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Closed Universe

  1. Nov 25, 2007 #1
    I have a question:

    I understand why the expansion of the universe is believed to be caused by the expansion of vacuum space, i.e. the Hubble flow.

    I understand that if the universe had mass/energy in excess of its critical density, then it would be a "closed" universe which would eventually start contracting (potentially after a long expansion phase). I understand that it is believed that in a contracting universe, space itself would be contracting (shrinking), bringing galaxies, etc. closer and closer together.

    Something strikes me as odd about such a contraction of space. If the universe has more mass/energy density than its critical density, won't normal gravitation cause the galaxies, etc. to clump together through peculiar motion (i.e., motion through space), first in many small clumps, then a few large clumps, and perhaps eventually in a single clump containing all of the mass/energy of the universe?

    If the latter were true, then at some point during the contraction, couldn't the single clump have a volume much smaller than the volume of space in the universe? Leaving an island of mass/energy surrounded by a potentially large (or infinite?) void of empty space? This seems entirely contradictory to the standard concept of a homogeneous, isotropic universe.

    Perhaps the answer is that the closed universe will eventually reach a spatial contraction rate far faster than the peculiar velocities of the mass/energy clumps, thus overtaking their gravitational clumping rate and, ultimately, reaching a singularity (whatever that means) before mass/energy has been able to concentrate in a single clump. I guess the question is whether that is mathematically compelled to be the case, or whether at some selected density value (as a factor of time), the "ultimate clump" could form, as I described. On the other hand, the faster space itself contracts, the more the rate of gravitational clumping through gravitation should also accelerate, due to the ever shrinking distances between clumps.

    Related question: Could the additional clumping occurring during the contraction phase cause the universe to become sufficiently inhomogeneous at large scales that the Friedmann equations no longer apply?

    Thanks, Jon
    Last edited: Nov 25, 2007
  2. jcsd
  3. Nov 25, 2007 #2


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    That is only true for a simple matter only case. There is no universal rule relating spatial curvature to the asymptotic fate of the Universe. A 'closed' Universe can expand forever, it depends on the nature of the material in the universe (summarized by its equation of state).

    Structure forms in an inhomogeneous universe regardless of the global curvature. The rate at which they do so does depend on the expansion rate, which in turn depends on the curvature. It depends on the details of the cosmology and the initial amplitude of the fluctuations.

    The last paragraph of the above is the crucial one. As you suggest, as the universe becomes more and more inhomogeneous then the Friedmann equations become a less and less accurate approximation. No one really knows how bad an approximation it is, and it is a subject of some debate at present.

    This issue does away with your other thought experiments, since if all the material in a closed and collapsing universe has formed into a single clump then the Universe has collapsed, there is no 'background' motion to speak of.

    Don't think of the Hubble flow as 'the expansion of space' in a way that imbues space with an ability to cause things to occur (I'm not suggesting you are, but it appears you may be). The expansion of space is a convenient way of describing the FRW solution of GR. What really matter though is what the mass/energy does. In a homogenous universe there is no problem, but it is not sensible to talk about a situation where all of the mass in a close universe has collapsed to a point, but 'space' continues to contract as might be suggested by a homogenous solution if that clump was smoothed over the whole universe. In the case the FRW solution no longer applies, and hence the intellectual shorthand of expanding space does not either.

    The whole issue of inhomogeneous universes is an interesting and very topical one at the moment. I can recommend some papers on this topic if you are interested.
    Last edited: Nov 25, 2007
  4. Nov 25, 2007 #3


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    Hi Jon, glad to see you considering the spatial closed case. Wallace makes the important point that spatial closed doesnt imply eventual collapse.

    For instance (I don't want to suggest that Omega ACTUALLY IS 1.011 as in Ned Wright's best fit LCDM but) suppose we are actually in a standard model (i.e. LCDM) universe with Omega = 1.011. That would fit all the data really well! As Wright indicates.
    And in that case you have expansion forever.

    Not much clumping would be expected in this case. Our local group of galaxies would eventually combine into a single galaxy and all the other galaxies could be expected to disappear from sight.

    Space, in this case, can be pictured as the 3D surface of an imaginary 4D ball, currently of radius about 130 billion LY. that is, the current circumference of 3D space being about 800 billion LY.

    As you may know, the math term for this is "3-sphere" or S3.
    I am not assuming any extra dimension (no evidence) so there would be no 4D ball inside and nothing outside. Space would simply be the 3-sphere, and it would expand indefinitely. In that sense it would be "closed" in the way that any sphere is closed---finite and boundaryless.

    I should give you the link to Wright's january 2007 paper, in case you want to glance at it. He doesnt make any big deal out of this Omega = 1.011 case. It is just one possible LCDM standard model case and the data is not yet statistically significant enough to decide with we have an infinite spatial flat or a finite spatial closed universe. but the data keeps getting better so someday we'll probably know or at least be pretty sure.

    Constraints on Dark Energy from Supernovae, Gamma Ray Bursts, Acoustic Oscillations, Nucleosynthesis and Large Scale Structure and the Hubble constant
    Edward L. Wright (UCLA)

    Wallace correct me if I'm mistook but I'd say basically what he did was take all the data he could get his hands on, which was of any decent quality, and put it all together to get the best handle on dark energy density (and other relevant parameters) that he could.
    He's primarily an observational cosmologist: to get an idea of his professional context, here are his papers on arxiv
    Last edited: Nov 25, 2007
  5. Nov 25, 2007 #4


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    That's pretty much the game of a lot of modern cosmology. I'm not sure what was novel about Ned Wright's paper as I haven't read it myself. You have to 'fit' all the cosmology parameters to the data at once, so he would have found constraints about the whole lot, not just the dark energy parameters, even if they were what the work focused on.

    The issue of measuring the curvature is tricky, since we know it is small and given the way uncertainty works, even if it is zero we would never measure it to be precisely zero. If we could measure the parameters to such an accuracy that we say that the curvature was non-zero, with the deviation from zero several times the uncertainty, then it would be interesting. I'm pretty sure even the 'next generation' cosmology probes aren't going to tie this down to that precision, so it might be 30 years or more before we can say for sure (and by then the whole game will probably have changed anyway).
  6. Nov 25, 2007 #5


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    The smallness of curvature is key. The universe is so close to flat it is impossible to distinguish which model is best. 1.011 looks compelling, but the error bars keep dead flat in play. I think it wobbles right on the edge [an uncertainty thing].
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