When calculating the amount of missing mass in a galaxy due to dark matter, do cosmologists take into account local effects of gravity on time?

Question

If I understand correctly, massive objects cause time dilation, and so time seems to pass more slowly for observers closer to a massive object than those who are farther away. Do cosmologists take the local gravitational effects of earth, the sun, and the Milky Way into account when calculating the expected speeds of stars orbiting distant galaxies? Do galaxies that are more distant rotate faster, by our observations, than those closer to us, even if comparing galaxies of similar mass?

Answer 1

The effects of gravitational time dilation are of order $10^{-6}$ (one part in a million) in galaxies; see for example this answer for our own galaxy. Gravitational time dilation due to the Sun and Earth is an even smaller effect, at $\sim 10^{-8}$ and $\sim 10^{-9}$, respectively. Meanwhile, the impact of dark matter on galactic rotation is an $\mathcal{O}(1)$ effect.

Also, any net gravitational redshift or blueshift of a galaxy would only affect our inference of its recession speed, relative to us. It wouldn't change our inference of its rotation.