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I think they measure cosmological redshift to use in the Law of Hubble-Lemaître together with the distance to calculate $H_0$. Is this correct, or do they use Doppler shift (too)?

$H_0$ indicates how fast the universe is expanding, so I find it logical that we measure cosmological redshift. But the galaxies are receding (due to the Hubble Flow) away from us, so do they gain an additional Doppler shift?

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The term "Hubble flow" refers to the homologous expansion of space and the resulting recession of all galaxies from each other (if they're not close enough to be gravitationally bound). This effect causes the "cosmological redshift", i.e. the redshift that light from distant galaxies attain as it travels through space.

In addition to this motion away from each other, galaxies have a so-called peculiar velocity, i.e. a motion through space. This motion adds an additional Doppler shift to the cosmological redshift, either to larger or smaller wavelengths, depending on the direction.

Whether or not you see these two types of redshift as fundamentally different, is not trivial I think. In most textbooks, they are described as two different things, the former having to do with the dynamics of the fabric of space, and the latter having to do with motions of emitters and observers. But in fact it may not be so different. For instance the Welsh cosmologist Geraint Lewis argues that, in some sense, the cosmological redshift can be interpreted as the sum of infinitely many infinitesimally small Doppler shift (Lewis 2016). On the other hand, the American physicist Sean Carroll argues that the notion of expanding space is nevertheless an extremely useful concept (Carroll 2008).

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