nLab FRW model







The Friedmann–Lemaître–Robertson–Walker models (often FRW-models) are class of models in cosmology. These are solutions to Einstein's equations describing a spatially 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. (In contrast to inhomogeneous 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

    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;


Named after Alexander Friedmann?, Georges Lemaître, Howard Robertson?, Arthur Walker.

Introduction and survey:

See also:

Discussion in Regge calculus:

  • Ren Tsuda, Takanori Fujiwara, Oscillating 4-Polytopal Universe in Regge Calculus (arXiv:2011.04120)

Last revised on June 7, 2024 at 09:11:30. See the history of this page for a list of all contributions to it.