting familiar with the prevailing mainstream ideas of space that working cosmologists use----the standard model in cosmology is called LCDM and the most common versions are either spatially flat-infinite, or "nearly flat" which could mean slight positive curvature and large but finite---similar to the 3D sphere analogy you mentioned.

I'd suggest you not rely on second hand information but go directly to primary sources on this. I will get a link to a basic paper by David Spergel et al.

Of course you can also have fun with alternatives. You don't have to BELIEVE the simple prosaic alternatives you find mainstream cosmologists focusing most of their attention on. Nobody knows so everybody is free to picture space as a donut or kleinbottle or hall-of-mirrors-without-the-mirrors, or whatever you like.

But I will get that source

http://arxiv.org/abs/astro-ph/0603449
Wilkinson Microwave Anisotropy Probe (WMAP) Three Year Results: Implications for Cosmology

I have to go out for the moment but I'll get back to this and find you a page reference to something I think you can understand,
it is not all gibberish there are some simple conclusions

 

[Chris Hillman]

When I think of the big bang, I naively think of these galaxies moving away from some big bang centre point in a sort of spherical space and that the universe expending is like the space getting stretched in the three usual dimension of space.

If it was like that in this sort of 3D classical space then I would presume that some of the galaxies would be on the peripheral of the universe and from looking out one side of such a peripheral galaxy one would see other galaxies on one side but looking out the other side there would be just darkness.

Of course, that is a bit naive as I should be thinking more in terms of a higher dimenional spacetime.

Actually, the real problems here are these:

1. no center,

2. no peripephery.

How about thinking in terms of a lower dimensional spacetime?

Consider a two dimensional spacetime: one "time dimension" plus one "space dimension". Since our model is homogeneous and isotropic, directions don't matter, so we need represent only one space dimension to get the general idea of the Hubble expansion. And for simplicity, let's try to envision a model sometimes called a "matter dominated universe which expands and recollapses", or as I prefer to say, "the FRW dust with S^3 hyperslices orthogonal to the world lines of the dust particles".

Then you can envision the geometry of this spacetime as the geometry induced when we consider a humble American football in a three dimensional "embedding space". Your intution will be a bit off since this flat embedding space is actually E^{1,2}, not E^3, but never mind that--- the main point is that the world lines of the galaxies correspond to the longitudes of the football, and "spaces at a time" correspond to the latitudes. These latitudes look like circles; really they are three dimensional spheres. The Big Bang is the lower end of the football and the Big Crunch is the upper end. As we proceed along the world line of the Milky Way galaxy, traveling at unit speed, the slices expand and then shrink, but there is no center and no periphery.

If your mind boggles at spacetime, try the "balloon analogy", in which we model the expanding three-dimensional spheres as expanding and recontracting two-dimensional spheres. Galaxies are like pennies glued to the surface of the balloon--- they don't expand, Brooklyn doesn't expand, but the galaxies do move apart from another during the expansion phase.

From any typical planet going around any typical sun like star in any typical galaxy would one expect the view to be more or less a typical as the view we have?

Yes, that's pretty much what we mean by homogeneous and isotropic. This is an idealization, of course.

Would it be better to think of 3D space being to spacetime like what
a 2D surface is to a regular sphere?

I am not sure I understand what you have in mind, but I should add a caveat concerning the embedding above: the flat embedding space has no physical significance, and is used only to obtain a bit of intuition for the geometry. There are always infinitely many ways of embedding a Lorentzian manifold in higher dimensional flat spaces (usually five dimensions does not suffice, though), but for physics only intrinsic properties matter, and these are precisely the properties which are independent of any embedding. At the opposite end of the spectrum, in something like knot theory, intrinsically all circles are the same, but the details of how we embedd a circle as a knotted curve in R^3 (really S^3) are crucial.

And another caveat: the particular FRW dust model I mentioned is no longer thought to be the most realistic one. Marcus already mentioned that current models are based on an FRW dust which has E^3 hyperslices orthogonal to the world lines of the dust and which in addition to a term from the dust has a somewhat mysterious "Lambda term" in the energy-momentum tensor. This energy-momentum tensor is how gtr represents the amount and motion of mass-energy, and the Einstein field equation equates it to a tensor called the Einstein tensor which is, roughly speaking, a kind of "average curvature".

