According to Steven Weinberg's book Gravitation and Cosmology : Principles and applications of the general theory of relativity" John Wiley Publishing, 1972 page 485, the exact equation relating the 'comoving radial coordinate' to redshift (z), Hubble Constant (H), and the ratio of the cosmic density to the critical density, Omega, is given by his Equation 15.3.23 as :
1/2
2 omega z + ( omega - 2 ) ( ( z omega + 1) - 1 )
r = c -------------------------------------------------------------
2
H omega ( 1 + z )
For H = 65 km/sec/mpc, and in terms of billions of light years you get:
1/2
( 2 omega z +( omega - 2)( ( omega z + 1) - 1)
r = 14.95 x -------------------------------------------------------
2
omega ( 1 + z)
Let's do an example.
Let's select a universe where Omega = 0.1 which means it has only 10 percent
of the critical density. Let's ask what r is for a quasar at a redshift of 4.0.
1/2
2 x 4 x 0.1 + (0.1 - 2) ( ( 4 x .1 + 1) - 1 )
r = 14.95 ------------------------------------------------------
2
(0.1) ( 1 + 4 )
From this you get
0.8 - 1.9 ( 0.18)
r = 14.95 x --------------------- light years
0.01 x 5
or 136.9 billion light years. For other combinations of redshift and
omega you get the following table:
Z Omega
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
------------------------------------------------------------------------------
0.5 52.9 27.9 19.6 15.4 12.9 11.2 10.0 9.0 8.3 7.7
1.0 80.2 42.7 30.0 23.7 19.8 17.2 15.3 13.9 12.8 11.9
1.5 97.2 52.0 36.7 29.0 24.3 21.1 18.8 17.0 15.6 14.5
2.0 109.0 58.6 41.5 32.8 27.5 23.9 21.2 19.2 17.6 16.3
2.5 117.8 63.6 45.2 35.7 29.9 25.9 23.1 20.8 19.1 17.6
3.0 124.7 67.6 48.0 38.0 31.8 27.6 24.5 22.1 20.2 18.7
3.5 130.4 70.9 50.4 39.9 33.4 28.9 25.7 23.2 21.2 19.5
4.0 135.1 73.7 52.5 41.5 34.7 30.1 26.6 24.0 21.9 20.2
4.5 139.2 76.1 54.2 42.9 35.8 31.0 27.5 24.8 22.6 20.8
5.0 142.8 78.2 55.7 44.1 36.8 31.8 28.2 25.4 23.1 21.3
5.5 146.0 80.0 57.1 45.1 37.7 32.6 28.8 25.9 23.6 21.7
6.0 148.8 81.7 58.3 46.1 38.5 33.2 29.4 26.4 24.0 22.1
6.5 151.4 83.3 59.4 46.9 39.1 33.8 29.9 26.8 24.4 22.5
7.0 153.8 84.7 60.4 47.7 39.8 34.3 30.3 27.2 24.8 22.8
7.5 156.0 85.9 61.3 48.4 40.3 34.8 30.7 27.6 25.1 23.0
limit 298.8 149.4 99.6 74.7 59.8 49.8 42.7 37.4 33.2 29.9
What this table shows is the length of the 'co-moving radial coordinate' of
the light source which emitted it light at a redshift of z at a time 't1' and
which arrives at our location at the current time. You can think of this as
the distance that the object is from us right now. The horizon distance in
these units is in the last row ( z = infinity) and this distance is always
greater than the distance to any object we are currently able to see. Note,
the only events occuring right now that will ever become visible to us are
those within a proper distance less than the indicated horizon distance.
For Omega = 1, and in these units, the horizon limit for H = 65 km/sec/mpc is 29.9 billion light years ( 2c/H = 2 x 300,000.0/65.0 = 29.9 billion light years). So, if we see a galaxy at a redshift on 4.0 in a universe with Omega = 1 and H = 65 km/sec/mpc, it is located at a distance of 20.2 billion light years just short of the horizon limit. Note, none of the above distances are actually observable today, but represent where objects are today given that we are just now receiving their light signals.
To make life even more complicated, we can calculate the time when the light was emitted by the quasar at a specific redshift, and for a specific Omega. Weinberg gives that formula in his book, equation 15.3.16 as:
1 -1.5
t = ----- Omega ( 1 - Omega ) ( sinh (Phi) - Phi )
H
where
2 2
sinh(Phi) = cosh(Phi) - 1
and
2 ( 1 - Omega)
cosh(Phi) = ------------------- + 1
Omega ( 1 + z )
for H = 65 km/sec/mpc, 1/H = 14.9 billion years. If you select a
z and Omage, solve for cosh(Phi), compute sinh(Phi) and then solve for t
in the first equation, you get the time when the light was first emitted in
billions of years for Omega < 1.0 which is the likely situation. The age of
the universe depends only on H and Omega and is found from the following
equation:
1 Omega -1 2
Age = 14.9 x ( ----------- - ------------------ x (cosh ( ------ - 1))
1 - Omega 1.5 Omega
2 ( 1 - Omega)
For Omega = 0.1 you get an age of 0.89 x 1/H = 13.26 billion years.
