Distance from redshift

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?

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

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