FRW model


Riemannian geometry




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 ta(t)t \mapsto a(t), the scale factor of the universe depending on coordinate time tt (fixed by some gauge condition).

The equations of motion of the FRW model are then

  1. H 2(a˙/a) 2=2ρk/a 2H^2 \coloneqq (\dot a / a)^2 = 2 \rho - k/a^2

  2. a¨/a=(ρ+3p)\ddot a/ a = -(\rho + 3 p)

  3. ρ˙+3H(ρ+p)=0\dot \rho + 3 H(\rho + p ) = 0


  • Ha˙/aH \coloneqq \dot a/a is called the Hubble parameter?;

  • and

    • pp is the pressure?

    • ρ\rho is the density

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

Moreover, the ratio

  • wp/ρw \coloneqq p/\rho

is part of the experimental/phenomenological input into the model, which describes which kind of matter/radiation is assumed to fill spacetime

wsource of energy-density filling spacetime
1/3radiation or relativistic matter
0dust matter
1stiff fluid
-1cosmological constant

The first equation may be rewritten as

ΩΩ R+Ω M+Ω Λ=1+ka 2H 2 \Omega \coloneqq \Omega_R + \Omega_M + \Omega_\Lambda = 1 + \frac{k}{a^2 H^2}

where the density parameter Ω\Omega consists of the contribution

  • Ω R=2ρ ra 2\Omega_R = 2 \frac{\rho_r}{a^2} of radiation;

  • Ω M=2ρ ra 2\Omega_M = 2 \frac{\rho_r}{a^2} of matter;

  • Ω Λ=Λa 2\Omega_\Lambda = \frac{\Lambda}{a^2} of cosmological constant;


Last revised on November 24, 2016 at 12:44:08. See the history of this page for a list of all contributions to it.