Engineering & Science
Friedmann Equation: Cosmological Critical Density & Hubble H(z)
Calculate the universe’s critical closure density rho_crit = (3 * H_0^2) / (8 * PI() * G) (in 10^-27 kg/m^3) and flat Lambda-CDM expansion rate H(z) = H_0 * SQRT(Omega_m * (1 + z)^3 + Omega_Lambda) at redshift z.
Compute cosmic critical density (~5 hydrogen atoms/m^3) and high-redshift Hubble expansion!
Alexander Friedmann (1922) derived from Einstein’s General Relativity field equations the master equation governing cosmic expansion! Setting spatial curvature k = 0 defines the Critical Closure Density today: rho_crit,0 = (3 * H_0^2) / (8 * PI() * G), where converting Hubble’s constant H_0 (km/s/Mpc) by 1 Mpc = 3.08567758e19 km gives H_0 in inverse seconds! For Planck’s CMB Hubble constant H_0 = 67.4 km/s/Mpc (G = 6.6743e-11 m^3/(kg*s^2)), rho_crit,0 = 8.5329 x 10^-27 kg/m^3—equivalent to just 5.1 protons per cubic meter of space!
Looking back in cosmic time to spectroscopic redshift z, non-relativistic matter density dilutes with expanding volume as (1 + z)^3 while dark energy (cosmological constant Omega_Lambda) remains constant, driving the flat Lambda-CDM Friedmann expansion rate H(z) = H_0 * SQRT(Omega_m * (1 + z)^3 + (1 - Omega_m))! With Planck parameters (H_0 = 67.4, Omega_m = 0.315, Omega_Lambda = 0.685), at redshift z = 1.0 the Hubble expansion parameter was H(1) = 120.66 km/s/Mpc, and at cosmic noon z = 2.0 it reached 204.32 km/s/Mpc!
| Cosmological Redshift Epoch (H0 = 67.4, Omega_m = 0.315) | Matter Term Omega_m*(1+z)^3 | Hubble Parameter H(z) (km/s/Mpc) | Cosmic Expansion Era |
|---|---|---|---|
| Present Universe (z = 0.0) | 0.315 (< 0.685 Lambda) | 67.40 km/s/Mpc | Dark-Energy Accelerated Era |
| Lookback ~7.8 Gyr (z = 1.0) | 2.520 (> 0.685 Lambda) | 120.66 km/s/Mpc | Matter-Dominated Deceleration |
| Cosmic Noon Starbursts (z = 2.0) | 8.505 | 204.32 km/s/Mpc | Rapid Early Expansion |
Present-Day Critical Density rho_crit,0 (in Units of 10^-27 kg/m^3)
Converts B2 (km/s/Mpc) into s^-1 (/ 3.08567758e19) inside (3 * H_0^2) / (8 * PI() * G) * 10^27.
=IF(OR(B2 <= 0, B3 < 0, B3 > 1, B4 < 0), "Invalid Cosmology Inputs", ROUND(((3 * ((B2 / 30856775800000000000) ^ 2)) / (8 * PI() * 0.000000000066743)) * (10 ^ 27), 4))Flat Lambda-CDM Hubble Expansion Rate H(z) (km/s/Mpc) at Redshift z
Evaluates H_0 * SQRT(Omega_m * (1 + z)^3 + (1 - Omega_m)) for flat spatial curvature Omega_Lambda = 1 - Omega_m.
=IF(OR(B2 <= 0, B3 < 0, B3 > 1, B4 < 0), "Invalid Cosmology Inputs", ROUND(B2 * SQRT(B3 * ((1 + B4) ^ 3) + (1 - B3)), 2))Try the example
Use the original values or change them to test your understanding.