Nobody has a good theory of time travel in Everrettian quantum mechanics. I don't think that you can claim right now that there is a once and for all sensible answer to that until you have some specific scenario in mind. There are theories of how to construct consistent quantum states in the presence of closed time like curves. Tara Lumagi says, considering the new nanograv results, do you think we may be able to detect a frequency that actually tells us what spacetime is made of? Any experiments that can help us rule out quantum gravity versus string theory may be even detect an echo of the Big Bang.