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I am trying to arrive to the following expression:

$1+z = (1+z_C)(1+z_G)(1+z_D)$

so the total redshift is the product of the cosmological redshift, the gravitational redshift and the Doppler redshift. I know that

$1+z = \frac{\lambda_{observed}}{\lambda_{emitted}}$

but I don't know why would you make the product of them in order to get the total redshift. How do you derive the formula of the total redshift?

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  • $\begingroup$ Add some context. What are $z,z_C,z_G,z_D$? $\endgroup$
    – glS
    Jan 4, 2015 at 11:46
  • $\begingroup$ $z_C$ is the cosmological redshift, $z_G$ is the gravitational redshift, $z_D$ is the Doppler redshift, and $z$ is the total redshift. $\endgroup$
    – Cosmo_1
    Jan 4, 2015 at 12:03

1 Answer 1

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Suppose light starts out with wavelength $\lambda_\mathrm{emit}$. If you were not moving with respect to the source in either a proper sense (Doppler shift) or comoving sense (cosmological redshift), but the light has to climb out of a potential well, it would have wavelength $\lambda_\mathrm{G} = (1 + z_\mathrm{G}) \lambda_\mathrm{emit}$.

Now consider a photon emitted with $\lambda_\mathrm{G}$, but cosmologically far away. You would observe the wavelength $\lambda_\mathrm{CG} = (1 + z_\mathrm{C}) \lambda_\mathrm{G}$.

Finally, suppose you are receiving photons of wavelength $\lambda_\mathrm{CG}$, but then you start moving very fast. Now those same photons will appear to you to have wavelength $\lambda_\mathrm{CGD} = (1 + z_\mathrm{D}) \lambda_\mathrm{CG}$.

With all three effects at work, you simply multiply the factors: $$ \lambda_\mathrm{CGD} = (1 + z_\mathrm{C}) (1 + z_\mathrm{G}) (1 + z_\mathrm{D}) \lambda_\mathrm{emit}. $$ The same argument holds for any combination of these effects in any order: $$ \lambda_\mathrm{obs} = \lambda_\mathrm{emit} \prod_i (1 + z_i). $$

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  • $\begingroup$ Thanks a lot! This is a really good intuitive answer. Do you know of any more mathematical derivation? Maybe something coming from the metric. $\endgroup$
    – Cosmo_1
    Jan 4, 2015 at 12:30
  • $\begingroup$ Why is using $v_{proper}+v_{hubble}$ (assuming non-relativistic) as the total velocity the source is moving from us and using $z=v/c$ to extract $z_{tot}=z_C+z_D$ not correct? Thanks. $\endgroup$ Mar 15, 2022 at 21:22

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