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Suppose we're in an unknown place in the universe, only knowing that the Milky Way is somewhere is the sky (but of course, younger that it actually is, due to unknown distance light has to travel to reach our position).

Under detailed observation, could we recognize our galaxy?

Our galaxy alone would probably look like just another younger spiral galaxy. But, its vicinity would also include younger Magellanic Clouds, Andromeda galaxy, etc. And maybe other properties that could hint that galaxy being ours.


Note: I severely edited the initial post to keep the core question, as comments and the proposed "duplicate question" kept focusing on the background, rather than the actual question. I know these kind of edits are discouraged as they can invalidate answers, but the only answer so far is not affected by the edit.

OxTaz
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The precise shape and medium details of our galaxy are very unknown: we are at the worst possible place to figure them, even if the first survey try to figure roughly our spirals and (vaguely) sub-spirals.

So we won't recognise our galaxy. But we might recognise it's neighborhood (close as you say, or far), and deduced it's us at the middle. Still if you have no idea where in the sky is supposed to be our "echo image" and under which angle, it will probably require to include a huge set of neighborhood constraints to do the matching.

Now, one more huge difficulty: the images will correspond to very different ages, so the look of every galaxies and their relative positions might be totally different. Here, it's depend the size of "your universe" and thus the age difference of each "echo" (and us). At this stage I would be more pessimistic (at least for huge age differences), excepted if you can either have good prediction of candidates, or have specific enough time-invariant characteristics of galaxies (I'm not sure we know enough yet). Or, preferably, both.

Note that as an equivalent situation occuring in our Universe, you might try to find us near the Einstein disk of gravitational lenses :-) (good luck !)