kev, this is good post, with points that are interesting to reply to
I see Muccasen already responded, and also that you said you had to go and would be away for a while. But I will reply to one or two things immediately.
kev said:
I have seen the balloon analogy which represents an unbounded surface to a 2 dimensional creature. That creature would assume a flat surface and no matter how far it kept going in one direction, it would neverfind an edge. However, if the balloon surface was tiled and it marked those tiles it would eventually work out that it was going round in circles and that the number of tiles (galaxies) was finite.
I am glad you are familiar with that "nearly flat" slight positive curvature alternative to the infinite case. Recently cosmology papers are using phrases like flat or nearly flat. The majority still think spatial infinite (with mass infinite of course) but they increasingly allow this S
3 finite volume case as a possibility.
I see you have a conceptual problem with the INFINITE case:
In a typical description of the evolution of the universe, it is explained that the expansion of the universe is accelerating "now", but in an earlier epoch the expansion was slowing down when gravity dominated the cosmological constant. The problem I have is that if the universe is infinite with an infinite amount of mass, then that slow down would have been impossible because every massive body would equally attracted in all directions by an equal amount of mass in all directions. There would be no preferred direction for anybody to move in.
This is a keen, insightful objection, but it fails because of General Relativity. So far GR is dominant as the theoretical basis of Cosmology because it predicts successfully to high accuracy the way no other theory of gravity does. Cosmology is based on a highly simplified model called the Friedmann equation which is derived from the Einstein GR field equation. It is one particular solution of GR. In the Friedmann model distances are either all increasing or all decreasing. With the parameters of our particular universe, as measured to date, we expect continued increase (no eventual collapse).
In the Friedmann model there IS NO PREFERRED DIRECTION to the expansion, and no preferred direction to any acceleration or deceleration of expansion. Expansion does NOT CONSIST OF GALAXIES MOVING THRU SPACE.
Expansion is just a percentage increase in the distance between stationary points.
GR teaches us to expect distance to change unless it is a distance locked in by atomic or other forces----like the crystal bonds of a metal rod, or the orbit dynamics of a planetary system. Within certain bound structures, forces keep distances from changing, but otherwise one should expect them to be expanding or contracting. The Friedmann equation is beautifully simple and you can see immediately how it governs the rate of expansion/contraction. It works either in the finite volume or infinite volume case and (as I said) doesn't need any force vector in any preferred direction. There's just this timedependent scalefactor a(t) and an ordinary differential equation that tells about the time derivative a'(t) and how it is determined. It is probably easier to understand in that form than in the form of words.
Currently the percentage rate of expansion or more exactly the fractional rate (which you could write a'(t)/a(t) if you want) is one percent every 140 million years. You can see why it has to be. The left hand side of the Friedmann equation is actually the square of that very quantity----(a'(t)/a(t))
2.
Actually there are two Friedmann equations, both are simple and you should look at both. But for some reason everybody always talks about THE equation as if there were just one. the other one is about the second derivative of the scalefactor: a''(t).
when you look at these two Friedmann equations the first thing you see is that they work for both the infinite volume case and the positive curved S
3 case. that is made explicit with a parameter k which can be either +1, 0, or -1
You are right that expansion, as far as observations can tell us by fitting the data, was DECELERATING until some 4 or 5 billion years ago! That was when matter was denser and its effect prevailed over the cosmological constant. And then with expansion matter thinned out and the cosmological constant effect of acceleration began to prevail.
But
all this stuff works equally well in the Friedmann model whether you pick the infinite case or the finite case!. This doesn't need any preferred direction. And the expansion is not outwards from any particular point. Basically it is just GR effects involving the dynamics by which distances change (GR's dynamic geometry).
I agree this is idiosyncratic view, but I just have difficulty imagining an infinite amount of mass.
Well, in that case just picture space as S
3! A large balloon is almost flat. This fits the data even slightly better than the infinite space model-----it is just that the infinite flat version is mathematically simpler and the difference in fit is not statistically significant at this point.
I believe some researchers have tried to find evidence of this wrapping around effect by looking for symmetries in the pattern of distant galaxies in different directions at the edge of the visible universe and never found that evidence.
That is right. I mentioned some of that research----papers by Neil Cornish and David Spergel----in an earlier post in this thread. They did not rule out the finite picture. They just proved that if it was finite it had to be BIG.
But that is what the curvature data already suggested anyway.
IIRC a 2007 Ned Wright paper presented a good fit for Omega as 1.011, and in a landmark 2006 paper by Spergel et al the errorbar all on the upside of 1------something like [1.010, 1.041] at 68 percent confidence. Ned Wright's 1.011 falls in that range.
https://www.physicsforums.com/showthread.php?p=1558402#post1558402
And that 1.011 would indicate an S
3 circumference of some 800 billion LY, which was way more than the lowerbound that Cornish and Spergel came up with in 2004 (and later refined in 2006). So that much agrees.
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What I have still not given you is INTUITION for why an infinite flat volume that starts off expanding should ever slow down! The real reason is that GR is the most accurate theory of gravity at large scale that we have and if you buy GR then you buy Friedmann derived from it and you buy that in the interesting cases expansion slows down unless there is a cosmological constant. But this is unsatisfying. One still wants some intuition for why in the hell it happens.
The puzzle is in the infinite flat case. There is no preferred direction, the net effect of everything should be zero---why should expansion be slowed down? I hope someone else has some intuition about this. When I think about it I come up short on the intuition end.
One thing to remember is that GR is a field theory and information only travels at a finite speed. So in some big region the local patch of geometry doesn't KNOW about the matter way out there. the effective distribution of matter in its past light cone could be very different from the instantaneous (approximately even) distribution. But I'm not sure enough about this. the problem of how you get intuition for the slowing down is interesting----maybe there is some obvious answer that I'm missing. We might hear from SpaceTiger or Garth about this.
Lol. But basically that's valid. A start time and infinite space.
But I favour 4 macro spatial + Loads o other macros not relevant to us but take no notice of me on that 1 !
