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If space were to expand at > $c$ (as in inflation) would that mean gravity would no longer have any effect on the curvature of space, since gravity can only propagate at $c$?

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Your question is ill-formed for several reasons.

First, inflation is driven by gravity: all stress-energy gravitates, and if this is a field with certain properties, you get inflation.

Second, "gravity propagates at c" refers to a perturbation on some background geometry, relative to that geometry. For example, if you wiggle some mass or whatever, the gravitational influence that produces won't be superluminal. But that does not mean that two sufficiently distant particles can't increase their distance faster than c due to expansion of space.

Third, inflation is not characterized by superluminal expansion per se. It refers to an exponential expansion in the early universe. It is completely possible for the proper distance between two galaxies to be superluminal without space expanding exponentially (although given dark energy, our universe is probably transitioning to an inflation-like exponential expansion).

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would that mean gravity would no longer have any effect on the curvature of space

Gravity is the curvature of spacetime.

Let me emphasize this: it isn't that gravity affects the curvature of spacetime; it is that gravity is the curvature of spacetime.

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Ok. Would the expansion of space at > c mean that there would nolonger be any curvature of space, given that the curvature of space can only propogate at c? – Lionel Doolan Jul 24 '13 at 3:51
It is curvature of spacetime that matters, not just curvature of space. An expanding universe has a curved spacetime, while space by itself may or may not be curved. – Michael Brown Jul 24 '13 at 4:48

I think the question is ill-stated for a simple dimensional reason. Expansion rate is measured by the Hubble constant which is the inverse of a time (speed/distance). It cannot be compared to an actual speed such as c, which is distance/time.

It is as meaningful as saying that the speed of inflation at Wall Street is 123 mph. The use of the word "speed" in place of "rate" is common, but ill-advised.

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