Redshift to Distance Calculator
Convert cosmological redshift to distance using Hubble law and standard cosmological parameters.
Formula
D_C = (c/H0) * integral from 0 to z of dz / E(z), where E(z) = sqrt(Om*(1+z)^3 + OL)
The comoving distance is computed by numerically integrating the inverse of the dimensionless Hubble parameter E(z) from redshift 0 to the observed redshift z. The Hubble distance c/H0 sets the overall scale. Luminosity distance = D_C * (1+z) and angular diameter distance = D_C / (1+z).
Worked Examples
Example 1: Nearby Galaxy Cluster
Problem: Calculate the distance to a galaxy cluster at redshift z = 0.05 using standard cosmological parameters (H0 = 70, Om = 0.3, OL = 0.7).
Solution: At z = 0.05, we use numerical integration of 1/E(z) from 0 to 0.05\nE(z) = sqrt(0.3*(1+z)^3 + 0.7)\nDH = c/H0 = 299792/70 = 4283 Mpc\nComoving distance = 4283 * integral = ~212 Mpc (~691 Mly)\nLuminosity distance = 212 * 1.05 = ~223 Mpc\nLookback time = ~0.68 Gyr
Result: Distance: ~212 Mpc (691 million light-years), lookback time: ~680 million years
Example 2: High-Redshift Quasar
Problem: Find the distance to a quasar at z = 3.0 with standard parameters.
Solution: At z = 3.0, we integrate 1/E(z) from 0 to 3.0\nE(z) = sqrt(0.3*(1+z)^3 + 0.7)\nDH = 4283 Mpc\nComoving distance = 4283 * integral = ~6,394 Mpc (~20.8 Gly)\nLuminosity distance = 6394 * 4 = ~25,576 Mpc\nAngular diameter distance = 6394 / 4 = ~1,599 Mpc\nLookback time = ~11.5 Gyr\nRecession velocity = ~88% of c
Result: Distance: ~6,394 Mpc (20.8 Gly), lookback time: ~11.5 billion years
Frequently Asked Questions
What is cosmological redshift and how does it relate to distance?
Cosmological redshift occurs when light from distant objects is stretched to longer, redder wavelengths as it travels through expanding space. Unlike Doppler redshift from motion, cosmological redshift results from the expansion of the universe itself stretching the wavelength of photons during their journey. The redshift value z tells us by what factor the universe has expanded since the light was emitted: a galaxy at z=1 emitted its light when the universe was half its current size. Higher redshift means greater distance and earlier cosmic time. For nearby objects (z less than 0.1), the relationship between redshift and distance is approximately linear following Hubble law, but for more distant objects the relationship becomes nonlinear and depends on cosmological parameters like the matter density and dark energy density.
What is the difference between comoving distance and luminosity distance?
Comoving distance is the proper distance to an object measured at the present time, accounting for the expansion of the universe. It represents the distance you would measure if you could freeze the universe and lay down rulers from here to the object right now. Luminosity distance is a different measure defined so that the inverse-square law for brightness still works in an expanding universe. Because photons lose energy to redshift and arrive at a slower rate, distant objects appear dimmer than their comoving distance alone would suggest. The luminosity distance is always larger than the comoving distance by a factor of (1 + z). Angular diameter distance, conversely, is the comoving distance divided by (1 + z), and it determines how large an object appears on the sky. These different distance measures converge for nearby objects where z approaches zero.
Can objects have a redshift that implies they are receding faster than light?
Yes, and this does not violate special relativity. Galaxies at redshifts greater than about z=1.5 are currently receding from us faster than the speed of light due to the expansion of space itself. Special relativity only prohibits objects from moving through space faster than light, but it places no limit on how fast space itself can expand. The most distant observed galaxies at z of 10 or higher are receding at several times the speed of light. We can still see them because the light they emitted in the past was emitted when they were closer and the expansion rate was different. The relativistic velocity formula used in Redshift to Distance Calculator, v = c times ((1+z) squared minus 1) over ((1+z) squared plus 1), gives the velocity of the object in special-relativistic terms, which always stays below c. The actual recession velocity in general relativity can and does exceed c for distant objects.
How is the distance to a star measured?
For nearby stars, astronomers use parallax: measuring the apparent shift in position as Earth orbits the Sun. One parsec (3.26 light-years) is the distance at which a star shows one arcsecond of parallax. For more distant objects, standard candles like Cepheid variables and Type Ia supernovae provide distance estimates.
What is redshift and how does it indicate motion?
Redshift occurs when light from an object moving away from us is stretched to longer (redder) wavelengths. It is measured as z = (observed wavelength - emitted wavelength) / emitted wavelength. Cosmological redshift indicates the expansion of the universe. A galaxy at z=1 is seen as it was when the universe was half its current size.
What is the cosmic distance ladder?
The cosmic distance ladder is a series of methods for measuring increasingly distant objects. Radar ranges within the solar system, parallax reaches a few thousand light-years, Cepheid variables extend to tens of millions of light-years, and Type Ia supernovae reach billions of light-years. Each method calibrates the next.