HTH

 

[marcus]

I'm back and can point to the page in that paper

http://arxiv.org/abs/astro-ph/0603449
Wilkinson Microwave Anisotropy Probe (WMAP) Three Year Results: Implications for Cosmology

Nasher, you asked whether spatially finite or infinite and in the simplest terms that turns on the value of a parameter Omega, whether it is exactly one or slightly more, like 1.01

this has been measured but there is uncertainty---there is an errorbar on the estimate of Omega derived from various sets of data.

If it is exactly 1.00, then we are in the spatial flat, most likely infinite case.
If it is 1.01, then there is overall positive curvature and the simplest thing to assume is what you, Nasher, suggested----the 3D sphere.

this is what Spergel et al refer to as the "nearly flat" case.

So let's cut to the chase. Look at page 50, where there is Figure 17 and it says
"The marginalized best fit values for the equation of state and curvature are
w = −1.08 ± 0.12 and
Omega_k [-0.026-0.015, -0.026+0.016] = [- 0.041, - 0.010]
at the 68% confidence level."

I had to unpack their notation a bit. Omega_k is another handle on Omega. The convention is Omega = 1 - Omega_k
so if Omega_k is negative, what we are interested is bigger than one. Omega is also called Omega_total (the "total Omega").
Sorry about all the convoluted notation.

But it is really pretty exciting. It says that there is a 68% confidence level for Omega which is [1.010, 1.041]

68% is not real confident. One would like 95% or 99% of course. But already this is saying with 68% confidence we can EXCLUDE THE FLAT INFINITE CASE.
This is not polite to say because a lot of people implicitly assume flat infinite in their work. So one has to quickly reassure everybody that the data is CONSISTENT with the flat infinite case. There is enough uncertainty so we can say that the case of Omega exactly = 1 is not ruled out!

This year, in January, one of the top cosmologists, Ned Wright, who was also a co-author of the Spergel et al paper I referred to here, came out with a paper using a number of different data sets and he gave a "best fit" value of Omega of 1.011----in other words if we are using LCDM (which is the prevailing standard model) then the best fit LCDM is not the flat case but rather the positive curved finite "nearly flat" case.

But Ned Wright made very sure to clearly state that the data was all CONSISTENT with the flat Omega exactly 1 case of LCDM that many working cosmologists assume.

The Ned Wright paper is
http://arxiv.org/abs/astro-ph/0701584
Constraints on Dark Energy from Supernovae, Gamma Ray Bursts, Acoustic Oscillations, Nucleosynthesis and Large Scale Structure and the Hubble constant

Nasher you asked is it finite or infinite. I'm trying to answer as directly as possible. We don't know but it comes down to one number which they can measure or constrain limits on by various sets of data. Is Omega > 1 or not? The latest data suggest that it just might be > 1. We don't know. But space just might be a bumpy star-dimpled black hole punctured version of that 3D sphere you mentioned in yr original post. It might in other words have a slight overall positive curvature

 

[xantox]

Would it be better to think of 3D space being to spacetime like what a 2D surface is to a regular sphere?

Yes, and you should note that a "regular sphere" S^2 is not a 3D object but is precisely a (curved) 2D surface. The 3D sphere S^3 is not the usual sphere but it has similarly no center and no boundary on it, and you may well use this picture to get some intuition (for the case of positive curvature).

So let's cut to the chase. Look at page 50, where there is Figure 17 and it says
"The marginalized best fit values for the equation of state and curvature are
w = −1.08 ± 0.12 and
Omega_k [-0.026-0.015, -0.026+0.016] = [- 0.041, - 0.010]
at the 68% confidence level."

But it is really pretty exciting. It says that there is a 68% confidence level for Omega which is [1.010, 1.041]

68% is not real confident. One would like 95% or 99% of course. But already this is saying with 68% confidence we can EXCLUDE THE FLAT INFINITE CASE.

This quote does not refer to the power-law lambdaCDM, but to a less likely quintessence model.

