What Is the Universe Expanding Into?

[Chronos]

...Since the Big Bang supposedly does not have a physical origin and began everywhere at once, you must consider that these most distant galaxies/quasars that we see in all directions are each at the very center of their own "observable universes" each with a unique sphere of observable space 27Gy in diameter. Every galaxy that is observable just inside the surface of last scattering of each of these spheres is also in the center of its own sphere. We can carry this model out forever, increasing the size of the universe with each iteration...

ohwilleke answered this nicely, I just wanted to add this. Suppose the light from a galaxy 12 billion light years distant included a map of the universe made by an ancient astronomer. How big do you think they observed the universe to be? I would guess 1.5 billion light years in all directions.

 

[turbo]

ohwilleke answered this nicely, I just wanted to add this.

Please read my post above carefully, taking into account the validity of the point of view of an observer in the distant galaxy today, then read his reply. There is a disconnect that I believe you have missed. Then read below:

Suppose the light from a galaxy 12 billion light years distant included a map of the universe made by an ancient astronomer. How big do you think they observed the universe to be? I would guess 1.5 billion light years in all directions.

Look at a galaxy 12Gly distant. We see it as it was 12Gy ago. If that galaxy still exists, today, what would an observer in such a galaxy see? He or she (or it or "the hive mind" or whatever) would look around themselves and see a Universe of apparent diameter 27Gly, just like we do. In addition, 'WAY off near their surface of last scattering, they would see our galactic neighborhood (if it were recognizable as such) as it appeared 12Gy ago.

We are forbidden by the finite speed of light from seeing the distant galaxy in "real time". We only get 12G-y-old EM waves to work with. The trick is, if you believe in the BB theory, you also have to accept a homogenous, isotropic Universe with NO preferred location. That galaxy on the edge of our Universe occupies a reference frame just as valid as our own. And since (by the BB model's extrapolation of redshift = cosmological expansion) we believe that we see that galaxy as it was 12Gy ago, we must believe that an observer in that galaxy today sees himself at the most mature location (center) of the Universe, surrounded by a 27Gly sphere of observable Universe, which he interprets as "younger" with increasing distance, due to the finite speed of light. There will be many objects in his observable sphere surrounding that galaxy (about half of them, in fact) that are not observable from ours, because they are beyond our particular "surface of last scattering". The fact that they are NOT observable from our location in no way negates their existence. In fact, if you wish to deny the existence of galaxies beyond our surface of last scattering or plead agnosticism to their existence, you must either deny the validity of the frame of reference of the observer on the 12Gly-distant galaxy (which violates the "no special location" concept), or you must amend your acceptance of "homogenous and isotropic" as definitions of our Universe, and establish some arbitrary boundaries on the Universe to enforce this.

JesseM pointed out a third possibility - that the Universe can be locally flat/Euclidean, but have a non-trivial topology that allows it to curve back on itself. The website he links uses a torus as an example, but others use Klein bottles, teacups, manifolds with multiple interconnedctions, etc, etc. Other than the possibility of seeing self-similar constructs on opposite sides of the Universe (which has never been done, even in the WMAP data) I am not aware of any testable predictions by which these mathematical curiosities might be falsified. In science, something that cannot be falsified (for instance a statement like "angels are pushing the Universe apart, causing expansion") has no standing. Just because it is mathematically possible to do geometry in a topologically non-trivial frame, that does not make the chance that our Universe has assumed that topology likely, nor even possible.

Using Occam's Razor (apparently flat universe, without assuming a complex non-trivial topology) I believe that you will have to accept a spacially infinite Universe with no center from which it all began, as JesseM stated earlier.

 

[Chronos]

Please read my post above carefully, taking into account the validity of the point of view of an observer in the distant galaxy today, then read his reply. There is a disconnect that I believe you have missed.

I don't think I missed a thing, except the 12 billion years it took me to get there [at near light speed] and hear his explanation of how the universe, to him, appears to be 24 billion years older than it was when he first sent the message.

 

[ohwilleke]

It is my understanding (and I stand to be corrected) that the BB theory does not posit ANY center from which the expansion originated.

I'd disagree. BB theory may not precisely require a "point source", probably a better analogy would be a compact glob of silly puddy that rapidly gets pulled apart in all directions with a "center" fuzzed out over time, but the whole point of BB theory is that the universe starts at a particular time in a relatively small geometric region.

Furthermore the model of a homogenous and isotropic Universe forbids us to exist in a "special location" or "special frame of reference" from which such a center could be sensed or from which the direction of such a center could be deduced.

I agree that there would be nothing special about a "center" of the universe (unless you get very Machian). In the same way, the only thing that makes "Four Corners" (CO, UT, AZ, NM) in the U.S.A. special is a little monument. Absent that there is nothing about the place that is significant.

In the BB universe, every observer (regardless of location) is at the most mature spot in the Universe, looking out (and back) toward more distant (and younger) galaxies in every direction.

Of course, every spot is equally "mature", it has just been stretched (imagine, e.g. mapping the rational number line onto the real number line) and of course, everything we see happened in the past. This doesn't however, make the Universe infinitely wide.

 

[ohwilleke]

we must believe that an observer in that galaxy today sees himself at the most mature location (center) of the Universe,

Nope. As you yourself explained, there is no such most mature place, identifying "most mature" with "center" doesn't make sense. Every location is equally mature, just "stretched". For example, a "spherical" region with a path from Earth to Alpha Centuri as a diameter (in round numbers 3 light years long), currently has a volume of perhaps 113 cubic light years. If you map that "spherical" region back 12.3 billion light years, the "source" region has a volume of 0.11 cubic light years. But, every point in the current region has an analogous point in the prior region, and hence every point in the current region is at least 12.3 billion light years old. This can be carried back all the way to the singularity.

 

[JesseM]

JesseM pointed out a third possibility - that the Universe can be locally flat/Euclidean, but have a non-trivial topology that allows it to curve back on itself. The website he links uses a torus as an example, but others use Klein bottles, teacups, manifolds with multiple interconnedctions, etc, etc. Other than the possibility of seeing self-similar constructs on opposite sides of the Universe (which has never been done, even in the WMAP data) I am not aware of any testable predictions by which these mathematical curiosities might be falsified. In science, something that cannot be falsified (for instance a statement like "angels are pushing the Universe apart, causing expansion") has no standing. Just because it is mathematically possible to do geometry in a topologically non-trivial frame, that does not make the chance that our Universe has assumed that topology likely, nor even possible.

Using Occam's Razor (apparently flat universe, without assuming a complex non-trivial topology) I believe that you will have to accept a spacially infinite Universe with no center from which it all began, as JesseM stated earlier.

