16.6 Horizons and the Size of the Observable Universe
259
For a source that emitted light near the beginning of the universe, t e = 0, we thus
have
σ =
3c
a 0
t 0 ≡ σ hor .
(16.23)
Any source beyond this coordinate distance would not be observable since its light
would not yet have reached us; σ hor is our cosmic horizon, beyond which we cannot
see. The coordinate horizon increases from a small value at early times to encompass
more and more of the universe as time passes.
It is important to also calculate how far away, in physical distance, is a source
now at the horizon. From (16.23) and the relation between coordinate and physical
distances we obtain the physical distance L,
L = a 0 σ hor = 3ct 0 .
(16.24)
That is, the source is three times as far away as the light from it has traveled to reach
us!
The above result is remarkable and may be somewhat counter-intuitive. Since the
scale factor approaches zero at very early times all the parts of the universe were
then very close together. How is it then that light emitted from an object very near
to our position has taken billions of years to reach us? It is because the expansion
of the universe was initially so rapid that the matter outran the speed of light! This
is evident from the fact that the Hubble function for the flat cold matter universe is
2/3t and diverges at early times. See also Example 15.1 for a discussion of recession
velocity greater than c.
As we noted above the assumption of zero pressure and curvature and cosmological constant is reasonable for much of the history of our universe. The same sort of
manipulations can be applied for the LCDM universe containing also dark energy but
the analog of the integral in (16.22) does not reduce to a simple function like (16.23)
for the horizon. Of course the relation nevertheless exists between the horizon and
the present time: it is defined by the integral.
In Exercise 16.5 you are asked to obtain the horizon for a de Sitter universe using
the same procedure as above. In the next section we will consider a different approach
to the horizon for a flat de Sitter universe, that is one containing only dark energy.
16.7 Conformal Time
In the preceding chapters we have used the FLRW metric, which has g 00 = 1 for a
universal time coordinate. There is another type of metric which is often useful; it is
one in which the line element is a multiple of the flat space metric of special relativity,
so the behavior of light is essentially the same as in special relativity and light cones
259
For a source that emitted light near the beginning of the universe, t e = 0, we thus
have
σ =
3c
a 0
t 0 ≡ σ hor .
(16.23)
Any source beyond this coordinate distance would not be observable since its light
would not yet have reached us; σ hor is our cosmic horizon, beyond which we cannot
see. The coordinate horizon increases from a small value at early times to encompass
more and more of the universe as time passes.
It is important to also calculate how far away, in physical distance, is a source
now at the horizon. From (16.23) and the relation between coordinate and physical
distances we obtain the physical distance L,
L = a 0 σ hor = 3ct 0 .
(16.24)
That is, the source is three times as far away as the light from it has traveled to reach
us!
The above result is remarkable and may be somewhat counter-intuitive. Since the
scale factor approaches zero at very early times all the parts of the universe were
then very close together. How is it then that light emitted from an object very near
to our position has taken billions of years to reach us? It is because the expansion
of the universe was initially so rapid that the matter outran the speed of light! This
is evident from the fact that the Hubble function for the flat cold matter universe is
2/3t and diverges at early times. See also Example 15.1 for a discussion of recession
velocity greater than c.
As we noted above the assumption of zero pressure and curvature and cosmological constant is reasonable for much of the history of our universe. The same sort of
manipulations can be applied for the LCDM universe containing also dark energy but
the analog of the integral in (16.22) does not reduce to a simple function like (16.23)
for the horizon. Of course the relation nevertheless exists between the horizon and
the present time: it is defined by the integral.
In Exercise 16.5 you are asked to obtain the horizon for a de Sitter universe using
the same procedure as above. In the next section we will consider a different approach
to the horizon for a flat de Sitter universe, that is one containing only dark energy.
16.7 Conformal Time
In the preceding chapters we have used the FLRW metric, which has g 00 = 1 for a
universal time coordinate. There is another type of metric which is often useful; it is
one in which the line element is a multiple of the flat space metric of special relativity,
so the behavior of light is essentially the same as in special relativity and light cones