TASI Lectures on In
ation
William H. Kinney
Department of Physics, University at Bualo, SUNY,
Bualo, NY 14260-1500, USA
This series of lectures gives a pedagogical review of the subject of cosmological in
ation. I discuss
Friedmann-Robertson-Walker cosmology and the horizon and
atness problems of the standard hot
Big Bang, and introduce in
ation as a solution to those problems, focusing on the simple scenario
of in
ation from a single scalar eld. I discuss quantum modes in in
ation and the generation of
primordial tensor and scalar
uctuations. Finally, I provide comparison of in
ationary models to
the WMAP satellite measurement of the Cosmic Microwave Background, and brie
y discuss future
directions for in
ationary physics. The majority of the lectures should be accessible to advanced
undergraduates or beginning graduate students with only a background in Special Relativity, al-
though familiarity with General Relativity and quantum eld theory will be helpful for the more
technical sections.
I. INTRODUCTION
Cosmology today is a vibrant scientic enterprise. New precision measurements are revealing a universe with
surprising and unexpected properties, in particular the Dark Matter and Dark Energy which are now believed to
be the dominant components of the cosmos. Galaxy surveys such as the Sloan Digital Sky Survey are making the
rst large-scale maps of the universe, and satellites such as WMAP are making exquisitely precise measurements of
the Cosmic Microwave Background (CMB), the haze of relic photons left over from the Big Bang. In turn, these
measurements are giving us clues which are helping to unravel one of the oldest and most profound questions people
have ever asked: Where did the universe come from? In these lectures, I discuss what is currently the best motivated
and most completely developed physical model for the rst moments of the universe: cosmological in
ation [1, 2, 3].1
In
ation naturally explains how the universe came to be so large, so old, and so
at, and provides a compellingly
elegant and predictive mechanism for generating the primordial perturbations which gave rise to the rich structure
we see in the universe today [7, 8, 9, 10, 11, 12, 13]. In
ation provides a link between the Outer Space of astrophysics
and the Inner Space of particle physics, and gives us a window to physics at energy scales far beyond the reach of
particle accelerators. Furthermore, in
ation makes testable predictions, which have so far proven to be an excellent
match to the data.
The lectures are organized as follows:
 Section I provides an introduction and a brief overview of General Relativity.
 Section II discusses the Friedmann-Robertson-Walker spacetime and the standard hot Big Bang picture of
cosmology, including the Cosmic Microwave Background.
 Section III explains unresolved issues in the standard cosmology, in particular the horizon and
atness problems.
 Section IV introduces in
ation in scalar eld theories.
 Section V discusses quantum
uctuations in in
ation and the generation of cosmological perturbations.
 Section VI discusses the observational predictions of in
ation, and current constraints from Cosmic Microwave
Background measurements.
 Section VII discusses conclusions and the future outlook for in
ationary physics.
 Appendix A describes in detail the generation of density perturbations during in
ation.
Electronic address: whkinney@bualo.edu
1 In
ation in its current form was introduced by Guth, but similar ideas had been discussed before [4, 5]. A short history of the early
development of in
ation can be found in Ref. [6].
ar
X
iv
:0
90
2.
15
29
v2
[
as
tro
-p
h.C
O]
1
6 J
un
20
09

2The lectures are at an advanced undergraduate or beginning graduate student level. Most of the the lectures should
be accessible with only a background in Special Relativity. A working knowledge of General Relativity and quantum
eld theory are helpful for Sections IV and V and for Appendix A. Where possible, I reference review articles for
further reading on related topics. For other reviews on in
ation, see Refs. [6, 14, 15, 16, 17, 18, 19, 20].
A. The Metric
The fundamental object in General Relativity is the metric, which encodes the shape of the spacetime. A metric is
a symmetric, bilinear form which denes distances on a manifold. For example, we can express Pythagoras' theorem
in a Euclidean three-dimensional space,
`2 = x2 + y2 + z2; (1)
as a matrix product over the identity matrix ij = diag (1; 1; 1),
`2 =
X
i;j=1;3
ijx
ixj : (2)
Therefore the identity matrix ij can be identied as the metric for the Euclidean space: if we wish to describe a
non-Euclidean manifold, we replace ij with a more complicated matrix gij , which in general can depend on the
coordinates xi. For an arbitrary path through the space, we express distances on the manifold in dierential form,
d`2 =
X
i;j
gijdx
idxj : (3)
The distance along any path in the spacetime, or world line, is then given by integrating d` along that path. A
familiar example of a non-Euclidean space frequently used in physics is the Minkowski Space describing spacetime in
Special Relativity. Distances along a world line in Minkowski Space are measured by the proper time, which is the
time as measured by an observer traveling on that world line. The proper time s along a world line is given by the
relation
ds2 = dt2 dx2
=
X
;=0;3
dx
dx ; (4)
where we take the speed of light c = 1. We express four-vectors as ~x = (t; x; y; z) = (x0; x1; x2; x3), and dx2 =
dx2 + dy2 + dz2 is the Euclidean distance along a spatial interval. The metric  for Minkowski Space is given by
 =
0
B
@
1
1
1
1
1
C
A : (5)
Anything traveling the speed of light has velocity d jxj =dt = 1. Photons therefore always travel along world lines
of zero proper time, ds2 = dt2 dx2 = 0, called null geodesics. Massive particles travel along world lines with real
proper time, ds2 > 0, called timelike geodesics. Causally disconnected regions of spacetime are separated by spacelike
intervals, with ds2 < 0. The set of all null geodesics passing through a given point (or event) in spacetime is called
the light cone (Fig. 1) The interior of the light cone, consisting of all null and timelike geodesics, denes the region
of spacetime causally related to that event.
B. General Relativity and the Einstein Field Equation
The Minkowski metric  of Special Relativity describes a Euclidean spacetime which is static, empty, and innite
in space and time. The addition of gravity to the picture requires General Relativity, which describes gravitational
elds as curvature in the spacetime. The fundamental object in General Relativity is the metric g (~x), which
describes the shape of the spacetime and in general depends on the spacetime coordinate ~x. As in Minkowski Space,

