<< стр. 18(всего 24)СОДЕРЖАНИЕ >>
d d R
= =в€’
Л™ Л™
R2
dt R dt R

and so the condition (16.1) can be written as

H в€’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

ВЁ
R = в€’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 < в€’1 (16.3)
3

In other words, we need the вЂ˜matterвЂ™ 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

Л™
R2 = R2 в€’ k
1
(16.4)
3
430 Inflationary cosmology

ВЁ
During a period of acceleration R > 0 , the scale factor must increase faster than
R t в€ќ t. Provided that p < в€’ 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 = в€’p and so would clearly satisfy the criterion (16.3).
However, we have chosen to omit such terms since, as we will see, if вЂ˜matterвЂ™
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 в€ќ t1/2 в€ќ (16.5)
Tt
where the вЂ˜temperatureвЂ™ T is related to the typical particle energy by T в€ј E/kB .
The basic scenario is as follows.

вЂў EP в€ј 1019 GeV > E > EGUT в€ј 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 в€ј 1019 GeV (or temperature TP в€ј 1032 K) and time scales tP в€ј
10в€’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 в€ј
1014 GeV (corresponding to TGUT в€ј 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 в€ј 10в€’36 s.
вЂў EGUT в€ј 1015 GeV > E > EEW в€ј 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 в€ј 100 GeV (corresponding to TGUT в€ј 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 в€ј 10в€’11 s.
вЂў EEW в€ј 100 GeV > E > EQH в€ј 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 в€ј 100 MeV (corresponding to TQH в€ј 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 quarkвЂ“hadron phase transition occurs
at tQH в€ј 10в€’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 вЂ˜inflatonвЂ™ 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 в€’V
1
432 Inflationary cosmology

The corresponding field equation for is obtained from the EulerвЂ“Lagrange
dV
+ =0
2
(16.6)
d

where 2 в‰Ў =g is the covariant dвЂ™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 KleinвЂ“Gordon
1
be V
equation,

+ m2 = 0
2

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

= в€’g в€’V
1
(16.7)
T 2

The energyвЂ“momentum tensor for a perfect fluid is

= + p u u в€’ 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 Л™ 2 +V +2 2
1 1
(16.8)
p = Л™ 2 в€’V в€’ 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 = в€’ 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
Л™
R
Л™ +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

ВЁ + 3H Л™ + 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

Л™ 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.
ВЁ
R > 0) provided that

Л™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 Л™ 2 V . On differentiating, this in turn implies that ВЁ dV/d
and so the ВЁ -term can be neglected in the equation of motion (16.9), to yield

dV
3H Л™ = в€’ (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 = в€’2 Л™ 2
Л™ 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
Л™ 2 V , it is easy to show that

2
1 V
в‰Ў 1 (16.15)
2 V

where V в‰Ў 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
в‰Ў 1 (16.16)
V
These two conditions make good physical sense in that they require the potential
to be sufficiently вЂ˜flatвЂ™ that the field вЂ˜rollsвЂ™ 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 Л™ , 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 в€ќ exp 1
3V t

16.6 Ending inflation
As the field value вЂ˜rollsвЂ™ 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 Л™ 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 = в€’ 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 в€ј tв€— . From (16.5), we thus have

1/2
Rв€— tв€— T0
в€ј в€ј (16.17)
R0 t0 Tв€—

where T0 в€ј 3 K is the present-day temperature of the cosmic microwave back-
ground radiation and t0 в€ј 1/H0 в€ј 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 tв€—
kв€—
= в€ј (16.18)
Hв€— Rв€— t0
k0

where we have used the fact that H0 /Hв€— в€ј tв€— /t0 . Assuming inflation to occur
at some time between the Planck era and the GUT phase transition, so that
tP < tв€— < tGUT , from Section 16.2 we find that the ratio (16.18) lies in the range
в€ј 10в€’60 вЂ“10в€’54 . Thus, if the present-day value k 0 is of order unity then the
required degree of fine-tuning of k в€— is extreme, in a standard cosmological
model. Since the ratio above depends on 1/R2 we thus find that, to solve the
в€—
flatness problem (in order that k в€— can also be of order unity), we require the
scale factor to grow during inflation by a factor в€ј 1027 вЂ“1030 . In terms of the
required number N of e-foldings of the scale factor, we thus have

60вЂ“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 в€— = 2ctв€—

which, taking tP < tв€— < tGUT , gives the size of a causally connected region at this
time as в€ј 10в€’33 вЂ“10в€’27 m. From (16.17), we see that the size of such a region
today would be only в€ј 10в€’3 вЂ“1 m. The current size of the observable universe,
however, is given approximately by the present-day Hubble distance,

