Re: What is the relation between recessional velocities and red shifts >>1?

I agree, this silence is a bit annoying. However, I think you'll see in a moment why introductory texts do not go into great detail regarding redshifts greater than 1.

First, the redshift itself is defined in terms of an observed wavelength of light, lambdao, and a "rest frame" wavelength of light, lambdarf, as

z = (lambdao - lambdarf)/lambdarf.

The rest frame wavelength is the wavelength of light that would be measured if one could be, as it were, "right next to" the object emitting the light. In practice, the rest frame wavelength is the wavelength measured in the laboratory for the atomic transition being observed.

When it comes to the Universe, we have to be careful how we specify distances. Define a luminosity distance, DL, such that doubling the luminosity distance of an object causes its apparent brightness to decrease by a factor of 4. (That is, the luminosity distance allows the inverse square law of light to hold.) Then the relation between the redshift and the luminosity distance is

      cz           z(1 - q)
 DL = -- [1 + ----------------------].
       H      sqrt(1 + 2qz) + 1 + qz

Here H is the Hubble parameter (current best estimates are that H ~ 70 kilometers per second per Megaparsec) and q is the deceleration parameter. If q < 0.5, the Universe is open and will expand forever; if q > 0.5, the Universe is closed and will recollapse in the (distant) future. More commonly cited today is the density parameter Omega, Omega = 2q.

The original question asked about the recessional velocity. The recessional velocity is v = HDL.

[It is left as an exercise for the reader :) to verify that if z is much much less than 1, then DL ~ cz/H or v = HDL ~ cz. Hint: Observationally, q ~ 0.5 so just set q = 0.5.]

An astute reader will have noted that I was careful in my definition of distance. There are, in fact, four definitions for distance in an expanding Universe: a luminosity distance, an angular diameter distance, a parallax distance, and a proper motion distance. We cannot yet measure the last two quantities directly over cosmological distances. The first two quantities are related by

DA = DL/(1 + z)2

These formulae are summarized in K. R. Lang, Astrophysical Formulae (1980, Springer-Verlag: Berlin, ISBN 3-540-09933-6).