Mattig formula

Mattig's formula is one of the most important formulae in observational cosmology and extragalactic astronomy which gives relation between radial coordinate and redshift of a given source. It depends on the cosmological model being used and is needed to calculate luminosity distance in terms of redshift.^[1]

Without dark energy

Derived by W. Mattig in a 1958 paper,^[2] the mathematical formulation of the relation is,^[3]

$r_1 = \frac{c}{R_0 H_0} \frac{q_0z+(q_0-1)(-1+\sqrt{1+2q_0z})}{q_0^2(1+z)}$

Where, $r_1=\frac{d_p}{R}=\frac{d_c}{R_0}$ is the radial coordinate distance (proper distance at present) of the source from the observer while $d_p$ is the proper distance and $d_c$ is the comoving distance.

$q_0=\Omega_0/2$ is the deceleration parameter while $\Omega_0$ is the density of matter in the universe at present.

$R_0$ is scale factor at present time while $R$ is scale factor at any other time.

$H_0$ is Hubble's constant at present and

$z$ is as usual the redshift.

This equation is only valid if $q_0 > 0$. When $q_0 \le 0$ the value of $r_1$ cannot be calculated. From this radius we can calculate luminosity distance using the following formula,

$D_L \ = \ R_0r_1(1+z) = \frac{c}{H_0q_0^2} \left[q_0z+(q_0-1)(-1+\sqrt{1+2q_0z})\right]$

When $q_0=0$ we get another expression for luminosity distance using Taylor expansion,

$D_L = \frac{c}{H_0}\left(z+\frac{z^2}{2}\right)$

But in 1977 Terrell devised a formula which is valid for all $q_0 \ge 0$,^[4]

$D_L = \frac{c}{H_0}z\left[1+\frac{z(1-q_0)}{1+q_0z+\sqrt{1+2q_0z}}\right]$

References

1. Observations in Cosmology, Cambridge University Press
2. Mattig, W. (1958), "Über den Zusammenhang zwischen Rotverschiebung und scheinbarer Helligkeit", Astronomische Nachrichten 284: 109, Bibcode:1958AN....284..109M, doi:10.1002/asna.19572840303
3. Bradley M. Peterson, "An Introduction to Active Galactic Nuclei", p. 149
4. Terrell, James (1977), "The luminosity distance equation in Friedmann cosmology", Am. J. Phys., 45(9): 869–870, Bibcode:1977AmJPh..45..869T, doi:10.1119/1.11065