Velocity of most distant galaxies

Question

Taking Hubble's law into account, what is the velocity of the most distant galaxies relative to Earth? Yes, I know that it is actually space that is expanding, (so, it is not their real velocity), but at what speed relative to Earth are they receding from us?

Answer 1

The most distant galaxy currently known is at a redshift $Z=11.1$ (Oesch et al. 2016). According to Ned Wright's cosmology calculator, for a set of concordance cosmological parameters, this corresponds to a comoving distance of 9.88 Gpc (32 billion light years) and a recession velocity (now) from Hubble's law of 2.28c (i.e. 684,000 km/s).

Of course galaxies could exist at greater distances, although they do need time to form after the big bang. Theoretical simulations (e.g. Bromm 2011) suggest star forming galaxies might be present at redshifts of 20, only 200 million years after the big bang. Such galaxies, if they exist, would have recession velocities of 2.5c. This is the maximum possible recession velocity for any galaxy that is observable by us now (or in the near future).

There is no possibility of observable galaxies having recession velocities as high as 3.3c, since that would place them at redshifts of $>1000$ and they would need to have formed even before the epoch of hydrogen recombination.

However, as Pela and Zephyr are pointing out, there are (likely) galaxies that exist now, but that we cannot observe now and never will be able to measure, that are more distant than this and which are receding faster. We are reasonably sure that the universe probed by current observations of the cosmic microwave background is isotropic to comoving distances of about 14.1 Gpc (46 billion light years; "the edge of the observable universe"). Thus galaxies that are currently there, but which are just perturbations in the CMB as we see them now, will have recession velocities of 3.3c. Equally, there could be more distant galaxies than that in a much larger, or possibly infinite, universe and these could have much larger recession velocities still (that we also cannot observe or measure), and indeed they, like the galaxies we do observe, are possibly accelerating away from us.

Answer 2

The speed relative to Earth is greater than the speed of light. This is consistent with Relativity as the space between us and the distant galaxies has stretched.

The speed is given by

$$(\mathrm{Hubble\ constant}) \times (\mathrm{radius\ of\ the\ observable\ universe\ in\ Mpc})$$

Which, as noted by Pela in a comment, is roughly 3.3 times the speed of light.

Answer 3

There is a general misunderstanding of Relativity. Relativity primarily uses 'Relative Time', (t) whereas the 'traveller' uses Proper time, (t0). (t) is invariably greater than (t0), and the ratio: (t)/(t0) is referred to as time dilation, or Gamma, and has the positive range only of unity to infinity. The realationship between (t) and (t0) is dependant upon a Right Triangle, wherein the base represents distance in Light-seconds, and the height represents Proper time, (t0) in seconds.
The Hypotenuse then represents Relative time (t) in seconds. If, in one second of Proper time, (t0), the vessel carrying the Proper clock covers a distance of one Light-second, then the perceived velocity, as observed by the pilot of the vessel, will be one Light-second per second. However, the 'static' observer will perceive relative time, which is here represented by the Hypotenuse of the Right Triangle, which under these circumstances will be root(2) seconds, so the Observed Relative velocity will be measured as 1/(root(2)) Light-seconds per second.

Einstein sets no limit on Proper velocity. When the Proper velocity approaches infinity, then, for any distance covered, the Height of the triangle approaches zero, so that the length of the Hypotenuse approaches the length of the base. Thus in the limiting case, where Height is zero, Base and Hypotenuse are equal, so the Reltive velocity as perceived by the static observer will be one light-second per second, or c.

Back to Hubble, and recessional velocity. Both Hubble, and Doppler work on Proper velocity.

The point I am trying to make is that space is not expanding, but Proper velocities are without limit. Only RELATIVE velocities are limited by the factor of 'c'. Dopper shift is wrt Proper velocity, and the CMBR indicates recessive Proper velocities in excess of 50,000 x c