# Link redshift with luminosity distance?

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1. Oct 25, 2015

### June_cosmo

1. The problem statement, all variables and given/known data
Plot luminosity distance and redshift z

2. Relevant equations
$$d_L(z)=(1+z)r(z)$$
where $d_L(z)$ is luminosity distance and r(z) is the comoving distance.
and we have
$$r(z)= \frac{H_0^{-1}}{\sqrt\Omega_K}*sinn[\sqrt{\Omega_K}\int_0^z\frac{dz'}{\sqrt{\Omega_M(1+z')^3}}]$$
where $\Omega_K$is a measure of openness or closedness of the universe, sinn(x)=x in flat universe.
Suppose we consider a universe that is both flat and matter dominant, where $\Omega_K=0$,and $\Omega_M=1$.
3. The attempt at a solution
From the information given we know that
$$H_0d_L(z)=(1+z)\int_0^6\frac{dz'}{\sqrt{(1+z')^3}}$$
but I don't know how do I deal with z' when I plot it in, for example python? Since I don't know what z' equals to

Last edited: Oct 25, 2015
2. Oct 26, 2015

### andrewkirk

z' is not equal to anything. It's what's called a dummy variable, like a looping variable in Python.
Nor do you need to program the integral calculation. Just work out the definite integral and you'll get a number that does not change with z. Work out the number once, then hard-code it into your program as a constant.

3. Oct 26, 2015

### June_cosmo

Oh so you mean just work out
$$\int_0^6\frac{dz'}{\sqrt{(1+z')^3}}$$
which is approximate 1.24, then just plot
$$H_0d_L(z)=1.24(1+z)?$$

4. Oct 26, 2015

### andrewkirk

Yes, that's it.