= =’

™ ™

R2

dt R dt R

and so the condition (16.1) can be written as

H ’1

d

<0

dt R

Thus, an equivalent condition for inflation is that the comoving Hubble distance

decreases with cosmic time. Hence, when viewed in comoving coordinates, the

characteristic length scale of the universe becomes smaller as inflation proceeds.

Let us suppose that, at some period in the early universe, the energy density

is dominated by some form of matter with density and pressure p. The first

cosmological field equation (14.36) (with = 0 and 8 G = c = 1) then reads

¨

R = ’1 + 3p R (16.2)

6

Thus, we see that in order for the universe to accelerate, i.e. for inflation to occur,

we require that

p < ’1 (16.3)

3

In other words, we need the ˜matter™ to have an equation of state with negative

pressure. In fact, the above criterion can also solve the flatness problem. The

second cosmological field equation (with = 0 and 8 G = c = 1) reads

™

R2 = R2 ’ k

1

(16.4)

3

430 Inflationary cosmology

¨

During a period of acceleration R > 0 , the scale factor must increase faster than

R t ∝ t. Provided that p < ’ 1 , the quantity R2 will increase during such a

3

period as the universe expands and can make the curvature term on the right-

hand side negligible, provided that the accelerating phase persists for sufficiently

long.

We could, of course, have included the cosmological-constant terms in the

two field equations, which would then be equivalent to those for a fluid with

an equation of state = ’p and so would clearly satisfy the criterion (16.3).

However, we have chosen to omit such terms since, as we will see, if ˜matter™

in the form of a scalar field exists in the early universe then this can act as an

effective cosmological constant. In order to show that the existence of such fields

is likely, we must consider briefly the topic of phase transitions in the very early

universe.

16.2 Scalar fields and phase transitions in the very early universe

The basic physical mechanism for producing a period of inflation in the very

early universe relies on the existence, at such epochs, of matter in a form that can

be described classically in terms of a scalar field (as opposed to a vector, tensor

or spinor field, examples of which are provided by the electromagnetic field, the

gravitational field and normal baryonic matter respectively). Upon quantisation,

a scalar field describes a collection of spinless particles.

It may at first seem rather arbitrary to postulate the presence of such scalar

fields in the very early universe. Nevertheless, their existence is suggested by our

best theories for the fundamental interactions in Nature, which predict that the

universe experienced a succession of phase transitions in its early stages as it

expanded and cooled. For the purposes of illustration, let us model this expansion

by assuming that the universe followed a standard radiation-dominated Friedmann

model in its early stages, in which case

1

R t ∝ t1/2 ∝ (16.5)

Tt

where the ˜temperature™ T is related to the typical particle energy by T ∼ E/kB .

The basic scenario is as follows.

• EP ∼ 1019 GeV > E > EGUT ∼ 1015 GeV The earliest point at which the universe can

be modelled (even approximately) as a classical system is the Planck era, corresponding

to particle energies EP ∼ 1019 GeV (or temperature TP ∼ 1032 K) and time scales tP ∼

10’43 s (prior to this epoch, it is considered that the universe can be described only

in terms of some, as yet unknown, quantum theory of gravity). At these extremely

431

16.3 A scalar field as a cosmological fluid

high energies, grand unified theories (GUTs) predict that the electroweak and strong

forces are in fact unified into a single force and that these interactions bring the

particles present into thermal equilibrium. Once the universe has cooled to EGUT ∼

1014 GeV (corresponding to TGUT ∼ 1027 K), there is a spontaneous breaking of the

larger symmetry group characterising the GUT into a product of smaller symmetry

groups, and the electroweak and strong forces separate. From (16.5), this GUT phase

transition occurs at tGUT ∼ 10’36 s.

• EGUT ∼ 1015 GeV > E > EEW ∼ 100 GeV During this period (which is extremely

long in logarithmic terms), the electroweak and strong forces are separate and these

interactions sustain thermal equilibrium. This continues until the universe has cooled

to EEW ∼ 100 GeV (corresponding to TGUT ∼ 1015 K), when the unified electroweak

theory predicts that a second phase transition should occur in which the electromagnetic

and weak forces separate. From (16.5), this electroweak phase transition occurs at

tEW ∼ 10’11 s.

• EEW ∼ 100 GeV > E > EQH ∼ 100 MeV During this period the electromagnetic, weak

and strong forces are separate, as they are today. It is worth noting, however, that

when the universe has cooled to EQH ∼ 100 MeV (corresponding to TQH ∼ 1012 K)

there is a final phase transition, according to the theory of quantum chromody-

namics, in which the strong force increases in strength and leads to the confine-

ment of quarks into hadrons. From (16.5), this quark“hadron phase transition occurs

at tQH ∼ 10’5 s.

In general, phase transitions occur via a process called spontaneous symmetry

breaking, which can be characterised by the acquisition of certain non-zero values

by scalar parameters known as Higgs fields. The symmetry is manifest when

the Higgs fields have the value zero; it is spontaneously broken whenever at

least one of the Higgs fields becomes non-zero. Thus, the occurrence of phase

transitions in the very early universe suggests the existence of scalar fields and

hence provides the motivation for considering their effect on the expansion of the

universe. In the context of inflation, we will confine our attention to scalar fields

present at, or before, the GUT phase transition (the most speculative of these phase

transitions).

16.3 A scalar field as a cosmological fluid

For simplicity, let us consider a single scalar field present in the very early

universe. The field is traditionally called the ˜inflaton™ field for reasons that will

become apparent shortly. The Lagrangian for a scalar field (see Section 19.6)

has the usual form of a kinetic term minus a potential term:

L = 2g ’V

1

432 Inflationary cosmology

The corresponding field equation for is obtained from the Euler“Lagrange

equations and reads

dV

+ =0

2

(16.6)

d

where 2 ≡ =g is the covariant d™Alembertian operator. A simple

example is a free relativistic scalar field of mass m, for which the potential would

= 2 m2 2 and the field equation becomes the covariant Klein“Gordon

1

be V

equation,

+ m2 = 0

2

For the moment, however, it is best to keep the potential function V general.

