Math 2.0

Relativity and gravitational theory Relativistic cosmology Problem

Flatness problem

Builds on Friedmann equationsBig BangScale factorFriedmann–Lemaître–Robertson–Walker metric

Problem

Consider a Friedmann–Lemaître–Robertson–Walker model with spatial curvature kk, Hubble parameter HH and scale factor aa. Its first Friedmann equation can be written as Ω(t)−1=ka(t)2H(t)2,\Omega(t) - 1 = \frac{k}{a(t)^2 H(t)^2}, where Ω\Omega is the total density divided by the critical density 3H2/(8πG)3H^2/(8\pi G). In the radiation era ∣Ω−1∣\abs{\Omega - 1} grows like tt, and in the matter era like t2/3t^{2/3}. Observations show that Ω\Omega is close to 11 today. So in the early universe Ω\Omega must have been equal to 11 with extreme precision. At the time of nucleosynthesis, a deviation of more than a tiny fraction would have led to a very different universe today. The problem is to explain this fine tuning of the initial conditions of the Big Bang model.

Remarks

The growth rates follow from aH=a˙aH = \dot a: for a∝t1/2a \propto t^{1/2} one has a˙∝t−1/2\dot a \propto t^{-1/2}, so 1/a˙2∝t1/\dot a^2 \propto t; for a∝t2/3a \propto t^{2/3} one has 1/a˙2∝t2/31/\dot a^2 \propto t^{2/3}. More generally ∣Ω−1∣\abs{\Omega - 1} grows whenever a˙\dot a decreases, that is, whenever the expansion decelerates. Cosmic inflation reverses this: while a¨>0\ddot a > 0, a˙\dot a grows and Ω\Omega is driven toward 11, which is the standard proposed solution. Some authors question whether the fine tuning is a real problem, for example on probabilistic grounds, so its status is partly a matter of interpretation.

Reading path

To understand this page from the ground up, read these 11 pages in order. The foundations come first.

Show the path as a graph
Big Bang, concept, no evidence levelBig BangconceptCosmological constant, concept, no evidence levelCosmological constantconceptCosmological principle, concept, no evidence levelCosmological principleconceptEinstein field equations, concept, no evidence levelEinstein field equati…conceptPerfect fluid, concept, no evidence levelPerfect fluidconceptScale factor, concept, no evidence levelScale factorconceptSpace form, concept, no evidence levelSpace formconceptSpacetime, concept, no evidence levelSpacetimeconceptFriedmann–Lemaître–Robertson–Walker metric, example, no evidence levelFriedmann–Lemaître–Ro…exampleFriedmann equations, statement, no evidence levelFriedmann equationsstatementFlatness problem, problem, no evidence levelFlatness problemproblem
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. Big Bang concept
  2. Cosmological constant concept
  3. Cosmological principle concept
  4. Einstein field equations concept
  5. Perfect fluid concept
  6. Scale factor concept
  7. Space form concept
  8. Spacetime concept
  9. Friedmann–Lemaître–Robertson–Walker metric example
  10. Friedmann equations statement
  11. Flatness problem problem

Further reading

  1. Web https://en.wikipedia.org/wiki/Flatness_problem
About this page
TagD7XJTSZZEA
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Kindproblem
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MSC83F05
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AboutBig Bang
UsesFriedmann equations, Big Bang, Scale factor, Friedmann–Lemaître–Robertson–Walker metric
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Changed3 November 2026

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