Mattig formula explained

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It assumes zero dark energy, and is therefore no longer applicable in modern cosmological models such as the Lambda-CDM model, (which require a numerical integration to get the distance-redshift relation).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, the mathematical formulation of the relation is,^[2]

r _{1} = (c)/(R _{0} H _{0}) (q _{0z+(q} _{0-1)(-1+\sqrt{1+2q} _{0z})/()}{q) ^{2 ( 1+ z)}} ^{0}

Where,

r _{1= (d _{p})/(R)} = (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 andz 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. That follows from the fact that the derivation assumes no cosmological constant and, with no cosmological constant, q _{0} is never negative.

From the radial coordinate we can calculate luminosity distance using the following formula,

D _{L} = R _{0r} _{1(1+z)} = (c)/(H _{0 q} ^{2} ^{0}) \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} = (c)/(\left(z + H _{0}) (z ^{2})/(2) \right)

But in 1977 Terrell devised a formula which is valid for all

q _{0} \ge 0 ,D _{L} = (c)/(z\left[ 1+ H _{0}) (z( 1- q _{0)})/(1+ q _{0z+\sqrt{1+2q} _{0z}) }\right]

References

1. http://ned.ipac.caltech.edu/level5/Peacock/Peacock3_4.html Observations in Cosmology
2. Bradley M. Peterson, "An Introduction to Active Galactic Nuclei", p. 149