The energy“momentum tensor T for a scalar field can be derived from

this variational approach (see Section 19.12), but in fact we can use our earlier

experience to anticipate its form. By analogy with the forms of the energy“

momentum tensor for dust and for electromagnetic radiation, we require that T

is (i) symmetric and (ii) quadratic in the derivatives of the dynamical variable ,

and (iii) that T = 0 by virtue of the field equation (16.6). It is straightforward

to show that the required form must be

= ’g ’V

1

(16.7)

T 2

The energy“momentum tensor for a perfect fluid is

= + p u u ’ pg

T

and by comparing the two forms in a Cartesian inertial coordinate system g =

in which the fluid is at rest, we see that the scalar field acts like a perfect

fluid, with an energy density and pressure given by

= 2 ™ 2 +V +2 2

1 1

(16.8)

p = ™ 2 ’V ’ 2

1 1

2 6

In particular, we note that if the field were both temporally and spatially

constant, its equation of state would be p = ’ and so the scalar field would

act as a cosmological constant with = V (with 8 G = c = 1). In general this

is not the case, but we will assume that the spatial derivatives can be neglected.

This is equivalent to assuming that is a function only of t and so has no spatial

variation.

433

16.4 An inflationary epoch

16.4 An inflationary epoch

Let us suppose that the scalar field does not interact (except gravitationally) with

any other matter or radiation that may be present. In this case, the scalar field

will independently obey an equation of motion of the form (14.39), namely

™

R

™ +3 +p =0

R

Substituting the expressions (16.8) and assuming no spatial variations, we quickly

find that the equation of motion of the scalar field is

¨ + 3H ™ + dV = 0 (16.9)

d

The form of this equation will be familiar to any student of classical mechanics

and allows one to develop an intuitive picture of the evolution of the scalar field.

If one thinks of the plot of the potential V versus as defining some curve, then

the motion of the scalar field value is identical to that of a ball rolling (or,

more precisely, sliding) under gravity along the curve, subject to a frictional force

proportional to its speed (and to the value of the Hubble parameter).

Let us assume further that there is some period when the scalar field dominates

the energy density of the universe. Moreover we will demand that the scalar field

energy density is sufficient large that we may neglect the curvature term in the

cosmological field equation (16.4) although this is not strictly necessary.2 Thus,

we may write (16.4) as

™ 2 +V

H2 = 1 1

. (16.10)

3 2

This equation and (16.9) thus provide a set of coupled differential equations in

and H that determine completely the evolution of the scalar field and the

scale factor of the universe during the epoch of scalar-field domination. From our

criterion (16.3) and the expressions (16.8), we see that inflation will occur (i.e.

¨

R > 0) provided that

™2 <V (16.11)

2

Note that, even if the curvature term is not negligible to begin with, the initial stages of inflation will soon

render it so.

434 Inflationary cosmology

16.5 The slow-roll approximation

The inflation equations (16.9) and (16.10) can easily be solved numerically, and

even analytically for some special choices of V . In general, however, an

analytical solution is only possible in the slow-roll approximation, in which it is

assumed that ™ 2 V . On differentiating, this in turn implies that ¨ dV/d

and so the ¨ -term can be neglected in the equation of motion (16.9), to yield

dV

3H ™ = ’ (16.12)

d

Moreover, the cosmological field equation (16.10) becomes simply

H2 = 1V (16.13)

3

It is worth noting that, in this approximation, the rate of change of the Hubble

parameter and the scalar field can be related very easily. Differentiating (16.13)

with respect to t and combining the result with (16.12), one obtains

H = ’2 ™ 2

™ 1

(16.14)

The conditions for inflation in the slow-roll approximation can be put into

a useful dimensionless form. Using the two equations above and the condition

™ 2 V , it is easy to show that

2

1 V

≡ 1 (16.15)

2 V

where V ≡ dV/d and the factor 2 is included according to the standard conven-

1

tion. Differentiating the above expression with respect to , one also finds that

V

≡ 1 (16.16)

V

These two conditions make good physical sense in that they require the potential

to be sufficiently ˜flat™ that the field ˜rolls™ slowly enough for inflation to

V

occur. It is worth noting, however, that these conditions alone are necessary but

not sufficient conditions for inflation, since they limit only the form of V and

not that of ™ , which could be chosen to violate the condition (16.11). Thus, one

must also assume that (16.11) holds.

It is worth considering the special case in which the potential V is sufficiently

flat that, during (some part of) the period of inflation, its value remains roughly

435

16.7 The amount of inflation

constant. From (16.12), we see that in this case the Hubble parameter is constant

and the scale factor grows exponentially:

R t ∝ exp 1

3V t

16.6 Ending inflation

As the field value ˜rolls™ down the potential V , the condition (16.11) will

eventually no longer hold and inflation will cease. Equivalently, in the slow-

roll approximation, the conditions (16.15, 16.16) will eventually no longer be

satisfied. If the potential V possesses a local minimum, which is usually the

case in most inflationary models, the field will no longer roll slowly downhill but

will oscillate about the minimum of the potential, the oscillation being gradually

damped by the 3H ™ friction term in the equation of motion (16.9). Eventually,

the scalar field is left stationary at the bottom of the potential. If the value of

the potential at its minimum is Vmin > 0 then clearly the condition (16.11) is

again satisfied and the universe continues to inflate indefinitely. Moreover, in this

case p = ’ and so the scalar field acts as an effective cosmological constant

= Vmin . If Vmin = 0, however, no further inflation occurs, the scalar field has

zero energy density and the dynamics of the universe is dominated by any other

fields present.

In fact, the scenario outlined above would occur only if the scalar field were

not coupled to any other fields, which is almost certainly not the case. In practice,

such couplings will cause the scalar field to decay during the oscillatory phase

into pairs of elementary particles, into which the energy of the scalar field is thus

converted. The universe will therefore contain roughly the same energy density

as it did at the start of inflation. The process of decay of the scalar field into

other particles is therefore termed reheating. These particles will interact with

each other and subsequently decay themselves, leaving the universe filled with

normal matter and radiation in thermal equilibrium and thereby providing the

initial conditions for a standard cosmological model.

16.7 The amount of inflation

Although the motivation for the introduction of the inflationary scenario was (in

part) to solve the flatness and horizon problems, we have not yet considered the

amount of inflation required to achieve this goal. From our present understanding

436 Inflationary cosmology

of particle physics, it is thought that inflation occurs at around the era of the GUT

phase transition, or earlier. For illustration, let us assume that the universe has

followed a standard radiation-dominated Friedmann model for (the majority of)

its history since the epoch of inflation at t ∼ t— . From (16.5), we thus have

1/2

R— t— T0

∼ ∼ (16.17)

R0 t0 T—

where T0 ∼ 3 K is the present-day temperature of the cosmic microwave back-

ground radiation and t0 ∼ 1/H0 ∼ 1018 s is the present age of the universe.

