As far as we can tell, the spatial geometry of our universe is Euclidean, which means, among other things, that it is infinite in extent. We can, of course, only see light that has had enough time to reach us since the beginning of the universe, i.e., light from bits that are less than 13.8 billion light years (light travel time) from here. These also happen to be the bits with a Doppler velocity less than the speed of light relative to us. (Because spacetime is curved, there are many different, equally valid ways to define distance and velocity, hence my use of these specific terms.)
But the universe does not end there. In fact, the standard cosmology tells us that it is infinite and the same (on average) as here. Now obviously we don’t know for sure that it is the same everywhere, but we do know that, unless the universe behaves in a spectacularly weird way, it will be roughly the same as it is here in a volume that’s millions of times bigger than the visible volume.
So yes, the so-called visible universe is just a part of a bigger universe, and while the rest we cannot see, we can infer that unless Nature is truly weird, it is roughly similar to the parts we can see, at least over a distance scale that is many times the size of the visible part.
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