Variable

Transformation in

terms of velocity

Transformation in

terms of redshift

Time interval $\frac{\Delta t}{\Delta t_0}$

$\frac{1}{1-\beta }$

z + 1 (1')

Length $\frac{l}{l_0}$

$\frac{1+\beta }{1-\beta }$ (2)

2z + 1 (2')

Matter density $\frac{{\rho }_M}{{\rho }_0}$

$\frac{1-\beta }{1+\beta }$ (3)

$\frac{1}{2z+1}$ (3')

Matter energy density $\frac{e_M}{e_0}$

$\frac{1-\beta }{1+\beta }{\beta }^2$ (4)

$\frac{z^2}{{\left(z+1\right)}^2\left(2z+1\right)}$ (4')

Dark matter density $\frac{{\rho }_{DM}}{{\rho }_0}$

$\frac{2\beta }{1+\beta }$ (5)

$\frac{2z}{2z+1}$ (5')

Dark matter energy density $\frac{e_{DM}}{e_0}$

$\frac{2{\beta }^3}{1+\beta }$ (6)

$\frac{2z^3}{{\left(z+1\right)}^2\left(2z+1\right)}$ (6')