Let us first consider the flatness problem. From (15.47), the ratio of the spatial

curvature density at the inflationary epoch to that at the present epoch is given by

2 2

H0 R0 t—

k—

= ∼ (16.18)

H— R— t0

k0

where we have used the fact that H0 /H— ∼ t— /t0 . Assuming inflation to occur

at some time between the Planck era and the GUT phase transition, so that

tP < t— < tGUT , from Section 16.2 we find that the ratio (16.18) lies in the range

∼ 10’60 “10’54 . Thus, if the present-day value k 0 is of order unity then the

required degree of fine-tuning of k — is extreme, in a standard cosmological

model. Since the ratio above depends on 1/R2 we thus find that, to solve the

—

flatness problem (in order that k — can also be of order unity), we require the

scale factor to grow during inflation by a factor ∼ 1027 “1030 . In terms of the

required number N of e-foldings of the scale factor, we thus have

60“70 (flatness problem)

N

We now turn to the horizon problem. If the universe followed a standard

radiation-dominated Friedmann model in its earliest stages, then (reinstating c for

the moment) the particle horizon at the inflationary epoch is

dp — = 2ct—

which, taking tP < t— < tGUT , gives the size of a causally connected region at this

time as ∼ 10’33 “10’27 m. From (16.17), we see that the size of such a region

today would be only ∼ 10’3 “1 m. The current size of the observable universe,

however, is given approximately by the present-day Hubble distance,

’1

dH 0 = cH0 ∼ 1026 m ∼ 3000 Mpc

437

16.8 Starting inflation

To solve the horizon problem, we thus require the scale factor to grow by a factor

of ∼ 1026 “1029 during the period of inflation. Expressing this result in terms of

the required number N of e-foldings, we once again find

60“70 (horizon problem)

N

We have thus found that both the flatness and the horizon problems can be

solved by a period of inflation, provided that the scale factor undergoes more than

around 60“70 e-foldings during this period. We may now consider the constraints

placed by this condition on the form of the scalar field potential V . In the

slow-roll approximation, the number of e-foldings that occur while the scalar field

˜rolls™ from 1 to 2 is given by

t2 H V

2 2

N= H dt = d =’ d

™ V

t1 1 1

If the potential is reasonably smooth then V ∼ V/ . Thus, if = start ’ end

is the range of -values over which inflation occurs, one finds N ∼ 2 . In

order to solve the flatness and horizon problems, one hence requires 1.

16.8 Starting inflation

The observant reader will have noticed that so far we have not discussed how

inflation may start. During the inflationary epoch, the scalar field rolls downhill

from start to end , but we have not yet considered how the universe can arrive at

an appropriate starting state. The details will depend, in fact, on the precise infla-

tionary cosmology under consideration, but there are generally two main classes

of model. In early models of inflation, the inflationary epoch is an ˜interlude™ in

the evolution of a standard cosmological model. In such models, the inflaton field

is usually identified with a scalar Higgs field operating during the GUT phase

transition. It is thus assumed that the universe was in a state of thermal equi-

librium from the very beginning and that this state was relatively homogeneous

and large enough to survive until the beginning of inflation at the GUT era; an

example of this sort is provided by the ˜new™ inflation model discussed below in

Section 16.9. In more recent models of inflation, the scalar field is not identified

with the Higgs field in the GUT phase transition but is some generic scalar field

present in the very early universe. In particular, in these models the universe may

inflate soon after it exits the Planck era, thereby avoiding the above assumptions

regarding the state of the universe prior to the inflationary epoch; an example of

such a model is the chaotic inflation scenario discussed in Section 16.10. We will

438 Inflationary cosmology

also discuss briefly the natural extension of the chaotic inflation model, called

stochastic inflation (or eternal inflation) in Section 16.11.

16.9 ˜New™ inflation

In the ˜new™ inflation model,3 the inflationary epoch occurs when the universe

goes through the GUT phase transition. As we will see, models of this general type

typically require a rather special form for the potential V in order to produce

an effective period of inflation. In particular, identifying the inflaton field with

the scalar Higgs field operating during the GUT spontaneous-symmetry-breaking

phase transition, considerations from quantum field theory suggest a form for the

potential V T which is actually also a function of temperature T . The typical

form for V T is shown in Figure 16.1 for several values of T . At very high

temperatures the potential is parabolic with a minimum at = 0, which is the

true vacuum state (i.e. the state of lowest energy). Thus at very high temperatures

we would expect the scalar field to have the value = 0. However, for lower

temperatures the form of the potential changes until at the critical temperature

V(φ)

T > Tc T ˜ Tc T < Tc

V(0)

φ

φ=σ

φ=0

Figure 16.1 The temperature-dependent potential function for a Higgs-like

scalar field .

3

The ˜new™ inflationary model is so called in order to distinguish it from the original ˜old™ inflation model of

Guth, in which the scalar Higgs field executed quantum mechanical tunnelling at T ∼ Tc , where Tc is the

critical temperature, from the metastable false ground state at = 0 through a potential barrier to the true

ground state with > 0. Although this model provided the genesis for the inflationary idea, it was quickly

shown to predict a universe very different to the one we observe. In short, the tunnelling process produces

bubble nucleation and it turns out that these bubbles are too small to be identified with the observable universe

and are carried apart too quickly by the intervening inflating space for them to coalesce, hence resulting in a

highly inhomogeneous universe, contrary to observations.

439

16.9 ˜New™ inflation

T = Tc the potential develops a lower energy state than that at = 0. Thus this

new non-zero value of is now the true vacuum state, and = 0 is now a false

vacuum state. For even lower values the new true vacuum state becomes more

pronounced until a final form is reached for ˜low™ temperatures.

Let us now consider the evolution of the scale factor R t , the radiation energy

density r and the scalar field .

Phase 1 When the temperature is very high, i.e. far above the GUT phase

transition scale of Tc ∼ 1027 K, from Figure 16.1 we would expect the scalar field

to have the value = 0 (i.e. at the true vacuum state for these temperatures),

and Figure 16.1 shows that it will remain at = 0. Since r ∝ R’4 , however, we

would expect the radiation to dominate over the scalar field at very early epochs.

