That first theory would be interesting, but wouldn't it be basically impossible to prove unless the finite repeating universe were also pretty small? We see distant objects as they were in the past, and I'm not sure we'd recognize our own galaxy a billion years ago and a billion light years away.
It's actually an interesting thought experiment. If that theory were true, and assuming we could see ourselves face-on (rather than edge-on), then I wonder how far away and far back could we recognize ourselves.
Yeah, I realize we're looking in the past and can't "see" ourselves (probably not even our own galaxy, depending how far out we have to look). But I guess the theory in itself is not any more "weird" or "outlandish" as the idea of an infinite universe.
And maybe if we actually look both ways, we could find another galaxy that could be proven to be the same (because of some particularly remarkable stars/quasars/supernovae etc. in them). Who knows, maybe the universe is smaller than we think and we just don't recognize the "same" galaxies on different images because they are billions of years apart?
It's actually an interesting thought experiment. If that theory were true, and assuming we could see ourselves face-on (rather than edge-on), then I wonder how far away and far back could we recognize ourselves.