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I was wondering if a person somehow managed to go 13.7 billion light-years from the center of the universe would he be able to see Big Bang itself (with a Hubble).
We can see the Big Bang (or at least very close to it: within about 380,000 years of it, before which the universe was opaque) by looking out with our telescopes from here on Earth. For each light year out in space that you look, you see back 1 year in time.
The universe also has no absolute center. Any point in space that you choose is the center of its observable
part of the universe, due to the finite age of the universe and the finite speed of light. For everyone, no matter where they are, the Big Bang is seen at a distance of (current age of universe)x(speed of light) away from them, in terms of light travel time.
Signals from shortly after the Big Bang are extremely stretched out (redshifted) by the expansion as well. For example, beyond the Cosmic Microwave Background (emitted 380,000 years after Big Bang when the universe became transparent to light), lies the Cosmic Neutrino
Background (emitted just a few seconds after Big Bang when the universe became transparent to neutrinos). The microwave background is stretched out by a factor of about 1100, while the neutrino background is stretched out by a factor of billions, making it extremely difficult to observe even aside from the problem of neutrinos already being hard to detect under normal conditions.
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As space-time is responsible for universal expansion, by the time the boundary of the universe reaches that person he would enter space-time and be carried along with the expansion
Just as the universe has no absolute center, it also has no absolute edge. If you could freeze the expansion and fly off in any direction at any arbitrary speed, you will either keep going forever without reaching an edge, or (if the space is curved like the surface of a sphere), you'll eventually wind up back where you started.