Rahul Sharma (Editor)

Mattig formula

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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.

Without dark energy

Derived by W. Mattig in a 1958 paper, the mathematical formulation of the relation is,

r 1 = c R 0 H 0 q 0 z + ( q 0 − 1 ) ( − 1 + 1 + 2 q 0 z ) q 0 2 ( 1 + z )

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.

This equation is only valid if q 0 > 0 . When q 0 ≤ 0 the value of r 1 cannot be calculated. From this radius we can calculate luminosity distance using the following formula,

D L   =   R 0 r 1 ( 1 + z ) = c H 0 q 0 2 [ q 0 z + ( q 0 − 1 ) ( − 1 + 1 + 2 q 0 z ) ]

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

D L = c H 0 ( z + z 2 2 )

But in 1977 Terrell devised a formula which is valid for all q 0 ≥ 0 ,

D L = c H 0 z [ 1 + z ( 1 − q 0 ) 1 + q 0 z + 1 + 2 q 0 z ]

References

Mattig formula Wikipedia
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