3FIG. 1: Light cones in Minkowski Space. The past light cone denes the causal past of the event P , and the future light cone
denes the causal future of P .
lengths in curved spacetime are measured by the proper time s, with the proper time along a world line determined
by the metric
ds2 =
X
;=0;3
g (~x) dx
dx : (6)
As in Special Relativity, photons travel along null geodesics, with ds2 = 0, and massive particles travel along timelike
geodesics, with ds2 > 0. However, unlike Special Relativity, null geodesics need not always be 45 lines dening light
cones, but can be curved by gravity.
In General Relativity, the distribution of mass/energy in the spacetime determines the shape of the metric, and the
metric in turn determines the evolution of the mass/energy. Electromagnetism provides a convenient analogy: in elec-
tromagnetism, the distribution of charges and currents determines the electromagnetic eld, and the electromagnetic
eld in turn determines the evolution of the charges and currents. Given a current four-vector J, Maxwell's Equations
are a set of linear, rst-order partial dierential equations that allow us to calculate the resulting electromagnetic eld
@F
 
X
=0;3
@F
 =
4
c
J: (7)
Here we have explicitly included the speed of light c to highlight its role as an electromagnetic coupling constant.
We also adopt the typical summation convention for relativity: repeated indices are implicitly summed over. In
General Relativity, we describe the distribution of mass/energy in a covariant way by specifying a symmetric rank-2
stress-energy tensor T , which acts as a source for the gravitational eld similar to the way the current four-vector J
sources electromagnetism. The analog of Maxwell's Equations is the Einstein Field Equation, which can be written
in the deceptively simple form
G = 8GT ; (8)

4where the coupling constant is Newton's gravitational constant G. The tensor G , called the Einstein Tensor, is a
symmetric 4 4 tensor consisting of the metric g and its rst and second derivatives. The Einstein Field Equation
therefore represents a set of ten coupled, nonlinear, second-order partial dierential equations of ten free functions,
which are the elements of the metric tensor g . However, only six of these equations are actually indpendent, leaving
four degrees of freedom. The physics of gravity is independent of coordinate system, and the additional degrees of
freedom correspond to a choice of a coordinate system, or gauge on the four-dimensional space. Gravity is much more
complicated than electromagnetism! As with any intractably complicated problem, we simplify the job by introducing
a symmetry. In General Relativity there are a number of symmetries which allow either exact or perturbative solution
to the Einstein Field Equations:
 Vacuum: T = 0. If we evaluate the Einstein Field Equations for small perturbations about an empty Minkowski
Space, we nd that they reduce at lowest order to a wave equation, and therefore General Relativity predicts
the existence of gravity waves.
 Spherical Symmetry. If we assume a spherically symmetric spacetime (also empty of matter, T = 0) the
Einstein Field Equation can be solved exactly, resulting in the Schwarzschild solution for black holes.
 Homogeneity and Isotropy. If we assume that the stress-energy is distributed in a fashion which is homogeneous
and isotropic, this is called a Friedmann-Robertson-Walker (FRW) space, and is the case of interest for cosmology.
Since the homogeneity and isotropy remove all spatial dependence, the Einstein Field Equations reduce from a
set of partial dierential equations to a set of nonlinear ordinary dierential equations in time. For particular
types of homogeneous, isotropic matter, these equations can be solved exactly, and perturbations about those
exact solutions can be handled self-consistently.
Continuing the analogy with electromagnetism, the equivalent of charge conservation,
@J
 =
@
@t
+r
j = 0; (9)
in General Relativity is stress-energy conservation
DT
 = 0; (10)
where D represents a covariant derivative, which is a generalization of the partial derivative to a curved manifold.
We will also denote covariant derivatives with a semicolon, for example T; = 0. Likewise, simple partial derivatives
are denoted with a comma, @f=@x  @f  f;. As in the case of electromagnetism, where the charge conservation
equation is not independent, but is instead a consequence of Maxwell's Equations, stress-energy conservation in General
Relativity is a consequence of the Einstein Field Equations and does not independently constrain the solutions. In
the next section, we discuss FRW spaces and their application to cosmology in more detail.
II. FRIEDMANN-ROBERTSON-WALKER SPACETIMES
A. The Friedmann Equation
A homogeneous space is one which is translationally invariant, or the same at every point. An isotropic space is one
which is rotationally invariant, or the same in every direction. The two are not the same: a space which is everywhere
isotropic is necessarily homogeneous, but a space which is homogeneous is not necessarily isotropic. (Consider, for
example a space with a uniform electric eld: it is translationally invariant but not rotationally invariant.) It is possible
to show [21] that the most general metric consistent with homogeneity and isotropy is obtained by multiplying a static
spatial geometry with a time-dependent scale factor a(t):
ds2 = dt2 a2 (t) dx2
= dt2 a2 (t)

dr2
1 kr2
+ r2d
2

; (11)
where we have expressed the spatial line element in terms of spherical coordinates r,,, and the solid angle is given
by the usual d
2 = sin dd. The constant k denes the curvature of the spacetime, with k = 0 corresponding
to
at (Euclidean) spatial sections, and k = 1 corresponding to positive and negative curvatures, respectively. A

5spacetime of this general form is called a Friedmann-Robertson-Walker (FRW) spacetime. Likewise, the most general
homogeneous, isotropic stress-energy is diagonal, with all of its spatial components identical,
T =
0
B
@
 (t)
p (t)
p (t)
p (t)
1
C
A ; (12)
where we identify the energy density  and the pressure p from the continuity equation arising from stress-energy
conservation,
T; = _+ 3

_a
a

(+ p) = 0: (13)
The Einstein eld equations then reduce to a set of two coupled, non-linear ordinary dierential equations,

