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But what is the coordinate system to the scale of the whole universe that can describe these inertial reference frames appropriately? Is the coordinate system expanding together with the whole universe.

But I think that locally, we use a constant (not changing, otherwise the system seems to be a accelerated RF, as universe is expanding at increasing rate) coordinate system which might contradict the changing coordinate system which varies with the whole universe.

For example, if we use the same system to describe galaxies and quazars, they are seemingly "moving" faster than light.