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

    Frequently Asked Questions

    Why is it called the critical density?

    It is the precise density at which the universe is geometrically flat. Below it, the universe is open and expands forever (even without dark energy). Above it, the universe is closed.

    How do I convert H0 from km/s/Mpc to 1/s?

    Divide by 3.086e19 (the number of km in a Mpc). So 70 km/s/Mpc becomes 70 / 3.086e19 ≈ 2.27e-18 s⁻¹.

    Is the observed density close to critical?

    Yes. Observations indicate the total density is within about 0.4% of the critical value, meaning the universe is very nearly flat.

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