Thus we have the standard early-time radiation-dominated Friedmann model, in

which we can neglect the curvature constant k. Thus, for T Tc ,

∝ t’2

R ∝ t1/2 =0

r

Phase 2 It is clear from the above equations that there will come a time when

the scalar-field energy density dominates over that of the radiation. Provided that

this occurs for T > Tc the scalar field remains at = 0, in which case it acts as an

effective cosmological constant of value = V 0 . Thus, in this phase, the scale

factor undergoes an exponential expansion:

1

R t ∝ exp V0t

3

As a result of the exponential expansion, however, there is a corresponding

exponential decrease in the temperature T , which results in a rapid change of

the potential function. Thus T ∼ Tc is reached very quickly, and so this phase

is extremely short-lived, and very little expansion is actually achieved. Indeed, if

T ∼ Tc is reached before the scalar-field energy density dominates over that of

the radiation then phase 2 does not occur at all.

Phase 3 Once T ∼ Tc , we see from Figure 16.1 that the scalar field is now able to

roll downhill away from = 0 and so the GUT phase transition occurs. Provided

that the potential is sufficiently flat, the slow-roll approximation holds and the

universe inflates, the evolution of the scalar field being determined by (16.12)

and the Hubble parameter by (16.13). If the potential is roughly constant then

the exponential expansion continues. The rapid growth of the scale factor once

again causes the evolution of the potential function as the temperature drops. The

440 Inflationary cosmology

duration of this period of inflation depends critically on the flatness and length

of the plateau of the V function for T < Tc . For certain ˜reasonable™ potentials

the universe can easily inflate in such a way that the number of e-foldings

N 60, and can be considerably larger. This is therefore the main inflationary

phase. According to detailed calculations, phase 3 occurs between t1 ∼ 10’36 s

and t2 ∼ 10’34 s and the scale factor increases by a factor of around 1050 .

Phase 4 Eventually, the slow-roll approximation fails and inflation ends. The

scalar field then rolls rapidly down towards the true vacuum state at = ,

oscillating about the minimum point, and follows the behaviour outlined in

= 0 then the universe will revert to the standard

Section 16.6. In particular, if V

radiation-dominated Friedmann model with

R t ∝ t1/2

Hence, at t ∼ 10’34 s, the universe starts a standard Friedmann expansion, albeit

with the desired ˜initial™ conditions. Thus, the inflationary model incorporates all

the observationally verified predictions of the standard cosmological models.

Although the ˜new™ inflation model still has its advocates, it suffers from

undesirable features. In particular, the scenario only provides an effective period

T has a very flat plateau near = 0, which is somewhat

of inflation if V

artificial. Moreover, the period of thermal equilibrium prior to the inflationary

phase (so one can speak sensibly of the universe having a particular temperature)

requires many particles to interact with one another, and so already one requires

the universe to be very large and contain many particles. Finally, the universe

could easily recollapse before inflation starts. As a result of these difficulties, new

inflation may not be a viable model, and so there are strong theoretical reasons

to believe that the inflaton field cannot be identified with the GUT symmetry-

breaking Higgs field. Thus, the hope that GUTs could provide the mechanism for

the homogeneity and flatness of the universe may have to be abandoned.

16.10 Chaotic inflation

In more recent models of inflation, the scalar field is not identified with

the Higgs field in the GUT phase transition but is regarded as a generic scalar

field present in the very early universe. In particular, these models invoke the

idea of chaotic inflation. In this scenario, as the universe exits the Planck era

at t ∼ 10’43 s the initial value of the scalar field start is set chaotically, i.e. it

acquires different random values in different regions of the universe. In some

regions, start is somewhat displaced from the minimum of the potential and

441

16.11 Stochastic inflation

V(φ)

φ

φ start

= 2 m2 2 for a free scalar field. The field is

1

Figure 16.2 The potential V

initially displaced from the minimum of the potential due to chaotic initial

conditions as the universe comes out of the Planck era.

so the field subsequently rolls downhill. If the potential is sufficiently flat, the

field is more likely to be displaced a greater distance from its minimum and

will roll slowly enough, and for a sufficiently prolonged period, for the region to

undergo an effective period of inflation. Conversely, in other regions start may

not be displaced sufficiently from the minimum of the potential for the region to

inflate. Thus, on the largest scales the universe is highly inhomogeneous, but our

observable universe lies (well) within a region that underwent a period of inflation.

According to this scenario, inflation may occur even in theories with very

∼ n , and is thus a very generic process that can

simple potentials, such as V

take place under a broad range of conditions. Indeed, the potential function need

not depend on the temperature T . A very simple example is a free scalar field,

= 2 m2 2 (see Figure 16.2). Moreover, in the chaotic scenario,

1

for which V

inflation may begin even if there is no thermal equilibrium in the early universe,

and it may even start just after the Planck epoch.

16.11 Stochastic inflation

A natural extension to the chaotic inflation model is the mechanism of stochastic

(or eternal) inflation. The main idea in this scenario is to take account of quantum

fluctuations in the evolution of the scalar field, which we have thus far ignored

by modelling the field entirely classically. If, in the chaotic assignment of initial

values of the scalar field, some regions have a large value of start then quantum

fluctuations can cause to move further uphill in the potential V . These

regions inflate at a greater rate than the surrounding ones, and the fraction of the

total volume of the universe containing the growing -field increases. Quantum

442 Inflationary cosmology

fluctuations within these regions lead in turn to the production of some new

inflationary regions that expand still faster. This process thus leads to eternal

self-reproduction of the inflationary universe.

16.12 Perturbations from inflation

We have seen that inflation can solve the horizon and flatness problems. Arguably

its greatest success so far, however, is to provide a mechanism by which the

fluctuations needed to seed the development of structure within the universe can

be generated. This topic is the subject of much current research, and we can give

only a limited treatment here. Nevertheless, by following through the equations for

structure generation and development in the simplest case, namely for a spatially

flat universe with a simple ˜gauge choice™ (see below), we hope that the reader

will be able to get a flavour of the physics involved.

