258
16 Some Properties of the LCDM Universe
Fig. 16.3 Co-moving objects remain at coordinate rest in the FLRW metric. Light follows the
indicated curve from the distant source to us, defining our past light cone
increases as time progresses the physical distance between two co-moving objects,
such as galaxies, increases.
Let us ask how far we can see in the expanding universe? That is, what is the
maximum distance, both coordinate and physical, that a source can be so that light
from it has reached us? The question is very fundamental because it is equivalent to
asking the size of the observable universe. Recall that light is characterized as having
a null trajectory, ds
2
= 0. We impose this on the FLRW metric in the form (13.12b)
to characterize the path of light so the coordinate distance differential obeys
ds
2
= c
2 dt
2
− a
2 dσ
2
= 0, dσ =
cdt
a
.
(16.20)
To obtain the total coordinate distance σ we need to integrate this relation. Since the
universe has been dominated by cold matter for most of its history, about 10 billion
years, we do this using the scale factor for the period of matter dominance,
a = a 0
t
t 0
2/3
, a 0 = present scale factor, t 0 = present time.
(16.21)
In this section we write explicitly the present value of the scale factor a 0 . From
(16.20) and (16.21) we obtain the total coordinate distance σ for light emitted at
t e and observed by us at t 0
dσ =
cdt
a 0
t 0
t
2/3
, ,σ =
c
a 0
t 0
t e
dt
t 0
t
2/3
=
3c
a 0
t 0 − t
2/3
0 t
1/3
e
.
(16.22)
16 Some Properties of the LCDM Universe
Fig. 16.3 Co-moving objects remain at coordinate rest in the FLRW metric. Light follows the
indicated curve from the distant source to us, defining our past light cone
increases as time progresses the physical distance between two co-moving objects,
such as galaxies, increases.
Let us ask how far we can see in the expanding universe? That is, what is the
maximum distance, both coordinate and physical, that a source can be so that light
from it has reached us? The question is very fundamental because it is equivalent to
asking the size of the observable universe. Recall that light is characterized as having
a null trajectory, ds
2
= 0. We impose this on the FLRW metric in the form (13.12b)
to characterize the path of light so the coordinate distance differential obeys
ds
2
= c
2 dt
2
− a
2 dσ
2
= 0, dσ =
cdt
a
.
(16.20)
To obtain the total coordinate distance σ we need to integrate this relation. Since the
universe has been dominated by cold matter for most of its history, about 10 billion
years, we do this using the scale factor for the period of matter dominance,
a = a 0
t
t 0
2/3
, a 0 = present scale factor, t 0 = present time.
(16.21)
In this section we write explicitly the present value of the scale factor a 0 . From
(16.20) and (16.21) we obtain the total coordinate distance σ for light emitted at
t e and observed by us at t 0
dσ =
cdt
a 0
t 0
t
2/3
, ,σ =
c
a 0
t 0
t e
dt
t 0
t
2/3
=
3c
a 0
t 0 − t
2/3
0 t
1/3
e
.
(16.22)