в€’1
dH 0 = cH0 в€ј 1026 m в€ј 3000 Mpc
437
16.8 Starting inflation

To solve the horizon problem, we thus require the scale factor to grow by a factor
of в€ј 1026 вЂ“1029 during the period of inflation. Expressing this result in terms of
the required number N of e-foldings, we once again find

60вЂ“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 60вЂ“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
вЂ˜rollsвЂ™ from 1 to 2 is given by

t2 H V
2 2
N= H dt = d =в€’ d
Л™ V
t1 1 1

If the potential is reasonably smooth then V в€ј V/ . Thus, if = start в€’ end
is the range of -values over which inflation occurs, one finds N в€ј 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 вЂ˜interludeвЂ™ 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 вЂ˜newвЂ™ 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 вЂ˜NewвЂ™ inflation
In the вЂ˜newвЂ™ 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(П†)
T > Tc T ˜ Tc T < Tc

V(0)

П†
П†=Пѓ
П†=0

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

3
The вЂ˜newвЂ™ inflationary model is so called in order to distinguish it from the original вЂ˜oldвЂ™ inflation model of
Guth, in which the scalar Higgs field executed quantum mechanical tunnelling at T в€ј 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 вЂ˜NewвЂ™ 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 вЂ˜lowвЂ™ 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 в€ј 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 в€ќ Rв€’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 ,

в€ќ tв€’2
R в€ќ 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 в€ќ 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 в€ј Tc is reached very quickly, and so this phase
is extremely short-lived, and very little expansion is actually achieved. Indeed, if
T в€ј 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 в€ј 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 вЂ˜reasonableвЂ™ 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 в€ј 10в€’36 s
and t2 в€ј 10в€’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

R t в€ќ t1/2

Hence, at t в€ј 10в€’34 s, the universe starts a standard Friedmann expansion, albeit
with the desired вЂ˜initialвЂ™ conditions. Thus, the inflationary model incorporates all
the observationally verified predictions of the standard cosmological models.

Although the вЂ˜newвЂ™ 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 в€ј 10в€’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(П†)

П†
П† 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
в€ј 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 вЂ˜gauge choiceвЂ™ (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 energyвЂ“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
tв†’ t+ (16.19)
tx
0

These perturbations will lead to a perturbed energyвЂ“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 FriedmannвЂ“RobertsonвЂ“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 вЂ˜gaugeвЂ™ (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 вЂ˜curvatureвЂ™ 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 вЂ“ one where
only one extra function (known here as a вЂ˜potential functionвЂ™) 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 energyвЂ“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 в€’ 1 в€’ 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 energyвЂ“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

= в€’R2 Л™ + 4H
0
ii
1
=
i
00 i
R2
=в€’
i
i

Л™
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
Л™ +H
G0 = в€’2 i
i

в€’ 3H Л™ в€’ 3H 2
G0 = в€’2 2 (16.21)
0

Gi = 2 ВЁ + 4 Л™ H + 2H + 3H 2
Л™
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 = в€’2 Л™ 2
Л™ 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 energyвЂ“
momentum tensor. Substituting (16.19) into (16.7) and working to first order in
, one quickly finds that

Ti0 = Л™ 0 i

T0 = в€’ Л™ 2 + Л™ 0 Л™ + V
0
(16.24)
0

Tii = Л™ 2 в€’ Л™ 0 Л™ + 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 energyвЂ“momentum tensor. Since the unperturbed part of the field
equations is automatically satisfied, one simply requires that G = в€’ 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

= Л™0
Л™ +H
2 (16.25)
i i

Remembering that H and have no spatial dependence, we can integrate this
0
immediately to obtain

= 2 Л™0
Л™ +H 1
(16.26)

i
One next equates the -components, which gives
i

= Л™ 2 в€’ Л™ 0 Л™ +V
в€’2 ВЁ + 4 Л™ H + 2H + 3H 2
Л™ (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 Л™ = 2 ВЁ0 + 2 Л™0 Л™
ВЁ +H
Л™ 1 1
(16.28)

then, using equations (16.9) and (16.23) to substitute for ВЁ 0 and H respectively,
Л™
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
Л™ 2 +2 = Л™2
2
(16.29)
Л™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 В· 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 в€’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 Л™0
Л™ k +H 1
k k

(16.30)
2k2 d k
1в€’ =
R2 Л™ 2 Л™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 вЂ˜slicesвЂ™
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 в†’ 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
Л™
f t = f t в€’ t в‰€ f t в€’f t (16.32)

where we do not have to specify whether it is f or f that is being differentiated
Л™
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 вЂ˜gauge transformationвЂ™ t в†’ t + t is given by4

Л™
f = в€’f t (16.33)

We may now evaluate the change in the perturbed spacetime metric corre-
sponding to the gauge transformation t в†’ 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 в€’ 1 в€’ 2 R2 t dx2 + dy2 + dz2 (16.34)

in which and are different functions. Nevertheless, for matter with no
вЂ˜anisotropic stressвЂ™ (so that all the off-diagonal components of the space part
of the stressвЂ“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 в†’ t + t we find that