The current opinion of how structure in the universe originated is that it was via

amplification, during a period of inflation, of initial quantum irregularities of the

scalar field that drives inflation. Thus what we need to do can be divided into two

broad categories. First, we need to work out the equations of motion for spatial

perturbations in the scalar field. This can be done classically, i.e. taking the scalar

field as a classical source linked self-consistently to the gravitational field via a

classical energy“momentum tensor. Second, we need to derive initial conditions

for these perturbations, and this demands that we understand the quantum field

theory of the perturbations themselves. This sounds formidable but actually turns

out to be no more complicated than considering the quantum physics of a mass

on a spring, albeit one in which the mass changes as a function of time. These

topics are discussed in detail in the remainder of this chapter.

16.13 Classical evolution of scalar-field perturbations

We assume that the scalar field , which hitherto has been a function of cosmic

time t only, now has perturbations that are functions of space and time. We can

thus write

t’ t+ (16.19)

tx

0

These perturbations will lead to a perturbed energy“momentum tensor, which we

shall derive shortly. The Einstein field equations then imply that the Einstein

tensor is also perturbed away from its background value. In turn, therefore, we

must have a metric different from the Friedmann“Robertson“Walker one assumed

so far. We thus need to assume a form for this metric in order to calculate the

new Einstein tensor. It is at this point we must make the choice of ˜gauge™ (i.e.

443

16.13 Classical evolution of scalar-field perturbations

coordinate system) referred to above. Once perturbations are present there is no

preferred way to define a spacetime slicing of the universe. The details of this are

quite subtle but amount simply to the fact that by choosing different coordinate

systems we can change the apparent character of the perturbations considerably.

For example, suppose that we choose, as a new time coordinate, one for which

surfaces of constant time have a constant value of the new perturbed scalar field

on them. This is always possible and, in such a gauge, the spatial fluctuations of

have apparently totally vanished!

To meet such problems, methods that deal only with gauge-invariant quantities

have been developed. We will make contact with such methods below, when

we introduce the so-called ˜curvature™ perturbations. These are gauge invariant

and therefore represent physical quantities. To reach this point, however, we first

work with a specific simple form of gauge known as the as the longitudinal

or Newtonian gauge, and indeed with a restricted form of this “ one where

only one extra function (known here as a ˜potential function™) is introduced.

The justification for using such a restricted form is that it leads to an Einstein

tensor with the correct extra degrees of freedom to match the extra terms in the

scalar-field energy“momentum tensor arising from the field perturbations.

For a spatially flat (k = 0) background FRW model, which is what we will

assume, we adopt Cartesian comoving coordinates and write the perturbed metric as

ds2 = 1 + 2 dt2 ’ 1 ’ 2 R2 t dx2 + dy2 + dz2 (16.20)

where is a general infinitesimal function of all four coordinates (and should

not to be confused with the scalar field ). Its assumed smallness means that we

will only need to consider quantities to first order in . A general discussion of

this linearising process is presented in the next chapter, but for the time being we

simply note that one can consider as representing the Newtonian potential of

the perturbations. For instance, for a spherically symmetric perturbation of mass

M and radius r, if we put = GM/rc2 then the first term of (16.20) recovers the

tt-term of the Schwarzschild metric.

The perturbed Einstein field equations

We now need to find both the new energy“momentum tensor of the scalar field

and the new Einstein tensor corresponding to our perturbed metric. Equating

them will link our two perturbation variables and and provide us with the

equations of evolution that we need. The first step is to calculate the connection

coefficients corresponding to the perturbed metric (16.20) to first order in .

=0 +

These are easily shown to take the form , where the first

444 Inflationary cosmology

term corresponds to the connection coefficients of the unperturbed metric (i.e.

with = 0) and the perturbation terms are given by

=

0

0

= ’R2 ™ + 4H

0

ii

1

=

i

00 i

R2

=’

i

i

™

In these expressions, H = R/R is the Hubble parameter of the unperturbed back-

ground, and no sum over repeated i indices is implied. The remaining perturbed

connection coefficients either follow from symmetry or are zero.

These connection coefficients yield a Riemann and hence an Einstein tensor.

Again working to first order in , the perturbed part of the Einstein tensor is

found to be

™ +H

G0 = ’2 i

i

’ 3H ™ ’ 3H 2

G0 = ’2 2 (16.21)

0

Gi = 2 ¨ + 4 ™ H + 2H + 3H 2

™

i

where again no sum over repeated i indices is implied and the remaining entries

either follow from symmetry or are zero. The symbol 2 here denotes the spatial

Laplacian, which in this simple flat case is given by

2 2 2

1

=2 + +

2

(16.22)

x2 y2 z2

R

It is worth noting that, in the entries of (16.21), the time derivative of the Hubble

parameter appears. From (16.14), this can be rewritten as

H = ’2 ™ 2

™ 1

(16.23)

0

remembering that this equation now applies to the background FRW spacetime.

We also need to evaluate the perturbed part of the scalar-field energy“

momentum tensor. Substituting (16.19) into (16.7) and working to first order in

, one quickly finds that

Ti0 = ™ 0 i

T0 = ’ ™ 2 + ™ 0 ™ + V

0

(16.24)

0

Tii = ™ 2 ’ ™ 0 ™ + V

0

445

16.13 Classical evolution of scalar-field perturbations

where V = dV/d 0 and the remaining components either follow from symmetry

or are zero.

We may now use the Einstein field equations to relate the Einstein tensor and

the scalar-field energy“momentum tensor. Since the unperturbed part of the field

equations is automatically satisfied, one simply requires that G = ’ T (since

= 8 G/c4 equals unity in our chosen system of units). We may thus equate,

with the inclusion of a minus sign, the components shown in equations (16.21)

and (16.24). At first sight, it is by no means obvious that we have allowed

ourselves enough freedom in including only one extra function, , in the metric.

Nevertheless, as we now show, everything in fact works out. Let us start with the

0

i -components, for which we have the equation

= ™0

™ +H

2 (16.25)

i i

Remembering that H and have no spatial dependence, we can integrate this

0

immediately to obtain

= 2 ™0

™ +H 1

(16.26)

i

One next equates the -components, which gives

i

= ™ 2 ’ ™ 0 ™ +V

’2 ¨ + 4 ™ H + 2H + 3H 2

™ (16.27)

0

but we may show that this contains no information beyond that already obtained

from the 0 -components. In particular, differentiating (16.26) with respect to time

i

gives

+H ™ = 2 ¨0 + 2 ™0 ™

¨ +H

™ 1 1

(16.28)

then, using equations (16.9) and (16.23) to substitute for ¨ 0 and H respectively,

™

one finds that (16.27) is satisfied if (16.26) holds, thus establishing consistency.

