nLab
FRW model

Idea

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.

Details

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

1. $H^2≔(\dot{a}/a{)}^2=2\rho -k/a^2$

2. $\ddot{a}/a=-(\rho +3p)$

3. $\dot{\rho }+3H(\rho +p)=0$

where

• $H≔\dot{a}/a$ is called the Hubble parameter?;

• and

  • $p$ is the pressure?

  • $\rho$ is the density

  of a “perfect fluid” of matter and radiation? filling the universe.

Moreover, the ratio

• $w≔p/\rho$

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

$$\Omega ≔{\Omega }_R+{\Omega }_M+{\Omega }_{\Lambda }=1+\frac{k}{a^2{H}^2}$$\Omega \coloneqq \Omega_R + \Omega_M + \Omega_\Lambda = 1 + \frac{k}{a^2 H^2}

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;

Related concepts

• big bang

• cosmic inflation

• standard model of cosmology

References

• Jorge L. Cervantes-Cota, George Smoot, Cosmology today – A brief review (2011)(arXiv:1107.1789)

• Wikipedia, Friedmann–Lemaître–Robertson–Walker metric