 

[marcus]

This quote does not refer to the power-law lambdaCDM, but to a less likely quintessence model.

indeed they did let w vary in that analysis---by contrast, Ned Wright's "best fit" Omega = 1.011 is explicitly for LCDM in other words with dark energy equation of state w = -1
I'd say the Spergel et al result includes, but is not restricted to, quintessence. The analysis also includes the case w = -1, the standard cosmological constant, as well. I don't know what precise error bar you'd get if you restricted to w = -1, in their case.
===============

but if you don't like referring to Spergel et al Figure 17, then I will hunt up the page reference for Ned Wright's January paper. Or maybe you have read it already?
http://arxiv.org/abs/astro-ph/0701584
Constraints on Dark Energy from Supernovae, Gamma Ray Bursts, Acoustic Oscillations, Nucleosynthesis and Large Scale Structure and the Hubble constant

Look at Wright's Table 4 on page 17
"The best fit model is slightly closed with Omega_tot = 1.011 and M = 0.315."

However he concludes that the data is still consistent with flat LCDM, and remarks that with the currently available data the fit with Omega = 1.011 is only a little bit better than the fit you get with Omega = 1 flat. Not enough better to prove anything.

But I've been seeing more papers about this. so I think there is a growing willingness to consider finite positive curved as a possibility.

 

[marcus]

Lets get a bit of brain-storming going...

Well Nasher, is this enough brain-storming for you or shall we step it up some?

 

[xantox]

the Spergel et al result is not only for quintessence, it includes the case w = -1, the standard cosmological constant, as well.

Yes, but for w=-1 the error bar does not exclude Ω=1 (or even Ω<1).

but if you don't like referring to Spergel et al Figure 17, then I will hunt up the page reference for Ned Wright's January paper. Or maybe you have read it already? Look at Wright's Table 4 on page 17 "The best fit model is slightly closed with Omega_tot = 1.011 and M = 0.315." However he concludes that the data is still consistent with flat LCDM, and remarks that with the currently available data the fit with Omega = 1.011 is only a little bit better than the fit you get with Omega = 1 flat. Not enough better to prove anything.

I read it, and you're referring here to the best fit non-flat model, however in that same paper the best fit flat model has practically the same likelihood (and it is actually the flat model which is a bit better, with about one more unit of chi squared, but this is indeed not significant).

But I've been seeing more papers about this. so I think there is a growing willingness to consider finite positive curved as a possibility.

I certainly agree that it is a possibility, but the above evidence is not enough to present it as a preferred possibility.

 

[FizX]

I went to a seminar a while back, and they now believe the universe is a hyper-spherical-torus. Parallel lines wrap around each other, and if you keep going in one direction forever, you will eventually pass by the same points over and over.

 

[smallphi]

Is there an observation that demands a global topology of torus?

 

[marcus]

Is there an observation that demands a global topology of torus?

no, and contrary to what FizX claims about what "they" believe, I don't know of any working cosmologist who favors the toroidal case.
however Neil Cornish and David Spergel have written a couple of papers where they work hard to EXCLUDE the toroidal and other similar cases.

the way they do this is to say that if the structure is periodic then their analysis of the data shows that the periodicity distance scale must be AT LEAST such and such, because they rule out any signs of repetition in a smaller scale.
And as the data get better they can push this distance scale farther and farther out.

you may know of someone who favors toroidal topology, but they would be in a tiny minority AFAIK. there is that French astronomer who has several times urged a dodecahedron-style "soccerball" structure, but that is also very much a minority interest AFAIK.

and I believe there are some string theorists who think they need a negative curvature model (perhaps because their theory doesn't cope very well with positive curvature) but the mainstream cosmology majority seems fairly happy with the flat (nontoroid) or the nearly flat (slight positive curved) cases.

if anyone has a different impression, I'd be glad to hear it. maybe with some arxiv references.

here are the Cornish and Spergel et al papers I mentioned:
1. arXiv:astro-ph/0604616 [ps, pdf, other] :
Title: Extending the WMAP Bound on the Size of the Universe
Authors: Joey Shapiro Key, Neil J. Cornish, David N. Spergel, Glenn D. Starkman
Comments: 9 pages, 16 figures

2. arXiv:astro-ph/0310233 [ps, pdf, other] :
Title: Constraining the Topology of the Universe
Authors: Neil J. Cornish, David N. Spergel, Glenn D. Starkman, Eiichiro Komatsu
Comments: Submitted to PRL
Journal-ref: Phys.Rev.Lett. 92 (2004) 201302

 

[Gib Z]

All astronomical information comes from our past light cone - a sphere of light of decreasing radius created at big bang and sweeping through space, and focusing on us right now. Everything outside that is unknown. The light cone has finite extent since the universe existed for finite time after the big bang.

