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I have heard of the "Big Rip" as one theory for the eventual end of the universe. If the speed at which the universe is expanding, and all the matter inside it is moving away from all other matter at an increasing rate, would the matter eventually have to move at the speed of light? Or would it have to slow down?

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If you look at Hubble's law $V=H_0 D$, you notice at far enough distances, galaxies are receding at velocities greater than the speed of light. In fact, the universe does not even need to be accelerating for this to be true.

This is not a violation of relativity, which prevents information from being transmitted faster than the speed of light. Since all galaxies are receding away from all other galaxies, there is no way that they could be transmitting info.

Now that you mention it, however, there are some problems with the big rip/relativity. In particular, for faster than exponential acceleration requires a "phantom like" dark energy with equation of state less than $-1$, which is inconsistent with some energy conditions in general relativity, but this is different from the problem you described.

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The Big Rip was a prediction that was put out around 2003, although speculation of these sorts of events have existed longer as anyone who has toyed with the cosmological constant in the can tell you. These specific types of predictions depend on the types of models that are used, and around the same time the idea emerged, it was quickly pointed out that the problem was avoidable in certain solutions that avoided instabilities caused by phantom energy.

These discussion are closely related to similar discussions involving vacuum instability and the idea of false vacuums. This question is very important because the recently observed properties of the Higgs Boson would indicate we are in a very borderline position in regards to the stability of the vacuum.

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