Friedmann model

A Friedmann model or Friedmann-Lemaître model (named after the Russian mathematician and meteorologist Alexander Friedmann and the Belgian astrophysicist Georges Lemaître ) is understood in cosmology to be a solution to the Friedmann equation , i.e. H. a solution of Einstein's field equations with constant curvature , which is spatially isotropic around every point .

Friedmann models differ in terms of the parameter from the Robertson-Walker metric ${\displaystyle k}$

• ${\displaystyle k=+1}$: positive curvature
• ${\displaystyle k=0}$: no curvature, flat space
• ${\displaystyle k=-1}$: negative curvature

and the value of the cosmological constant . ${\displaystyle \Lambda }$

Special cases of the Friedmann models

Einstein cosmos

It is a non-expanding or contracting, static (unstable towards small changes) universe with

${\displaystyle k=+1,\quad \Lambda =\Lambda _{c}\ ,}$

where is. ${\displaystyle \Lambda _{c}=4/(\kappa M)^{2}}$

Lemaître Universe

${\displaystyle k=+1,\quad \Lambda =\Lambda _{c}(1+\epsilon )\ ,}$

where is a very small parameter. By choosing a suitable one , the time scale of the expansion of the universe is stretched so that an almost static universe exists between two expanding time phases. ${\displaystyle \epsilon }$ ${\displaystyle \epsilon }$

De-Sitter model

→ Main article : De-Sitter model

${\displaystyle \rho =0,\quad \Lambda >0}$

The three different values ​​for result in three possible models, which are, however, only different cuts of the same space-time . ${\displaystyle k}$

Einstein de Sitter model

The Einstein-de-Sitter universe results with

${\displaystyle k=0,\quad \Lambda =0\ .}$

For this flat, infinitely expanded universe, the parameter of the Robertson-Walker metric is currently developing with it . ${\displaystyle R}$ ${\displaystyle R\sim t^{2/3}}$

Individual evidence

1. ↑ Hubert Goenner: Einstein's theories of relativity: space, time, mass, gravitation . CH Beck, 1999, ISBN 978-3-406-45669-5 , p. 96 (accessed April 9, 2012).
2. ↑ ^{a b c d} R. Sexl, H. Urbantke: Gravitation and cosmology . 3rd, corrected edition. BI-Wissenschaftsverlag, Mannheim 1987, ISBN 3-411-03177-8 .