The expansion of our universe is much like the inflation of a balloon. The distances to remote galaxies are increasing. Astronomers casually say that distant galaxies are "receding" or "moving away" from us, but the galaxies are not traveling through space away from us. They are not fragments of a big bang bomb. Instead the space between the galaxies and us is expanding. Individual galaxies move around at random within clusters, but the clusters of galaxies are essentially at rest. The term "at rest" can be defined rigorously. The microwave background radiation fills the universe and defines a universal reference frame, analogous to the rubber of the balloon, with respect to which motion can be measured.
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