a2 H 2 3H 2

This illustrates that there is a simple relation between the curvature k and

the deviation from the critical density ρc . The classi¬cation of cosmological

models as open (in¬nite), ¬‚at or closed (¬nite), which is summarized in

Table 10.2, follows from this equation.27

ρ „¦ spatial curvature k type of Universe

’1

< ρc <1 open

= ρc =1 0 ¬‚at

> ρc >1 1 closed

Table 10.2: The classi¬cation of cosmological models.

The Friedmann and acceleration equations imply the continuity or ¬‚uid

equation, which expresses energy conservation

ρtot + 3H(ρtot + ptot ) = 0 .

™ (10.274)

26 The reduced Planck mass has a numerical value MP = 2.436 — 1018 GeV and di¬ers by a factor

√

8π from the alternative de¬nition mp = 1.22 — 1019 GeV.

27 The value of Λ has been absorbed into „¦ in this table.

530 Flux compacti¬cations

If there is a single cosmic ¬‚uid, with equation of state given by Eq. (10.264),

one obtains from here the following dependence of ρ on the FRW scale-factor

1

ρ∼ . (10.275)

a3(w+1)

This relation, valid for any value of k, is displayed in Table 10.1 for the

most important cosmic ¬‚uids. The acceleration equation implies that a < 0

¨

for ¬‚uids with ρ + 3p > 0, and hence the associated FRW cosmologies

describe decelerating Universes. Under the general assumption that the

energy density ρ is positive, one can show that a FRW cosmology implies

an initial singularity. This forms the basis for the SBB model of cosmology

in which a FRW Universe starts from an initial it Big-Bang singularity.

The SBB model of cosmology

Let us now brie¬‚y summarize the successes and remaining puzzles of the

SBB model of cosmology. In the cosmological time period starting at the

time of nucleosynthesis, when protons and neutrons bound together to form

atomic nuclei (mostly of hydrogen and helium), the SBB model is very well

con¬rmed by three main observations. These are

• The Hubble redshift law: by extrapolation of the measured velocities of

galaxies of the nearby galaxy cluster, Hubble made the bold conjecture

that the Universe is undergoing a uniform expansion, so that galaxies that

are separated by a distance L recede from one another with a velocity

v = H0 L, where H0 is the present Hubble parameter. This relation and

deviations from it are well understood.

• Nucleosynthesis: the relative abundance of the light elements, such as 75%

H, 24% 3 He and smaller fractions of Deuterium and 4 He, is explained by

the theory of nucleosynthesis and constitutes the earliest observational

con¬rmation of the SBB model.

• The cosmic microwave background (CMB): most of the radiation con-

tained in the Universe at present is nearly isotropic and has the form of

a blackbody spectrum with temperature about 2.7 o K. It is known as the

Cosmic Microwave Background (CMB). The discovery of this radiation in

1964 by Penzias and Wilson constitutes one of the great triumphs of the

SBB model, which predicts a black-body distribution for the CMB. The

measurement of the CMB™s temperature ¬‚uctuations, δT /T , whose spa-

tial variation is decomposed into a power spectrum, provides information

on the energy-density ¬‚uctuations δρ/ρ in the early Universe. This is im-

portant for understanding the potential microscopic origin of the observed

large-scale structure of the Universe.

10.7 Fluxes and cosmology 531

However, puzzles still remain in the SBB model. Some of the most im-

portant ones are

• The horizon problem: the observed CMB is isotropic. However, when we

follow the evolution of the Universe backwards in time according to the

SBB model the sky decomposes into lots of causally disconnected patches.

It needs to be explained why opposite points in the sky look so similar

even though they cannot have been in causal contact since the Big Bang.

• The ¬‚atness problem: observation shows that „¦ = ρtot /ρc 1 at the

current epoch. From the SBB evolution one ¬nds that the comoving

Hubble length 1/(aH) increases in time. Hence the Friedmann equation

Eq. (10.273) shows that „¦ would have to be ¬ne-tuned to a value extremely

close to one at earlier times in order to comply with present observation.

• Unwanted relics: the SBB model does not explain why some relics, that

could in principle be abundant, are so rare. Examples of such relics are

magnetic monopoles, which would be produced when the gauge group of

a grand-uni¬ed theory is broken to a smaller group. Other examples are

domain walls, cosmic strings or the gravitino. Perhaps not all of these

objects exist, but some of them probably do. The presence of unwanted

relics would be dramatic, since some of them could quickly dominate the

evolution of the Universe.

• The origin of the CMB anisotropies: the SBB does not explain the ob-

served CMB anisotropies occurring at a relative magnitude of about 10’5 .

These four puzzles are successfully addressed by an in¬‚ationary phase in

the early Universe (taking place prior to the Big Bang), as discussed in the

next section. There are more puzzles, which may or may not be connected

to in¬‚ation, such as

• Dark matter: rotation curves of galaxies and the application of the virial

theorem to the dynamics of clusters of galaxies indicate that there must be

some form of invisible matter, called dark matter, which clusters around

galaxies and is responsible for explaining the large-scale structure of the

Universe. This dark matter should be predominantly cold, meaning that it

is composed of particles that were nonrelativistic at the time of decoupling

with no signi¬cant random motion.

• Dark energy: measurements of high red-shift Type I supernovas imply

that our Universe is undergoing an accelerated expansion in the present

epoch. A positive a requires an unusual equation of state with sources

¨

of negative pressure appearing in the energy“momentum tensor, as the

inequality ρ + 3p < 0 needs to be satis¬ed. The presence of a positive

532 Flux compacti¬cations

cosmological constant on the right-hand side of the acceleration equation

Eq. (10.269) would give such a repulsive force.

• Why four dimensions?: Critical M-theory or string theory predicts 11 or

ten dimensions, respectively. The answer to the question of why we only

observe four large dimensions might be provided within the context of

cosmology.

These last three problems seem to require new physics beyond the SBB for

their solution. For example, supersymmetry can provide viable dark mat-

ter candidates such as the lightest supersymmetric partner of the standard

model particles (LSP). A thorough understanding of quantum gravity may

be required to solve the latter two questions. On the other hand, as is dis-

cussed in the next subsection, there is a simple mechanism within the FRW

cosmology framework that solves the ¬rst set of four puzzles.

Basics of in¬‚ation

In¬‚ationary cosmology was introduced in the 1980s to solve some of the

previously mentioned problems of the SBB model. This theory does not

replace the SBB model, rather it describes an era in the evolution of our

Universe prior to the Big Bang, without destroying any of its successes.

De¬nition of in¬‚ation

Very generally, a period of in¬‚ation is de¬ned as a period in which the

Universe is accelerating and thus the scale factor satis¬es

a(t) > 0.

¨ (10.276)

Equivalently, this condition can be rephrased as

d1

<0. (10.277)

dt aH

This equation states that the comoving28 Hubble length 1/aH, which is the

most important characteristic scale of an expanding Universe, decreases in

time. From the acceleration equation Eq. (10.269), one ¬nds that in¬‚ation

implies

ρtot + 3ptot < 0, (10.278)

so that, assuming ρ > 0, the e¬ective pressure of the material driving the

expansion has to be negative. Scalar (spin-0) particles have this property,