If neither an infinite universe with the simplest topology nor a finite universe with a nontrivial topology make any distinct predictions, how can it possibly be a scientific question which one is actually true? An argument which uses "Occam's razor" is a purely metaphysical one if it is impossible in principle to test whether your conclusion is correct (unlike, say, the theory that the laws of physics work differently on a single planet in the Andromeda galaxy, an idea which seems very implausible by Occam's razor, but which could in principle be tested directly). Plus, some people might argue that a finite universe is inherently simpler than an infinite one, and is therefore favored by Occam's razor even if it requires a slightly more complicated topology. As it happens, it could actually be possible to find experimental evidence for a finite universe by looking for repeating patterns in the cosmic microwave background radiation (see http://www.hep.upenn.edu/~angelica/topology.html ), but this would only work if the radius of the universe is smaller than the maximum distance we can observe.

 

[JesseM]

I'd disagree. BB theory may not precisely require a "point source", probably a better analogy would be a compact glob of silly puddy that rapidly gets pulled apart in all directions with a "center" fuzzed out over time, but the whole point of BB theory is that the universe starts at a particular time in a relatively small geometric region.

Isn't the whole point just that the density goes to infinity as you approach the big bang, and the distance between any two points in the universe which are today some finite distance apart goes to zero? This isn't the same as saying that "the universe starts at a particular time in a relatively small geometric region", unless you're just talking about the observable universe. Would you agree that in the standard Friedmann-Robertson-Walker cosmological model, if the universe is flat or open then its volume is infinite at all finite times after the big bang?

 

[ohwilleke]

Isn't the whole point just that the density goes to infinity as you approach the big bang, and the distance between any two points in the universe which are today some finite distance apart goes to zero? This isn't the same as saying that "the universe starts at a particular time in a relatively small geometric region", unless you're just talking about the observable universe. Would you agree that in the standard Friedmann-Robertson-Walker cosmological model, if the universe is flat or open then its volume is infinite at all finite times after the big bang?

I think the distinction you are coming to is a semantic one. In other words, how the phrase "volume of the universe" defined determines the answer, and I believe that at least two different definitions of that phrase are being used in this case.

I would intuitively define "volume of the universe" operationally as something on the order of: "(1) select two points at which matter or energy arising from the Big Bang are present, which are as distant or more distant from each other than any other two points in the universe; (2) call the magnitude of the distance between them d; and (3) the volume of the universe in the space-like dimensions is then defined to equal pi*d/6".

In a conventional Big Bang scenario with radiation emitting in all directions from day one and outpacing everything else, one would expect that d would be approximately equal to 2*c*t, or in speed of light units simply 2t, so long as the universe is not contracting. Hence, in a Big Bang scenario, the 3-D volume of the universe, if this definition is adopted, is a function of the time elapsed since the Big Bang (defined as t=0 and hence the volume of the universe overall through point t in four dimensions would be the integral from zero to t of f(t) with respect to t. Hence, this definition would produce a finite 3-D volume of the universe at any given time t, and a 4-D volume of the universe that is infinite or finite depending on the form of f(t) (which depends on the values you put into the Friedman-Robertson-Walker equation in standard GR). (Of course, one would have to be quite clever in defining "t" in the equations above in a way that makes sense).


This isn't quite the same as the "observable universe" (and certainly less elegant) although it is pretty close.

It sounds like the definition of "volume of the universe" you are using is something on the order of V=pi*d/6 for the value of d (defined as above) that is the maxima of the function d(t)= for t between 0 and infinity. This would be infinite given the proper inputs into F-R-W.

The implicit issue that hinges between the two definitions of "what is the universe whose volume we are measuring" is whether empty space should be included when you are defining what the universe is. In a non-aether theory, it would seem to make sense not to include that empty space. In an aether theory, it is vital to do so. General relativity, is basically a non-aether theory that gets a close to an aether theory as it is possible to do, because its geometrical elements are very aether-like.

 

[JesseM]

I think the distinction you are coming to is a semantic one. In other words, how the phrase "volume of the universe" defined determines the answer, and I believe that at least two different definitions of that phrase are being used in this case.

I would intuitively define "volume of the universe" operationally as something on the order of: "(1) select two points at which matter or energy arising from the Big Bang are present, which are as distant or more distant from each other than any other two points in the universe; (2) call the magnitude of the distance between them d; and (3) the volume of the universe in the space-like dimensions is then defined to equal pi*d/6".

But in the Friedmann-Robertson-Walker model of a flat or open universe, there would be no upper limit on d. Matter and energy are distributed evenly throughout all of space in these models, so in a flat or open universe, for any finite distance d you can find two bits of matter/energy which are separated by a distance greater than d (although if the distance is too large there will be no possibility of causal interaction between these points since the big bang).

n a conventional Big Bang scenario with radiation emitting in all directions from day one and outpacing everything else, one would expect that d would be approximately equal to 2*c*t, or in speed of light units simply 2t, so long as the universe is not contracting. Hence, in a Big Bang scenario, the 3-D volume of the universe, if this definition is adopted, is a function of the time elapsed since the Big Bang (defined as t=0 and hence the volume of the universe overall through point t in four dimensions would be the integral from zero to t of f(t) with respect to t. Hence, this definition would produce a finite 3-D volume of the universe at any given time t, and a 4-D volume of the universe that is infinite or finite depending on the form of f(t) (which depends on the values you put into the Friedman-Robertson-Walker equation in standard GR). (Of course, one would have to be quite clever in defining "t" in the equations above in a way that makes sense).

I think you are fundamentally misunderstanding how the "conventional Big Bang scenario" (ie the Friedmann-Roberston-Walker model) actually works. There is no central position in space where the explosion originated, so the notion of radiation "outpacing everything else" doesn't make sense--rather, at all finite times matter and energy are evenly distributed throughout all of space, so there is no empty region where radiation hasn't reached yet.

 

[ohwilleke]

Various Quotations:

One of the most persistently asked questions has been: How was the universe created? Many once believed that the universe had no beginning or end and was truly infinite. Through the inception of the Big Bang theory, however,no longer could the universe be considered infinite. The universe was forced to take on the properties of a finite phenomenon, possessing a history and a beginning.

About 15 billion years ago a tremendous explosion started the expansion of the universe. This explosion is known as the Big Bang. At the point of this event all of the matter and energy of space was contained at one point. What exisisted prior to this event is completely unknown and is a matter of pure speculation. This occurance was not a conventional explosion but rather an event filling all of space with all of the particles of the embryonic universe rushing away from each other. The Big Bang actually consisted of an explosion of space within itself unlike an explosion of a bomb were fragments are thrown outward. The galaxies were not all clumped together, but rather the Big Bang lay the foundations for the universe.