_a
a
2
+
k
a2
=
8
3m2Pl
;

a
a

=
4
3m2Pl
(+ 3p) : (14)
The rst is called the Friedmann Equation, and the second is called the Raychaudhuri Equation. Note that the
equations for the evolution of the scale factor depend not only on the energy density , but also the pressure p:
pressure gravitates! The continuity equation (13) is not independent of the Einstein Field Equations (14), but can
be derived directly from the Friedmann and Raychaudhuri Equations. The expansion rate _a=a is called the Hubble
parameter H:
H 
_a
a
; (15)
and has units of inverse time. A positive Hubble parameter H > 0 corresponds to an expanding universe, and a
negative Hubble parameter H < 0 corresponds to a collapsing universe. (Since our actual universe is expanding, we
will specialize to that case.) Minkowski Space can be recovered by assuming a
at geometry k = 0, and no expansion,
_a = 0. The Hubble parameter sets the fundamental scale of the spacetime, i.e. a characteristic time is the Hubble
time t  H1, and likewise the Hubble length is d  H1. We will see later that the Hubble time sets the scale for
the age of the universe, and the Hubble length sets the scale for the size of the observable universe.
The coordinate system (t;x) is called a comoving coordinate system, because observers with constant comoving
coordinates are at rest relative to the expansion, i.e. two observers with constant separation in comoving coordinates

x have a physical, or proper, separation which increases in proportion to the scale factor

xprop = a (t)
xcom: (16)
An important kinematic eect of cosmological expansion is the phenomenon of cosmological redshift: we will see later
that solutions to the wave equation in an FRW space have constant wavelength in comoving coordinates, so that the
proper wavelength of (for example) a photon increases in time in proportion to the scale factor
 / a (t) : (17)
For a photon emitted at time tem and detected at time t0, the redshift z is dened by:
(1 + z) 
0
em
=
a (t0)
a (tem)
: (18)
(Here we introduce the convention used frequently in cosmology that a subscript 0 refers to the current time, not
an initial time.) Note that the cosmological redshift is not a Doppler shift caused by the relative velocity of the
source and detector, but is an expansion eect: the wavelength of a photon traveling through the spacetime increases
because the underlying spacetime is expanding. Another way to look at this is that a photon traveling through an
FRW spacetime loses momentum with time,
p = h / a1(t): (19)

6By the equivalence principle, this momentum loss must apply to massive particles as well as photons: any particle
moving in an expanding FRW spacetime will lose momentum as p / a1. For massless particles like photons, this is
manifest as a redshift in the wavelength, but it means that a massive particle will asymptotically come to rest relative
to the comoving coordinate system. Thus, comoving coordinates represent a preferred reference frame reminiscent of
Aristotelian physics: any free body with a \peculiar" velocity relative to the comoving frame will eventually come to
rest in that frame.
There are three possibilities for the curvature of the universe:
at (k = 0), positively curved (k = +1), or negatively
curved (k = 1). The current value of the Hubble parameter is (from the Hubble Space Telescope Key Project [22]),
H0 = 72 8 km=s=Mpc: (20)
Therefore, we can see from the Friedmann Equation (14) that, given the expansion rate H, the curvature is determined
by the density:
k = a2

8
3m2Pl
H2

: (21)
Note that only the sign of k is physically important, since any rescaling of k is equivalent to a rescaling of the scale
factor a. We dene the critical density as the density for which k = 0, corresponding to a geometrically
at universe,
c 
3m2Pl
8
H2 ) k = 0: (22)
For  > c, the universe is positively curved and closed, with nite volume, and for  < c, the universe is negatively
curved and open, with innite volume. We express the ratio of the actual density  to the critical density c as the
parameter
:




c

=
8
3m2Pl

H2
: (23)
(Do not confuse the density parameter
with the solid angle d
in Eq. 11!) Table I summarizes the relation between
density, curvature, and geometry. The density parameter
is not in general constant in time, and we can re-write
TABLE I: Cosmological density and curvature
density curvature geometry

= 1 k = 0
at

> 1 k = 1 closed

< 1 k = 0 open
the Friedmann Equation as

(t) = 1 +
k
(aH)2
: (24)
Since the Hubble parameter is proportional to the inverse time H / t1, we see that the time-dependence of
is
determined by the time dependence of the scale factor a (t). In the next section, we tackle the problem of solving for
a (t).
B. Solving the Friedmann Equation
In the previous section, we considered the form and kinematics of FRW spaces, but not the dynamics, that is, how
does the stress-energy of the universe determine the expansion history? The answer to this question depends on what
kind of matter dominates the cosmological stress-energy. In this section, we consider three basic types of cosmological
stress-energy: matter, radiation, and vacuum.
The simplest kind of cosmological stress-energy is generically referred to as matter. Imagine a comoving box with
sides of length L. By comoving box, we mean a box whose corners are at rest in a comoving coordinate system,
and whose proper dimension is therefore increasing proportional to the scale factor, Lprop / a. That is, the box is

7FIG. 2: A comoving box full of matter. The energy density in matter scales inversely with the volume of the box.
FIG. 3: A comoving box full of radiation. The number density of photons scales inversely with the volume of the box, but the
photons also increase in wavelength.
growing with the expansion of the universe. Now imagine the box lled with N particles of mass m, also at rest in the
comoving reference frame (Fig. 2). In units where c = 1, the relativistic energy density of such a system of particles
is given by
m =
MN
V
; (25)
FIG. 4: A comoving box full of vacuum. The energy density of vacuum does not scale at all!

8where V is the proper volume of the box, V = L3prop / a
3. Since neither M nor N change with expansion, we have
immediately that
m =
MN
L3a3
/ a3; (26)
where L is the comoving size of the box. So the proper energy density of massive particles at rest in a comoving
volume evolves as the inverse cube of the scale factor. Now imagine the same box lled with N photons with frequency
 (Fig. 3). The energy per photon is h, so that the energy density in the box is then

=
Nh
V
: (27)
As in the case of massive particles, the number density of photons in the box redshifts inversely with the proper
volume of the box n = N=V / a3. But each photon also loses energy through cosmological redshift,  / a1 (19),
so that the total energy density in photons or other massless degrees of freedom, which we generically refer to as
radiation, redshifts as

