28 In general, a comoving point is de¬ned as a point moving with the expansion of the Universe,

that is, a point with vanishing momentum density.

10.7 Fluxes and cosmology 533

The in¬‚aton

The scalar particles used to construct di¬erent in¬‚ationary models are called

in¬‚atons. When there is just one such in¬‚aton, it is described by the La-

grangian

1

L = ’ g µν ‚µ φ‚ν φ ’ V (φ), (10.279)

2

where φ is the in¬‚aton and V (φ) is its potential. Di¬erent in¬‚ationary

models are described by di¬erent potentials, which ultimately should be

derived from a fundamental theory, such as string theory. The components

of the energy“momentum tensor following from Eqs (10.279), (10.83) and

(10.263) determine the expressions for the density and pressure to be

1™

ρφ = φ2 + V (φ), (10.280)

2

1 ™2

φ ’ V (φ).

pφ = (10.281)

2

Here spatial gradients are assumed to be negligible, so that φ can be regarded

to be a function of t only.

™

We conclude from this that in¬‚ation takes place as long as φ2 < V (φ),

which is generally the case for potentials that are ¬‚at enough. Neglecting k,

Λ and other forms of matter, these expressions can be substituted into the

Friedmann equation Eq. (10.268) and the continuity equation Eq. (10.274)

to get the equations of motion

1 1™

H2 = [V (φ) + φ2 ] (10.282)

2 2

3MP

and

dV

¨ ™

φ + 3H φ = ’ . (10.283)

dφ

One observes that the ¬eld equation for the in¬‚aton looks like a harmonic

oscillator with a friction term given by the Hubble parameter. Di¬erent

models of in¬‚ation can be obtained by solving these two equations for a

variety of potentials V (φ). Some examples are discussed below. Before

doing so, let us ¬rst explain why in¬‚ation solves some of the problems not

explained within the context of the SBB model.

Solution to some problems of the SBB model

From the form of the Friedmann equation, it becomes evident why in¬‚ation

can solve some of the unanswered questions of the SBB model. According

534 Flux compacti¬cations

to Eq. (10.277), the comoving Hubble length decreases in time during in¬‚a-

tion, and this is just what is needed to solve the ¬‚atness problem. Whereas

usually „¦ is driven away from 1, the opposite happens during in¬‚ation, as

we can see from Eq. (10.273) (the Friedmann equation), with the cosmo-

logical constant term set to zero or absorbed into „¦. The curvature term

become negligible once the comoving Hubble length increases. Hence, if in-

¬‚ation lasts for a long enough time, it brings „¦ very close to 1 without the

necessity for ¬ne-tuning „¦. The horizon problem is solved as the distance

between comoving points gets drastically stretched during in¬‚ation. This

allows the entire present observable Universe to lie within a region that was

well inside the Hubble radius before in¬‚ation. Since the Hubble radius is a

good proxy for the particle horizon size, that is, the size over which massless

particles can causally in¬‚uence each other, the whole currently observable

Universe could have been causally connected before in¬‚ation. Likewise, this

stretching dilutes the density of any undesired relic particles, provided they

are produced before the in¬‚ationary era.

Di¬erent in¬‚ationary models

Cosmologists have considered a large number of models and studied their

in¬‚ationary behavior. The models studied in the literature can be classi¬ed

according to three independent criteria.

• Initial conditions for in¬‚ation: many in¬‚ationary models are based on the

assumption that the Universe was in a state of thermal equilibrium with

a very high temperature at the beginning of in¬‚ation. The in¬‚aton was

at the minimum of its temperature dependent e¬ective potential V (φ, T ).

The main idea of chaotic in¬‚ation is to study all possible initial conditions

for the Universe including those where the Universe is outside of thermal

equilibrium and the scalar is no longer at its minimum.

• Behavior of the model during in¬‚ation: there are various possibilities for

the time dependence of the scale factor a(t). Power law in¬‚ation is one

example that is discussed next.

• End of in¬‚ation: there are basically two possibilities for ending the in-

¬‚ationary era, slow roll or a phase transition. In the ¬rst type of model

the in¬‚aton is a slowly evolving (or ”rolling”) ¬eld, which at the end of

in¬‚ation becomes faster and faster. Phase transition models contain at

least two scalar ¬elds. One of the ¬elds becomes tachyonic at the end

of in¬‚ation, which generally signals an instability, where a phase transi-

tion takes place. Hybrid in¬‚ation is an example. This type of in¬‚ation is

10.7 Fluxes and cosmology 535

of particular interest in recent attempts to make contact between string

theory and in¬‚ation.

Power-law in¬‚ation

It is hard to ¬nd the exact solution of Eqs (10.282) and (10.283) for a

generic in¬‚aton potential V (φ), so approximations or numerical studies have

to be made. However, there is one known analytic solution called power-law

in¬‚ation. For power-law in¬‚ation the potential is

2φ

V (φ) = V0 exp ’ , (10.284)

p MP

where V0 and p are constants. The scale factor and in¬‚aton that solve the

spatially ¬‚at equations of motion are

a(t) = a0 tp , (10.285)

√ V0 t

φ(t) = 2pMP log . (10.286)

p(3p ’ 1) MP

The scale factor is in¬‚ationary as long as p > 1.

Slow-roll approximation

As stated above, ¬nding exact solutions to Eqs (10.282) and (10.283) is

di¬cult, so approximations need to be made. The so-called slow roll ap-

proximation neglects one term in each equation

V (φ)

H2 ≈ 2, (10.287)

3MP

™

3H φ ≈ ’V (φ), (10.288)

where primes are derivatives with respect to the in¬‚aton. A necessary con-

dition for the slow-roll approximation to be valid is that the two slow-roll

parameters µ and · are small

12

MP (V /V )2

µ(φ) = 1, (10.289)

2

2

|·(φ)| = MP |V /V | 1. (10.290)

The parameter µ is positive by de¬nition, but the absolute value is required

on the left-hand side of the second equation, since · can be negative. Ob-

taining a solution to the slow-roll conditions is su¬cient to achieve in¬‚ation,

536 Flux compacti¬cations

but not necessary. This can be seen by rewriting the condition for in¬‚ation

Eq. (10.276) as

a¨ ™

= H + H 2 > 0, (10.291)

a

where a > 0 needs to be taken into account. This is obviously satis¬ed

™

for H > 0. From the Friedman and acceleration equations this requires in