I am looking for a exact derivation of a relation between redshift $z$ and distance $d$.

What I know is the definition $$z=\frac{\lambda_{\text{observed}}}{\lambda_{\text{unshifted}}}-1=\sqrt{\frac{1+\frac{v}{c}}{1-\frac{v}{c}}}-1$$ and that the Hubble constant $H$ as a function of $z$ is: $$H^2=H_0^2\left(\Omega_m\left(1+z\right)^3+\Omega_{\Lambda}\right)$$

How can I use this to derive the distance?

  • $\begingroup$ Which distance measure are using: co-moving, transverse, light-travel, etc.? $\endgroup$ – Kyle Kanos Jul 20 '14 at 16:01
  • $\begingroup$ I'm not entirely certain, but isn't that what Wikipedia does here? $\endgroup$ – ACuriousMind Jul 20 '14 at 16:04
  • $\begingroup$ @Kyle_Kanos it must be light-travel.. actually I have the redshift of a galaxy and I want to know the distance to it e.g. in Mpc $\endgroup$ – Andy Jul 20 '14 at 16:04
  • $\begingroup$ @ACuriousMind in Wikipedia they make an approximation and have: $d=\frac{zc}{H_0}$ which is only valid for low redshifts $\endgroup$ – Andy Jul 20 '14 at 16:06
  • $\begingroup$ This is the better Wikipedia article to refer to for this problem. $\endgroup$ – Kyle Kanos Jul 20 '14 at 16:14

Depending on the shape of the universe the luminosity distance is given by :

\begin{equation} d_L(z) = \left\{ \begin{array}{rl} \frac{(1 + z) c}{H_0 \sqrt{|\Omega_k|}} \sin \left[ \sqrt{|\Omega_k|} \int _0 ^z \frac{dz'}{H(z')/H_0} \right] & \mbox{for $k = 1$} \\ \frac{(1 + z) c}{H_0} \int _0 ^z \frac{dz'}{H(z')/H_0} & \mbox{for $k = 0$} \\ \frac{(1 + z) c}{H_0 \sqrt{|\Omega_k|}} \sinh \left[ \sqrt{|\Omega_k|} \int _0 ^z \frac{dz'}{H(z')/H_0} \right] & \mbox{for $k = -1$} \end{array} \right. \end{equation}

  • $\begingroup$ what is the difference between the luminosity distance and light-travel distance? which one should I use if my objects are galaxies with a determined redshift (spectroscopic or photometric)? $\endgroup$ – Andy Aug 3 '14 at 16:38
  • $\begingroup$ "my objects are galaxies" the type of distance measure you should use doesn't depend on the object you're looking at. All distance measurements apply to all objects. What you should be telling me instead is what use you'll be making of that measurement. $\endgroup$ – ticster Aug 3 '14 at 16:42
  • $\begingroup$ from the distance I want to calculate the volume and apply the Vmax-method (for the stellar mass function) $\endgroup$ – Andy Aug 3 '14 at 16:46
  • $\begingroup$ The distance between what and what ? In what volume ? You have to be clearer. As far as I can tell from the literature the Vmax-method usually employs the luminosity distance. $\endgroup$ – ticster Aug 3 '14 at 16:50
  • $\begingroup$ the volume should be the maximum volume in which the object can be detected; thus I assumed to take the distance between the observer and the galaxy. Oh, ok, do you by chance know which literature that was? $\endgroup$ – Andy Aug 3 '14 at 16:55

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