Calculator
Choose a mode, enter values, then press Calculate.
Example Data Table
Sample outputs using H0 = 70, Ωm = 0.30, ΩΛ = 0.70.
| z | Proper today (Mpc) | Proper at emission (Mpc) | Lookback (Gyr) |
|---|---|---|---|
| 0.50 | 1888.63 | 1259.08 | 5.04 |
| 1.00 | 3303.83 | 1651.91 | 7.72 |
| 2.00 | 5179.86 | 1726.62 | 10.24 |
| 3.00 | 6355.69 | 1588.92 | 11.35 |
Formula Used
In a homogeneous FLRW model, the dimensionless expansion function is: E(z) = √(Ωr(1+z)^4 + Ωm(1+z)^3 + Ωk(1+z)^2 + ΩΛ).
The comoving radial coordinate integral is: χ(z) = ∫₀ᶻ dz′ / E(z′), and the Hubble distance is D_H = c / H0.
Curvature enters through S_k(χ): S_k(χ)=χ (flat), S_k(χ)=sinh(√Ωk χ)/√Ωk (open), S_k(χ)=sin(√|Ωk| χ)/√|Ωk| (closed).
Proper distance today (a₀=1) is: D_p(today) = D_H · S_k(χ). Proper distance at emission uses a = 1/(1+z): D_p(emission) = D_p(today)/(1+z).
Lookback time is computed by: t_L = (1/H0) ∫₀ᶻ dz′ /((1+z′)E(z′)).
How to Use
- Select a mode: redshift, scale-factor, or local separation.
- Enter values in the visible fields for that mode.
- Choose output units, integration method, and steps as needed.
- Press Calculate to show results above the form.
- Use the export buttons to download CSV or PDF.