Use the responsive grid below. Large screens show three fields, medium screens show two, and mobile screens show one.
These sample cases show present-day, past, and future-like expansion conditions.
| Case | H₀ | Ωm | ΩΛ | Ωr | z | ρc,0 (kg/m³) | ρc(z) (kg/m³) | ρDE (kg/m³) | Dark Energy Fraction at z |
|---|---|---|---|---|---|---|---|---|---|
| Planck-like present universe | 67.4 km/s/Mpc | 0.315 | 0.685 | 0.00009 | 0 | 8.533e-27 | 8.534e-27 | 5.845e-27 | 0.6849 |
| Higher local H₀ example | 73.0 km/s/Mpc | 0.300 | 0.700 | 0.00009 | 0 | 1.001e-26 | 1.001e-26 | 7.007e-27 | 0.6999 |
| Matter-dominated past snapshot | 70.0 km/s/Mpc | 0.300 | 0.700 | 0.00009 | 2 | 9.204e-27 | 8.106e-26 | 6.443e-27 | 0.0795 |
| Future-like dark energy era | 70.0 km/s/Mpc | 0.300 | 0.700 | 0.00009 | -0.5 | 9.204e-27 | 6.788e-27 | 6.443e-27 | 0.9491 |
1. Critical density today:
ρc,0 = 3H₀² / (8πG)
2. Expansion function:
E(z)² = Ωr(1+z)⁴ + Ωm(1+z)³ + Ωk(1+z)² + ΩΛ
3. Hubble rate at redshift z:
H(z) = H₀ × √E(z)²
4. Critical density at redshift z:
ρc(z) = 3H(z)² / (8πG)
5. Dark energy density for a cosmological constant:
ρDE = ΩΛ × ρc,0
6. Vacuum energy density:
uDE = ρDE × c²
If Ωk is left blank, the calculator uses Ωk = 1 − Ωm − ΩΛ − Ωr.
- Enter the Hubble constant and choose its unit.
- Set the redshift for the epoch you want.
- Fill in Ωm, ΩΛ, and Ωr values.
- Leave Ωk blank to auto-complete curvature.
- Keep default G and c unless needed.
- Click Calculate to view densities and graph.
- Use the export buttons for CSV or PDF files.
1. What is critical density?
Critical density is the density needed for a spatially flat universe. It depends on the expansion rate, so it changes when Hubble rate changes.
2. What does dark energy density represent?
Dark energy density describes the energy linked to cosmic acceleration. In the cosmological constant model, its physical density stays constant over time.
3. Why can the dark energy fraction change with redshift?
The absolute dark energy density can stay constant while critical density changes. That makes the ratio ρDE / ρc(z) shift across cosmic time.
4. Why does the calculator ask for Ωr?
Radiation matters most at very early times. Including Ωr improves high-redshift calculations and keeps the expansion model more complete.
5. What happens if I leave Ωk empty?
The calculator derives curvature from the closure relation. It sets Ωk equal to one minus the other listed density parameters.
6. Why are SI units used in results?
SI units make the physics consistent. They also help compare densities, expansion rate, and vacuum energy without hidden conversion errors.
7. Does this calculator model evolving dark energy?
No. This version assumes a cosmological constant. That means dark energy density remains constant instead of following a changing equation of state.
8. Why is critical density much higher in the past?
The early universe expanded faster and had denser matter and radiation content. Since critical density scales with H(z)², it rises strongly backward in time.