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]

${\displaystyle r_{1}={\frac {c}{R_{0}H_{0}}}{\frac {q_{0}z+(q_{0}-1)(-1+{\sqrt {1+2q_{0}z}})}{q_{0}^{2}(1+z)}}}$

Where, ${\displaystyle r_{1}={\frac {d_{p}}{R}}={\frac {d_{c}}{R_{0}}}}$ 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.

${\displaystyle q_{0}=\Omega _{0}/2}$ 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 ${\displaystyle q_{0}>0}$. When ${\displaystyle q_{0}\leq 0}$ the value of $r_{1}$ cannot be calculated. From this radius we can calculate luminosity distance using the following formula,

${\displaystyle D_{L}\ =\ R_{0}r_{1}(1+z)={\frac {c}{H_{0}q_{0}^{2}}}\left[q_{0}z+(q_{0}-1)(-1+{\sqrt {1+2q_{0}z}})\right]}$

When ${\displaystyle q_{0}=0}$ we get another expression for luminosity distance using Taylor expansion,

${\displaystyle D_{L}={\frac {c}{H_{0}}}\left(z+{\frac {z^{2}}{2}}\right)}$

But in 1977 Terrell devised a formula which is valid for all ${\displaystyle q_{0}\geq 0}$,^[4]

${\displaystyle D_{L}={\frac {c}{H_{0}}}z\left[1+{\frac {z(1-q_{0})}{1+q_{0}z+{\sqrt {1+2q_{0}z}}}}\right]}$

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