The origin of the Big Bang theory can be credited to Edwin Hubble. Hubble made the observation that the universe is continuously expanding. He discovered that a galaxys velocity is proportional to its distance. Galaxies that are twice as far from us move twice as fast. Another consequence is that the universe is expanding in every direction. This observation means that it has taken every galaxy the same amount of time to move from a common starting position to its current position. Just as the Big Bang provided for the foundation of the universe, Hubbles observations provided for the foundation of the Big Bang theory.

Since the Big Bang, the universe has been continuously expanding and, thus, there has been more and more distance between clusters of galaxies. This phenomenon of galaxies moving farther away from each other is known as the red shift. As light from distant galaxies approach Earth there is an increase of space between Earth and the galaxy, which leads to wavelengths being stretched.

http://www.umich.edu/~gs265/bigbang.htm

What a cosmic can of worms. Turns out even the word "universe" is elusive, having three meanings (two of which depend on whether or not you hit the shift key). So we start with the basics.

What does the word "universe" mean?

The "observable universe," Sweitzer explained, "is the one astrophysicists generally talk about because it's the one open to empirical measurements. In fact it's the only one we can or ever will be able to talk with any certainty about."

He goes on to explain that "universe" (sans the word "observable") is a larger concept that scientists think "conforms to our laws of physics and all the assumptions that go with them." Comprehending this universe, Sweitzer said, "requires a leap of faith into unobservable realms."

Finally, there is "the Universe," which, by virtue of its capital "U," includes "absolutely everything, even possibilities of dimensions, modes and regions that obey laws of physics we don't know or maybe even can't know."

Is the universe finite or infinite?

"The observable universe is finite," Sweitzer said, which is to say that it had boundaries -- physical limits. Sort of. "It's a boundary to the events we can see directly, but not a boundary in the sense that New York State has a boundary."

And in an expanding universe, this boundary is constantly moving, as is everything within it. Cosmologists typically invoke a balloon with spots on its surface, representing galaxies, to explain the expanding universe. As the balloon is inflated, the spots grow farther apart. If you stood within one of these spots, you'd see all the others moving away from you, and the most distant spots would move appear to move the fastest.

Sweitzer goes on to say that the observable universe is probably part of a much larger universe, "which could be finite or infinite. Any global statements about the universe, such as overall extent, are speculative because they require extrapolating local mathematical theories and measurements beyond the observable universe."

So if the observable universe is finite, like the space occupied by a car or a house is finite, then there must be a brick wall or something up there, holding it all together. Right?

Does the observable universe have an edge?

No, said Livio.

Argh. First you guys tell us the universe is expanding. Then you say it is finite. Now you say it has no edge! We need a visual here.

Livio is up to the task. He dredges up the old expanding balloon as his prop. "An ant traveling on the surface of a balloon will never reach an edge," Livio explains. "In the worst case it will return to its starting point."

http://www.space.com/scienceastronomy/astronomy/universe_overview_010605-1.html

Before we discuss which of these three pictures describe our universe (if any) we must make a few disclaimers:

Because the universe has a finite age (~13.7 billion years) we can only see a finite distance out into space: ~13.7 billion light years. This is our so-called horizon. The Big Bang Model does not attempt to describe that region of space significantly beyond our horizon - space-time could well be quite different out there.

http://map.gsfc.nasa.gov/m_uni/uni_101bb2.html

More specifically in response to the statement that:

But in the Friedmann-Robertson-Walker model of a flat or open universe, there would be no upper limit on d.

Because FRW pertains to the "observable universe", there would be a limit on d, because the observable universe is finite, although it does not have an edge. FRW does not model a universe of infinite dimension.

Most cosmologists agree that the observable universe is well approximated by an almost FLRW model, that is, a model which follows the FLRW metric apart from primordial density fluctuations. In a strictly FLRW model, there are no clusters of galaxies, stars or people, since these are objects much denser than a typical part of the universe.

However, for brevity, the almost FLRW model is often referred to simply as the FLRW model (or the FRW model).

http://en.wikipedia.org/wiki/Friedmann-Lema%EEtre-Robertson-Walker

And with regard to this statement:

I think you are fundamentally misunderstanding how the "conventional Big Bang scenario" (ie the Friedmann-Roberston-Walker model) actually works. There is no central position in space where the explosion originated, so the notion of radiation "outpacing everything else" doesn't make sense--rather, at all finite times matter and energy are evenly distributed throughout all of space, so there is no empty region where radiation hasn't reached yet.

While I would agree that there is no "center" where the explosion originated (or more usefully, due to the "stretching of space" we are all still in the center), the speed of radiation does define the distance of the big bang "horizon" which defines the "observable universe" which is the object described by FRW. Put another way, there is no "d" in the universe described by FRW at the present time with a length of 50 billion light years.

See also: http://odin.physastro.mnsu.edu/~eskridge/astr225/week14.html

As I understand it, even in FRW with a flat topology, "stuff" only exists out to the "horizon" even though "stuff" will expand infinitely with the horizon.

Neither FRW, nor any mainstream Big Bang theory makes any statement about the "Universe" as opposed to the "universe".

 

[turbo]

This is a tough crowd, JesseM, with lots of moves. I would like to try one more time with pure logic - stated very simply so questions regarding semantics cannot cloud the issue.

The standard model assumes that the Big Bang occurred ~13.7Gy ago, and it attributes three very basic qualities to the universe, that it is homogeneous and isotropic, and that there is NO privileged or special frame of reference in this universe. These are non-controversial aspects of the standard model, and I will confine the logical proof to these qualities.

Stipulation 1: We observe ourselves and our surroundings, including things beyond Earth. We are Observer "A".

Stipulation 2: Due to the finite speed of light, we see things as they were when light impinging our instruments left those objects. For instance, we see a star 10 Ly distant as it was 10 years ago. If it goes nova NOW, we will not know it for 10 more years. The most mature point in our observable universe is right here, in the very center of our observable universe, 13.7Gy from the surface of last scattering as it appears to us.

Stipulation 3: Judging from their redshifts, we see some distant objects as they were 13Gy ago, less than a billion years after the surface of last scattering.

Now for the logical proof:
Choose a quasar or galaxy at an apparent distance of 13Gly. Given the concordance assumptions of homogeneity, isotropy, and no special frame of reference, what can we say with certainty about a theoretical observer "B" who exists at that distant position right NOW?

We can say:

1. Since the universe is homogeneous and isotropic, and because "B's" frame of reference is no more or less special than ours, our theoretical observer looks out at his universe and sees a universe that is identical in its basic qualities to the one we see. He sees his own neighborhood, and due to the finite speed of light, he sees distant objects as they appeared in the past. Like us, he can only see objects out to about 13 billion light years distant. Anything much further, and he is looking at his surface of last scattering, just like we look out at our own. Just like us, "B" has a visible universe about 27 billion light years in diameter. We are on one edge of his visible universe, just as he is on one edge of our visible universe.

