10.7 Fluxes and cosmology

Superstring theory and M-theory have implications for cosmology, some of

which are addressed in this section. The main conceptual issues arise when

the classical space-time description derived from general relativity breaks

down, and the curvature of space-time diverges. This happens at the be-

ginning of the Universe in the SBB, when the classical space-time becomes

singular and the energy density becomes in¬nite. Here, one might hope

that string theory smoothes out the singularity, due to the ¬nite size of the

string, so that there could be a sensible cosmology before the Big Bang.

When the curvature of space-time and the string coupling become large,

the perturbative formulation of string theory becomes unreliable, and one

needs to turn to other techniques, such as the Matrix-theory proposal for

M-theory,24 which is an interesting area of current research.

Some basic cosmology

Before discussing string-theory cosmology, some basic features of the stan-

dard model of cosmology, including its successes and shortcomings, are pre-

24 Matrix theory is introduced in Chapter 12.

10.7 Fluxes and cosmology 527

sented. The next two subsections are intended to present a basic “tool kit”

of cosmology for the string-theory student. The interested student should

consult cosmology textbooks for a more detailed and complete explanation.

The perfect-¬‚uid description

Let us consider four-dimensional general relativity in the presence of a per-

fect ¬‚uid, which describes the energy content of the Universe. By de¬ni-

tion, a perfect ¬‚uid is described in terms of a stress-energy tensor that is

a smoothly varying function of position and is isotropic in the local rest

frame. The perfect-¬‚uid description is suggested by the fact that the mat-

ter and radiation distribution of the Universe looks remarkably homoge-

neous and isotropic on very large cosmological scales. For instance, most

of the radiation contained in the Universe is accounted for by the cosmic

microwave background (CMB), which is isotropic up to tiny ¬‚uctuations

of order 10’5 once the dipole moment due to the motion of the Sun and

Earth is subtracted. Furthermore, galaxy surveys indicate a homogeneous

distribution at scales greater than 100 Mpc (1 pc = 3.086 — 1016 m). The

energy“momentum tensor of a perfect ¬‚uid takes the form

T00 = ρ, Tij = pgij . (10.263)

This tensor is characterized by three quantities: the mass-energy density ρ,

the pressure p and the spatial components of the metric gij . In addition, it

is generally assumed that there is a simple relation between the mass-energy

density ρ and pressure p given by the equation of state

p = wρ , (10.264)

where w is a constant that depends on whether the Universe is dominated

by relativistic particles (termed radiation), nonrelativistic particles (collec-

tively called matter) or vacuum energy. Some of the cosmologically relevant

gravitating sources are listed in Table 10.1.

ρ ∼ a’3(w+1) a(t) ∼ t2/3(w+1)

type of ¬‚uid w

1/a4 t1/2

radiation 1/3

1/a3 t2/3

matter 0

√

e Λ/3t

’1

vacuum energy const.

Table 10.1: Cosmologically most relevant gravitating sources. The time

dependence of the scale factor a is given for k = 0.

528 Flux compacti¬cations

Friedmann“Robertson“Walker Universe

The homogeneity and isotropy of the D = 4 space-time uniquely determines

the metric to be of the following Friedmann“Robertson“Walker (FRW) type

dr2

2 2 2

+ r2 (dθ2 + sin2 θdφ2 ) .

ds = ’dt + a (t) (10.265)

2

1 ’ kr

The only functional freedom remaining in this metric is the time-dependent

scale-factor a(t) which determines the radial size of the Universe. It is

determined by the Einstein equations

1

Gµν = Rµν ’ gµν R = 8πGTµν ’ Λgµν , (10.266)

2

and therefore by the dynamics of the theory. Here G denotes Newton™s con-

stant. A cosmological constant has been included in this equation, since re-

cent astronomical observations indicate that it has a positive (nonvanishing)

value Λ = 10’120 MP = (10’3 eV)4 . In addition, the metric is characterized

4

by the discrete parameter k, which characterizes the spatial curvature25

Rcurv = a|k|’1/2. (10.267)

It takes the values ’1, 0, 1 depending on whether there is enough gravitating

energy in the Universe to render it closed, ¬‚at or open. The precise de¬nition

of these terms is given below. For the ¬‚at case, k = 0, the time-dependence

of the scale factor for various cosmic ¬‚uids is displayed in Table 10.1.

Friedmann and acceleration equations

The Einstein ¬eld equations, which determine a(t), reduce for the FRW

ansatz to the Friedmann and acceleration equations, respectively

1 k Λ

H2 = ρtot ’ 2 + , (10.268)

2 a 3

3MP

a

¨ 1 Λ

=’ (ρtot + 3ptot ) + , (10.269)

2

a 3

6MP

where

H(t) = a(t)/a(t)

™ (10.270)

de¬nes the Hubble parameter, which determines the rate of expansion of the

Universe. Furthermore,

ρtot = ρi , ptot = pi (10.271)

i i

√

’g = a3 .

25 In these conventions r is dimensionless and a(t) is a length. For k = 0,

10.7 Fluxes and cosmology 529

are the total energy density and pressure, while MP = (8πG)’1/2 denotes

the reduced Planck mass.26 The index i labels di¬erent contributing ¬‚uids,

as listed in Table 10.1. Sometimes the cosmological constant is regarded as

a time-independent contribution to the energy density and pressure of the

2

vacuum ρvac = ’pvac = Mp Λ. It does not appear explicitly in the previous

equations.

Open, ¬‚at and closed Universes

It follows from the Friedmann equation Eq. (10.268) that (for Λ = 0) the

Universe is ¬‚at, k = 0, when the energy density equals the critical density

ρc = 3H 2 MP .

2

(10.272)

This is a time-dependent function that at present has the value ρc,0 = 1.7 —

10’29 g/cm3 .

It is customary to de¬ne the energy density of the various ¬‚uids that are

present in units of ρc by introducing the density parameter „¦i = ρi /ρc for

the ith ¬‚uid. In terms of the sum over all such contributions, „¦ = i „¦i =

ρtot /ρc , the Friedmann equation takes the simple form

k Λ