to compute the time t e at which the light from the distant galaxy was emitted:
t e =
t 0
(1 + z) 3(1+w)/2
=
2
3(1 + w)H 0
1
(1 + z) 3(1+w)/2
. (5.52)
The current proper distance to the galaxy is
d p (t 0 ) = c Z
t 0
t e
dt
a(t)
= ct 0
3(1 + w)
1 + 3w
[1 − (t e /t 0 ) (1+3w)/(3+3w) ] , (5.53)
when w 6 = − 1/3. In terms of H 0 and z rather than t 0 and t e , the current
proper distance is
d p (t 0 ) =
c
H 0
2
1 + 3w
[1 − (1 + z) −(1+3w)/2 ] .