The elongation of light waves occurs not because galaxies are moving away from each other, but because light waves are stretching along with the fabric of expanding space (cosmological redshift). Although the Milky Way is held together by gravity, space expansion occurs everywhere, including within the Milky Way. Since space expansion occurs in the Milky Way and light stretches due to space expansion, why don't we observe cosmological redshift within the Milky Way?
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space expansion occurs everywhere
It doesn't. There are significant parts of the Universe that is not expanding. The Milky Way is one of them, because the local density of matter within the Milky Way is greater than that of dark energy. Your body is not expanding either, for the same reason.
See this question and also this question on Astronomy SE.
Allure
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On short distance scales, the overall expansion of the universe is too small to detect. This is why, for example, we can't detect it within the bounds of our own solar system or our neighborhood in the galaxy.
niels nielsen
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