The only new information must therefore come from equating the 0 -components.

0

Using (16.28) and eliminating V again then yields

d

™ 2 +2 = ™2

2

(16.29)

™0

0 0

dt

Perturbation equations in Fourier space

The results (16.26) and (16.29) are the basic equations relating and . To

make further progress, however, it is convenient to work instead in terms of

the Fourier decomposition of these quantities and analyse what happens to a

perturbation corresponding to a given comoving spatial scale. Thus, we assume

446 Inflationary cosmology

that and are decomposed into a superposition of plane-wave states with

comoving wavevector k, so that

1

x= exp ik · x d3 k

k

3/2

2

where (with a slight abuse of notation) x = x y z and a similar expression

holds for . The evolution of a mode amplitude k depends only on the

comoving wavenumber k = k ; the corresponding actual physical wavenumber

is k/R t . We thus work simply in terms of k and k . In terms of these

k by ’k /R t , whereas

2 will be just to multiply 2 2

variables, the action of

the time derivatives remain unchanged. Equations (16.26) and (16.29) therefore

become

= 2 ™0

™ k +H 1

k k

(16.30)

2k2 d k

1’ =

R2 ™ 2 ™0

k

dt

0

Thus, we see that we have obtained two coupled first-order differential equations

for the quantities k and k , which are the amplitudes of the plane-wave

perturbations of comoving wavenumber k in the metric and in the scalar field

respectively. Clearly, what we could do next is to eliminate one quantity in terms

of derivatives of the other and then obtain a single second-order equation in

terms of just one of them (plus the background quantities, of course, but the

evolution of these is assumed known). In fact, this leads to rather messy equations

and, moreover, in terms of the discussion given above the results are not gauge

invariant, since neither k nor k is gauge invariant on its own.

16.14 Gauge invariance and curvature perturbations

As mentioned above, gauge invariance is related to how we define spatial ˜slices™

of the perturbed spacetime. By transforming to a new time coordinate, one can

apparently make the perturbations in the scalar field come and go at will. There

are two ways to take care of this difficulty. First, one can choose variables that are

insensitive to such changes and therefore definitely describe something physical.

These are called gauge-invariant variables. Second, one can use variables which

would change if one altered the slicing but which are defined relative to a particular

slicing that can itself be defined physically. These are then also physical variables

and are, perhaps confusingly, also sometimes called gauge invariant, although this

is not really a good description. Note that changing spatial coordinates within a

447

16.14 Gauge invariance and curvature perturbations

particular slicing also induces changes, but these are not relevant to our discussion

here and we concentrate just on changes in time coordinate.

Let us start our discussion by taking the first of the two routes outlined above,

namely describing the perturbations in terms of truly gauge-invariant quantities.

For any scalar function f in spacetime, consider the effects upon it of the change

in time coordinate t ’ t = t + t. We may define a new, perturbed, function by

f t =f t (16.31)

where, as just stated, we suppress the x-dependences in what follows. Thus, to

first order in t, we may write

™

f t = f t ’ t ≈ f t ’f t (16.32)

where we do not have to specify whether it is f or f that is being differentiated

™

with respect to time to obtain f or whether the latter is evaluated at t or t , since

these would be second-order differences. Hence the perturbation in the scalar

function due to the ˜gauge transformation™ t ’ t + t is given by4

™

f = ’f t (16.33)

We may now evaluate the change in the perturbed spacetime metric corre-

sponding to the gauge transformation t ’ t + t. To do this, however, one must

distinguish between the two occurrences of the -variable in (16.20). For an

arbitrary scalar perturbation, the general form of the perturbed metric in fact

takes the form

ds2 = 1 + 2 dt2 ’ 1 ’ 2 R2 t dx2 + dy2 + dz2 (16.34)

in which and are different functions. Nevertheless, for matter with no

˜anisotropic stress™ (so that all the off-diagonal components of the space part

of the stress“energy tensor are zero), the two functions may be taken as equal;

this is the case for a perfect fluid or a scalar field and hence leads to (16.20).

Even in this case, however, the two functions behave differently under the gauge

transformation. We need consider only the -function above, which clearly takes

the role of a spatial curvature term since it modifies the space part of the metric

by a multiplicative factor. Under t ’ t + t we find that

’ 2R2 ™

™

R2 1 ’ 2 ’ R2 1 ’ 2 + 2RR 1 ’ 2 (16.35)

t

4

This is the simplest version of the ˜Lie derivative™, which describes the change in a (possibly tensor) function

when ˜dragged back™ along ˜flow lines™ in parameter space; see, for example, B. Schutz, Geometrical Methods

of Mathematical Physics, Cambridge University Press, 1980.

448 Inflationary cosmology

Since both and t are infinitesimal, we may employ the same arguments that

led to (16.33). Then, to first order, we have

=H t (16.36)

™

where we have also used the fact that RR = R2 H. Thus, for any scalar function

f with perturbations f , we see that the combination

Hf

= + (16.37)

™

f

f

is gauge invariant under the gauge transformation t ’ t + t that we are consid-

ering, since to first order we have

™

H f ’f t

’ = +H t+ = (16.38)

™

f f

f

f

Thus, for the specific example of our scalar-field perturbation , we may

identify the corresponding gauge-invariant quantity as

H

= + (16.39)

™0

We will therefore use this variable (or its Fourier transform) in our subsequent

discussion in later sections. In the literature this quantity is called the curvature

perturbation, for reasons that will become clear shortly.

Before going on to consider the evolution of these curvature perturbations,

let us first discuss briefly the second route outlined at the start of this section

for defining a physically meaningful perturbation variable. This route can be

illustrated directly with the -function, and one begins by defining the quantity

≡’ (16.40)

co

where the subscript indicates that is to be evaluated on comoving slices. By

˜comoving™ we mean a time-slicing that is orthogonal to the worldlines of the

˜fluid™ that makes up the matter. For an ordinary fluid, this would amount to

choosing frames in which, at each instant and position, the fluid appears to be

at rest. The same applies here and, because the frame involved is physically

defined, the variable , which measures the spatial curvature in the given frame,

is itself physically well defined. Thus the quantity is also called the ˜curvature

449

16.15 Classical evolution of curvature perturbations

perturbation™ in the literature. As we now show, it is in fact equal to minus the

variable defined in (16.39), and so both may be described as such.

