Jeff Root said:
Hey there! This is my very first post on Physics Forums!
So the figures are all for the maximum curvature / minimum size possible
given the inability to detect any overall curvature?
That's right. There's a 95% confidence interval or error bar that includes zero. Komatsu et al take the top end, max curvature, of that interval and convert it to a "radius of curvature" for the hypersphere case. That is the most common model of finite spatial volume. (There is also a toroidal model analogous to the 2D surface of a donut, but not much discussed.)
They get a radius of curvature of about 100 billion LY. This is a lower bound.
What if there actually is no overall curvature? What can be said about the
minimum possible size in that case?
If average largescale curvature is zero then finite volume is still not ruled out, for a curious reason. The toroidal or 'pac man' case can actually be spatially flat according to the definitions. People don't consider this much or discuss it. They usually assume if curvature is zero then space extends indefinitely (like conventional 3D Euclidean). I'm certainly of that mind. If the errorbar shrinks tenfold and still contains zero, then I would tend to think of the volume as infinite.
But you have to also consider the 'pac man' case. Our space models are not embedded in any higher dimension surround space. So take a 2D analog. A 2D torus surface can be construced just by taking a 2D flat square and 'identifying' opposite edges. So if pac man runs off the east edge of the screen he reappears coming in from the west edges. A cylinder is flat. As curvature is calculated, it has zero curvature. And when you see him run off the north edge and reappear coming from the south edge then you know he lives in a flat 2D torus.
The same construction can be made with a 3D cube by identifying (joining) opposite faces.
So mathematically you can describe a zero-curvature spatial flat finite volume universe.
So then you ask
what can be said about minimum size? Well then we have papers by Neil Cornish, David Spergel and Glenn Starkman, where they looked for periodic patterns in the stars. They got some lower bound. They did not find any periodicity so they said, in effect, "if we are in a Pac Man type situation then it must be at least so and so big". Because if it weren't that big we would have noticed some repetitions. I'm probably misrepresenting somewhat here. I don't take these periodic models as seriously as my betters do. Cornish Spergel Starkman are all very major cosmology scholars. You can look up their papers on arxiv.
The short answer to your question is that I can't give you a lower bound size estimate in the flat case, but some people people have cranked one out.
Ich says it very concisely. He refers to the Toroidal model, and other periodic stuff, as having "non-trivial topology". A Pac Man universe does not literally "have holes in it" unless you embed it in some higher dimensional surround, which there is no evidence for.
But there are loops which cannot be shrunk to a point. (Loops going all the way around the donut.) This can be interpreted as "having holes" that prevent the loop from being shrunk.
Ich said:
Then the universe is infinite, except for the possibility that curvature does not necessarily dictate the shape of the universe. It could also have so-called non-trivial topolgy (with holes in it) and still be finite.
Jeff Root said:
...it is clear to me that the matter/energy that is participating in the cosmic expansion must be finite. An infinite amount of matter spread throughout an infinite volume of space could not all acquire the same properties in finite time. Clearly, everything participating in the expansion must have been in causal contact at some time,
or it would not be participating in the expansion.
I don't see how to rebut. There are philosophical problems with infinity, I guess. The fact is observational cosmologists are practical people and they like to work with the spatially flat infinite volume model. It is mathematically simple.
They don't ask "where did it ultimately come from?" They just want to fit the data.
If "Universe" means "everything participating in the expansion", and curvature does not necessarily dictate the shape of the Universe, is the minimum possible size just the size of the visible Universe?
Personally I can't answer with confidence. There are competing Occam Razor demands. Say we want a model that obeys either General Relativity or the simplest most straightforward quantization of it. We don't want some weird Baroque modification of our theory of how geometry behaves.
That means no boundaries, because GR doesn't describe what the behavior would be. If there was no existence beyond some distance...well it just doesn't make sense. The dynamics would have to be cooked up.
The simplest thing is assume uniformity, space and matter more or less the same everywhere.
Then you get a mathematical model with the fewest variables, basically very simple, with a good fit to the data. Occam is happy.
The model cosmologists use is the Friedman model---formulated around 1923 by Alex Friedman. And recently also a quantized version of the Friedman model, which when put on a computer and run, predicts a bounce. But otherwise looks the same after a small interval of time. The Friedman model comes in the space-infinite version or a positive curve space-finite version (depending on a curvature parameter, which we are still trying to determine.) In all cases the Friedman model indicates a size which is beyond what we can actually see. It is the simplest way to get a satisfactory dynamic geometry based on GR, which is well tested at accessible scales like in the solar system.
So what do you do? You either accept that the universe extends beyond what we can see or you need a more complicated, and untested, theory of gravity.
The most distant matter which we can see, according to the standard picture, is now about 45 billion lightyears from us. That means, if we could freeze expansion, and send a flash of light, the flash would take 45 billion years to reach that matter.
That matter is what the CMB is the glow from, and when it emitted the light we now receive as CMB it was over 1000 times closer, estimated 41 million lightyears.
If you want to model some universe that is radically curtailed, no existence either of space or matter beyond 41 million LY, back then...somehow expanding...it seems very hard. How would it work? Why are we at the very center? What is physics like out there at the boundary? What do "they" see, the hypothetical observers out there. To me, it seems to get thorny and unsimple.
