Hubble's Law and Recession Velocity Calculator

Hubble's law states that the recession velocity of a galaxy is proportional to its distance: v = H0 x d. Discovered by Edwin Hubble in 1929, it was the first observational evidence that the universe is expanding. The constant of proportionality H0 (the Hubble constant) gives the current rate of expansion in km/s per megaparsec.

There are two competing measurements. The Planck 2018 CMB analysis gives H0 = 67.4 km/s/Mpc with an uncertainty of about 0.5 km/s/Mpc. Direct local distance-ladder measurements (SH0ES) give H0 = 73.0 plus or minus 1.0 km/s/Mpc. The tension between these two values is one of the biggest open problems in cosmology.

Yes. Recession velocities in Hubble's law represent the rate at which space expands between two points, not the motion of an object through space. Special relativity limits the speed of objects through space to c, but there is no such restriction on the expansion rate of space itself. Galaxies beyond the Hubble distance (c/H0, about 4,280 Mpc for H0 = 70) are currently receding faster than light.

The Hubble time is 1/H0, the inverse of the Hubble constant. For H0 = 70 km/s/Mpc it equals about 13.97 billion years. In a universe with no acceleration or deceleration it would be the exact age of the universe, but in the real LCDM universe the actual age (13.8 Gyr) is close but not identical.

The lookback time is the light travel time approximation: t = d/c, where d is the distance in Mpc and c is the speed of light. This is accurate to a few percent for z less than 0.5. For large redshifts, the true lookback time requires integrating over the Friedmann equation with matter, radiation, and dark energy density parameters.

Redshift z is the fractional change in wavelength of light emitted by a receding galaxy. For small recession velocities (v much less than c), z equals v/c, which is the formula this calculator uses. For z greater than about 0.3, the relativistic Doppler formula or the full cosmological treatment gives more accurate results.

At large distances two effects matter. First, the linear relationship v = H0 x d is a first-order Taylor expansion of the full scale factor evolution, so it fails at high z. Second, the expansion rate itself has changed over cosmic time (it decelerated due to matter and then accelerated due to dark energy), so the current H0 does not describe the expansion rate in the distant past.

Recession velocities are inferred from spectroscopic redshift. Astronomers compare the observed wavelengths of known spectral lines (hydrogen Lyman-alpha at 121.6 nm, calcium H and K at 397 and 393 nm, etc.) against laboratory wavelengths. The redshift z = (lambda_obs - lambda_rest) / lambda_rest directly gives v = cz for nearby galaxies.

The Hubble tension is the 5-sigma discrepancy between early-universe CMB measurements (H0 approximately 67 km/s/Mpc) and late-universe distance-ladder measurements (H0 approximately 73 km/s/Mpc). If the tension is real and not due to systematic errors, it implies new physics beyond the standard LCDM model, possibly involving early dark energy or modified gravity.

The Hubble constant H0 is measured in km/s/Mpc (kilometres per second per megaparsec). This means for every additional megaparsec of distance, the recession velocity increases by H0 km/s. In SI units, H0 has dimensions of inverse time (s to the power minus 1), with 70 km/s/Mpc corresponding to about 2.27 times 10 to the power minus 18 per second.

Proper distance is the actual physical separation between two points at a given cosmic time. Comoving distance factors out the expansion so that two objects in the Hubble flow have a fixed comoving separation even as the universe expands. The distance d in Hubble's law v = H0 x d refers to the current proper distance.

Using d = v/H0: d = 1000 / 70 = 14.29 Mpc, or about 46.6 million light-years. This is roughly the distance to the Virgo Cluster, whose member galaxies do indeed show recession velocities near 1000 km/s.