Explore expansion history with adjustable matter and radiation. Switch units and compare flat or curved. Download clean tables in CSV or PDF anytime now.
| H0 (km/s/Mpc) | Ωm | Ωr | ΩΛ | z | H(z) (km/s/Mpc) |
|---|---|---|---|---|---|
| 70 | 0.3 | 8.4E-5 | 0.7 | 0 | 70.00000 |
| 70 | 0.3 | 8.4E-5 | 0.7 | 1 | 123.2678 |
| 70 | 0.3 | 8.4E-5 | 0.7 | 3 | 312.4240 |
The expansion rate is set by the Friedmann equation for a homogeneous universe. Using density parameters today, the normalized form is:
H(z) = H0 · √( Ωr(1+z)^4 + Ωm(1+z)^3 + Ωk(1+z)^2 + ΩΛ )
We also compute the present critical density:
ρc0 = 3H0² / (8πG)
Component densities follow the scaling laws ρm ∝ (1+z)^3, ρr ∝ (1+z)^4, and ρΛ is constant. Curvature is inferred when enabled:
Ωk = 1 − (Ωm + Ωr + ΩΛ)
Tip: If E(z)2 becomes negative, choose a different parameter set or redshift.
The Friedmann equation connects the universe’s expansion rate to its contents. It links the Hubble parameter H(z) to matter, radiation, curvature, and dark energy through their density parameters. By adjusting these inputs, you can model how expansion changes from early radiation domination to late-time acceleration.
The calculator reports H(z) in both km/s/Mpc and s−1. The dimensionless factor E(z)=H(z)/H0 isolates how expansion evolves independent of the chosen H0. For example, E(z)>1 indicates faster expansion in the past, while E(z)<1 would imply slower expansion relative to today.
Each term grows differently with (1+z). Radiation scales as (1+z)4 because photon energy and number density both increase going backward in time. Matter scales as (1+z)3 from volume dilution. Curvature scales as (1+z)2, and the dark-energy term stays constant in this standard model.
Curvature is controlled by Ωk. If Ωk≈0, the model is spatially flat. If Ωk>0, space is open; if Ωk<0, it is closed. When you enable “infer curvature,” the calculator enforces the closure condition Ωm+Ωr+ΩΛ+Ωk=1, which is common in parameter studies.
The present critical density ρc0=3H02/(8πG) sets the scale for translating dimensionless Ω values into kg/m3. The calculator then applies the redshift scaling to return ρm(z), ρr(z), and a constant ρΛ. This is useful when comparing to simulations or converting to alternative unit systems.
The deceleration parameter q(z)=0.5Ωm(z)+Ωr(z)−ΩΛ(z) summarizes whether expansion is slowing or speeding up. Positive q implies deceleration, typical at high redshift when matter or radiation dominate. Negative q implies acceleration, which becomes possible once ΩΛ(z) is large enough compared to the other components.
Reasonable “today” values often place Ωm around a few tenths, ΩΛ near the remainder, and Ωr near 10−4. Large negative ΩΛ or extreme curvature can make E(z)2 negative at some redshifts, which the calculator flags. If that happens, reduce the magnitude of the conflicting term or choose a smaller redshift.
This tool computes H(z), density evolution, curvature classification, and helpful derived quantities like dt/dz at the selected redshift. It does not perform full integrals for distance measures or the cosmic age unless you add numerical integration. Still, it is a strong starting point for sanity checks, quick comparisons, and teaching.
Redshift measures how much the universe has expanded since light was emitted. Larger z generally means earlier cosmic time and denser conditions. The calculator uses z to scale each Friedmann term appropriately.
They are connected by a=1/(1+z) when a=1 today. Turning off the z checkbox lets you input a directly; the calculator converts it to z internally.
It enforces Ωk=1−(Ωm+Ωr+ΩΛ). This is convenient when you want a self-consistent parameter set without manually tuning curvature. Disable it if you want to explore explicit curvature values.
E(z)2 is a weighted sum of terms. Certain combinations, such as strongly negative ΩΛ with large z, can force the sum below zero. That corresponds to an unphysical model for that redshift.
Not directly. It reports the integrand dt/dz at your chosen z, which is useful for building an age integral. A full age requires integrating dt/dz over a redshift range numerically.
Critical and component densities are shown in kg/m3. They are derived from H0 in s−1 and your chosen G. This keeps the output consistent with SI-based calculations.
Ωr is small at present because radiation energy density is low compared to matter and dark energy. Values around 10−4 are commonly used for quick calculations, but your study may require a more precise choice.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.