black hole spacetimes | vanishing angular momentum | positive angular momentum |
---|---|---|
vanishing charge | Schwarzschild spacetime | Kerr spacetime |
positive charge | Reissner-Nordstrom spacetime | Kerr-Newman spacetime |
A Friedmann–Lemaître–Robertson–Walker model is a model in cosmology. It is a solution to Einstein's equations describing an homogeneous and isotropic expanding or contracting spacetime. Hence these are solutions used as models in cosmology. Indeed, an FRW-model is part of the standard model of cosmology.
The plain FRW model parameterizes a homogenous and isotropic spacetime after diffeomorphism gauge fixing with a single parameter $t \mapsto a(t)$, the scale factor of the universe depending on coordinate time $t$ (fixed by some gauge condition).
The equations of motion of the FRW model are then
$H^2 \coloneqq (\dot a / a)^2 = 2 \rho - k/a^2$
$\ddot a/ a = -(\rho + 3 p)$
$\dot \rho + 3 H(\rho + p ) = 0$
where
$H \coloneqq \dot a/a$ is called the Hubble parameter?;
and
of a “perfect fluid” of matter and radiation filling the universe.
Moreover, the ratio
is part of the experimental/phenomenological input into the model, which describes which kind of matter/radiation is assumed to fill spacetime
w | source of energy-density filling spacetime |
---|---|
1/3 | radiation or relativistic matter |
0 | dust matter |
1 | stiff fluid |
-1 | cosmological constant |
The first equation may be rewritten as
where the density parameter $\Omega$ consists of the contribution
$\Omega_R = 2 \frac{\rho_r}{a^2}$ of radiation;
$\Omega_M = 2 \frac{\rho_r}{a^2}$ of matter;
$\Omega_\Lambda = \frac{\Lambda}{a^2}$ of cosmological constant;
Matthias Blau, chapter 33 and 34 of Lecture notes on general relativity (web)
Jorge L. Cervantes-Cota, George Smoot, Cosmology today – A brief review (2011)(arXiv:1107.1789)
Wikipedia, Friedmann–Lemaître–Robertson–Walker metric