How does the distance $r$ scale with the expansion of the universe?For example if $r_{o}$ is the distance between us and a galaxy and $V(r_{o})$ is the rate of expansion of the universe at that distance , what is $V(r_{0})$ equal to?
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1$\begingroup$ Define "distance". en.wikipedia.org/wiki/Distance_measure $\endgroup$– ProfRobCommented Nov 13, 2023 at 8:24
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$\begingroup$ The coordinate distance , not the proper distance $\endgroup$– RootCommented Nov 13, 2023 at 8:24
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2$\begingroup$ "coordinate distance" isn't a term I recognise, can you define it? $\endgroup$– ProfRobCommented Nov 13, 2023 at 8:30
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The relationship between proper distance and the rate of change of proper distance is Hubble's law. So $$ v(r_0) = H_0 r_0 \ .$$ Note that $H_0$ is the Hubble parameter at the present epoch. The Hubble parameter changes with time.
There are however different ways to define velocity and distance... https://en.wikipedia.org/wiki/Distance_measure and these will have more complex relationships between them that depend on other cosmological parameters.