2. Over half of the volume of our visible universe (a ~27Gly diameter sphere) is outside the observer "B's" visible universe and is invisible to him. Over half the volume of "B's" visible universe is outside our visible universe and cannot be detected by us. It may help to imagine these visible universes as a pair of interconnected spheres that overlap one another just a bit more than one radius (~13.7Gy)

3. If observer "B" looks in the direction opposite that of our galaxy, he will be able to see other galaxies ~13Gly distant, and a hypothetical observer "C" in one of those galaxies will be able to look out and see a universe that is identical in its basic properties to the universes that observers "A" and "B" see. This is guaranteed by the three basic properties of the BB universe assumed in the introduction. Except for a very tiny intersecting volume centered on the location of observer "B", no part of the visible universe of observer "C" is in our visible universe (we are at observer position "A"), and except for same that tiny (lenticular, obviously) slice of space, observer "C" can see no part of our visible universe.

4. In a BB universe that is homogeneous, isotropic, and devoid of preferred reference frames, this logical iteration can be carried out forever, projecting to an infinite number of "visible universes" each centered on a unique observer. Therefore, if the BB universe is flat or open (and most adherents of standard cosmology are solidly wedded to flat at a minimum, and perhaps open), it must also be spacially infinite.

This is a logical proof derived from the principles of the standard model. I would attempt to simplify it further, but refrain for fear of loss of coherence.

How could the BB universe possibly be finite? To model a finite BB universe, either at least one the three assumptions made by the standard model about the basic qualities of the universe must be wrong, OR the universe must assume a complex topology that somehow both keeps the universe flat/Euclidean locally AND bends space in such a way that one can set off in one direction and come back upon one's previous location without deviating from a straight path. Such theoretical topologies are apparently not falsifiable by any means, and absent any compelling reason to embrace them (apart from sheer revulsion at the thought of infinities ) there is presently no need to regard them as anything more than mathematical curiosities.

I welcome any logical refutation of this proof. "Carpet-bombing" this post with citations that do not address the logic of the proof and simple nay-saying will be cheerfully ignored. Is there a logical failure in this proof? I would love to see it.

 

[JesseM]

ohwilleke, you are conflating two different issues--one is what the FRW model says about spacetime as a whole, the other is to what extent observational evidence can tell us which model is the best one for the actual universe. The FRW models describe spacetime as a whole, it's just that we can never really be sure which model actually describes the real world. For example, in the FRW model of an open universe with zero cosmological constant, the universe will expand forever (see this page of Ned Wright's cosmology tutorial, for example); however, even if we find that the observable universe has overall negative curvature and zero cosmological constant, that doesn't prove that our universe really will expand forever, because we could just be in a local region that has negative curvature, while most of the rest of the universe has positive curvature, so that the universe will actually collapse at some finite future time. If the FRW models didn't apply to the universe as a whole, you wouldn't be able to say anything for certain about what the models predict about the long-term future of the universe.

On p. 135 of The Large-Scale Structure of Spacetime by Stephen Hawking and George Ellis they explain that the FRW models assume the universe is isotropic at every point in space, so that there is exact spherical symmetry about every point:

It is possible to write down and examine the metrics of all space-times which are spherically symmetric; particular examples are the Schwarzschild and Reissner-Nordstrom solutions (see 5.5); however these are asymptotically flat spaces. In general, there can exist at most two points in a spherically symmetric space from which the space looks spherically symmetric. While these may serve as models of space-time near a massive body, they can only be models of the universe consistent with the isotropy of our observations if we are located near a very special position. The exceptional cases are those in which the universe is isotropic about every point in space time; so we shall interpet the Copernican principle as stating that the universe is approximately spherically symmetric about every point (since it is approximately spherically symmetric about us).

As has been shown by Walker (1944), exact spherical symmetry about every point would imply that the universe is spatially homogeneous and admits a six-parameter group of isometries whose surfaces of transitivity are spacelike three-surfaces of constant curvature. Such a space is called a Robertson-Walker (or Friedmann) space (Minkowski space, de Sitter space and anti-de Sitter space are all special cases of the general Robertson-Walker spaces). Our conclusion, then, is that these spaces are a good approximation to the large scale geometry of space-time in the region that we can observe.

Note that last "in the region that we can observe"--they are making no claim that the real universe matches the model outside the region that we can observe. But the model itself assumes a universe which is spatially homogeneous everywhere, and for a flat or open universe with the simplest topology, this must mean the model is of a spatially infinite universe with matter/energy distributed evenly throughout all of space.

Here are some pages which say that in the FRW model with zero cosmological constant, a universe with zero or negative curvature is spatially infinite:

http://www.phys-astro.sonoma.edu/people/faculty/tenn/FriedmannModels.html
http://tinyurl.com/68beu
http://www.321books.co.uk/encyclopedia/cosmology/standard-models.htm
http://www.jb.man.ac.uk/~jpl/cosmo/RW.html

This is also discussed on pages 724-725 of Gravitation by Misner, Thorne and Wheeler, where they show the metric of each "hypersurface of homogeneity" (a way of slicing spacetime into spacelike hypersurfaces such that the universe is spatially homogeneous in each hypersurface) for a flat and open universe, and conclude in both cases that "the volume of the hypersurface is infinite". They do qualify this with a remark about the topology though:

Warning: Although the demand for homogeneity and isotropy determines completely the local geometric properties of a hypersurface of homogeneity up to the single disposable factor K, it leaves the global topology of the hypersurface undetermined. The above choices of topology are the most straightforward. But other choices are possible.

They then go on to show how a flat universe with the topology of a torus wouldn't have to be infinite in volume.

One final example is Roger Penrose's book The Emperor's New Mind where on p. 321 he writes:

This expanding balloon provides quite a good picture of one of the three standard so-called Friedmann-Robertson-Walker (FRW) models of the universe--namely the spatially closed positively curved FRW-model. In the other two FRW-models (zero or negative curvature), the universe expands in the same sort of way, but instead of having a spatially finite universe, as the surface of the balloon indicates, we have an infinite universe with an infinite number of galaxies.

As I understand it, even in FRW with a flat topology, "stuff" only exists out to the "horizon" even though "stuff" will expand infinitely with the horizon.

Neither FRW, nor any mainstream Big Bang theory makes any statement about the "Universe" as opposed to the "universe".

The models do apply to the "Universe" as a whole. But physicists don't claim there is any scientific evidence to favor the hypothesis that one of these models is actually the correct one for our own "Universe" as a whole, the most we can do is say that our local region (our 'universe') resembles one model or another.

 

[JesseM]

This is a tough crowd, JesseM, with lots of moves. I would like to try one more time with pure logic - stated very simply so questions regarding semantics cannot cloud the issue.

