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If you were to divide the present-day universe into cubes with sides 10 million light-years long, each cube would contain, on average, about one galaxy similar in size to the Milky Way. Now suppose you travel back in time, to an era when the average distance between galaxies is one quarter of its current value, corresponding to a cosmological redshift of z = 3. How many galaxies similar in size to the Milky Way would you expect to find, on average, in cubes of that same size? In order to simplify the problem, assume that the total number of galaxies of each type has not changed between then and now. Based on your answer, would you expect collisions to be much more frequent at that time or only moderately more frequent?

I am very confused with this question. From my intuition, I take that volume would equal to distance^3. They have given us that the past distance would be 1/4 of current. Therefore, do I cube 1/4distance? Also how does the redshift of 3 play a part in the solution. Thanks for any help! I do not need an answer, just how to start.

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Comoving separations go with the scale factor. Comoving volumes scale as $(1+z)^{-3}$.

At $z=3$, that volume was 64 times smaller.

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