The table below gives the time when the light was first emitted for a given
redshift and Omega. The last row gives the age of the universe for the given
Omega. All times are in billions of years:
*********************** Omega ***************************
Z 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
--------------------------------------------------------------------
0.5 8.45 7.74 7.22 6.82 6.49 6.21 5.97 5.76 5.57
1.0 6.06 5.42 4.98 4.64 4.37 4.14 3.95 3.78 3.64
1.5 4.66 4.09 3.70 3.42 3.20 3.01 2.86 2.73 2.61
2.0 3.74 3.23 2.90 2.66 2.47 2.32 2.19 2.08 1.99
2.5 3.10 2.64 2.35 2.14 1.98 1.85 1.75 1.66 1.58
3.0 2.63 2.22 1.96 1.78 1.64 1.53 1.44 1.36 1.30
3.5 2.27 1.89 1.66 1.50 1.38 1.29 1.21 1.14 1.09
4.0 1.99 1.64 1.44 1.29 1.19 1.10 1.03 0.98 0.93
4.5 1.76 1.44 1.26 1.13 1.03 0.96 0.90 0.85 0.81
5.0 1.58 1.28 1.11 1.00 0.91 0.84 0.79 0.75 0.71
5.5 1.42 1.15 0.99 0.89 0.81 0.75 0.70 0.66 0.63
6.0 1.29 1.04 0.89 0.80 0.73 0.67 0.63 0.59 0.56
6.5 1.18 0.94 0.81 0.72 0.66 0.61 0.57 0.53 0.51
7.0 1.08 0.86 0.74 0.66 0.60 0.55 0.52 0.49 0.46
7.5 1.00 0.79 0.68 0.60 0.55 0.51 0.47 0.44 0.42
Age= 13.36 12.59 12.03 11.58 11.21 10.89 10.61 10.35 10.13
---------------------------------------------------------------------
Example, for a redshift of 5.0, and a universe with Omega = 0.5, its age since
the Big Bang is 11.21 billion years, and the light we now see from the object
began its journey when the universe was only 0.91 billion years old.
We can also subtract the age of the universe from each of the entries above to get the time-of-flight for the light to reach us ( age - t):
*********************** Omega ***************************
Z 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
-------------------------------------------------------------------
0.5 4.91 4.86 4.81 4.76 4.72 4.68 4.63 4.59 4.56
1.0 7.30 7.17 7.06 6.95 6.84 6.75 6.66 6.57 6.49
1.5 8.70 8.51 8.33 8.16 8.02 7.88 7.75 7.63 7.52
2.0 9.62 9.36 9.13 8.93 8.74 8.57 8.41 8.27 8.14
2.5 10.26 9.95 9.68 9.44 9.23 9.03 8.86 8.69 8.54
3.0 10.73 10.38 10.07 9.81 9.57 9.36 9.17 8.99 8.83
3.5 11.09 10.70 10.37 10.08 9.83 9.60 9.40 9.21 9.04
4.0 11.37 10.95 10.60 10.29 10.02 9.79 9.57 9.38 9.20
4.5 11.60 11.15 10.78 10.46 10.18 9.93 9.71 9.50 9.32
5.0 11.78 11.31 10.92 10.59 10.30 10.04 9.82 9.61 9.42
5.5 11.94 11.44 11.04 10.70 10.40 10.14 9.90 9.69 9.50
6.0 12.07 11.55 11.14 10.79 10.48 10.22 9.98 9.76 9.56
6.5 12.18 11.65 11.22 10.86 10.55 10.28 10.04 9.82 9.62
7.0 12.28 11.73 11.29 10.93 10.61 10.34 10.09 9.87 9.67
7.5 12.36 11.80 11.35 10.98 10.66 10.38 10.13 9.91 9.71
Age= 13.36 12.59 12.03 11.58 11.21 10.89 10.61 10.35 10.13
We see that for Omega = 0.5 and z = 5.0, the light has been traveling for
10.30 billion years.
So how far away is the object at a redshift of 5.0 for a cosmological model
with Omega = 0.5? If you use the 'lookback time' you get 10.30 billion light
years. If you use the 'proper' comoving radial coordinate distance, you get
36.8 billion light years. I prefer using 'look back' distance because even
though it is not a real physical distance, it gives you an immediate
sense for how close to the 'horizon' you are looking which we can define by
multiplying the age of the universe by the speed of light. For Omega = 0.5,
the current age of the universe is 11.21 billion years or a 'distance' of
11.21 billion light years. The object is then seen at a look-back time of
10.30 billion years and a 'distance' of 10.30 billion light years. It is this
distance that astronomers often mention, however, if you listen very closely
to interviews with them, they speak only in terms of either redshift, z,
or in look-back time ' we see this galaxy the way it was only 1 billion years
after the Big Bang". If you ask them to tell you a distance, they often
preface the answer with a particular H and Omega, and then revert to
a 'look back' distance. The actual metric, or co-moving proper distance
is not observable, even though it can be exactly calculated.