far away when the light left them?
Author: William Keel <keel@bildad.astr.ua.edu>

Distance is indeed a slippery thing in an expanding universe such as ours.
There are at least three kinds of distances:

* angular-diameter distance---the one you need to make the usual
  relation
      sine(angular size) = linear size/distance    
  work;

* luminosity distance---makes the typical relationship
      observed flux = luminosity / 4 pi (distance**2)    
  work; and

* proper distance---the piece-by-piece distance the light actually
  travelled.

Of the three, the proper distance is perhaps the most sensible of the
three. In this case, distance doesn't mean either when the light was
emitted or received, but how far the light travelled. Since the
Universe expands, we have been moving away from the emitting object so
the light is catching up to us (at a rate set by the rate of expansion
and our separation from the quasar or whatever at some fiducial
time). You can of course turn this distance into an extrapolated
distance (where the quasar or it descendant object is "today") but
that gets very slippery.  Both special and general relativity must be
taken into account, so simultaneity, i.e., "today," has only a limited
meaning.  Nearby galaxies are pretty much where we see them; for
example, the light from the Andromeda galaxy M31 has been travelling
only about 0.01% of the usually estimated age of the Universe, so its
distance from us would have changed by about that fraction, if nothing
but the Hubble expansion affected its measured distance (which is not
the case, because gravitational interactions between the Andromeda
galaxy and our Galaxy affect the relative velocity of the two
galaxies).

To muddy the waters further, observers usually express distances (or
times) not in light-years (or years) but by the observable quantity
the redshift.  The redshift is, by definition, the amount by which
light from an object has been shifted divided by the emitted or
laboratory wavelength of the light and is usually denoted by z.  For
an object with a redshift z, one can show that (1+z) is the ratio of
the scale size of distances in the Universe between now and the epoch
when the light was given off.  Turning this into an absolute distance
(i.e., some number of light-years) requires us to plug in a rate for
the expansion (the Hubble constant) and its change with time (the
deceleration parameter), neither of which is as precisely known as we
might like.

As a result ages and distances are usually quoted in fairly round
numbers. If the expansion rate has remained constant (the unrealistic
case of an empty Universe), the age of the Universe is the reciprocal
of the Hubble constant. This is from 10--20 billion (US, 10^9) years
for the plausible range of Hubble constants.  If we account for the
matter in the Universe, the Universe's age drops to 7--15 billion
years.  A quick estimate of the look-back time (i.e., how long the
light from an object has been travelling to us) for something at
redshift z is 
             t = (z/[1+z])*1/H0
for Hubble constant H0. For example, the author has published a paper
discussing a cluster of galaxies at z=2.4.  For the press release we
quoted a distance of 2.4/3.4 x 15 billion light-years (rounded to 11
since that 15 is fuzzy).