/ a
4: (28)
Note also that cosmological redshift immediately gives us a rule for the behavior of a black-body spectrum of radiation
with temperature T . Since all photons redshift at exactly the same rate, a system with starts out as a black-body
stays a black-body, with a temperature that decreases with expansion,
T
/ a
1: (29)
The third type of stress-energy which is important in cosmology dates back to Einstein's introduction of a \cosmo-
logical constant" to his eld equations. If we take the stress-energy T and add a term proportional to the metric,
the identity Dg = 0 means the stress-energy conservation equation (10) is unchanged:
DT
 ! D (T
 + g) = 0: (30)
In our analogy with electromagnetism, this is like adding a constant to the electromagnetic potential, V 0(x) = V (x)+.
The constant  does not aect local dynamics in any way, but it does aect the cosmology. From Eq. (12), stress-
energy of the form T = g corresponds to an equation of state
p = : (31)
The continuity equation (13) then reduces to
_+ 3

_a
a

(+ p) = _ = 0; (32)
so that vacuum has a constant energy density,  = const: A cosmological constant is also frequently referred to
as vacuum energy, since it is as if we are assigning an energy density to empty space. With this interpretation, a
comoving box full of vacuum contains a total amount of energy which grows with the expansion of the universe (Fig.
4). This highlights the curious property of General Relativity that, while energy is conserved in a local sense, it is
not conserved globally. We are creating energy out of nothing!
It is straightforward to solve the Einstein Field Equations for the three basic types of stress-energy. Consider rst
a matter-dominated universe. We can write the time derivative of the energy density as:
m / a
3 ) _m = 3

_a
a

: (33)
From the continuity equation (13), we have
_+ 3

_a
a

(+ p) = 3

_a
a

p = 0: (34)
We then have that the pressure of matter vanishes, pm = 0. The matter-dominated Friedmann Equation becomes

_a
a
2
+
k
a2
=
8
3m2Pl
 / a3: (35)

9FIG. 5: Schematic diagram of how the three types of stress-energy scale with redshift 1 + z / a: at early time, radiation
dominates, followed by matter, and nally the universe is dominated by vacuum energy.
In the case of a
at universe, k = 0, the solution is especially simple:

_a
a
2
/ a3 ) a (t) / t2=3: (36)
Similarly, for a radiation dominated universe, the continuity equation implies that

/ a
4 ) p
= 
=3: (37)
Again assuming a
at geometry,

_a
a
2
/ a4 ) a (t) / t1=2: (38)
Finally, solving the the Friedmann Equation for the vacuum case gives

_a
a
2
/  = const: ) a (t) / e
Ht; (39)
so that the universe expands exponentially quickly, with a time constant given by the Hubble parameter
H =
s
8
3mPl
2
 = const: (40)
Such a spacetime is called de Sitter space.
Note in particular that the energy density in radiation redshifts away more quickly than the energy density in
matter, and vacuum energy does not redshift at all, so that a universe with a mix of radiation, matter and vacuum
will be radiation-dominated at early times, matter-dominated at later times, and eventually vacuum-dominated (Fig.
5). Note also that for either matter- or radiation-domination, the universe is singular as t! 0: the universe has nite
age! Since the scale factor vanishes at t = 0, and the density scales as an inverse power of a, the initial singularity
consists of innite density. Likewise, since temperature also scales inversely with a, the initial singularity is also a
point of innite temperature. We therefore arrive at the standard hot Big Bang picture of the universe: a cosmological
singularity at nite time in the past, followed by a hot, radiation-dominated expansion, during which the universe
gradually cools as T / a1 and the radiation dilutes, followed by a period of matter-dominated expansion during

11
FIG. 6: Schematic diagram of recombination. At early time, the temperature of the universe is above the ionization energy of
hydrogen and helium, so that the universe is full of an ionized plasma, and the mean free path for photons is short compared
to the Hubble length. At late time, the temperature drops and the nuclei capture the electrons and form neutral atoms. Once
this happens, the universe becomes transparent to photons, which free stream from the surface of last scattering.
The most recent result from the WMAP satellite gives
bh2 = 0:022730:00062 [29]. Recombination happens quickly
(i.e., in much less than a Hubble time t  H1), but it is not instantaneous. The universe goes from a completely
ionized state to a neutral state over a range of redshifts
z  200. If we dene recombination as an ionization fraction
Xe = 0:1, we have that the temperature at recombination TR = 0:3 eV.
What happens to the photons after recombination? Once the gas in the universe is in a neutral state, the mean free
path for a photon becomes much larger than the Hubble length. The universe is then full of a background of freely
propagating photons with a blackbody distribution of frequencies. At the time of recombination, the background
radiation has a temperature of T = TR = 3000 K, and as the universe expands the photons redshift, so that the
temperature of the photons drops with the increase of the scale factor, T / a(t)1. We can detect these photons
today. Looking at the sky, this background of photons comes to us evenly from all directions, with an observed
temperature of T0 ' 2:73 K. This allows us to determine the redshift of recombination,
1 + zR =
a (t0)
a (tR)
=
TR
T0
' 1100: (45)
This is the cosmic microwave background. Since by looking at higher and higher redshift objects, we are looking
further and further back in time, we can view the observation of CMB photons as imaging a uniform \surface of last
scattering" at a redshift of 1100 (Fig. 7).
To the extent that recombination happens at the same time and in the same way everywhere, the CMB will be
of precisely uniform temperature. While the observed CMB is highly isotropic, it is not perfectly so. The largest
contribution to the anisotropy of the CMB as seen from earth is simply Doppler shift due to the earth's motion through
space. (Put more technically, the motion is the earth's motion relative to a comoving cosmological reference frame.)
CMB photons are slightly blueshifted in the direction of our motion and slightly redshifted opposite the direction of
our motion. This blueshift/redshift shifts the temperature of the CMB so the eect has the characteristic form of a
\dipole" temperature anisotropy (Fig. 8). The dipole anisotropy, however, is a local phenomenon. Any intrinsic, or
primordial, anisotropy of the CMB is potentially of much greater cosmological interest. To describe the anisotropy of
the CMB, we remember that the surface of last scattering appears to us as a spherical surface at a redshift of 1100.
Therefore the natural parameters to use to describe the anisotropy of the CMB sky is as an expansion in spherical