The standard model assumes that the Big Bang occurred ~13.7Gy ago, and it attributes three very basic qualities to the universe, that it is homogeneous and isotropic, and that there is NO privileged or special frame of reference in this universe. These are non-controversial aspects of the standard model, and I will confine the logical proof to these qualities.

Stipulation 1: We observe ourselves and our surroundings, including things beyond Earth. We are Observer "A".

Stipulation 2: Due to the finite speed of light, we see things as they were when light impinging our instruments left those objects. For instance, we see a star 10 Ly distant as it was 10 years ago. If it goes nova NOW, we will not know it for 10 more years. The most mature point in our observable universe is right here, in the very center of our observable universe, 13.7Gy from the surface of last scattering as it appears to us.

Stipulation 3: Judging from their redshifts, we see some distant objects as they were 13Gy ago, less than a billion years after the surface of last scattering.

Now for the logical proof:
Choose a quasar or galaxy at an apparent distance of 13Gly. Given the concordance assumptions of homogeneity, isotropy, and no special frame of reference, what can we say with certainty about a theoretical observer "B" who exists at that distant position right NOW?

We can say:

1. Since the universe is homogeneous and isotropic, and because "B's" frame of reference is no more or less special than ours, our theoretical observer looks out at his universe and sees a universe that is identical in its basic qualities to the one we see. He sees his own neighborhood, and due to the finite speed of light, he sees distant objects as they appeared in the past. Like us, he can only see objects out to about 13 billion light years distant. Anything much further, and he is looking at his surface of last scattering, just like we look out at our own. Just like us, "B" has a visible universe about 27 billion light years in diameter. We are on one edge of his visible universe, just as he is on one edge of our visible universe.

2. Over half of the volume of our visible universe (a ~27Gly diameter sphere) is outside the observer "B's" visible universe and is invisible to him. Over half the volume of "B's" visible universe is outside our visible universe and cannot be detected by us. It may help to imagine these visible universes as a pair of interconnected spheres that overlap one another just a bit more than one radius (~13.7Gy)

This assumption is not necessarily correct, because it may be that due to the topology of the universe, the right edge of B's observable universe overlaps with the left edge of our visible universe. Or it may be that B sees C's galaxy, and C sees D's galaxy, and D sees our galaxy. In the FRW model, if the universe has positive curvature then space must eventually loop around this way (as an analogy, just think of observers on a globe); if the universe has zero or negative curvature, it can still loop around this way if it has a nontrivial topology (think of the video game 'asteroids' which I mentioned earlier).

3. If observer "B" looks in the direction opposite that of our galaxy, he will be able to see other galaxies ~13Gly distant

Yes, but these galaxies may be the same galaxies that we see when we look in the direction opposite to B.

4. In a BB universe that is homogeneous, isotropic, and devoid of preferred reference frames, this logical iteration can be carried out forever, projecting to an infinite number of "visible universes" each centered on a unique observer. Therefore, if the BB universe is flat or open (and most adherents of standard cosmology are solidly wedded to flat at a minimum, and perhaps open), it must also be spacially infinite.

As I explained above, this doesn't logically follow. The universe can be flat, homogeneous, isotropic and devoid of preferred reference frames, but can still only have a finite volume if it has a nontrivial topology.

How could the BB universe possibly be finite? To model a finite BB universe, either at least one the three assumptions made by the standard model about the basic qualities of the universe must be wrong, OR the universe must assume a complex topology that somehow both keeps the universe flat/Euclidean locally AND bends space in such a way that one can set off in one direction and come back upon one's previous location without deviating from a straight path. Such theoretical topologies are apparently not falsifiable by any means

No, but it would be possible to have positive evidence for these topologies, if the radius of the universe is smaller than the radius of the observable universe. Conversely, there is no way to have positive evidence for the simplest topology, and it can only be falsified if we find positive evidence for one of these other topologies.

and absent any compelling reason to embrace them (apart from sheer revulsion at the thought of infinities ) there is presently no need to regard them as anything more than mathematical curiosities.

There is also no need to regard an infinite-volume universe with the simplest topology as anything more than a mathematical curiosity. Only philosophical prejudices should lead us to prefer one over the other, as long as there is no evidence whatsoever for or against either one.

 

[Chronos]

This is a tough crowd, JesseM, with lots of moves. I would like to try one more time with pure logic - stated very simply so questions regarding semantics cannot cloud the issue.

The standard model assumes that the Big Bang occurred ~13.7Gy ago, and it attributes three very basic qualities to the universe, that it is homogeneous and isotropic, and that there is NO privileged or special frame of reference in this universe. These are non-controversial aspects of the standard model, and I will confine the logical proof to these qualities.

Stipulation 1: We observe ourselves and our surroundings, including things beyond Earth. We are Observer "A".

Stipulation 2: Due to the finite speed of light, we see things as they were when light impinging our instruments left those objects. For instance, we see a star 10 Ly distant as it was 10 years ago. If it goes nova NOW, we will not know it for 10 more years. The most mature point in our observable universe is right here, in the very center of our observable universe, 13.7Gy from the surface of last scattering as it appears to us.

Stipulation 3: Judging from their redshifts, we see some distant objects as they were 13Gy ago, less than a billion years after the surface of last scattering.

Now for the logical proof:
Choose a quasar or galaxy at an apparent distance of 13Gly. Given the concordance assumptions of homogeneity, isotropy, and no special frame of reference, what can we say with certainty about a theoretical observer "B" who exists at that distant position right NOW?

We can say that observer B's 'now', is in our future.

1. Since the universe is homogeneous and isotropic, and because "B's" frame of reference is no more or less special than ours, our theoretical observer looks out at his universe and sees a universe that is identical in its basic qualities to the one we see. He sees his own neighborhood, and due to the finite speed of light, he sees distant objects as they appeared in the past.

He sees distant objects as they appeared to be in his past, not ours.

Like us, he can only see objects out to about 13 billion light years distant.

Incorrect. Observer B's universe is younger and smaller

and he is looking at his surface of last scattering, just like we look out at our own. Just like us, "B" has a visible universe about 27 billion light years in diameter.

By the time we receive observer B's report on the size of the universe, it will be 12 billion years older than we perceive it to be.

We are on one edge of his visible universe, just as he is on one edge of our visible universe.

But we are in his future and he is in our past. There is no simulataneity.

2. Over half of the volume of our visible universe (a ~27Gly diameter sphere) is outside the observer "B's" visible universe and is invisible to him.

Not according to his reference frame. You are imposing your reference frame on his reference frame using your 'here and now' coordinate system. That is invalid.

Over half the volume of "B's" visible universe is outside our visible universe and cannot be detected by us. It may help to imagine these visible universes as a pair of interconnected spheres that overlap one another just a bit more than one radius (~13.7Gy)

It will not help. B's visible universe is not observable until I arrive at his planet.

