Math 2.0

Relativity and gravitational theory Relativistic cosmology Example

de Sitter universe

Builds on Friedmann–Lemaître–Robertson–Walker metricFriedmann equationsCosmological constantDe Sitter space

Example

Let Λ>0\Lambda > 0 be a cosmological constant and H=Λ/3H = \sqrt{\Lambda/3}, in units with c=1c = 1. The de Sitter universe is the spatially flat Friedmann–Lemaître–Robertson–Walker metric with no matter and scale factor a(t)=eHta(t) = e^{Ht}: ds2=−dt2+e2Ht(dx2+dy2+dz2),t∈ℝ.ds^2 = -dt^2 + e^{2Ht} \left( dx^2 + dy^2 + dz^2 \right), \qquad t \in \R.

Check: here k=0k = 0, ρ=p=0\rho = p = 0 and a˙/a=H\dot a / a = H is constant. The first Friedmann equation reads H2=Λ/3H^2 = \Lambda/3, which holds by the choice of HH. The second reads a¨/a=H2=Λ/3\ddot a / a = H^2 = \Lambda/3, which also holds. So the metric solves the vacuum Einstein equations with cosmological constant Λ\Lambda.

Remarks

The Hubble parameter is constant and the deceleration parameter is q=−1q = -1, so the expansion accelerates forever. There is a cosmological event horizon at proper distance 1/H1/H from every comoving observer, and no particle horizon. These coordinates cover only half of the maximally symmetric de Sitter space; slicings with k=+1k = +1, a(t)=H−1cosh⁡(Ht)a(t) = H^{-1}\cosh(Ht), and with k=−1k = -1, a(t)=H−1sinh⁡(Ht)a(t) = H^{-1}\sinh(Ht), describe the same spacetime or parts of it. The model is the late-time limit of the Lambda-CDM model and the approximate geometry of the inflationary epoch.

Reading path

To understand this page from the ground up, read these 14 pages in order. The foundations come first.

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Cosmological constant, concept, no evidence levelCosmological constantconceptCosmological principle, concept, no evidence levelCosmological principleconceptEinstein field equations, concept, no evidence levelEinstein field equati…conceptEinstein manifold, concept, no evidence levelEinstein manifoldconceptMinkowski space, concept, no evidence levelMinkowski spaceconceptPerfect fluid, concept, no evidence levelPerfect fluidconceptScale factor, concept, no evidence levelScale factorconceptSectional curvature, concept, no evidence levelSectional curvatureconceptSpace form, concept, no evidence levelSpace formconceptSpacetime, concept, no evidence levelSpacetimeconceptDe Sitter space, example, no evidence levelDe Sitter spaceexampleFriedmann–Lemaître–Robertson–Walker metric, example, no evidence levelFriedmann–Lemaître–Ro…exampleFriedmann equations, statement, no evidence levelFriedmann equationsstatementde Sitter universe, example, no evidence levelde Sitter universeexample
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Cosmological constant concept
  2. Cosmological principle concept
  3. Einstein field equations concept
  4. Einstein manifold concept
  5. Minkowski space concept
  6. Perfect fluid concept
  7. Scale factor concept
  8. Sectional curvature concept
  9. Space form concept
  10. Spacetime concept
  11. De Sitter space example
  12. Friedmann–Lemaître–Robertson–Walker metric example
  13. Friedmann equations statement
  14. de Sitter universe example

Further reading

  1. Web https://en.wikipedia.org/wiki/De_Sitter_universe
About this page
Tag55ZQST8RG5
URIm2://55ZQST8RG5
Kindexample
WritingW0 stub
MSC83F05, 83C15
Instance ofExact solutions in general relativity
UsesFriedmann–Lemaître–Robertson–Walker metric, Friedmann equations, Cosmological constant, De Sitter space
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Changed3 November 2026

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