How this calculator works
This model contains pressureless matter and a cosmological constant, with zero curvature and negligible radiation. Flatness fixes the dark-energy fraction to 1 minus the matter fraction. Numerical integration gives comoving distance and lookback time; distance duality gives luminosity and angular-diameter distances. Radiation and neutrino evolution are omitted, so this tool is limited to redshifts from 0 through 20 and is unsuitable for recombination-era calculations. The chosen Hubble constant is an input, not a fitted measurement.
Formula
E(z) = sqrt(Omega_m (1+z)^3 + 1-Omega_m); H(z)=H0 E(z); a=1/(1+z). D_C=(c/H0) integral_0^z dz/E(z); t_L=(1/H0) integral_0^z dz/[(1+z)E(z)]. D_L=(1+z)D_C; D_A=D_C/(1+z).
Worked example
In a matter-only flat universe, z = 3 gives a = 0.25 and H(z) = 8 H0. With H0 = 70 km/s/Mpc, H(z) is 560 km/s/Mpc and comoving distance is about 4282.75 Mpc.
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Frequently asked questions
Which distance should I use for apparent brightness?
Use luminosity distance, along with a source luminosity and any necessary spectral or extinction corrections.
Can expansion accelerate while H falls?
Yes. Acceleration concerns the second derivative of scale factor; H is its fractional first derivative.
Can I use a redshift of 1100?
No. Radiation is important then. This deliberately limited model accepts z up to 20.