The cosmological constant

In 1917, attempting to apply his general theory of relativity (GR) to cosmology, Einstein sought a static solution of the field equations for a universe filled with dust of constant density and zero pressure. The general static solution of (1.34) and (1.36) has

p= ^~ (_A _ _ p)

(1.68) 3 41£GN

and

k 81£GN A

- = - - p + - (1.69)

R2 3 3·

With zero cosmological constant (A = 0), the only solution of these equations, apart from an empty, flat universe, requires that either the energy density p or the pressure p is negative. It was this unphysical result that led him to introduce the cosmologicalterm. Then the solution for pressureless dust is

A (1.70)

p

=

^41£GN

and

R2 =A. k (1.71)

Assuming that p is positive requires that A is positive, so that

k

=

+1 (1.72)

and

R

= ../K.

I ^{( 1.73)}

Hence, the universe is closed and has the geometry of S3 with volume V and mass M given by

v =

^{21£2 R3}

=

^{21£2 A -3/2} M = 2GN"/x. ^1£ ( 1.74) A non-zero cosmological constant also allows non-trivial static (de Sitter) solutions of the Einstein field equations with no matter (p = 0 = p) at all. It was, therefore, a considerable relief in the 1920s when the redshifts of distant

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 (TJlv ) 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

Pvac

=

^(p)

+

81l'GN A =

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 .... 10⁴² 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.

The first evidence suggesting this came from measurements of the redshifts of type la supernovae. Such supernovae arise as remnants of the explosion of white dwarfs which accrete matter from neighbouring stars. Eventually the white dwarf mass exceeds the Chandrasekhar limit and the supernova is born after the explosion. The intrinsic luminosity of such supernovae is considered to be a constant. That is, they are taken as standard candles and any variation in their apparent luminosity as measured on earth must be explicable in terms of their differing distances from the earth. In a Euclidean space, the apparent luminosity 1 of a source with intrinsic luminosity L at a distance D from the observer is given by

1= 41rD2· _L (1.8 I)

We may, therefore, define the 'luminosity distance' DL of a source from the observer by

(1.82) DL

== J4~"

In GR we must be more careful. So consider the circular mirror, area A, of a telescope at the origin, nonnal to the line of sight to a source at r ^I. Light emitted from the source at time 11 and arriving at the mirror at time 10 is bounded by a cone with solid angle

w=---:"""":" A (1.83) 41r R(lo)2rl

as measured in the locally inertial frame at the source. The emitted photons have their energy redshifted by a factor

R(tl)

= (1.84)

R(IO)

1+ z

as explained in section 1.2, (see (I. I 8». Also, photons emitted at time intervals of 1511 reach the mirror at time intervals 1510 = 1511 R(IO)/ R(II). Thus, the total power

P received at the mirror is given by

P

=

^{L (R(II»)2} _(1.85)

R(to) w and the apparent luminosity by

1 = P A. (1.86)

Then, using (1.27), the luminosity distance defined in (1.82) is

DL

= Ha

^l(1

+

^{z) [z -}

~(l

^+QO)z2

+ ... ]

^(1.87)

=

^Ho

I [I

z

+

2(1 - qo)z

2] _{+... .}

(1.88)

25

etQ 2. ,

\

~ ^\

°t ~23

..

_{- - ­} ^°M·OA 0.25 0.75

- - - ­ - - - 0.25 0.00

22 1.00 0.00

03 (... _0.' ~.--~.

z ^{0.5 0.6 07 0.8 o.e} """

etQ 20

~ '8

't ~

..

16

I";-'/~~~~~~~-~~~~·..

00' 0'

7

Figure 1.1. Hubble diagram giving the effective magnitude versus redshift for the supernovae in the primary low-extinction subset. The full line is the best-fit flat-universe cosmology from the low-extinction subset. the broken and dotted lines represent the indicated cosmologies.

Hence. for nearby supernovae the luminosity distance is proportional to the redshift of the source.

Astronomers measure the apparent magnitude m of the various supernovae sources. The difference m - M. where M .... -19.5. is the (assumed constant) intrinsic magnitude of the source. is just the logarithm of the luminosity distance.

So the apparent magnitude is predicted to be linear in In;z for small;z. This is consistent with the data for

z ;S

0.1. see figure 1.1 taken from [2]. For more distant supernovae the linear relationship between DL and ;z is distorted by quadratic terms depending on the present deceleration parameter qO of the universe. The data for 0.7

;S

;z

;S

1 do display such a distortion. see figure 1.1 [2].

For an FRW universe. it follows from (1.36) and the definition (1.22) of qO that. in general. the deceleration may be written as

qO =

! L(I ⁺

^{3Wi)S'2i (1.89)}

for a universe with components labelled by ; having energy density Pi and pressure Pi

==

WjPi; here S'2j

==

Pi/Pc where Pc

==

3HJ/87rGN is the critical density. In particular. for a universe with just (pressureless) matter and

2 2

o o

1

1

o o

^{1 1 2 2 3 3}

Figure

Figure 1.21.2. . 68%, 68%, 90%, 90%, 95%, 95%, and and 99% 99% confidence confidence regions regions for!lm for!lm and and !lA. !lA.

a

a cosmological cosmological constant, constant, we we get get

qO qO = = !!lm !!lm -!lA -!lA ( ( 1.90) 1.90) where!lm

where!lm

== ==

Pm/Pc Pm/Pc is is the the matter matter contribution contribution and and !lA !lA

== ==

Pvec/Pc Pvec/Pc

= =

^AAeetf/3HJ. tf/3HJ.

