Calculated Cosmological Outputs
Results appear here after submission and remain available for export.
Input Parameters
Example Data Table
| Scenario | H₀ (km/s/Mpc) | ΩΛ | Ωm | z | Use Case |
|---|---|---|---|---|---|
| Planck-like | 67.4 | 0.685 | 0.315 | 0 | Reference modern cosmology benchmark |
| Flat ΛCDM | 70.0 | 0.700 | 0.300 | 0.5 | Balanced instructional example |
| Higher H₀ | 73.0 | 0.720 | 0.280 | 1.0 | Expansion sensitivity comparison |
Formula Used
Λ = 3 ΩΛ H₀² / c²
ρΛ = Λ c² / (8 π G)
εΛ = ρΛ c²
RdS = √(3 / Λ)
E(z) = √(Ωm(1+z)³ + ΩΛ)
Here, H₀ is converted from km/s/Mpc to s⁻¹, c is the speed of light, and G is the gravitational constant. Flat-universe behavior is approximated when Ωm + ΩΛ ≈ 1.
How to Use This Calculator
- Enter the Hubble constant in km/s/Mpc.
- Provide dark energy and matter density parameters.
- Set a reference redshift to inspect expansion scaling.
- Choose a preset if you want fast population of values.
- Press Submit to calculate the cosmological constant and related quantities.
- Review the results shown above the form, then export them as CSV or PDF.
Article
Λ as a Curvature Measure
The cosmological constant represents vacuum-driven curvature in Einstein’s equations and is commonly estimated near 1.1 × 10-52 m-2 in a flat ΛCDM universe. For calculator work, its tiny size is important because small input changes in Hubble expansion or density fractions alter the inferred vacuum term. This tool converts those changes into readable outputs so curvature, vacuum density, and expansion sensitivity can be reviewed without manual unit conversion.
Dependence on the Hubble Constant
Λ scales with H02, so modest shifts in the Hubble constant produce measurable changes. Raising H0 from 67.4 to 73.0 km/s/Mpc increases the squared expansion term and therefore increases Λ when ΩΛ is held constant. The graph included in this calculator shows that response clearly. Users can compare baseline cosmology with higher-expansion scenarios and quickly judge how strongly the vacuum term responds to observed Hubble differences.
Influence of the Dark Energy Fraction
ΩΛ is the direct weighting factor on the vacuum contribution. At fixed H0, increasing ΩΛ from 0.685 to 0.720 produces a proportional increase in Λ, vacuum density, and energy density. Because Λ = 3ΩΛH02/c2, the dependence is linear in ΩΛ. That makes the calculator useful for scenario testing, classroom demonstrations, and checking consistency between parameter sets taken from different references.
Vacuum Density and Energy Meaning
The calculator converts Λ into mass density using ρΛ = Λc2/(8πG), then into energy density with ε = ρc2. These values are tiny in ordinary laboratory terms but dominant across cosmic scales. Reporting kg/m3, J/m3, and eV/m3 helps users compare relativity-based outputs with physically intuitive and particle-style units.
de Sitter Radius Interpretation
The de Sitter radius, √(3/Λ), provides a geometric length scale for vacuum-dominated spacetime. When Λ increases, the radius decreases, indicating stronger curvature influence. Presenting the result in meters and gigalight-years helps connect abstract curvature with astronomical distance and gives a practical benchmark for reports, lectures, and model comparison.
Use in Validation and Review
These outputs help validate spreadsheets, reproduce worked examples, and verify whether chosen Ωm and ΩΛ values are close to flat geometry. The added E(z), H(z), and acceleration indicator extend interpretation beyond one constant. Together, the graph, article, FAQs, and exports support transparent and repeatable cosmology analysis. This improves traceability during audits, teaching, and documentation.
FAQs
1. What does the cosmological constant represent?
It represents the vacuum term in Einstein’s equations and models the energy density associated with empty space, which contributes to accelerated cosmic expansion.
2. Why does Hubble input strongly affect Λ?
Because Λ depends on H02. Squaring the expansion rate magnifies even moderate changes in Hubble constant input.
3. Why are both mass density and energy density shown?
They describe the same vacuum component in different physical forms, helping users compare relativity-based quantities with common density and energy interpretations.
4. What is the meaning of the de Sitter radius?
It is a characteristic curvature scale for a vacuum-dominated universe. A smaller radius indicates a stronger influence from the cosmological constant.
5. Can this calculator test flat-universe assumptions?
Yes. The Ωm + ΩΛ output helps you see whether the chosen parameters are close to unity, which is a simple flatness check.
6. When should I use presets instead of custom values?
Use presets for quick benchmarking and demonstrations. Use custom values when reproducing published studies or testing sensitivity to alternative cosmological inputs.