For any scalar density perturbation , one can write the spatial curvature in

the comoving slice as

H co

=’ + (16.41)

™

Let us therefore consider what happens for the particular case of a perturbation in

a scalar field. As shown in (16.24), the 0 -components of the perturbed stress“

i

energy tensor read

Ti0 = ™ 0 (16.42)

i

In the comoving frame, this momentum density must vanish, by definition, and

so the scalar-field perturbation cannot depend on the spatial coordinates and thus

vanishes. Hence, for a scalar field, we have

=’ (16.43)

16.15 Classical evolution of curvature perturbations

We now consider the evolution of the Fourier transform of the gauge-invariant

perturbation (16.39), namely

k

≡ k +H (16.44)

™0

k

which is clearly itself gauge invariant. Using (16.30), the second-order differential

equation satisfied by this quantity is quite simply shown to be

™2 2¨0 2

™k + k

¨k + + + 3H =0

0

(16.45)

™0 k

R2

H

Given a potential V 0 and some initial conditions for H and 0 , we can integrate

the background evolution equations numerically and obtain H and 0 as functions

of cosmic time t. If we simultaneously integrate k using (16.45), we can thereby

trace the evolution of the curvature perturbation over the time period of interest.

An example of the results of this procedure is shown in Figures 16.3 and 16.4,

450 Inflationary cosmology

“10

Comoving Hubble distance 1/(RH)

“20

“30

“40

8 10 12 14 16

ln t

Figure 16.3 Evolution of the logarithm of the comoving Hubble distance

ln 1/ RH versus ln t (solid line) in a chaotic inflation model driven by a free

scalar field of mass m ∼ 2 — 10’6 , the initial values of H and 0 being chosen in

such a way that there is a period of inflation lasting approximately for the period

ln t ≈ 11“16. Also shown (broken line) is the fixed comoving scale 1/k, where

k = 104 is the comoving wavenumber of the perturbation shown in Figure 16.4.

Note that all quantities are in Planck units.

4

Curvature perturbation ζ k (—10“10)

2

0

8 10 12 14 16

ln t

“2

“4

Figure 16.4 Evolution of the curvature perturbation k versus ln t for k = 104

in a chaotic inflation model driven by a free scalar field of mass m ∼ 2 — 10’6 .

Note that all quantities are in Planck units.

451

16.15 Classical evolution of curvature perturbations

for the particular choice of potential V 0 = 2 m2 2 (chaotic inflation) with

1

0

’6 (a typical value in such theories). The initial conditions for H and

m ∼ 2 — 10

0 were chosen to give inflation over the period ln t ≈ 11“16, and the comoving

wavenumber of the perturbation5 was chosen as k = 104 .

From Figure 16.3, one can verify that the universe is indeed inflating during the

period ln t ≈ 11“16, since the comoving Hubble distance 1/ RH is decreasing

with cosmic time (see Section 16.1). In inflationary theory, this quantity is loosely

called the ˜horizon™ but must be distinguished from the ˜particle horizon™, as

discussed in Section 15.12. The broken line in Figure 16.3 is the natural logarithm

of the reciprocal of the comoving wavenumber k, which is of course constant for

a given perturbation. This reciprocal, 1/k, gives another dimensionless scale and

(ignoring possible factors of 2 that, one could argue, should be introduced) can

be thought of as the comoving wavelength scale of the perturbation itself.

The behaviour of the curvature perturbation k is shown in Figure 16.4 for

k = 104 and can be understood from the behaviour of the comoving Hubble

distance (or horizon) in Figure 16.3.6 Whilst the perturbation scale 1/k is less

than the horizon radius 1/ RH the curvature perturbation k just oscillates. Once

the comoving horizon radius has dropped below 1/k, however, we see that (at

ln t ∼ 13) the perturbation suddenly ˜freezes™ and no longer oscillates. We speak

of this moment, when 1/k becomes greater than 1/ RH , as the perturbation

˜leaving the horizon™ and, in intuitive terms, we can understand that beyond this

point the perturbation is no longer able to feel its own self-gravity, since it is larger

than the characteristic scale over which physical processes in the universe operate

coherently. The curvature perturbation thereafter remains frozen at whatever value

it has reached at this point until much later in the history of the universe, when

the comoving horizon scale eventually catches up with 1/k again. At this point,

the perturbation is said to ˜re-enter the horizon™, and oscillations will begin again

(though at this stage it is not expected that these will be in the scalar field itself,

since the latter is thought to decay into other particles, via the process of reheating,

shortly after inflation ends “ see Section 16.6).

The key point to note is that, via inflation, one has produced ˜super-horizon™

scale fluctuations in the early universe. These fluctuations later go on to provide

the seeds for galaxy formation and the perturbations in the cosmic microwave

5

Note that all quantities here are measured in Planck units, e.g. the masses are in inverse Planck lengths and

the times in Planck times.

6

The initial conditions used for examining the classical behaviour of k can of course be chosen arbitrarily. The

starting values of k and its time derivative used in Figure 16.4 in fact correspond to ˜quantum™ conditions,

where field-theoretic values for the initial fluctuation are set. This is discussed in Section 16.6 below, where a

new variable k , related to k , is introduced. The specific values used correspond to evaluating the imaginary

part of equation (16.51) and its time derivative, followed by a global phase shift such that the initial phase

is zero.

452 Inflationary cosmology

background radiation that we observe today. By studying the distribution of

galaxies and CMB fluctuations as a function of scale, it is possible to obtain an

idea of the underlying primordial spectrum of perturbations that produced them.

Thus, by predicting this primordial spectrum, we can perform a test of the whole

inflationary picture for the origin of fluctuations. This is obviously an area of great

current interest. We can give only a simplified treatment, but the basic equations

are within our reach, as we now discuss.

16.16 Initial conditions and normalisation of curvature perturbations

The key concept we need for predicting the primordial spectrum of perturbations

produced during inflation can be stated in the following question: what sets the

initial conditions for the perturbation k itself? If we knew this for each k, then,

since the evolution of k through to the point where it freezes would be known,

given the evolution of the background model we could compute a spectrum of

curvature perturbations as a function of k.

