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Hubble Rate, Distance and Cosmic Recession Speed

Relate an assumed Hubble rate, positive proper distance and cosmic recession speed, with signed rates, zero-case handling and independent speed/distance units.

Formula & worked example
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Calculator inputs

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How to use this calculator

Enter the known values in the units shown. Results update as you type.
Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
Use the worked example to check the method. Reset restores the starting fields.

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Formula and method

With the retained assumed rate 70.3 km/s/Mpc, a proper distance of 10 Mpc gives 703 km/s of recession speed.

v = H D; D = v/H when H ≠0; H = v/D for positive D.

Worked example

Enter these known values and leave the other values blank.

Proper distance: 10 Mpc

Cosmic recession speed: 703 km/s

Assumptions and limitations

This is the homogeneous cosmic-flow relation v=H D at a common cosmological time, using proper distance and the specified expansion rate. It is not a local special-relativistic relative speed between neighboring observers.
The captured starting rate 70.3 km/s/Mpc remains locked by default as a legacy input assumption. It is not presented as a new measurement or a universally agreed current value. Unlock and enter the rate appropriate to the stated model.
The Hubble-rate unit has a speed numerator and an independent distance denominator. Changing either unit converts the same physical rate; it does not change the expansion assumption.
Distance must be strictly positive. Positive H gives recession; a negative H gives the signed contraction case in the retained mathematical model. A galaxy’s local peculiar motion is not inferred from H or included in this calculation.
With a known positive distance, H=0 gives v=0 and v=0 gives H=0 when the rate is unlocked. If H and v are both zero but distance is unknown, every positive distance is compatible and no unique distance is returned.
Parsecs and megaparsecs use the IAU exact 648000/π astronomical-unit convention; the astronomical unit is 149597870700 m. Light-years use exact light speed and a Julian year of 365.25 days, and mega-light-years are one million of those units.
The model may return recession speed greater than c at sufficiently large proper distance. Cosmic recession is not a local signal or material speed, so imposing a special-relativistic c limit on this relation would change the model.
This tool does not infer observed redshift, luminosity distance, lookback time, peculiar velocity, a time-varying expansion history or cosmological parameters. Those require a specified cosmological model and additional calculations.

Common questions

Is the starting Hubble rate a verified latest measurement?

No. The retained 70.3 km/s/Mpc is an explicit input assumption from the captured calculator. Choose the value appropriate to your data and cosmological model; different measurement methods remain under study.

Why is distance not calculated when both the rate and speed are zero?

The equation then reads 0 = 0 × D for any positive D. Dividing 0 by 0 cannot select a unique distance. Supply a known distance or a nonzero compatible rate and speed.

Can cosmic recession speed exceed the speed of light?

In this proper-distance relation it can. This is an expansion-of-space quantity, not the locally measured speed of matter or information through space.

References

Preserve v=H D and both nonsingular inverse formulas, but leave distance undetermined when H is exactly zero instead of calculating the undefined zero/zero inverse. Valid zero-rate forward calculations remain available with a known positive distance. Replace rounded parsec/megaparsec and Julian light-year/mega-light-year factors in both distance and inverted Hubble-rate denominator selectors, including base/default units. Use positive-sign distance and exact zero/nonzero sign pairing while retaining signed Hubble rates, valid zero forward cases and the disclosed captured70.3 default.

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