“For once you have tasted flight you will walk the earth with your eyes turned skywards, for there you have been and there you will long to return.” -Leonardo da Vinci
The Universe is an awfully big place, but how certain are we that we couldn’t travel in a straight line for a very long time and simply return to our starting position? Just because we can travel through the Universe for an arbitrarily, perhaps infinitely long distance-or-time, doesn’t mean the Universe itself is infinite. It’s quite possible that it closes in on itself, and that any straight-line path will eventually return you to your beginning.
In a hypertorus model of the Universe, motion in a straight line will return you to your original location. Image credit: ESO and deviantART user InTheStarlightGarden, under a c.c.-by-s.a. 4.0 license.
If we wanted to test this possibility, there are a few ways to do it. The best would be to look for repeating structures at a specific distance in the Universe. While we wouldn’t recognize a younger version of our Milky Way, the large-scale structure or the cosmic microwave background would exhibit correlations if the Universe repeated on the scale of what’s observable ...