Mattig formula

Mattig's formula was an important formula 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 used to calculate luminosity distance in terms of redshift. ^[1]

It assumes zero dark energy, and is therefore no longer applicable in modern cosmological models such as the Lambda-CDM model, (which require a numerical integration to get the distance-redshift relation). However, Mattig's formula was of considerable historical importance as the first analytic formula for the distance-redshift relationship for arbitrary matter density, and this spurred significant research in the 1960s and 1970s attempting to measure this relation.

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 ${\displaystyle d_{p}}$ is the proper distance and ${\displaystyle d_{c}}$ is the comoving distance.

${\displaystyle q_{0}=\Omega _{0}/2}$ is the deceleration parameter while ${\displaystyle \Omega _{0}}$ is the density of matter in the universe at present.

${\displaystyle R_{0}}$ is scale factor at present time while ${\displaystyle R}$ is scale factor at any other time.

${\displaystyle H_{0}}$ is Hubble's constant at present and

${\displaystyle z}$ is as usual the redshift.

This equation is only valid if ${\displaystyle q_{0}>0}$. When ${\displaystyle q_{0}\leq 0}$ the value of ${\displaystyle r_{1}}$ cannot be calculated. That follows from the fact that the derivation assumes no cosmological constant and, with no cosmological constant, ${\displaystyle q_{0}}$ is never negative.

From the radial coordinate 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 (3): 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