в€’ 2R2 Л™
Л™
R2 1 в€’ 2 в†’ R2 1 в€’ 2 + 2RR 1 в€’ 2 (16.35)
t

4
This is the simplest version of the вЂ˜Lie derivativeвЂ™, which describes the change in a (possibly tensor) function
when вЂ˜dragged backвЂ™ along вЂ˜flow linesвЂ™ 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)

Л™
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)
Л™
f
f

is gauge invariant under the gauge transformation t в†’ t + t that we are consid-
ering, since to first order we have

Л™
H f в€’f t
в†’ = +H t+ = (16.38)
Л™
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)
Л™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

в‰Ўв€’ (16.40)
co

where the subscript indicates that is to be evaluated on comoving slices. By
вЂ˜comovingвЂ™ we mean a time-slicing that is orthogonal to the worldlines of the
вЂ˜fluidвЂ™ 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 вЂ˜curvature
449
16.15 Classical evolution of curvature perturbations

perturbationвЂ™ 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
=в€’ + (16.41)
Л™

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 stressвЂ“
i

Ti0 = Л™ 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

=в€’ (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
в‰Ў k +H (16.44)
Л™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

Л™2 2ВЁ0 2
Л™k + k
ВЁk + + + 3H =0
0
(16.45)
Л™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

вЂ“10

Comoving Hubble distance 1/(RH)

вЂ“20

вЂ“30

вЂ“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 в€ј 2 Г— 10в€’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 в‰€ 11вЂ“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 О¶ k (Г—10вЂ“10)

2

0
8 10 12 14 16
ln t

вЂ“2

вЂ“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 в€ј 2 Г— 10в€’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
в€’6 (a typical value in such theories). The initial conditions for H and
m в€ј 2 Г— 10
0 were chosen to give inflation over the period ln t в‰€ 11вЂ“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 в‰€ 11вЂ“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 вЂ˜horizonвЂ™ but must be distinguished from the вЂ˜particle horizonвЂ™, 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 в€ј 13) the perturbation suddenly вЂ˜freezesвЂ™ and no longer oscillates. We speak
of this moment, when 1/k becomes greater than 1/ RH , as the perturbation
вЂ˜leaving the horizonвЂ™ 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 вЂ˜re-enter the horizonвЂ™, 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 вЂ“ see Section 16.6).
The key point to note is that, via inflation, one has produced вЂ˜super-horizonвЂ™
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 вЂ˜quantumвЂ™ 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 вЂ˜classicalвЂ™ 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.

вЂў Convert from cosmic time t to a new dimensionless time variable known as вЂ˜confor-
mal timeвЂ™ and defined by d /dt = c/R.
вЂў Convert from the curvature perturbation k to a new variable k given by k = k ,
where = R Л™ 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 вЂ˜classicalвЂ™ 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, 203вЂ“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 в€’ =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 = в€’ / . The explicit expression
for this effective mass is given in terms of the background quantities by
2 4
2
0 0
m= в€’ в€’ в€’
2 0 0 0
(16.47)
2R2 H 2
2 RH 0
...
Л™2 Л™4 Л™0ВЁ0 3ВЁ0 0
= в€’2R2 H 2 1 + + + + +
0 0
(16.48)
2H Л™ 0 2H 2 Л™ 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 вЂ˜slow-rollвЂ™ approximation, ВЁ 0 and higher derivatives were
neglected. Furthermore, here we shall assume that Л™ 0 H during the periods of
interest. In this case m2 в€ј в€’2R2 H 2 and (16.46) becomes

+ k2 в€’ 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 вЂ˜leaving the horizonвЂ™. 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 в€ќ . 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
в€’i в€’ =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
= в€љ exp в€’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
= = = в€’ в€’
R2 H RH 2
R RH
Л™
1 H dR
= в€’ в€’
H 2 R2 H
RH
Л™ 2 dt
1
= в€’ + 0
2H 2 R
RH

Again ignoring a term in Л™ 2 /H 2 , we can thus write
0

1
= end в€’ (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 в€’ =0
2
(16.53)
k
end в€’
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 в€’
1
eв€’ik
k=в€љ (16.54)
end в€’
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 в‰€ 0 64).
455
16.16 Initial conditions and normalisation of curvature perturbations

0.006

0.005
Conformal time О·

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 в€ј 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=в€љ
k + iRH eik/RH в‰€ в€љ (16.55)
2k3 2k3
where the final expression is valid for k RH. Thus, for such modes,
H2
i
k
= в‰€в€љ (16.56)
2k3 Л™ 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
 << стр. 18(всего 24)СОДЕРЖАНИЕ >>