Appendix 2: Newtonian View of Dark Energy
243
v
2
=
m
r
−
r
2
R
2
d
c
2
,
(15.26)
where the brackets indicate an average (Goldstein 1980). The last equation implies
that the relative importance of the dark energy repulsive force to the attractive
Newtonian force may be estimated as
r
2
R
2
d
/
m
r
=
r ch
m
r ch
R d
2
= ε ch , r
3
ch =
r
2
1
r
.
(15.27)
Note that, with r ch as defined in the above, the ratio ε ch has the same form as ε in
(15.25). This ratio may be evaluated for galactic clusters, the largest bound structures
in the universe, to determine the relative effects of dark energy, which are small. For
an example see Exercise 15.9.
It is interesting to see how dark energy might fit ab initio into classical gravitational
theory as discussed in Chap. 7. As we emphasized above, dark energy is characterized
by a constant density, so we consider Poisson’s equation with a constant source
∇
2
φ = s, s = constant source
(15.28)
The spherically symmetric solution to this is
φ =
s
6
r
2
.
(15.29)
Thus we see that Poisson’s equation produces the same potential as general relativity
in the classical limit if we take the source s to be
s = −
3c
2
R
2
d
= −c
2
= −8π G(ρ DE /c
2
), ρ DE = dark energy density. (15.30)
That is, the source is negative 4π G times twice the dark energy mass density. In the
context of classical physics it is hard to justify such a negative source, whereas in
general relativity theory it arises naturally. See Exercise 15.10 as to how the factor
of −2 explicitly arises. The negative sign is a peculiar and important feature of dark
energy because of its effect on the accelerated expansion of the universe.
Finally we observe that the sort of correspondence between general relativity and
classical gravity theory we have discussed in Chap. 7 and in this appendix allows a
pedagogical development of cosmological theory based largely on classical physics
(Liddle 2003). However one needs to postulate the sign of the repulsive force due to
dark energy.
243
v
2
=
m
r
−
r
2
R
2
d
c
2
,
(15.26)
where the brackets indicate an average (Goldstein 1980). The last equation implies
that the relative importance of the dark energy repulsive force to the attractive
Newtonian force may be estimated as
r
2
R
2
d
/
m
r
=
r ch
m
r ch
R d
2
= ε ch , r
3
ch =
r
2
1
r
.
(15.27)
Note that, with r ch as defined in the above, the ratio ε ch has the same form as ε in
(15.25). This ratio may be evaluated for galactic clusters, the largest bound structures
in the universe, to determine the relative effects of dark energy, which are small. For
an example see Exercise 15.9.
It is interesting to see how dark energy might fit ab initio into classical gravitational
theory as discussed in Chap. 7. As we emphasized above, dark energy is characterized
by a constant density, so we consider Poisson’s equation with a constant source
∇
2
φ = s, s = constant source
(15.28)
The spherically symmetric solution to this is
φ =
s
6
r
2
.
(15.29)
Thus we see that Poisson’s equation produces the same potential as general relativity
in the classical limit if we take the source s to be
s = −
3c
2
R
2
d
= −c
2
= −8π G(ρ DE /c
2
), ρ DE = dark energy density. (15.30)
That is, the source is negative 4π G times twice the dark energy mass density. In the
context of classical physics it is hard to justify such a negative source, whereas in
general relativity theory it arises naturally. See Exercise 15.10 as to how the factor
of −2 explicitly arises. The negative sign is a peculiar and important feature of dark
energy because of its effect on the accelerated expansion of the universe.
Finally we observe that the sort of correspondence between general relativity and
classical gravity theory we have discussed in Chap. 7 and in this appendix allows a
pedagogical development of cosmological theory based largely on classical physics
(Liddle 2003). However one needs to postulate the sign of the repulsive force due to
dark energy.