™

(10.281). If H < 0, then the following inequality has to be satis¬ed

™

H

’ < 1. (10.292)

H2

This can be rewritten in terms of µ using the slow-roll approximation

™ 2

MP V 2

H

’ 2≈ = µ. (10.293)

H 2V

By the slow-roll approximation, µ 1, we observe that this condition leads

to a > 0 and in¬‚ation. The second restriction ·

¨ 1 guarantees the friction

term dominates in Eq. (10.283) so that in¬‚ation lasts long enough. The

above conditions provide a straightforward method to check if a particular

potential is in¬‚ationary. For the simple example of V (φ) = m2 φ2 /2, the

slow-roll approximation holds for φ2 > 2MP , and in¬‚ation ends once the

2

scalar ¬eld gets so close to the minimum that the slow-roll conditions break

down.

Exit from in¬‚ation

From the previous discussion, one concludes that the slow-roll conditions

provide a way to characterize the exit from in¬‚ation. The in¬‚ationary pro-

cess comes to an end when the approximations break down, which happens

for a value of φ for which µ(φ) = 1. A simple calculation shows that, for

power-law in¬‚ation, the slow-roll parameters are given by constants

µ = ·/2 = 1/p, (10.294)

so that in¬‚ation never ends. In principle, this is a problem. One way of

solving it could be provided by embedding this model into string theory,

where additional dynamics might provide an end to the in¬‚ationary era.

Hybrid in¬‚ation

An in¬‚ationary model that has played a role in recent string-cosmology de-

velopments, called hybrid in¬‚ation, was constructed in the early 1990s. This

model is based on two scalar ¬elds: the in¬‚aton ψ, whose potential is ¬‚at and

10.7 Fluxes and cosmology 537

satis¬es the slow-roll conditions, and another scalar φ, whose mass depends

on the in¬‚aton ¬eld. In¬‚ation ends in this model not because the slow-roll

approximation breaks down, but because the ¬eld φ becomes tachyonic, that

is, its mass squared becomes negative. This signals an instability, where a

phase transition takes place. During this phase transition topological de-

fects, such as cosmic strings29 , can be formed. The explicit form of the

potential for hybrid in¬‚ation is

V (φ, ψ) = a(ψ 2 ’ 1)φ2 + bφ4 + c, (10.295)

where a, b, c are positive constants. From the form of V (φ, ψ), one easily

observes that, for ψ 2 > 1, the ¬eld φ has a positive mass squared, it becomes

massless at ψ = 1 and φ is tachyonic for ψ 2 < 1. Since φ is driven to zero

for ψ > 1, the potential in the ψ direction is ¬‚at and satis¬es the slow-

roll conditions, so that ψ is identi¬ed with the in¬‚aton, while φ is called

the tachyon. As discussed in the next section, precisely such a tachyon

appears in brane“antibrane in¬‚ation, which is how hybrid in¬‚ation makes

its appearance in string theory. After in¬‚ation ψ 2 < 1, φ acquires a vev and

ψ becomes massive.

Number of e-foldings

There are various model-dependent quantities that can be compared with

cosmological observations, and which can eventually be used to rule out

some of the in¬‚ationary models. The amount of in¬‚ation that occurs after

time t is characterized by the ratio of the scale factors at time t and at the

end of in¬‚ation. This ratio determines number of e-foldings N (t)

a(tend )

N (t) = log , (10.296)

a(t)

where tend is the time when in¬‚ation ends. This quantity measures the

amount of in¬‚ation that remains to take place at any given time t. Using

the slow-roll approximation, N can be conveniently rewritten in terms of

the in¬‚aton and its potential

tend tend φ

a

™ 1 V

Hdt ≈ 2

N (t) = dt = dφ. (10.297)

a V

MP

t t φend

Here φend is the value of the in¬‚aton at the end of in¬‚ation, which satis-

¬es (φend ) = 1 when in¬‚ation ends through a breakdown of the slow-roll

approximation. To solve the ¬‚atness and horizon problems, the number of

29 The existence of cosmic strings would be extraordinary, as a direct experimental evidence of

string theory would be provided. This subject is nevertheless beyond the scope of this book.

538 Flux compacti¬cations

e-foldings has to be larger than 60, a criterion that can be used to rule out

some in¬‚ationary models.

Gravitational waves and density perturbations

In¬‚ation not only explains the homogeneity and isotropy of the Universe,

but it also predicts the spectrum of gravitational waves (also called tensor

perturbations) as well as the density perturbations (also called scalar pertur-

bations) of the CMB. Density perturbations create anisotropies in the CMB

and are responsible for the formation and clustering of galaxies. The size

of these irregularities depends on the energy scale at which in¬‚ation takes

place. The observed scalar perturbations are in excellent agreement with

the predictions of in¬‚ation. Gravitational waves do not a¬ect the forma-

tion of galaxies but lead to polarization of the CMB, which is beginning to

show up in the WMAP (Wilkinson Microwave Anisotropy Probe) satellite

experiment and will be measured better in future missions.

Without entering into much detail, let us mention that such ¬‚uctuations

in the energy density of the Universe can be explained in the context of

in¬‚ation as originating from the quantum ¬‚uctuations of the in¬‚aton. In-

¬‚ation produces density perturbations at every scale. The amplitude of

these perturbations depends on the form of the in¬‚aton potential V . More

precisely, the spectrum for density perturbations δH (k) ∼ δρ/ρ and gravita-

tional waves AG (k) are given by the expressions

512π V 2/3

δH (k) = , (10.298)

3

75 MP V k=aH

32 V 1/2

AG (k) = . (10.299)

2

75 MP k=aH

Here k is the comoving wave number, appearing because the ¬‚uctuations

are typically analyzed in a Fourier expansion into comoving modes δφ =

Σδφk eikx . The right-hand side of these equations is to be evaluated at a

particular time during in¬‚ation for which k = aH, which for a given k

corresponds to a particular value of φ.

Comparison with cosmological data

Cosmological data lead to δH = 1.91 — 10’5 , provided that AG << δH .

To compare with observational data, it is useful to express the spectrum in

terms of observable quantities and to make a power-law approximation

δH (k) ≈ k n’1 , A2 (k) ≈ k nG . (10.300)

G

10.7 Fluxes and cosmology 539

Here n and nG are called the spectral indices for scalar and tensor pertur-

bations, respectively

2 d ln A2

d ln δH G

n’1= , nG = . (10.301)

d ln k d ln k

The spectral indices can be expressed in terms of the slow-roll parameters

n = 1 ’ 6µ + 2·, (10.302)

nG = ’2µ, (10.303)

which shows that, in the slow-roll approximation, the spectrum is almost

scale invariant n ≈ 1. Because spectral indices are measurable quantities,