FRW model (changes) in nLab

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Contents

Idea

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.)

Details

The plain FRW model parameterizes a homogenous and isotropic spacetime after diffeomorphism gauge fixing with a single parameter t↦a(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=2ρ−k/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

where

• H≔a˙/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

• w≔p/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

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

Ω≔Ω 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;

• Hubble's law

• big bang

• comoving time

• cosmic inflation

• cosmic topology

• standard model of cosmology

• structure formation

• inhomogeneous cosmology

References

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

Introduction and survey:

• 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)

• Konstantinos Xenos, An Introduction to FRW Cosmology and dark energy models (arXiv:2101.06135)

  (emphasis on dark energy and inflation)

See also:

• Wikipedia, Friedmann–Lemaître–Robertson–Walker metric

Discussion in Regge calculus:

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