The basic idea for setting the initial conditions for the perturbations is that they

come from quantum-field-theoretic fluctuations in the value of the scalar field .

Thus the ˜classical™ perturbations discussed above need to be quantized, in a field

theory sense, and this will allow their initial values to be set. A rigorous way

of performing this quantisation has been developed7 and, although the process is

complicated, the final result in our case is very simple. To apply the result, we

must first make two changes of variable in our discussion above.

• Convert from cosmic time t to a new dimensionless time variable known as ˜confor-

mal time™ and defined by d /dt = c/R.

• Convert from the curvature perturbation k to a new variable k given by k = k ,

where = R ™ 0 /H.

The formal procedure then shows that the correct quantisation may be achieved

simply by treating k as a free complex scalar field and quantising it in the standard

fashion. The evolution equation for the quantum perturbations turns out to be

identical to the ˜classical™ equation for k . Thus, having fixed initial conditions

for k quantum mechanically, one may follow the classical evolution.

Let us first derive the classical evolution equation for k . Making the trans-

formation of variables noted above, equation (16.45) becomes even simpler. In

particular, the intermediate variable was chosen in order to remove the first-

derivative term in (16.45), so as to make it more like a simple harmonic oscillator

7

See, for example, V. F. Mukhanov, H. A. Feldman & R. H. Brandenberger, Theory of cosmological pertur-

bations, Physics Reports 215, 203“333, 1992.

453

16.16 Initial conditions and normalisation of curvature perturbations

equation. Using a prime to denote a derivative with respect to conformal time,

we obtain

+ k2 ’ =0 (16.46)

k

k

It is now clear that we are dealing with the equation for the kth mode of a scalar

field with a time-variable mass given by m2 = ’ / . The explicit expression

for this effective mass is given in terms of the background quantities by

2 4

2

0 0

m= ’ ’ ’

2 0 0 0

(16.47)

2R2 H 2

2 RH 0

...

™2 ™4 ™0¨0 3¨0 0

= ’2R2 H 2 1 + + + + +

0 0

(16.48)

2H ™ 0 2H 2 ™ 0

2H 2 4H 4 H3

where, in the last line, we have re-expressed the result in terms of derivatives with

respect to cosmic time t rather than conformal time, which we will find useful

momentarily. Perhaps surprisingly, it is the 0 / 0 term in (16.47) that gives rise

to the leading-order term 2R2 H 2 in (16.48)!

To set the initial conditions for k , we will study the variable-mass term m2 in

the form (16.48). In the ˜slow-roll™ approximation, ¨ 0 and higher derivatives were

neglected. Furthermore, here we shall assume that ™ 0 H during the periods of

interest. In this case m2 ∼ ’2R2 H 2 and (16.46) becomes

+ k2 ’ 2R2 H 2 =0 (16.49)

k

k

In this form, we can see the origin of the behaviour discussed above in terms of

a perturbation ˜leaving the horizon™. When k RH the perturbation length scale

is within the horizon (since 1/k 1/ RH ) and we have oscillatory behaviour.

When k RH, however, the perturbation length scale exceeds the horizon and

we have exponential growth in k . Moreover, in the latter case we see directly

from (16.46) that, if k can be neglected, we may immediately deduce the solution

k ∝ . Since k = k , this means that the curvature perturbation k is constant,

which is exactly the behaviour seen in Figure 16.4.

Let us now consider further the regime k RH, when the perturbations are well

inside the horizon, which is where the initial conditions for k can be set. In this

regime, (16.49) becomes simply the harmonic oscillator equation k + k2 k = 0,

the quantisation of which is well understood. This quantisation demands that the

norm of any state evaluates to unity in Planck units, or equivalently that the

conserved current of the field is unity, so that

’i ’ =1 (16.50)

454 Inflationary cosmology

It is this condition that sets the absolute scale of the perturbations. Hence, the

properly normalised positive-energy solution in the regime k RH is given (up to

a constant phase factor) by

1

= √ exp ’ik (16.51)

k

2k

which is therefore the form to which any solution of (16.49) must tend well within

the horizon.

We may now attempt to obtain a full solution to (16.49) and can in fact achieve

this quite simply. Consider the following series of manipulations concerning the

conformal time , in which we carry out an integration by parts:

1

dt dR dH

= = = ’ ’

R2 H RH 2

R RH

™

1 H dR

= ’ ’

H 2 R2 H

RH

™ 2 dt

1

= ’ + 0

2H 2 R

RH

Again ignoring a term in ™ 2 /H 2 , we can thus write

0

1

= end ’ (16.52)

RH

where end is the value at which the conformal time saturates at the end of

inflation (that it does indeed saturate is obvious from the facts that d /dt = 1/R

and that R is increasing exponentially during inflation). Figure 16.5 shows that

(16.52) is indeed a good approximation during inflation in our current numerical

example. Equation (16.49) now becomes

2

k+ k ’ =0

2

(16.53)

k

end ’

2

which finally is exactly soluble. There is a unique solution (up to a constant phase

factor) that tends to (16.51) for small ; it is given by

i+k end ’

1

e’ik

k=√ (16.54)

end ’

2k3

By inspection this has the correct property for end provided that k end 1.

Comparison with Figure 16.5 shows that this is indeed the case for k-values of

interest (for the figure, k = 104 and end ≈ 0 64).

455

16.16 Initial conditions and normalisation of curvature perturbations

0.006

0.005

Conformal time ·

0.004

0.003

0.002

0.001

8 10 12 14 16

ln t

Figure 16.5 Evolution of conformal time for the same numerical case as that

illustrated in Figures 16.3 and 16.4 (solid curve). The broken curve shows the

approximation given in equation (16.52), which is seen to be very good once

inflation starts, around ln t ∼ 11.

Now that we have a correctly normalised general solution for k , let us consider

the regime k RH at which the perturbation length scale exceeds the horizon.

We use (16.52) to rewrite the solution just found as

1 iRH

k=√

k + iRH eik/RH ≈ √ (16.55)

2k3 2k3

where the final expression is valid for k RH. Thus, for such modes,

H2

i

k

= ≈√ (16.56)

2k3 ™ 0

k

Since we have demonstrated that k is constant after the mode has left the horizon,

this means we are free to evaluate the right-hand side at the horizon exit itself.

We therefore write schematically

H2