12
FIG. 7: Cartoon of the last scattering surface. From earth, we see blackbody radiation emitted uniformly from all directions,
forming a \sphere" at redshift z = 1100.
harmonics Y`m:

T
T
=
1X
`=1
X`
m=`
a`mY`m (; ): (46)
If we assume isotropy, there is no preferred direction in the universe, and we expect the physics to be independent of
the index m. We can then dene
C` 
1
2`+ 1
X
m
ja`mj
2: (47)
The ` = 1 contribution is just the dipole anisotropy,


T
T

`=1
 103: (48)
The dipole was rst measured in the 1970's by several groups [30, 31, 32]. It was not until more than a decade after
the discovery of the dipole anisotropy that the rst observation was made of anisotropy for `  2, by the dierential
microwave radiometer aboard the Cosmic Background Explorer (COBE) satellite [33], launched in in 1990. COBE
observed that the anisotropy at the quadrupole and higher ` was two orders of magnitude smaller than the dipole:


T
T

`>1
' 105: (49)
Fig. 8 shows the dipole and higher-order CMB anisotropy as measured by COBE. This anisotropy represents
intrinsic
uctuations in the CMB itself, due to the presence of tiny primordial density
uctuations in the cosmological
matter present at the time of recombination. These density
uctuations are of great physical interest, since these
are the
uctuations which later collapsed to form all of the structure in the universe, from superclusters to planets
to graduate students. While the physics of recombination in the homogeneous case is quite simple, the presence of
inhomogeneities in the universe makes the situation much more complicated. I describe some of the major eects in
a qualitative way here, and refer the reader to the literature for a more detailed technical explanation of the relevant
physics [23, 24, 25, 26, 27]. In these lectures, I primarily focus on the current status of the CMB as a probe of
in
ation, but there is much more to the story.

13
FIG. 8: The COBE measurement of the CMB anisotropy [33]. The top oval is a map of the sky showing the dipole anisotropy

T=T  103. The bottom oval is a similar map with the dipole contribution and emission from our own galaxy subtracted,
showing the anisotropy for ` > 1,
T=T  105. (Figure courtesy of the COBE Science Working Group.)
FIG. 9: The WMAP measurement of the CMB anisotropy [34]. (Figure courtesy of the WMAP Science Working Group.)
WMAP measured the anisotropy with much higher sensitivity and resolution than COBE.

14
The simplest contribution to the CMB anisotropy from density
uctuations is just a gravitational redshift, known
as the Sachs-Wolfe eect [35]. A photon coming from a region which is slightly denser than the average will have
a slightly larger redshift due to the deeper gravitational well at the surface of last scattering. Conversely, a photon
coming from an underdense region will have a slightly smaller redshift. Thus we can calculate the CMB temperature
anisotropy due to the slightly varying Newtonian potential  from density
uctuations at the surface of last scattering:
T
T
=
1
3
[em obs] ; (50)
where em is the potential at the point the photon was emitted on the surface of last scattering, and obs is the
potential at the point of observation, which can be treated as a constant. The factor 1=3 is a General Relativistic
correction. This simple kinematic contribution to the CMB anisotropy is dominant on large angular scales, corre-
sponding to multipoles ` < 100. However, the amount of information we can gain from these multipoles is limited
by an intrinsic source of error called cosmic variance. Cosmic variance is a result of the statistical nature of the
primordial power spectra: since we have only one universe to measure, we have only one realization of the random
eld of density perturbations, and therefore there is an inescapable 1=
p
N uncertainty in our ability to reconstruct
the primordial power spectrum, where N is the number of independent wave modes which will t inside the horizon
of the universe! On very large angular scales, this problem becomes acute, and we can write the cosmic variance error
on any given C` as

C`
C`
=
1
p
2`+ 1
; (51)
which comes from the fact that any C` is represented by 2`+1 independent amplitudes a`m. Even a perfect observation
of the CMB can only approximately measure the true power spectrum | the errors in the WMAP data, for example,
are dominated by cosmic variance out to `  400 (Fig. 10).
For
uctuation modes on smaller angular scales, more complicated physics comes into play. The dominant process
that occurs on short wavelengths is acoustic oscillations in the baryon/photon plasma. The idea is simple: matter
tends to collapse due to gravity onto regions where the density is higher than average, so the baryons \fall" into
overdense regions. However, since the baryons and the photons are still strongly coupled, the photons tend to
resist this collapse and push the baryons outward. The result is \ringing", or oscillatory modes of compression and
rarefaction in the gas due to density
uctuations. The gas heats as it compresses and cools as it expands, which
creates
uctuations in the temperature of the CMB. This manifests itself in the C` spectrum as a series of peaks and
valleys (Fig. 10). The specic shape and location of the acoustic peaks is created by complicated but well-understood
physics, involving a large number of cosmological parameters. The presence of acoustic peaks in the CMB was rst
suggested by Sakharov [36], and later calculated by Sunyaev and Zel'dovich [37, 38] and Peebles and Yu [39]. The
complete linear theory of CMB
uctuations was worked out by Ma and Bertschinger in 1995 [40]. The shape of the
CMB multipole spectrum depends, for example, on the baryon density
b, the Hubble constant H0, the densities
of matter
m and cosmological constant
, the amplitude of primordial gravitational waves, and the redshift zri
at which the rst generation of stars ionized the intergalactic medium. This makes interpretation of the spectrum
something of a complex undertaking, but it also makes it a sensitive probe of cosmological models.
In addition to anisotropy in the temperature of the CMB, the photons coming from the surface of last scattering
are expected to be weakly polarized due to the presence of perturbations [41, 42]. This polarization is much less
well measured than the temperature anisotropy, but it has been detected by WMAP and by a number of ground-
and balloon-based measurements [43, 44, 45, 46, 47]. Measurement of polarization promises to greatly increase the
amount of information it is possible to extract from the CMB. Of particular interest is the odd-parity, or B-mode
component of the polarization, the only primordial source of which is gravitational waves, and thus provides a clean
signal for detection of these perturbations. The B-mode has yet to be detected by any measurement.
III. THE FLATNESS AND HORIZON PROBLEMS
We have so far considered two types of cosmological mass-energy { matter and radiation { and solved the Friedmann
Equation for the simple case of a
at universe. What about the more general case? In this section, we consider non-
at
universes with general contents. We introduce two related questions which are not explained by the standard Big
Bang cosmology: why is the universe so close to
at today, and why is it so large?
We can describe a general homogeneous, isotropic mass-energy by its equation of state
p = w; (52)

