The cosmological constant
11
galaxies were observed, the presumption of a static universe could be abandoned
and there was no need for a cosmological constant.
However, anything that contributes to the energy density of the vacuum (p)
acts just like a cosmological constant. This is because the Lorentz invariance
of the vacuum requires that the energy-momentum tensor in the vacuum (T Jlv )
satisfies
(TJlv ) = (p)gJlv,
( 1.75)
Then, by inspection of (1.32), we see that the vacuum energy density contributes
81l' G N (p) to the effective cosmological constant
Aeff = A + 81l'GN(P)·
(1.76)
Equivalently, we may regard the cosmological constant as contributing A/81l'G N
to the effective vacuum energy density
A
Pvac = (p) + 81l'GN = AeffM~.
( 1.77)
Thus, a cosmological constant is often referred to as 'dark energy', not to be
confused with dark matter which contributes to the non-vacuum energy density
(and has zero pressure).
A priori, in any quantum theory of gravitation, we should expect the scale
of the vacuum energy density to be set by the Planck scale Mp. Since A has
the dimensions of M2, it follows that we should have expected that A/M~ .... I.
We shall see that, in reality, the scale of any such energy density must be much
smaller. We noted in section 1.5 that the effect of the cosmological constant is
negligible at sufficiently early times, because the energy density p scales as a
negative power of R for radiation or matter domination. Thus, the most stringent
bounds arise from cosmology when the expansion of the universe has diluted the
matter energy density sufficiently. From the observation that the present universe
is of at least of size Hr; I, we may conclude that
IAeffl $ 3H6
(1.78)
where
HO-I", 1010 yr .... 10 42 GeV- 1
( 1.79)
from (1.67). Then, in Planck units,
IAeffl < 10- 120
(1.80)
M2 p '"
For many years, this tiny ratio was taken as evidence that the cosmological
constant is indeed zero. However, during the past few years, evidence has
accumulated that A is, in fact, non-zero.
11
galaxies were observed, the presumption of a static universe could be abandoned
and there was no need for a cosmological constant.
However, anything that contributes to the energy density of the vacuum (p)
acts just like a cosmological constant. This is because the Lorentz invariance
of the vacuum requires that the energy-momentum tensor in the vacuum (T Jlv )
satisfies
(TJlv ) = (p)gJlv,
( 1.75)
Then, by inspection of (1.32), we see that the vacuum energy density contributes
81l' G N (p) to the effective cosmological constant
Aeff = A + 81l'GN(P)·
(1.76)
Equivalently, we may regard the cosmological constant as contributing A/81l'G N
to the effective vacuum energy density
A
Pvac = (p) + 81l'GN = AeffM~.
( 1.77)
Thus, a cosmological constant is often referred to as 'dark energy', not to be
confused with dark matter which contributes to the non-vacuum energy density
(and has zero pressure).
A priori, in any quantum theory of gravitation, we should expect the scale
of the vacuum energy density to be set by the Planck scale Mp. Since A has
the dimensions of M2, it follows that we should have expected that A/M~ .... I.
We shall see that, in reality, the scale of any such energy density must be much
smaller. We noted in section 1.5 that the effect of the cosmological constant is
negligible at sufficiently early times, because the energy density p scales as a
negative power of R for radiation or matter domination. Thus, the most stringent
bounds arise from cosmology when the expansion of the universe has diluted the
matter energy density sufficiently. From the observation that the present universe
is of at least of size Hr; I, we may conclude that
IAeffl $ 3H6
(1.78)
where
HO-I", 1010 yr .... 10 42 GeV- 1
( 1.79)
from (1.67). Then, in Planck units,
IAeffl < 10- 120
(1.80)
M2 p '"
For many years, this tiny ratio was taken as evidence that the cosmological
constant is indeed zero. However, during the past few years, evidence has
accumulated that A is, in fact, non-zero.