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There is a very similar question to this one: Is dark matter really matter?. But the particular aspect I'm asking about seems not to be mentioned there. So, here we go:

Does dark matter have to exist as matter? Or, is it possible that space-time is just naturally not flat, but slightly crinkled? Obviously heavy objects can bend space-time, and if, as a layman, I understand it correctly, this is what we perceive as gravity. What now, if space-time is naturally not quite flat, and we perceive this non-flatness as dark matter?

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user1264
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Everything we do understand about space-time, under the framework of General Relativity, suggests that it only bends in response to matter, radiation, and some sort of dark energy. In this understanding, space will "crinkle" if and only if one or more of these components is present. With that in mind, there are myriad observations of space-time bending due to the influence of an unidentified type of matter concentrated in clusters of galaxies. The answer to your question is "yes."

kleingordon
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    Purely being the Devil's advocate here: we haven't observed the matter, we've just seen spacetime being affected in ways similar to how it would when matter would be there. It is likely that it actually is some form of matter, but until we have actual observations and a solid theoretical basis for it, it could be anything. – Rody Oldenhuis Aug 27 '12 at 07:23
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    Occam says its matter. Granted, we don't know the details. – kleingordon Aug 27 '12 at 20:59
  • @RodyOldenhuis If you want to simultaneously explain gravitational lensing observations, the bullet cluster, and CMB anisotropy measurements, you have to go through epic contortions of reasoning to arrive at any other conclusion than the presence of matter. – kleingordon Aug 27 '12 at 21:28
  • Yes, you don't have to convince me :) I'm just pointing out that you should keep in mind that dark matter and dark energy remain to this point somewhat "magical" explanations to our observations -- matter is only a model that fits the data really well. – Rody Oldenhuis Aug 28 '12 at 03:55
  • @kleingordon Wouldn't Occam rather say that it's not matter? It's not as mathematically or aesthetically pleasing, nor is it as interesting from a physics point of view, but from my understanding "That's just the shape that space-time came out of the big bang" (paraphrased) seems to be simpler than saying "There must be some invisible stuff that bends it into the shape we observe". No? – user1264 Sep 24 '12 at 05:59
  • @user1264 No. The crinkling of space-time you are envisioning without the presence of matter would be a more radical departure from what we do understand about the universe than the existence of an unidentified type of matter. If someone were able to make a compelling theoretical case for some alternative explanation of all the observations that relate to the existence of dark matter, then the scientific community would need to pay attention. But no such compelling alternative exists at the moment. Theories of modified newtonian dynamics are valiant attempts, but so far have come up short. – kleingordon Sep 24 '12 at 08:00
  • @user1264 Let me try to put this one more way: from general relativity, which has so far survived every experimental test thrown at it, we have some idea of how matter interacts with space-time, even if we do not know what that matter is. If we then make a hypothesis that some sort of unidentified matter exists concentrated in galaxy clusters, we find that this hypothesis does a good job tying various observations together, including gravitational lensing surveys, the bullet cluster, and measurements of the cosmic microwave background anisotropy. No other hypothesis is as successful. – kleingordon Sep 24 '12 at 08:14
  • @kleingordon So, if I understand you correctly the statement that "space-time is naturally flat" is not just an assumption, but there is some evidence to that (directly or indirectly)? – user1264 Oct 02 '12 at 05:10
  • Basically, yes, although I'm not sure I would put it exactly like that. What we understand about space-time from GR tells us that it reacts in a specific, experimentally verifiable manner with respect to the relativistic stress-energy tensor. This borders on tautology because just about anything you can imagine (a quantum field, vacuum energy, dark matter) will show up in the stress-energy tensor, so all that this statement is really saying is that "space-time reacts in response to stuff that makes it react." It's almost like Newton's first law. – kleingordon Oct 02 '12 at 05:24
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    But the dark matter hypothesis buys you more than just a wave of the hands. It allows us to understand the dynamics of the unknown stuff in to a framework with which we can make theoretical predictions, and does quite well in that regard (see also http://adsabs.harvard.edu/abs/2012Natur.487..202D). It allows us to consider different equations of state for the matter, and allows us to conclude based on observations and simulations that the dark matter is non-relativistic. This explanation tis more compelling than a statement that space-time just curves on its own. – kleingordon Oct 02 '12 at 05:34