Math 2.0

Astronomy and astrophysics Definition

Spherical collapse model

Builds on Einstein–de Sitter universe

Definition

Consider an Einstein–de Sitter universe, a flat universe that contains only pressureless matter, with mean density ρˉ(t)=1/(6πGt2)\bar\rho(t) = 1/(6\pi G t^2), where GG is the gravitational constant. The spherical collapse model describes a region of mass MM by the following assumptions.

  1. The region is a sphere of uniform density, larger than the mean (a top hat), and it has no internal pressure.
  2. Each spherical shell moves only under the gravity of the mass inside it, and shells do not cross before the final collapse.
  3. The sphere has negative total energy, so its radius RR follows the cycloid R=A(1−cos⁡θ),t=B(θ−sin⁡θ),A3=GMB2,R = A (1 - \cos\theta), \qquad t = B (\theta - \sin\theta), \qquad A^3 = G M B^2, with a parameter θ∈[0,2π]\theta \in [0, 2\pi] and constants A,B>0A, B > 0.

Remarks

The sphere reaches its largest radius (turnaround) at θ=π\theta = \pi and collapses to a point at θ=2π\theta = 2\pi. The density contrast at turnaround is 1+δ=9π2/16≈5.551 + \delta = 9\pi^2/16 \approx 5.55. The overdensity predicted by linear theory at the time of collapse is δc=320(12π)2/3≈1.686,\delta_c = \frac{3}{20} (12\pi)^{2/3} \approx 1.686 , independent of MM. In practice the collapse stops when the matter virializes. By the virial theorem the final radius is half of the turnaround radius. The mean density is then 18π2≈17818\pi^2 \approx 178 times the background density at the collapse time.

Example: a sphere whose linear overdensity, extrapolated to today, is 22 has exceeded δc≈1.686\delta_c \approx 1.686, so the model predicts it has already collapsed.

Non-example: an ellipsoidal overdensity violates clause 1. Tidal shear makes it collapse first along its shortest axis, and the collapse threshold then depends on its shape and on MM.

Reading path

To understand this page from the ground up, read these 11 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…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 equationsstatementEinstein–de Sitter universe, example, no evidence levelEinstein–de Sitter un…exampleSpherical collapse model, definition, no evidence levelSpherical collapse mo…definition
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  1. Cosmological constant concept
  2. Cosmological principle concept
  3. Einstein field equations concept
  4. Perfect fluid concept
  5. Scale factor concept
  6. Space form concept
  7. Spacetime concept
  8. Friedmann–Lemaître–Robertson–Walker metric example
  9. Friedmann equations statement
  10. Einstein–de Sitter universe example
  11. Spherical collapse model definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Structure_formation
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MSC85A40, 85A05
DefinesSpherical collapse model
UsesEinstein–de Sitter universe
Changed3 November 2026

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