16.5 Density Ratios and the Shape of the Universe
257
Consider for a moment the earliest times for which the LCDM model, neglecting
radiation, could be roughly valid, which is for z = a 0 /a ≈ 10
3
. At that time the
vacuum energy density was negligible compared to the matter density. The present
measured value for k0 is consistent with zero but could be as large as about 10
−2 .
Thus from (16.17) the total density ratio at the beginning of the LCDM era must
have been quite close to unity, as is clear from
∼ = 1 −
k0
a 0
a
2
m0
a 0
a
3 = 1 −
k0
m0
a
a 0
∼ 1 ± 10
−5
.
(16.18)
Thus at the beginning of the matter era all the curves in Fig. 16.2 are quite close to 1.
Finally, recall from Chap. 13 that the curvature parameter k was first encountered
as the inverse square of the radius of hypersphere. We therefore express k as the
inverse square of a characteristic radius, k ≡ 1/R
2
c . From the definition of k0 in
(16.12) we can make a rough lower estimate of that radius by
c 2
R 2
c
= kc 2 = | k0 |a 2
0 H 2
0 = | k0 |
1
T 2
H
,
R c =
cT H
| k0 |
∼ 10 11 ly for k0 ∼ 10 −2 .
(16.19)
Thus the characteristic radius is at least of order 10 times the Hubble distance cT H ,
so our observable universe lies well within it. This serves as a reasonable definition
of “almost flat” (Adler 2005).
In summary, based on current observations it appears that the total energy density
of the universe is close to critical, and the universe is spatially flat or nearly so. This
is the currently favored theoretical case, the flat LCDM universe or standard model.
However we again emphasize that cosmology is like all of science in that observations
are the primary facts of life and are continually changing and improving.
16.6 Horizons and the Size of the Observable Universe
Let us next study how different events in the universe can influence each other. In
special relativity this problem is easy since no influence can move faster than the
speed of light: obviously we can be influenced only by events within our past light
cone. In general relativity and cosmology the answer is similar and nearly as simple,
but the past light cone in the spacetime of cosmology is just a little more subtle and
interesting.
Recall that the motion of co-moving objects in the FLRW metric is very simple:
they remain at coordinate rest, as indicated in Fig. 16.3. Since the scale factor
257
Consider for a moment the earliest times for which the LCDM model, neglecting
radiation, could be roughly valid, which is for z = a 0 /a ≈ 10
3
. At that time the
vacuum energy density was negligible compared to the matter density. The present
measured value for k0 is consistent with zero but could be as large as about 10
−2 .
Thus from (16.17) the total density ratio at the beginning of the LCDM era must
have been quite close to unity, as is clear from
∼ = 1 −
k0
a 0
a
2
m0
a 0
a
3 = 1 −
k0
m0
a
a 0
∼ 1 ± 10
−5
.
(16.18)
Thus at the beginning of the matter era all the curves in Fig. 16.2 are quite close to 1.
Finally, recall from Chap. 13 that the curvature parameter k was first encountered
as the inverse square of the radius of hypersphere. We therefore express k as the
inverse square of a characteristic radius, k ≡ 1/R
2
c . From the definition of k0 in
(16.12) we can make a rough lower estimate of that radius by
c 2
R 2
c
= kc 2 = | k0 |a 2
0 H 2
0 = | k0 |
1
T 2
H
,
R c =
cT H
| k0 |
∼ 10 11 ly for k0 ∼ 10 −2 .
(16.19)
Thus the characteristic radius is at least of order 10 times the Hubble distance cT H ,
so our observable universe lies well within it. This serves as a reasonable definition
of “almost flat” (Adler 2005).
In summary, based on current observations it appears that the total energy density
of the universe is close to critical, and the universe is spatially flat or nearly so. This
is the currently favored theoretical case, the flat LCDM universe or standard model.
However we again emphasize that cosmology is like all of science in that observations
are the primary facts of life and are continually changing and improving.
16.6 Horizons and the Size of the Observable Universe
Let us next study how different events in the universe can influence each other. In
special relativity this problem is easy since no influence can move faster than the
speed of light: obviously we can be influenced only by events within our past light
cone. In general relativity and cosmology the answer is similar and nearly as simple,
but the past light cone in the spacetime of cosmology is just a little more subtle and
interesting.
Recall that the motion of co-moving objects in the FLRW metric is very simple:
they remain at coordinate rest, as indicated in Fig. 16.3. Since the scale factor