Critical Density Calculator Formula

Understand the math behind the critical density calculator. Each variable explained with a worked example.

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Formulas Used

Critical Density

rho_crit = 3 * pow(hubble_constant_si, 2) / (8 * pi * 6.674e-11)

Variables

VariableDescriptionDefault
hubble_constant_siHubble Constant (H0)(1/s)2.27e-18

How It Works

Critical Density of the Universe

The critical density is the dividing line between a universe that expands forever and one that eventually re-collapses.

Formula

rho_crit = 3 H0² / (8 pi G)

  • *H0* must be in SI units of 1/s (about 2.27 x 10⁻¹⁸ s⁻¹ for 70 km/s/Mpc)
  • *G* = 6.674 x 10⁻¹¹ N m² kg⁻²

    • The observed value is roughly 9.5 x 10⁻²⁷ kg/m³, equivalent to about 5.7 hydrogen atoms per cubic metre.

    Worked Example

    Using H0 = 2.27e-18 s⁻¹ (≈ 70 km/s/Mpc).

    hubble_constant_si = 2.27e-18
    1. 01rho_crit = 3 H0² / (8 pi G)
    2. 02H0² = (2.27e-18)² = 5.153e-36
    3. 038 pi G = 1.676e-9
    4. 04rho_crit = 3 * 5.153e-36 / 1.676e-9
    5. 05rho_crit ≈ 9.22e-27 kg/m³

    Ready to run the numbers?

    Open Critical Density Calculator