As

As noted noted previously, previously, a a negative negative value value of of qO, qO, corresponding corresponding to to an an accelerating accelerating universe,

universe, can can only only arise arise with with a a positive positive cosmological cosmological constant. constant. The The data data shown shown in in figures

figures 1.1 1.1 and and 1.2 1.2 taken taken from from [2] [2] suggest suggest that that this this is is indeed indeed the the case. case.

The

The determination determination of of !lm !lm and and !lA !lA requires requires at at least least one one further further input. input. The The recent

recent data data on on the the temperature temperature anisotropies anisotropies of of the the cosmic cosmic microwave microwave background background provide

provide just just such such a a constraint. constraint. Photons Photons originating originating at at the the 'last 'last scattering scattering surface', surface', when

when matter matter and and radiation radiation decouple decouple (see (see section section I. I. 10), 10), having having a a redshift redshift

z z ... ...

1300,

1300, are are seen seen now now as as the the microwave microwave background. background. Quantum Quantum fluctuations fluctuations in in the

the early early universe universe give give rise rise to to fluctuations fluctuations in in the the energy energy density density of of the the radiation radiation and

and these these appear appear as as temperature temperature fluctuations fluctuations in in the the microwave microwave background background (see (see section

section 7.7). 7.7). These These fluctuations fluctuations may may be be analyzed analyzed by by multipole multipole moments, moments, labelled labelled

by I, and are characterized by their power spectrum. The multi pole number lpeak of the first peak in the power spectrum is determined by the total matter content of the universe. In fact, lpeak '" 22000, where 00

==

po/Pc measures the total energy density PO relative to the critical density. The measured position of the first peak yields the value (1.42). Thus, for a universe with just matter and a cosmological constant, we get

Om

+

OA '" I. (1.91)

When this result is combined with the supernova and other data, it is found that Om '" 0.3 OA '" 0.7. ( 1.92) In Planck units, this means that

Aeff

=

^{PvIM: _ n} Pc '" 0 8 10-120

2 4 - UA 4 - . x . ^{( 1.93)}

Mp Mp Mp

There is currently no known explanation of this extremely small number. It corresponds to ~t: ~ 10-³ eV. It is generally believed that the particle physics vacuum is the minimum of an effective potential in which the electroweak gauge symmetry SU(2)L x U(I)y is spontaneously broken (see section 2.5). The value of the effective potential at this minimum (p) has no effect on the particle physics.

By adding a constant Vo to the tree-level potential (2.93), it is easy to arrange that the potential, including any radiative and temperature-dependent corrections, has any desired value at the minimum. However, to do so requires the fine tuning of Vo to ensure that the value (1.93) is obtained and it is this fine tuning that is regarded as unnatural and for which an explanation is sought. The obvious first approach to the problem is to seek a symmetry that requires A

=

^{0 and then to explore}

mechanisms that break the symmetry only slightly. The only known symmetry that requires a vanishing cosmological constant is global supersymmetry. The (fermionic) supersmmetry generator Q satisfies the anticommutation relation

{Q,

Q} =

2yl' PI' ( 1.94) where PI' is the energy-momentum vector. It follows [3] that. for any state 11/1),

(1/II PoI1/l)

=

(1/IIQaQ:

+

Q: Qa 11/1) ~ O. ( 1.95) Thus, the energy of any non-vacuum state is positive and the vanishing of the vacuum energy defines a unique, supersymmetric vacuum state 10) that satisfies

(OIPoIO)

=

^{0 ~} QaIO) =

o.

^(1.96)

In a supersymmetric theory. all particles have supersymmetric partners (called 'sparticles') having opposite statistics. That is to say. the sparticle associated with a fermi on is a hoson and the sparticle associated with a boson is a fermion. The sparticles associated with the quarks and leptons, called respectively 'squarks'

and 'sleptons', are (spin-O) scalar particles and, in a supersymmetric theory, they must have the same mass and quantum numbers as the original particles. This has the important consequence that the vanishing cosmological constant result is unaffected by quantum effects, because supersymmetry ensures that any quantum corrections arising from fermion loops, say, are cancelled by those that arise field generating diffeomorphisms of spacetime. Then, in a supersymmetric theory incorporating GR, the supersymmetry generators too become local fields: this is

Equilibrium thermodynamics in the expanding universe anthropic principle accounts for the value of the (positive) cosmological constant.

then we should expect Pvac ,.., (10 - 100)Pm because there is no anthropic

Im Dokument Supersymmetric Gauge Field Theory and String Theory (Seite 23-30)