If we set aside the physical impossibilities and consider a hypothetical situation where you could travel fast enough to outrun the expansion of the universe, several possibilities can be speculated upon:
Observational effects: As you move faster than the expansion of the universe, you might witness changes in the cosmic background radiation and the distribution of galaxies. The farther you travel, the more pronounced these effects could become. Objects that were previously visible might appear to fade away or redshift into invisibility as their light becomes increasingly stretched due to the expansion.
Separation from the universe: If you were able to continually outrun the expansion, you would eventually move far away from other galaxies and structures in the universe. Over time, the density of matter around you would diminish, resulting in a relatively empty and isolated environment.
Cosmic microwave background radiation: The cosmic microwave background radiation, which is the remnant of the early universe's thermal radiation, would appear differently as you move away from the regions affected by the expansion. It would likely become more redshifted and potentially less uniform in all directions.
Unexplored regions: By outpacing the expansion, you would access regions of the universe that are beyond our observable horizon. These regions would contain galaxies, structures, and potentially unknown phenomena that have not been observed by astronomers in our local universe.
New cosmic evolution: It is possible that you might encounter different stages of cosmic evolution or witness the formation of new structures and galaxies that have not yet formed in our observable universe. This would allow for the exploration of cosmic processes and phenomena that are currently inaccessible to us.
Unknown consequences: Since this scenario exceeds our understanding of physics and the limitations imposed by the current cosmological models, it is difficult to predict the full extent of the consequences. The behavior of matter, the stability of physical laws, and other fundamental aspects might be vastly different from what we know.
It's important to reiterate that this hypothetical scenario assumes the ability to travel faster than the expansion of the universe, which is currently considered impossible based on our current understanding of physics. However, exploring such speculative ideas can foster curiosity and help us contemplate the nature of the universe and the limitations of our knowledge.