15
FIG. 10: The C` spectrum for the CMB as measured by WMAP, showing the peaks characteristic of acoustic oscillations. The
gray shaded region represents the uncertainty due to cosmic variance. (Figure courtesy of the WMAP Science Working Group.)
so that pressureless matter corresponds to w = 0, and radiation corresponds to w = 1=3. We will consider only the
case of constant equation of state, w = const: From the continuity equation, we have
_+ 3 (1 + w)
_a
a
 = 0; (53)
with solution
 / a3(1+w): (54)
The Friedmann Equation for a
at universe is then

_a
a
2
/ a3(1+w); (55)
so that the scale factor increases as a power-law in time,
a (t) / t2=3(1+w): (56)
What about the evolution of a non-
at universe? Analytic solutions for a(t) in the k 6= 0 case can be found in
cosmology textbooks. For our purposes, it is sucient to consider the time-dependence of the density parameter
.
From Eqs. (14, 23, 24) it is not too dicult to show that the density parameter evolves with the scale factor a as:
d

d ln a
= (1 + 3w)
(
1) : (57)
Proof is left as an exercise for the reader. Note that a
at universe,
= 1 remains
at at all times, but in a non-
at
universe, the density parameter
is a time-dependent quantity, with the evolution determined by the equation of
state parameter w. For matter (w = 0) or radiation (w = 1=3), the prefactor in Eq. (57) is positive,
1 + 3w > 0; (58)
which means a
at universe is an unstable xed point:
d j
1j
d ln a
> 0; (1 + 3w) > 0: (59)

16
Any deviation from a
at geometry is amplied by the subsequent cosmological expansion, so a nearly
at universe
today is a highly ne-tuned situation. The WMAP5 CMB measurement tells us the universe is
at to within a few
percent, j
0 1j < 0:02 [29, 48]. If we are very conservative and take a limit on the density today as
0 = 1 0:05,
that means that at recombination, when the CMB was emitted,
rec = 1  0:0004, and at the time of primordial
nucleosynthesis,
nuc = 1 1012. Why did the universe start out so incredibly close to
at? The standard Big Bang
cosmology provides no answer to this question, which we call the
atness problem.
There is a second, related problem with the standard Big Bang picture, arising from the nite age of the universe.
Because the universe has a nite age, photons can only have traveled a nite distance in the time since the Big Bang.
Therefore, the universe has a horizon: the further out in space we look, the further back in time we see. If we could
look far enough out in any direction, past the surface of last scattering, we would be able to see the Big Bang itself,
and beyond that we can see no further. Every observer in an FRW spacetime sees herself at the center of a spherical
horizon which denes her observable universe. To calculate the size of our horizon, we use the fact that photons travel
on paths of zero proper length:
ds2 = dt2 a2 (t) jdxj2 = 0; (60)
so that the comoving distance jdxj traversed by a photon in time dt is
jdxj =
dt
a (t)
: (61)
Therefore, the size of the cosmological horizon at time t after the Big Bang is
dH (t) =
Z t
0
dt0
a (t0)
: (62)
To convert comoving length to proper length, we just multiply by a (t), so that the proper horizon size is
dpropH (t) = a (t) d
com
H (t) : (63)
Normalizing a (t0) = 1, the horizon size of a 14-billion year-old
at, matter-dominated universe is dH = 3t0  13 Gpc.
To see why the presence of a horizon is a problem for the standard Big Bang, we examine the causal structure of
an FRW universe. Take the FRW metric
ds2 = dt2 a2 (t) jdxj2 ; (64)
and re-write it in terms of a redened clock, the conformal time  :
ds2 = a2 ()
h
d2 jdxj2
i
: (65)
Conformal time is a \clock" which slows down with the expansion of the universe,
d =
dt
a (t)
; (66)
so that the comoving horizon size is just the age of the universe in conformal time
dH (t) =
Z t
0
dt0
a (t0)
=
Z 
0
d 0 = : (67)
The conformal metric is useful because the expansion of the spacetime is factored into a static metric multiplied by a
time-dependent conformal factor a (), so that photon geodesics are simply described by d jxj = d . In a diagram of 
versus jxj, photons travel on 45 angles. (Note that this is true even for curved spacetimes!) We can draw light cones
and infer causal relationships with the expansion factored out, in a manner identical to the usual case of Minkowski
Space.
There is one major dierence between FRW and Minkowski: an FRW spacetime has a nite age. Therefore, unlike
the case of Minkowski Space, which has an innite past, an FRW spacetime is \chopped o" at some nite past time
 = 0 (Fig.11). The initial singularity is a surface of constant conformal time, and it is easy to see from Eq. (67) that
our horizon size is the width of our past light cone projected on the surface dened by the initial singularity. This is
a very dierent picture from the notion many people (even scientists) have of the Big Bang, which is something akin
to an explosion, with the universe initially a cosmic \egg" of zero size. On the contrary, in the case of a
at or open

