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Redshift to distance calculator

For small redshifts velocity is simply cz. Past about z = 0.1 the relativistic Doppler formula matters, and past z = 1 you need a cosmological model.

km/s per Mpc
Hubble distance
214.137Mpc
= Exact Redshift and distance
Distance in million light-years
698.42
Recession velocity, cz
14 989.6 km/s
Relativistic Doppler velocity
14 615.3 km/s
Wavelength stretch factor
1.0500
Rough light travel time
0.698 billion years
Hubble time
13.968 billion years
Show the working
  1. Redshift z = 0.05
  2. Hubble constant = 70
  3. Recession velocity, cz = 0.05 * 299792.458 = 14 989.6
  4. Relativistic Doppler velocity = 299792.458 * ((1+0.05)^2 - 1) / ((1+0.05)^2 + 1) = 14 615.3
  5. Hubble distance = 14 989.6 / 70 = 214.137
  6. Distance in million light-years = 214.137 * 3.2615637769 = 698.423
  7. Rough light travel time = 698.423 / 1000 = 0.698 423
  8. Hubble time = 977.79222 / 70 = 13.9685
  9. Wavelength stretch factor = 1 + 0.05 = 1.05

Values are shown to six significant figures here; the arithmetic itself runs on exact fractions and is only rounded where a number is printed.

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    The formula

    • v = cz for small z
    • relativistic: v = c((1+z)^2 - 1)/((1+z)^2 + 1)
    • distance = v / H0

    Hubble law is a local approximation. Beyond z of about 0.1 the answer depends on the cosmological model, and the naive distance is increasingly wrong.

    The Hubble constant itself is disputed: local measurements give about 73 and cosmic microwave background fits give about 67.

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