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]

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

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 $q_{0}\leq 0$ 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)={\frac {c}{H_{0}q_{0}^{2}}}\left[q_{0}z+(q_{0}-1)(-1+{\sqrt {1+2q_{0}z}})\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 $q_{0}\geq 0$ ,^[4]

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

↑ Observations in Cosmology, Cambridge University Press
↑ Mattig, W. (1958), "Über den Zusammenhang zwischen Rotverschiebung und scheinbarer Helligkeit", Astronomische Nachrichten, 284: 109, Bibcode:1958AN....284..109M, doi:10.1002/asna.19572840303
↑ Bradley M. Peterson, "An Introduction to Active Galactic Nuclei", p. 149
↑ 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

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