Luminosity Distance Calculator

Luminosity distance (d_L) is the distance inferred from an object's observed flux and its intrinsic luminosity: d_L = sqrt(L / (4*pi*F)). In an expanding universe, d_L = (1+z) * d_C, where d_C is the comoving distance and z is redshift. Luminosity distance is larger than the physical distance at the time of emission because the universe was expanding while light was traveling.

Cosmological redshift (z) is the stretching of light wavelengths caused by the expansion of the universe. z = (lambda_observed - lambda_emitted) / lambda_emitted. For a source at redshift z = 1, the universe was half its current size when the light was emitted. The most distant galaxies have z > 10. Redshift is not the same as Doppler shift, although both change wavelength.

The Hubble constant H0 characterizes the current expansion rate of the universe: v = H0 * d, where v is recession velocity and d is proper distance. The Planck 2018 result is H0 = 67.4 +/- 0.5 km/s/Mpc. The local measurement (Cepheids and Type Ia supernovae) gives H0 approximately 73 km/s/Mpc. This discrepancy (the Hubble tension) is an active area of research.

Proper distance is the physical distance between two objects at a given moment in time, expanding with the universe. Comoving distance is defined to factor out the expansion: it remains constant for objects following the Hubble flow. At z = 0 (today), proper and comoving distance are equal. At the time of emission of light from a source at redshift z, the proper distance was d_C / (1+z).

The Hubble radius (or Hubble length) is c/H0 = the distance at which recession velocity equals the speed of light. For H0 = 67.4 km/s/Mpc: c/H0 = 2.998e5 / 67.4 = 4,450 Mpc = 14.5 billion light-years. Objects beyond the Hubble radius are receding faster than light (but we can still see them because the universe was smaller when the light was emitted). The observable universe extends about 46.5 billion light-years.