3. If observer "B" looks in the direction opposite that of our galaxy, he will be able to see other galaxies ~13Gly distant, and a hypothetical observer "C" in one of those galaxies will be able to look out and see a universe that is identical in its basic properties to the universes that observers "A" and "B" see. This is guaranteed by the three basic properties of the BB universe assumed in the introduction. Except for a very tiny intersecting volume centered on the location of observer "B", no part of the visible universe of observer "C" is in our visible universe (we are at observer position "A"), and except for same that tiny (lenticular, obviously) slice of space, observer "C" can see no part of our visible universe.

4. In a BB universe that is homogeneous, isotropic, and devoid of preferred reference frames, this logical iteration can be carried out forever, projecting to an infinite number of "visible universes" each centered on a unique observer. Therefore, if the BB universe is flat or open (and most adherents of standard cosmology are solidly wedded to flat at a minimum, and perhaps open), it must also be spacially infinite.

This is a logical proof derived from the principles of the standard model. I would attempt to simplify it further, but refrain for fear of loss of coherence.

How could the BB universe possibly be finite? To model a finite BB universe, either at least one the three assumptions made by the standard model about the basic qualities of the universe must be wrong, OR the universe must assume a complex topology that somehow both keeps the universe flat/Euclidean locally AND bends space in such a way that one can set off in one direction and come back upon one's previous location without deviating from a straight path. Such theoretical topologies are apparently not falsifiable by any means, and absent any compelling reason to embrace them (apart from sheer revulsion at the thought of infinities ) there is presently no need to regard them as anything more than mathematical curiosities.

I welcome any logical refutation of this proof. "Carpet-bombing" this post with citations that do not address the logic of the proof and simple nay-saying will be cheerfully ignored. Is there a logical failure in this proof? I would love to see it.

Your argument is fundamentally flawed. Simultaneity does not exist between our reference frames. By the time observer B travels to Earth to reveal his findings, the universe will be 12 billion years older [not counting inflation].

 

[JesseM]

We can say that observer B's 'now', is in our future.He sees distant objects as they appeared to be in his past, not ours.

His now is not in our future light cone, since there is a spacelike separation between us...do you just mean we won't receive information about his now until the future? Of course, if the universe has negative curvature, we may never receive information about his observations.

Incorrect. Observer B's universe is younger and smaller

If a "hypersurface of homogeneity" (as described in the section of Gravitation I quoted from in my last post to ohwilleke) contains both the event of me making my observation and B making his observation, then each of us should observe the universe to be the same size and age.

 

[Chronos]

His now is not in our future light cone, since there is a spacelike separation between us...do you just mean we won't receive information about his now until the future?

My 'now' will always be in his future light, until we meet. And yes, I do mean I won't receive information from his 'now' until my future.

Of course, if the universe has negative curvature, we may never receive information about his observations. If a "hypersurface of homogeneity" (as described in the section of Gravitation I quoted from in my last post to ohwilleke) contains both the event of me making my observation and B making his observation, then each of us should observe the universe to be the same size and age.

But you will not agree when you made that observation.

 

[JesseM]

My 'now' will always be in his future light, until we meet. And yes, I do mean I won't receive information from his 'now' until my future.

Huh? The event of you making your observation is not in the future light cone of the event of his making his own observation, there is a spacelike separation between these two events. What do you mean by the phrase "My 'now'", anyway? That phrase doesn't seem to refer to a unique event (a unique point in spacetime), since every point along my worldline is called "now" by the version of me at that point.

But you will not agree when you made that observation.

Sure we will, there is only a single unique way to slice up spacetime into a series of "hypersurfaces of homogeneity", and all observers will agree on this unique slicing (foliation). Thus all observers will agree on whether two events occur in the same hypersurface of homogeneity or two different hypersurfaces of homogeneity. And this particular foliation can be used to define a global notion of time (although of course you could also slice up spacetime in such a way that each slice was not spatially homogeneous, and this would define a different global time-coordinate with a different notion of simultaneity).

If you just mean that I won't know about the results of his observation until after I make my observation, and he won't know about the results of my observation until after he makes his, I agree. But in relativity "simultaneity" doesn't have anything to do with when you actually learn about a given pair of events, it just has to do with what time-coordinates you assign the events in retrospect.

 

[turbo]

Only philosophical prejudices should lead us to prefer one over the other, as long as there is no evidence whatsoever for or against either one.

Good science should lead us to prefer a simple topology over a non-trivial (i.e. complex, manifold, etc) one. Alternate topologies cannot be seriously considered unless they can be falsified by some means. If I were to tell you that a "demon" will intervene in your linear path through the universe and bring you back to your original location, you would scoff and dismiss the idea. If I couch the idea in mathematical possibilities, you will likely be a bit more receptive, but good science should prompt you to ask me to predict what effects we might see if the complex topology is real. If I cannot give you testable predictions, you should not waste time pursuing that model. If it cannot be falsified, it is no more scientifically significant than the idea that "demons" intervene.

Mathemeticians may be able to demonstrate Euclidean geometries in complex n-diminesional frames, but that ability does not address the likelihood nor even the possibility that our universe can assume the topologies of their models.

 

[turbo]

This assumption is not necessarily correct, because it may be that due to the topology of the universe, the right edge of B's observable universe overlaps with the left edge of our visible universe. Or it may be that B sees C's galaxy, and C sees D's galaxy, and D sees our galaxy. In the FRW model, if the universe has positive curvature then space must eventually loop around this way (as an analogy, just think of observers on a globe); if the universe has zero or negative curvature, it can still loop around this way if it has a nontrivial topology (think of the video game 'asteroids' which I mentioned earlier). Yes, but these galaxies may be the same galaxies that we see when we look in the direction opposite to B. As I explained above, this doesn't logically follow. The universe can be flat, homogeneous, isotropic and devoid of preferred reference frames, but can still only have a finite volume if it has a nontrivial topology. No, but it would be possible to have positive evidence for these topologies, if the radius of the universe is smaller than the radius of the observable universe. Conversely, there is no way to have positive evidence for the simplest topology, and it can only be falsified if we find positive evidence for one of these other topologies. There is also no need to regard an infinite-volume universe with the simplest topology as anything more than a mathematical curiosity. Only philosophical prejudices should lead us to prefer one over the other, as long as there is no evidence whatsoever for or against either one.

JesseM, each of your objections addresses a possible complex topology that may allow looping. Assume a trivial topology in a flat or open universe and parse the proof in post #41. I do not believe you can find fault with it, although I would be delighted to learn something deeper from this exercise.

