meteor said:
I really think that is not a coincidence that the value of omega is so close to 1. It really has to be 1!
I sympathize very much about this. It is so close that it seems as if it really should be exactly 1.
But there may be some other explanation, something in somebody's inflation scenario, something speculative that might later be verified.
Maybe future cosmologists will find some reason why, even though it is not exactly 1, it is force to be very close.
IIRC the apparent flatness was one of the motivations for Guth to invent inflation scenarios. It would be an explanation, he hoped, for why it is so close to spatially flat.
it is all too frustrating. We humans do not know enough yet. though I half-believe and want it to be flat, I must not let myself believe this, because it is not known
But why flatness implies infiniteness? Why can't the universe suddenly stops at a certain distance?
the unspoken assumption is that it is a differentiable manifold---the kind of model for space given us by Riemann around 1840-1850.
It is in that context that we say "curvature = zero" and "flat". the idea of manifold gives these terms meaning and it is the basic model in cosmology.
a 3D flat manifold must be infinite, or it must be analogous to a donut, in some sense.
Manifolds have been "classified" by mathematicians. there are only limited possibilities. If it does not loop around and join itself
The donut I am thinking of is a flat piece of paper with the right edge "identified" with the left edge. and the top edge with the bottom. all the metric geometry stuff you do on the paper is familiar flat euclidean---except when the pencil runs off the right ede it reappears on the left etc.
that is what i mean by a donut and there is a flat compact 3D space analogous to that.
let us call it a "toroidal" flat 3D space
the mathematicians prove to their complete satisfaction that if a 3D manifold is flat it must be toroidal or infinite.
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whoah! you also want to consider manifolds with BOUNDARY. I was not thinking of manifolds that have some kind of edge. that makes it more complicated. I cannot picture the universe having an edge.
maybe someone else has considered that case. normally I don't think one does but maybe someone can talk about it. I am limited to discussing universes that don't have any boundary.
so positive curvature, for me, means like a sphere (and it can come from a point initially I guess)
and zero curvature, in my simple view, means toroidal or more likely infinite euclidean 3D space (I can't see either one coming from a point)
I agree that a universe that started in a point singularity, and being flat now, cannot be infinite. But all the articles and books that I've read declare:
a)the universe started in a point singularity
or
b)the universe started in a singularity
no one says "the universe started in an infinite volume singularity".
this could be raised as a criticism of those articles and books
they talk about the universe starting at a singularity but they take for granted that the reader understands that a singularity does not have to be a point but can have extent, so they neglect to say it.
this carelessness or negligence on the authors part is a big nuisance
and causes a lot of misunderstanding
If only they would make it a custom to always remind people that a singularity can be infinite extending in various dimensions. It can be a 3D singularity, or a 1 D singularity (a long line where the equations blow up at every point on that line). Singularities that one meets in Gen Rel can be various sizes and shapes--so why can't the cosmology-writers regularly point this out to us?
maybe you would be satisfied if ONE authoritative source mentioned the possibility, I think perhaps the Ned Wright Cosmology FAQ does. Let me go look. I will be right back.