We have no idea what is outside of our light cone, so questions like weather the universe is finite and what shape it has cannot be answered based on the current observational data. The simplest is to assume it is infinite which we currently do.

I would have thought that existing for a finite time after the big bang would imply an finite time for it to expand, hence a finite volume?

 

[marcus]

... The simplest is to assume it is infinite which we currently do.

who is we? Don't you think your statement needs some qualification?
Not all mainstream professional cosmologists find it simplest to assume space infinite. And not all assume that it is.
There is currently some controversy about whether it introduces errors in the analysis of data to make this assumption.

What I read in recent (2006-2007) papers is that the data is CONSISTENT with flat (infinite space) LCDM. So the spatially flat LCDM model is not ruled out.

But the data is also consistent with a spatial-finite positive curve "nearly flat" case of LCDM, with Omega around 1.01.

Ned Wright and Sean Carroll are both prominent cosmologists who exemplify what I am talking about. I greatly prefer Ned Wright's style, which is observational and numerical-quantitative------not so theoretical and speculative as Carroll, but it takes all kinds. Anyway both are highly visible figures in cosmology and both have been talking finite space cases lately.
Wright's january paper had a "best fit" LCDM with Omega = 1.011, which means a radius of curvature of 130 billion LY-----basically it is the balloon-surface analog but in 3D. And just this week Carroll was explaining a reproductive multiverse picture in which "universes like ours" emerge from finite patches of space and pinch off----clearly finite. Not that I advocate Carroll's picture---just that he finds it simplest to assume that universes like ours are spatial finite. so he is not part of your "we" who assume infinite.

We have no idea what is outside of our light cone,..

smallphi that is too simplistic. In their routine work cosmologists conceptualize and map the universe outside our lightcone. they make assumptions like everybody else. we can't see where Andromeda galaxy is at this moment because its present situation consists of events outside of our lightcone, but we have a pretty good idea of where it is and what it is doing nevertheless.

the basic FRW model doesn't work unless you have a pretty good idea of the distribution of matter and shape of space everywhere outside our lightcone.

the basic notion of the Hubble flow doesn't make sense unless you have a definite idea of the universe outside our lightcone----for example the average distance between galaxies at the present moment

the Hubble parameter H₀ itself refers by definition to stuff outside our lightcone-----it is the ratio of the DISTANCE AT THIS MOMENT to some galaxy and the recession SPEED AT THIS MOMENT
since none of that stuff is in our lightcone, it would be awkward to have to change how we talk about everything so as to eliminate references to stuff outside the cone and remove logical dependence on conceptualizing the rest of the universe.

of course we can only get observational confirmation from events which are within our cone----but that is different from the question of whether we have "no idea" or some idea of what is outside it.

one gets ideas about what one cannot see by empirically fitting a global model to the part one can see (and after rigorous testing, placing cautious trust in the model)

 

[xantox]

I would have thought that existing for a finite time after the big bang would imply an finite time for it to expand, hence a finite volume?

Not necessarily, as in the case the universe would be infinite, then it would have been infinite at every past time.

 

[marcus]

I would have thought that existing for a finite time after the big bang would imply an finite time for it to expand, hence a finite volume?

If I understand you, that is perfectly correct. If the universe was finite volume when it started expanding, as is one reasonable possibility to assume, then it continues to have finite volume.

to put it in topological terms, if it started with a compact topology (at least as soon as a definite topology emerged from the confusion ) then one would expect it to continue so.

I don't see how anyone could object to that.

But at present there is no scientific reason to assume that the universe is infinite or that it is finite. We have good models of both kinds that fit the data.

I would say that anyone who claims otherwise is pretending to knowledge he does not possess.