 

[turbo]

We can say that observer B's 'now', is in our future.He sees distant objects as they appeared to be in his past, not ours.Incorrect. Observer B's universe is younger and smallerBy the time we receive observer B's report on the size of the universe, it will be 12 billion years older than we perceive it to be.But we are in his future and he is in our past. There is no simulataneity.Not according to his reference frame. You are imposing your reference frame on his reference frame using your 'here and now' coordinate system. That is invalid.It will not help. B's visible universe is not observable until I arrive at his planet.Your argument is fundamentally flawed. Simultaneity does not exist between our reference frames. By the time observer B travels to Earth to reveal his findings, the universe will be 12 billion years older [not counting inflation].

I have lumped all your quotes because they carry a common theme - you have invalidated the reference frames of observers "B" and "C" by imposing the limitiations of our reference frames upon them. This is not acceptable as per the "no special frame of reference".

If we see ourselves as existing at a time 13.7Gy after the BB, then every other presently-existing observer in the BB universe will see themselves as existing at a time 13.7Gy after the BB. The fact that we may not become aware of the existence of such a theoretical observer until we travel to his planet or EM from his location reaches us is irrelevant. EVERY observer presently existing in a BB universe sees himself as existing 13.7Gy after the BB. The fact that our observable universes may or may not intersect is of no consequence to the ultimate validity of the respective reference frames of each of the observers. If every reference frame is equally valid, every presently-existing observer in the BB universe will see essentially what we do, except in the minor details (the homogeneous and isotropic nature of the universe demands it).

 

[whatzzupboy]

Ok could it be that our nearow range of undestanding of our universe be doing to us now what it did to us when Christopher Columbus said the world is round? Oh and what makes the border of our universe and another?

 

[JesseM]

Good science should lead us to prefer a simple topology over a non-trivial (i.e. complex, manifold, etc) one. Alternate topologies cannot be seriously considered unless they can be falsified by some means. If I were to tell you that a "demon" will intervene in your linear path through the universe and bring you back to your original location, you would scoff and dismiss the idea. If I couch the idea in mathematical possibilities, you will likely be a bit more receptive, but good science should prompt you to ask me to predict what effects we might see if the complex topology is real. If I cannot give you testable predictions, you should not waste time pursuing that model. If it cannot be falsified, it is no more scientifically significant than the idea that "demons" intervene.

This is all just philosophical prejudice, not science. Someone else might find the mathematical model of an infinite universe as philosophically unsettling as you apparently find the mathematical model of a universe with an unusual topology; perhaps this person might make a similar argument against it by imagining a "demon" who keeps on creating new lands as you roam the Earth so that it appears to you that no matter how far you travel you never return to your starting point, and then he might say "If I couch the idea in mathematical possibilities, you will likely be a bit more receptive, but good science should prompt you to ask me to predict what effects we might see if the universe is really infinite". What is the difference between your argument and his? In both cases, you are asking us to rule out one of two theories which make precisely the same physical predictions, based on verbal arguments and appeals to intuition which have nothing whatsoever to do with science.

JesseM, each of your objections addresses a possible complex topology that may allow looping. Assume a trivial topology in a flat or open universe and parse the proof in post #41. I do not believe you can find fault with it, although I would be delighted to learn something deeper from this exercise.

No, of course I agree that with the simplest topology a flat or open homogeneous universe would have to be infinite, but I don't think there's anyone who disagrees with this.

 

[setAI]

This is all just philosophical prejudice, not science. Someone else might find the mathematical model of an infinite universe as philosophically unsettling as you apparently find the mathematical model of a universe with an unusual topology; perhaps this person might make a similar argument against it by imagining a "demon" who keeps on creating new lands as you roam the earth

untrue- these models are not simply differentiated by philosophy- they get to the heart of the Scientific method and Occam’s Razor-

the difference is this:

if the observable universe is continuous and flat-

a conjecture that it is spatially infinite requires NO UNOBSERVED MECHANISMS/ENTITIES- it simply posits that what you see is what you get and that the universe continues beyond the horizon of observable space- there is no reason to suggest that a demon creates new space- since observation shows that space is continuous and isotropic and never curves back on itself there is nothing to prevent it from being infinite- no boundary conditions are observed or implied by observations

a conjecture that states that the Universe is finite REQUIRES such an unobserved mechanism/entity to create a boundary condition!- it requires nontrivial topology- or a demon that makes you go back- or some form of boundary- yes these mechanisms are certainly possible- but they have no observational evidence- you had to invent a mechanism to provide a finite boundary that is entirely hypothetical: an epicycle to fit the observed with ones own notions-

 

[JesseM]

untrue- these models are not simply differentiated by philosophy- they get to the heart of the Scientific method and Occam’s Razor-

the difference is this:

if the observable universe is continuous and flat-

a conjecture that it is spatially infinite requires NO UNOBSERVED MECHANISMS/ENTITIES- it simply posits that what you see is what you get and that the universe continues beyond the horizon of observable space- there is no reason to suggest that a demon creates new space- since observation shows that space is continuous and isotropic and never curves back on itself there is nothing to prevent it from being infinite- no boundary conditions are observed or implied by observations

a conjecture that states that the Universe is finite REQUIRES such an unobserved mechanism/entity to create a boundary condition!- it requires nontrivial topology- or a demon that makes you go back- or some form of boundary- yes these mechanisms are certainly possible- but they have no observational evidence- you had to invent a mechanism to provide a finite boundary that is entirely hypothetical: an epicycle to fit the observed with ones own notions-

No, a nontrivial topology doesn't require a new "mechanism" or "boundary conditions", any more than the Earth requires a "mechanism" to insure that if you walk far enough along its surface you'll return to your point of origin--in both cases, it's just a question of shape. Don't be misled by the fact that I've chosen to call other topologies "nontrivial"--fundamentally, it's just a choice between different possible shapes, there's no clear reason why one is any more "complicated" than any other aside from some arbitrary human intuitions. (do you think a round Earth is a more complex hypothesis than an infinite flat earth? In the absence of evidence for either one, should we automatically prefer the flat hypothesis?)

 

[dhearn]

I'm no expert here, but it seem to me that referring to another vantage point at the edge of our observable universe has no real meaning according to GR. If the universe becomes non-observable because of some horizon past which objects are moving away from us faster than the speed of light, then there is no horizon. Objects in any reference frame cannot move faster that the speed of light wrt any other reference frame.

Let me try to explain what I mean. If we take our theoretical galaxy (observer B) who is at a distance such that his frame of reference is moving away from us at .99999... times the speed of light, and an observer (C) at a distance such that he is moving away from B at .99999... times the speed of light then C will be moving away from us at .99999999... (a few more 9's than B was) we can still see C. There is no way for C to be moving faster than the speed of light away from us. If C can be causally related to B and B can be causally related to us (A) then there is no other possibility than C can be causally related to us. It's like a closed set. We are all in the same observable universe no matter where we are in it. That's how the math works out.

Observers B and C would actually seem to be right next to each other since, at that rate of speed, the length dilation would be so great.

