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Which type of redshift does the equation: $$\Delta \lambda / \lambda = v/c$$ apply to, that which results from the Doppler effect, or that which results from cosmological expansion?

Does it even apply to a particular type of the two? What exactly are the conditions of its applicability?

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It applies to both so long as $v \ll c$ and if you are interpreting the observed redshift of an object as a line of sight velocity.

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