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 one's using and is needed to calculate luminosity distance.

Without dark energy

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

$r_1 = \frac{c}{R_0 H_0} \frac{q_0z+(q_0-1)(-1+\sqrt{1+2q_0z})}{q_0^2(1+z)}$

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.

q₀ = 2Ω₀ is the deceleration parameter while Ω₀ is the density of matter in the universe at present.

R₀ is scale factor at present time while R is scale factor at any other time.

H₀ is Hubble's constant at present and

z is as usual the redshift.

This equation is only valid if q₀ > 0. When $q_0 \le 0$ the value of r₁ cannot be calculated. From this radius we can calculate luminosity distance using the following formula,

$D_L \ = \ R_0r_1(1+z) = \frac{c}{H_0q_0^2} \left[q_0z+(q_0-1)(-1+\sqrt{1+2q_0z})\right]$

When q₀ = 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 $q_0 \ge 0$,

$D_L = \frac{c}{H_0}z\left[1+\frac{z(1-q_0)}{1+q_0z+\sqrt{1+2q_0z}}\right]$

With dark energy

External links

• Observations in Cosmology, Cambridge University Press