17
FIG. 11: A conformal diagram of a Friedmann-Robertson-Walker space. The FRW space is causally identical to Minkowski
Space, except that it is not past-innite, so that past light cones are \cut o" at the Big Bang, which is a spatially innite
surface at time t = 0.
universe, the universe is spatially innite an innitesimal amount of time after the initial singularity: the Big Bang
happens everywhere at once in an innite space! Our observable universe is nite because we can only see a small
patch of the much larger cosmos.3 Closed universes are spatially nite, but are still much larger in extent than our
observable patch. The key point is that two events on the conformal spacetime diagram are causally connected only
if they share a causal past: that is, if their past light cones overlap.
Consider two points on the CMB sky 180 degrees apart (Fig. 12). Their past light cones do not overlap, and the
two points are causally disconnected. Those two points on the surface of last scattering occupy completely separate,
disconnected observable universes. How did these points reach the observed thermal equilibrium to a few parts in
105 if they never shared a causal past? This apparent paradox is called the horizon problem: the universe somehow
reached nearly perfect equilibrium on scales much larger than the size of any local horizon. From the Friedmann
Equation, it is easy to show that the horizon problem and the
atness problem are related: consider a comoving
length scale . It is is easy to show that for w = const:, the ratio of  to the horizon size dH is related to the curvature
by a conservation law


dH
2
j
1j = const: (68)
Proof is left as an exercise for the reader. Therefore, for a universe evolving away from
atness,
d j
1j
d ln a
> 0; (69)
3 Of course, this is an idealization, and the actual universe could well have a nontrivial global topology, even if it is locally
at, as long
as the scale of the overall manifold is much larger than our horizon size [49, 50].

18
FIG. 12: A conformal diagram of the Cosmic Microwave Background. Two points on opposite sides of the sky are causally
separate, since their past light cones do not intersect.
the horizon size gets bigger in comoving units
d
d ln a


dH

< 0: (70)
That is, more and more space \falls into" the horizon, or becomes causally connected, at late times.
What would be required to have a universe which evolves toward
atness, rather than away from it? From Eq.
(57), we see that having 1 + 3w negative will do the trick,
d j
1j
d ln a
< 0; (1 + 3w) < 0: (71)
Therefore, if the energy density of the universe is dominated not by matter or radiation, but by something with
suciently negative pressure, p < =3, a curved universe will become
atter with time. From the Raychaudhuri
Equation (14), we see that the case of p < =3 is exactly equivalent to an accelerating expansion:
a
a
/ (1 + 3w) > 0; (1 + 3w) < 0: (72)
If the expansion of the universe is slowing down, as is the case for matter- or radiation-domination, the curvature
evolves away from
atness. But if the expansion is speeding up, the universe gets
atter. From Eq. (68), we see that
this negative pressure solution also solves the horizon problem, since accelerating expansion means that the horizon
size is shrinking in comoving units:
d
d ln a


dH

> 0; (1 + 3w) < 0: (73)
When the expansion accelerates, distances initially smaller than the horizon size are \redshifted" to scales larger than
the horizon at late times. Accelerating cosmological expansion is called in
ation.

19
The simplest example of an accelerating expansion from a negative pressure
uid is the case of vacuum energy we
considered in Section (II B), for which the scale factor increases exponentially,
a / eHt: (74)
For such expansion, the universe is driven exponentially toward a
at geometry,
d ln

d ln a
= 2 (1
) : (75)
We can see that the horizon problem is also solved by looking at the conformal time:
d =
dt
a (t)
= eHtdt; (76)
so that
 =
1
H
eHt =
1
aH
: (77)
The conformal time during the in
ationary period is negative, tending toward zero at late time. Therefore, if we have
a period of in
ationary expansion prior to the early epoch of radiation-dominated expansion, in
ation takes place in
negative conformal time, and conformal time  = 0 represents not the initial singularity but the transition from the
in
ationary expansion to radiation domination. The initial singularity is pushed back into negative conformal time,
and can be pushed arbitrarily far depending on the duration of in
ation. Figure 13 shows the causal structure of an
in
ationary spacetime. The past light cones of two points on the CMB sky do not intersect at  = 0, but in
ation
provides a \sea" of negative conformal time, which allows those points to share a causal past. In this way, in
ation
solves the horizon problem.
In more realistic models of in
ation in the early universe, the energy density is approximately, but not exactly,
constant, and the expansion is approximately, but not exactly, exponential. In such quasi-de Sitter spaces, the quali-
tative picture above still holds, and in
ation provides a clean and compelling explanation for the peculiar boundary
conditions for our universe. In the next section, we discuss how to construct more detailed models of in
ation in eld
theory.
IV. INFLATION FROM SCALAR FIELDS
The example of de Sitter evolution we considered in Section III gives a good qualitative picture of how in
ation,
or accelerated expansion, solves the horizon and
atness problems of the standard Big Bang cosmology. However,
this leaves open the question: what physics is responsible for the accelerated expansion at early times? It cannot
be Einstein's cosmological constant, simply because a universe dominated by vacuum energy stays dominated by
vacuum energy for the innite future, since in a de Sitter background matter ( / a3) and radiation ( / a4) are
diluted exponentially quickly. Therefore, we will never reach a radiation-dominated phase, and we will never see a
hot Big Bang. In order to transition from an in
ating phase to a thermal equilibrium, radiation-dominated phase,
the vacuum-like energy during in
ation must be time-dependent. We model this dynamics with a scalar eld , for
which we assume the following action:
S =
Z
d4x
p
gL; (78)
where g  Det (g) is the determinant of the metric and the Lagrangian for the eld  is
L =
1
2
g@@ V () : (79)
Comparing the action (78) and the Lagrangian (79) with their Minkowski counterparts illustrates how we generalize
a classical eld theory to curved spacetime:
SMinkowski =
Z
d4x

1
2
@@ V ()

: (80)
The metric appears in two places in the curved-spacetime action: First, it appears in the measure of volume in
the four-space, d4x, where the determinant of the metric takes the role of the Jacobian for arbitrary coordinate