That is the thing about GR that took me the longest to get. If you want to go to alpha centari (which is 4 LYs away) you can only go at speeds lower than the speed of light. That does not mean that you couldn't get there in 5 hour, because you could. You just have to go at such a speed that the length dilation shrinks the distance to less than 5 light-hours. Forget about getting back, you would also go through time dilation and that would play havoc on your return time.

The same thing applies to B and C. They appear to be moving at near the speed of light away from us. If we could measure their velocities we would calculate that they are moving away from each other at very near the speed of light, but they would both be visible to us no matter how far away from us they were.

None of this has anything to do with the edge of the expanding universe as it relates to the BB. No matter how far the edge is from us, we can see it (theoretically of course). If the question is what lies beyond the shockwave of the big bang, then I like the answer someone else gave as the future.

Please feel free to correct me if I am missing something, but this is what the math says to me.

 

[Chronos]

I have lumped all your quotes because they carry a common theme - you have invalidated the reference frames of observers "B" and "C" by imposing the limitiations of our reference frames upon them. This is not acceptable as per the "no special frame of reference".

I have not invalidated anyones reference frame, quite the contrary. I have affirmed we can only communicate at the speed of light. It is acceptable because they can only tell me what they observed up to the instant the signal was sent. And what they saw was in my past, not my present.

If we see ourselves as existing at a time 13.7Gy after the BB, then every other presently-existing observer in the BB universe will see themselves as existing at a time 13.7Gy after the BB. The fact that we may not become aware of the existence of such a theoretical observer until we travel to his planet or EM from his location reaches us is irrelevant.

It's not irrelevant after the 12 billion years it takes to get there.

EVERY observer presently existing in a BB universe sees himself as existing 13.7Gy after the BB. The fact that our observable universes may or may not intersect is of no consequence to the ultimate validity of the respective reference frames of each of the observers. If every reference frame is equally valid, every presently-existing observer in the BB universe will see essentially what we do, except in the minor details (the homogeneous and isotropic nature of the universe demands it).

And none of them can communicate what they see instantaneously.

 

[Garth]

I'm no expert here, but it seem to me that referring to another vantage point at the edge of our observable universe has no real meaning according to GR. If the universe becomes non-observable because of some horizon past which objects are moving away from us faster than the speed of light, then there is no horizon. Objects in any reference frame cannot move faster that the speed of light wrt any other reference frame.

Let me try to explain what I mean. If we take our theoretical galaxy (observer B) who is at a distance such that his frame of reference is moving away from us at .99999... times the speed of light, and an observer (C) at a distance such that he is moving away from B at .99999... times the speed of light then C will be moving away from us at .99999999... (a few more 9's than B was) we can still see C. There is no way for C to be moving faster than the speed of light away from us. If C can be causally related to B and B can be causally related to us (A) then there is no other possibility than C can be causally related to us. It's like a closed set. We are all in the same observable universe no matter where we are in it. That's how the math works out.
/////////////////////////////////
Please feel free to correct me if I am missing something, but this is what the math says to me.

Hi! and welcome to these Forums dhearn!

Actually, "Please feel free to correct me if I am missing something", you are! Velocities between widely separated objects in GR cosmology are not the same as Lorentzian transformations for boost in SR. It is quite possible for objects to be moving apart with mutual velocities greater than c, because it is space-time that is expanding rather than their peculiar motion within space-time. Astonishingly enough you can actually see objects that are receeding from us at speeds greater than c, although that does depend how you define the velocity of an object at a cosmological distance. There has been a thread discussing this on these Forums.

Garth

 

[JesseM]

I'm no expert here, but it seem to me that referring to another vantage point at the edge of our observable universe has no real meaning according to GR. If the universe becomes non-observable because of some horizon past which objects are moving away from us faster than the speed of light, then there is no horizon. Objects in any reference frame cannot move faster that the speed of light wrt any other reference frame.

Let me try to explain what I mean. If we take our theoretical galaxy (observer B) who is at a distance such that his frame of reference is moving away from us at .99999... times the speed of light, and an observer (C) at a distance such that he is moving away from B at .99999... times the speed of light then C will be moving away from us at .99999999... (a few more 9's than B was) we can still see C. There is no way for C to be moving faster than the speed of light away from us. If C can be causally related to B and B can be causally related to us (A) then there is no other possibility than C can be causally related to us. It's like a closed set. We are all in the same observable universe no matter where we are in it. That's how the math works out.

You are thinking purely in SR terms. In GR, the space between ourselves and a distant galaxy can be expanding at such a rate that the distance between ourselves and that galaxy can be increasing faster than light, at least in Hubble coordinates--see this page of Ned Wright's cosmology tutorial, for example. Wright points out that we can also plot the movement of distant galaxies in special relativistic coordinates where nothing can move faster than light, but these coordinates are pretty counterintuitive since even in a universe with an infinite number of equally-spaced galaxies ('equally-spaced' in the sense that if two observers took off at the same speed from one galaxy in opposite directions, they'd measure the same amount of time to reach the neighboring galaxy in that direction), every galaxy would be a finite coordinate distance from every other galaxy, and more distant galaxies would be packed closer and closer together in these coordinates. In Hubble coordinates, I'm pretty sure that equally-spaced galaxies would always have equal coordinate distance between them.

edit: I see Garth already addressed the issue of space expanding faster than light in GR...

 

[JesseM]

It's not irrelevant after the 12 billion years it takes to get there.

It is irrelevant, because turbo-1 never said anything about B sharing his observations with us, nor did he say anything about B traveling to meet us. In general relativity you can choose a coordinate system where two events can happen at the "same time" even though no observer will ever have knowledge about both events, because they lie outside one another's event horizons due of the expansion of the universe.

 

[turbo]

It is irrelevant, because turbo-1 never said anything about B sharing his observations with us, nor did he say anything about B traveling to meet us. In general relativity you can choose a coordinate system where two events can happen at the "same time" even though no observer will ever have knowledge about both events, because they lie outside one another's event horizons due of the expansion of the universe.

Thank you JesseM - you beat me to the punch. Requiring communication (either one-way or two-way) between B or C's reference frame and ours to "validate" B's or C's observations is absurd. It violates the "no special frame of reference" rule. An observer presently existing at the location that we percieve from it's ME emitted 12Gy ago currently sees a BB universe that is about 13.7Gy old, just as we do.

We cannot rashly generalize about size of the BB Universe from our observations of the "visible universe" with its limitation of the speed of light. If the expansion of the BB universe is accelerating, as some propose, we should expect that objects will disappear from visible universe, with the most distant objects disappearing from view one by one. (And before the chorus starts, NO, they won't wink out and disappear with a Poof!, but they will be redshifted into indetectability.) Will we then deny that those objects ever existed because they are no longer in our visible universe? Not likely.