Cosmological Distance Calculator

Below z = 0.1 the code uses the simple linear Hubble law, d ≈ (c/H₀)×z, which is accurate enough when the universe's expansion rate hasn't changed appreciably during the light's travel. Past that threshold it switches to a second-order correction, d ≈ (c/H₀)×z×[1 + 0.5(1 − Ωm + ΩΛ)z], which folds in matter and dark energy density so the growing error from ignoring how the expansion rate has evolved doesn't accumulate at larger redshifts.

They answer different physical questions from the same redshift. Luminosity distance (used here as the base calculation) reflects how faint a standard-brightness object appears; comoving distance divides by (1+z) to give the "distance today if expansion froze"; angular diameter distance divides by (1+z) again, which is why very distant objects can appear larger on the sky than the raw comoving distance alone would suggest. All three converge to the same value only in the limit of very small redshift.

The calculator scales lookback time proportionally against the universe's 13.8-billion-year age using your redshift relative to 0.1, rather than numerically integrating the full expansion history the way a rigorous cosmological calculation would. That proportional scaling is reasonable at low redshift but becomes progressively less accurate as z grows, since it doesn't account for how the expansion rate itself has changed over time.

It affects nearly everything: luminosity distance, comoving distance, angular diameter distance, and recessional velocity are all computed directly from your chosen H₀, since it sets the overall distance scale of the universe. A lower H₀ (like Planck's 67.4) stretches all these distances larger than a higher H₀ (like SH0ES's 73.0) for the same redshift — this is the substance of the ongoing "Hubble tension" debate in cosmology.