WIMP Thermal Freeze-Out: Dark Matter Relic Abundance

WIMP Parameters

Mass m_χ 100 GeV

Annihilation ⟨σv⟩ 3.0×10⁻²⁶ cm³/s

Relativistic dof g_* 86.25

Playback speed 1.0×

Live Freeze-Out State

x = m/T

1.0

Freeze-out x_f

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Y(x) comoving

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Y_eq(x)

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Predicted relic density Ω_χh²

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How it works

A WIMP stays in chemical equilibrium with the hot plasma while annihilations χχ↔SM SM keep pace with the Hubble expansion. As the universe cools past T ≈ m_χ the equilibrium density falls exponentially, but annihilations become too rare to track it — the comoving abundance freezes out at a constant value. The Boltzmann equation for the comoving number density Y = n/s, with x = m_χ/T, is:

dY/dx = -(λ/x²)(Y² - Y_eq²)
λ = √(π/45) · √g* · M_Pl · m_χ · ⟨σv⟩
Y_eq(x) ≈ 0.145 · (g/g*) · x^1.5 · e^-x   (x ≫ 1, g = 2)

This simulator integrates that ODE with RK4 from x = 0.3 to x = 60 every time you move a slider, then plays the result back. Solving analytically for x beyond freeze-out (dropping the small Y_eq² term) gives the asymptotic abundance:

1/Y(x) ≈ 1/Y(x_f) + (λ/x_f)(1 - x_f/x)
Ω_χh² ≈ 2.755×10⁸ · (m_χ/GeV) · Y_∞

Smaller ⟨σv⟩ or larger mass means annihilation shuts off earlier (smaller x_f), leaving more particles frozen in — a larger relic density. This inverse relation is the "WIMP miracle": a weak-scale mass (~100 GeV) and a weak-scale cross-section (⟨σv⟩ ≈ 3×10⁻²⁶ cm³/s, the value a particle with SU(2) coupling naturally has) lands almost exactly on the observed Ω_DMh² ≈ 0.12 — no fine-tuning required. The cube on the right shows this directly: each sphere is a comoving WIMP; watch the population thin out and then hold steady once freeze-out (the white dashed marker on the graph) is reached.

x = 1.00

Comoving WIMPs (cube) Y(x) — actual abundance Y_eq(x) — equilibrium value

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