Mattig's formula was an important formula in

observational cosmology Observational cosmology is the study of the structure, the evolution and the origin of the universe through observation, using instruments such as telescopes and cosmic ray detectors. Early observations The science of physical cosmology as it is ...

and

extragalactic astronomy Extragalactic astronomy is the branch of astronomy concerned with objects outside the Milky Way galaxy. In other words, it is the study of all astronomical objects which are not covered by galactic astronomy. The closest objects in extragalactic ...

which gives relation between radial coordinate and

redshift In physics, a redshift is an increase in the wavelength, and corresponding decrease in the frequency and photon energy, of electromagnetic radiation (such as light). The opposite change, a decrease in wavelength and increase in frequency and e ...

of a given source. It depends on the

cosmological model Physical cosmology is a branch of cosmology concerned with the study of cosmological models. A cosmological model, or simply cosmology, provides a description of the largest-scale structures and dynamics of the universe and allows study of fu ...

being used and is used to calculate

luminosity distance Luminosity distance ''DL'' is defined in terms of the relationship between the absolute magnitude ''M'' and apparent magnitude ''m'' of an astronomical object. : M = m - 5 \log_\!\, which gives: : D_L = 10^ where ''DL'' is measured in parsecs. ...

in terms of redshift. It assumes zero

dark energy In physical cosmology and astronomy, dark energy is a proposed form of energy that affects the universe on the largest scales. Its primary effect is to drive the accelerating expansion of the universe. It also slows the rate of structure format ...

, and is therefore no longer applicable in modern cosmological models such as the

Lambda-CDM model The Lambda-CDM, Lambda cold dark matter, or ΛCDM model is a mathematical model of the Big Bang theory with three major components: # a cosmological constant, denoted by lambda (Λ), associated with dark energy; # the postulated cold dark mat ...

, (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,

r_1 = \frac \frac

Where,

r_1=\frac=\frac

is the radial coordinate distance (proper distance at present) of the source from the observer while

d_p

is the

proper distance Proper length or rest length is the length of an object in the object's rest frame. The measurement of lengths is more complicated in the theory of relativity than in classical mechanics. In classical mechanics, lengths are measured based on t ...

and

d_c

is the

comoving distance In standard cosmology, comoving distance and proper distance (or physical distance) are two closely related distance measures used by cosmologists to define distances between objects. ''Comoving distance'' factors out the expansion of the univ ...

. ::

q_0=\Omega_0/2

is the

deceleration parameter The deceleration parameter q in cosmology is a dimensionless measure of the cosmic acceleration of the expansion of space in a Friedmann–Lemaître–Robertson–Walker universe. It is defined by: q \ \stackrel\ -\frac where a is the scale ...

while

\Omega_0

is the density of matter in the universe at present. ::

R_0

scale factor In affine geometry, uniform scaling (or isotropic scaling) is a linear transformation that enlarges (increases) or shrinks (diminishes) objects by a '' scale factor'' that is the same in all directions ( isotropically). The result of uniform sc ...

at present time while

R

is scale factor at any other time. ::

H_0

Hubble's constant Hubble's law, also known as the Hubble–Lemaître law, is the observation in physical cosmology that galaxies are moving away from Earth at speeds proportional to their distance. In other words, the farther a galaxy is from the Earth, the faster ...

at present and ::

z

is as usual the

. 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) = \frac \left_0z+(q_0-1)(-1+\sqrt)\right /math>

When q_0=0 we get another expression for luminosity distance using

Taylor expansion In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...

, :

D_L = \frac\left(z+\frac\right)

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

q_0 \ge 0

, :

D_L = \fracz\left +\frac\right /math>

References

{{DEFAULTSORT:Mattig Formula Observational cosmology