 

[marcus]

... The simplest is to assume it is infinite which we currently do.

smallphi have a look at
http://arxiv.org/abs/astro-ph/0701584
Constraints on Dark Energy from Supernovae, Gamma Ray Bursts, Acoustic Oscillations, Nucleosynthesis and Large Scale Structure and the Hubble constant
Edward L. Wright
19 pages, 8 figures. To be published by the Astrophysical Journal

I think this is an important, up-to-date paper. It takes a whole lot of different kinds of data and fits the leading cosmological models to it.

It looks to me as if the two main candidates are FLAT LCDM
and nearly flat positive CURVED LCDM.
The latter (curved LCDM) gives a slightly better fit than the former, but the difference is not significant. One can say that the latest best data is CONSISTENT WITH BOTH infinite and finite. NEITHER IS RULED OUT by the data.

So people who subscribe to one or the other do not have a scientific reason to do so and are claiming more than they actually know. It is important to realize this, so as not to be misled by false certitude.

Even though the difference is not significant, Ned Wright did take the trouble to provide us with a BEST FIT LCDM model, which is spatially finite with Omega = 1.011.
This corresponds to a 3D sphere with radius of curvature equal to 130 billion lightyears. The total 3D volume of space, at the present time, can be easily calculated from this---I have the calculation somewhere if you want it. Wright did not give details, he just gave the "best fit LCDM" Omega.

 

[marcus]

I certainly agree that it is a possibility, but the above evidence is not enough to present it as a preferred possibility.

I agree with you. We are talking about the spatial-finite "nearly flat" version of LCDM. In Ned Wright's analysis, which you read, the spatial finite LCDM is only slightly better fit to the data than the spatial infinite flat LCDM.

AFAICS neither is preferred by the data. One would be over-interpreting to take a preferred side at this point, don't you agree?

 

[xantox]

AFAICS neither is preferred by the data. One would be over-interpreting to take a preferred side at this point, don't you agree?

Yes. Lack of knowledge may be frustrating, but taking a side at all costs is an even weaker position.

 

[marcus]

Yes. Lack of knowledge may be frustrating, but taking a side at all costs is an even weaker position.

I strongly agree! BTW I have enjoyed looking at the (small but exquisite) science graphics category called "Gallery"
English version:
http://strangepaths.com/category/gallery/en/
French version:
http://strangepaths.com/category/galerie/fr/
the Galileo animation is an elegant idea and the Mars landscape is chosen very carefully---it reminded me of some Ansel Adams.

 

[xantox]

I strongly agree! BTW I have enjoyed looking at the (small but exquisite) graphics category called "Gallery"
the Galileo animation is an elegant idea and the Mars landscape is chosen very carefully---it reminded me of some Ansel Adams.

Thank you!

 

[morson]

The simplest is to assume it is infinite which we currently do.

This notion I can never digest. People propose that the universe is boundless and ever-stretching, but doesn't the concept of the universe continuing to stretch refute the theory that it is infinite? Only a finite structure can be stretching and growing at a point in time. Just because we can not see beyond a certain point, we should not fall back on the assumption that it never stops. It's analagous to proving the irrationality of numbers; we're not satisfied with assuming its decimal expansion is infinite and using the fact that its 5 billionth digit has been recorded to back it up.

 

[xantox]

People propose that the universe is boundless and ever-stretching, but doesn't the concept of the universe continuing to stretch refute the theory that it is infinite? Only a finite structure can be stretching and growing at a point in time.

No, it does not refute it. Just think at the two infinite series 1, 2, 3.. and 10, 20, 30.. where in the second "distances" have grown by a factor 10. The constraints of general relativity are actually stronger: if space is infinite, then it is impossible that it does not also expand.

 

[morson]

I merely have a problem with one person saying that the universe is infinite while it is extending at the same time. If it was infinite, it would not have a bound, would it not?

 

[xantox]

I merely have a problem with one person saying that the universe is infinite while it is extending at the same time. If it was infinite, it would not have a bound, would it not?

You don't need a bound to extend in order to make it expand. Just add new "atoms of space" between any two existing "atoms of space".

 

[chrisina]

I don't see why Omega being 1 or 1.01 has an influence on the universe being finite or infinite. Let's not confuse "being able to expand infinitely" with "being infinite".
As a matter of fact, a small positive cosmological constant sets an upper bound on the maximum entropy of the universe, which I prefer much more as a definition of whether it is infinite or not...