The thing is, the density of the Universe seems to be more or less uniform. If the Universe is infinite in size but contains a finite amount of matter, then the average density must be zero, and we living in a region where it’s not would have to be a very special region of the Universe indeed. The assumption that we occupy a special place in the Universe has had a very bad history, and it seems to go against everything that we can observe now.
And such a universe would not, incidentally, undergo a Big Rip. It (like our apparently uniform-density Universe) would expand forever at an ever-increasing rate, but the expansion would be exponential, with a fixed doubling time. A Big Rip depends not on any of the properties of the normal matter in the Universe, but on the dark energy: In a Big Rip model, the dark energy gets stronger with time, such that the doubling time of the size of the universe gets shorter and shorter without limit. In a normal Cosmological Constant universe (as ours appears to probably be), the expansion of space is never significant on the scale of people, planets, or even galactic clusters, while in a Big Rip universe, eventually even atoms and subatomic particles are torn apart by the expansion.
This is getting rather far afield of the OP, though. If you want to discuss it further, perhaps you could start a new thread.