Recent results from WMAP indicate that n ∼ .95.

Fluxes and in¬‚ation

The embedding of in¬‚ation into string theory is di¬cult in conventional

Calabi“Yau compacti¬cation. Even though such compacti¬cations contain

many scalar ¬elds that could potentially serve as in¬‚atons, namely the mod-

uli ¬elds, these ¬elds are generically either massless or have a potential with

a runaway behavior, which makes their interpretation as in¬‚atons rather

di¬cult. This situation has changed quite a bit with the development of a

better nonperturbative understanding of string theory and ¬‚ux compacti¬-

cations.

Brane“brane in¬‚ation

One of the ¬rst attempts to embed in¬‚ation into string theory (developed in

the late 1990s) makes use of D-branes. In this approach a pair of D-branes

is considered and the in¬‚aton is identi¬ed with the scalar ¬eld describing

the separation of the branes, that is, it is the lowest mode of the open string

that connects the two D-branes. If supersymmetry is preserved, there is no

net force between the branes and no potential for the in¬‚aton. This has been

veri¬ed by a one-loop string amplitude calculation, which is not presented

here. The intuitive argument is that, for a BPS brane con¬guration, the

gravitational attraction between the branes is compensated by the repulsive

Coulomb forces between the two branes coming from various NS“NS and

R“R ¬elds. However, when supersymmetry is broken (in a certain way),

there is a net attractive force between the branes. This leads to a potential

for the in¬‚aton ¬eld.

Even though this was the ¬rst proposal that demonstrated the possibility

540 Flux compacti¬cations

of making connections between string theory or brane physics and in¬‚ation,

the concrete model had some problems, such as a drastic ¬ne tuning required

to reproduce the experimental values of the density perturbations or the lack

of a satisfactory explanation for the end of in¬‚ation.

Brane“antibrane in¬‚ation

Some of these problems were solved in the context of brane“antibrane in-

¬‚ation. Consider instead a D3/anti-D3 system located at speci¬c points of

a Calabi“Yau three-fold. For a D3/anti-D3 system supersymmetry is bro-

ken, and there is a net attractive force between the branes and antibranes,

whose explicit form is given by the potential (for a large distance between

the brane and the antibrane)

1 T3

V (r) = 2T3 1 ’ , (10.304)

8

2π 3 M10 r4

where M10 is the ten-dimensional Planck mass, T3 is the D3-brane tension

and r is the separation between the brane and the antibrane. One can write

1/2

this potential in terms of the canonically normalized scalar φ = T3 r, where

it takes the form

3

1 T3

V (φ) = 2T3 1 ’ 3 8 4 . (10.305)

2π M10 φ

Using this potential, one can compute the slow-roll parameters appearing in

Eqs (10.289) and (10.290)

L6

12 2

= MP (V /V ) ∼ 10 . (10.306)

2 r

L6

2

/V ) ∼ 6 .

·= MP (V (10.307)

r

MP is the four-dimensional Planck mass appearing in Eq. (10.290) which

is related to the ten-dimensional Planck mass by MP = L6 M10 . Here L6

2 8

approximately represents the volume of the Calabi“Yau three-fold. The

D3 and anti-D3-branes are localized at speci¬c points on the Calabi“Yau

manifold, that is, they cannot be separated by more than L. As a result, it

is not possible to achieve |·| << 1, as needed for slow-roll in¬‚ation. Di¬erent

proposals for solving this problem have been presented in the literature, such

as D3- and anti-D3-branes in a warped geometry (this is discussed next),

branes at angles or collisions of multiple branes 30 .

30 A more recent proposal is to give up the slow roll condition.

10.7 Fluxes and cosmology 541

In¬‚ation and ¬‚uxes

In the previous treatment of the D3/anti-D3 system the size L was treated

as a constant. However, it is known that in string theory the size of the

internal manifold is a modulus. The potential (in the four-dimensional Ein-

stein frame) for this ¬eld (again for a D3/anti-D3-brane distance large as

compared to the string scale) is

2T3

V (φ, L) ≈ . (10.308)

L12

This potential is very steep for small L. As a result, treating L as a dy-

namical ¬eld causes the Calabi“Yau size to become large too fast to realize

slow-roll in¬‚ation. This issue could in principle be avoided if the radial mod-

ulus of the internal manifold is stabilized. In Section 10.3 a mechanism was

described to stabilize the radial modulus of a D3/anti-D3 system in terms of

¬‚uxes and nonperturbative e¬ects. The stabilization of the radial modulus

using nonperturbative corrections to the superpotential does not solve this

problem (unless some degree of ¬ne tuning is allowed), but it puts it into a

new perspective.

As discussed in Section 10.3, to analyze the stabilization of the moduli of

a D3/anti-D3 system on an internal warped geometry the scalar potential

for the radial modulus of the internal manifold and the scalars describing

the positions of the branes need to be derived. N = 1 supersymmetry

dictates that this potential is determined in terms of a K¨hler potential and

a

a superpotential.

Consider ¬rst a single D3-brane position modulus φ and the radial mod-

ulus of the internal space ρ. The K¨hler potential is given by

a

¯ ¯

K(ρ, ρ, φ, φ) = ’3 log ρ + ρ ’ k(φ, φ) .

¯ ¯ (10.309)

Here the real part of ρ is related to the size L by

¯

2L = ρ + ρ ’ k(φ, φ),

¯ (10.310)

¯

while the imaginary part of ρ is the axion χ. Furthermore, k(φ, φ) is the

canonical K¨hler potential for the inter-brane distance, which is given by

a

¯ ¯

k(φ, φ) = φφ.

The other quantity that determines the form of the low-energy e¬ective

action is the superpotential W . As explained in Section 10.3, W takes the

form

W (ρ) = W0 + Ae’aρ . (10.311)

542 Flux compacti¬cations

Here W0 is the perturbative superpotential

G3 § „¦,

W0 = (10.312)

where „¦ is the holomorphic (3, 0) form. The exponential contribution de-

pending on ρ comes from nonperturbative e¬ects, as discussed in Chapter 10.

Further contributions to the scalar potential involving the radial modulus

come from corrections to the K¨hler potential, which will be ignored in the

a

following. The complete form of these corrections is not known at present.

The above results for the K¨hler potential and the superpotential can be

a

used to compute the scalar potential for the Calabi“Yau volume and the

brane position, which is determined by supersymmetry

¯ ¯