What i really want to ask how much has the Milky Way moved, relative to where it was "at the big bang" or the soonest time that makes sense (since i doubt "at the big bang" makes much sense in this question). I suppose the galaxies have non-zero impulse, otherwise we wouldn't see things like galaxy collisions. So, relative to where our galaxy or whatever was there (dust cloud?) "in the beginning", how much did we move?
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The cosmic microwave background gives us a convenient rest frame in that it represents the average distribution of matter in the universe. So if the CMB looks isotropic, i.e. there is no Doppler shift in different directions then this is a plausible definition of stationary. A velocity relative to the CMB would then indicate a peculiar velocity.
The Milky Way is moving at about 550km/sec relative to the CMB, and it probably hasn't interacted strongly with any large galaxies because it seems to be reasonably unperturbed (it's devoured many small galaxies!) so relative to the CMB the distance we've moved is just 500km/sec times 13.7 billion years. I make this about $2 \times 10^{23}$m, which is about 25 million light years or about ten times the distance to the Andromeda galaxy.
But please be aware of what we're calculating here. This is a somewhat arbitrary distance using an arbitrary reference frame. You shouldn't attach too much physical significance to it. Still it's quite interesting to note that we're expected to collide with the Andromeda galaxy in about 4 billion years, so our calculated figure seems a plausible distance to have travelled relative to the averaged positions of all the other galaxies.
John Rennie
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