Equations used for the distance modulus

An equation for the calculation of the distance modulus is not given by Carroll, Press & Turner [1] but they give an equation for the proper motion distance (from which the distance modulus can easily be calculated). The proper motion distance is:

$$d_M = \frac{1}{H_0 \mid\Omega_K\mid^{1/2}} sinn \left\{ \mid\... ...\Omega_M\, z)-z\, (2+z)\, \Omega_\Lambda]^{-1/2}\, dz \right\}$$ (3)

where sinn is sinh if $\Omega_K > 0$ and sin if $\Omega_K < 0$. If $\Omega_K = 0$ the sinn and $\Omega_K$s disappear leaving only the integral. This equation, however, isn't used for the calculation but a modified version of it which makes use of the w parameter:

$$d_M = \frac{1}{H_0 \vert\Omega_K\vert^{1/2}} sinn \left\{ \ve... ...Omega_\Lambda\, (1+z)^{1+3\, w}+\Omega_K]^{-1/2}\, dz \right\}$$ (4)

If w = -1 this equation reduces to eq.  3. In the function, $\Omega_K$ is calculated from $\Omega_M + \Omega_\Lambda + \Omega_K = 1$.
The relation between the proper motion distance d_M and the luminosity distance d_L is:

d_L = (1+z) d_M (5)

The distance modulus is then:

$$\mu = 5 \log d_L + 25$$ (6)

So the distance modulus in terms of the proper motion distance is

$$\mu = 5 \log \left(d_M (1+z)\right)+ 25$$ (7)

where d_M is taken from eq  4.