20
FIG. 13: A conformal diagram of light cones in an in
ationary universe. In
ation ends in reheating at conformal time  = 0,
which is the onset of the radiation-dominated expansion of the hot Big Bang. However, in
ation provides a \sea" of negative
conformal time, which allows the past light cones of events at the last scattering surface to overlap.
transformations, x ! x0. Second, the metric appears in the kinetic term for the scalar eld, where we replace the
Minkowski metric  with the general metric g .
The action (78) is not the most general assumption we could make, as we can see by writing the full action including
gravity,
Stot =
Z
d4x
p
g

m2Pl
16
R+ L

: (81)
Here R is the Ricci Scalar, composed of the metric and its derivatives. Variation of the rst term in the action
results in the Einstein Field Equation (8). Such a minimally coupled theory assumes that there is no direct coupling
between the eld and the metric, which would be represented in a more general action by terms which mix R and
. In practice, many such non-minimally coupled theories can be transformed to a minimally coupled form by a eld
redenition. We could also write a more general theory by modifying the scalar eld Lagrangian (79) to contain
non-canonical kinetic terms,
L = F (; g
@@) V () : (82)
where F () is some function of the eld and its derivatives. Such Lagrangians appear frequently in models of in
ation
based on string theory, and are a topic of considerable current research interest. We could also complicate the
gravitational sector by replacing the Ricci scalar R with a more complicated function f (R). An example of such a
model is the in
ation model of Starobinsky [51], which can be reduced to the form (78) through a eld redenition.
We could also introduce multiple scalar elds.
Here we will conne ourselves for simplicity to a canonical Lagrangian (79) of a single scalar eld, for which the

21
only adjustable quantity is the choice of potential V (). For simplicity, we assume a
at spacetime,
g =
0
B
B
B
@
1
a2(t)
a2(t)
a2(t)
1
C
C
C
A
; (83)
and the equation of motion for the eld  with a Lagrangian given by Eq. (79) is:
+ 3H _r2+
V

= 0; (84)
where an overdot indicates a derivative with respect to the coordinate time t, and H = _a=a is the Hubble parameter.
We will be particularly interested in the homogeneous mode of the eld, for which the gradient term vanishes, r = 0,
so that the the functional derivative V= simplies to an ordinary derivative, and the equation of motion simplies
to4
+ 3H _+ V 0 () = 0: (85)
The stress-energy for a scalar eld is given by
T = @@ gL; (86)
and, for a homogeneous eld, it takes the form of a perfect
uid with energy density  and pressure p, with
 =
1
2
_2 + V () ;
p =
1
2
_2 V () : (87)
We see that the de Sitter limit, p ' , is just the limit in which the potential energy of the eld dominates the
kinetic energy, _2  V (). This limit is referred to as slow roll, and under such conditions the universe expands
quasi-exponentially,
a (t) / exp
Z
Hdt

 eN ; (88)
where it is conventional to dene the number of e-folds N with the sign convention
dN  Hdt; (89)
so that N is large in the far past and decreases as we go forward in time and as the scale factor a increases.
This can be made quantitative by plugging the energy and pressure (87) into the Friedmann Equation
H2 =

_a
a
2
=
8
3m2Pl

1
2
_2 + V ()

; (90)
and the Raychaudhuri Equation, which we write in the convenient form

a
a

=
4
3m2Pl
(+ 3p) = H2 (1 ) : (91)
4 The astute reader may well ask: if we are claiming in
ation is a solution to the problems of
atness and homogeneity in the universe,
why are we assuming
atness and homogeneity from the outset? The answer is that, as long as in
ation gets started somehow and
goes on for long enough, the late-time behavior of the eld  will always be described by Eq. (85). We will see later that we only
have observational access to the end of the in
ationary period, and therefore a consistent theory of initial conditions is not required for
investigating the observational consequences of in
ation.

22
Here H2 is given in terms of  by the Friedmann Equation (90), and the parameter  species the equation of state,
 
3
2

p

+ 1

=
4
m2Pl

_
H
!2
: (92)
It is a straightforward exercise to show that  is related to the evolution of the Hubble parameter by
 =
d lnH
d ln a
=
1
H
dH
dN
; (93)
where N is the number of e-folds (89). This is a useful parameterization because the condition for accelerated
expansion a > 0 is simply equivalent to  < 1. The de Sitter limit p!  is equivalent to ! 0, so that the potential
V () dominates the energy density, and
H2 '
8
3m2Pl
V () : (94)
We make the additional approximation that the friction term in the equation of motion (85) dominates,
 3H _; (95)
so that the equation of motion for the scalar eld is approximately
3H _+ V 0 () ' 0: (96)
Equation (96) together with the Friedmann Equation (94) are together referred to as the slow roll approximation.
The condition (95) can be expressed in terms of a second dimensionless parameter, conventionally dened as
 

H _
= +
1
2
d
dN
: (97)
The parameters  and  are referred to as slow roll parameters, and the slow roll approximation is valid as long as
both are small, ; jj  1. It is not obvious that this will be a valid approximation for situations of physical interest:
 need not be small for in
ation to take place. In
ation takes place when  < 1, regardless of the value of . We later
demonstrate explicitly that slow roll does in fact hold for interesting choices of in
ationary potential. In the limit of
slow roll, we can use Eqs. (94, 96) to write the parameter  approximately as
 =
4
m2Pl

_
H
!2
'
m2Pl
16

V 0 ()
V ()
2
: (98)
The in
ationary limit,  1 is then just equivalent to a eld evolving on a
at potential, V 0 () V (). The second
slow roll parameter  can likewise be written approximately as:
 =

H _
'
m2Pl
8
"
V 00 ()
V ()

1
2

V 0 ()
V ()
2
#
; (99)
so that the curvature V 00 of the potential must also be small for slow roll to be a valid approximation. Similarly, we
can write number of e-folds as a function N () of the eld as:
N =
Z
Hdt =
Z
H
_
d =
2
p

mPl
Z
d
p

'
8
m2Pl
Z 
e
V ()
V 0 ()
d; (100)
The limits on the last integral are dened such that e is a xed eld value, which we will later take to be the end
of in
ation, and N increases as we go backward in time, representing the number of e-folds of expansion which take
place between eld value  and e.

23
FIG. 14: A schematic of the potential for in
ation. In
ation takes place on the region of the potential which is suciently
\
at", and reheating takes place near the true vacuum for the eld.
The qualitative picture of scalar eld-driven in
ation is that of a phase transition with order parameter given by
the eld . At early times, the energy density of the universe is dominated by the eld  which is slowly evolving
on a nearly constant potential, so that it approximates a cosmological constant (Fig. 14). During this period, the
universe is exponentially driven toward
atness and homogeneity. In
ation ends as the potential steepens and the
eld begins to oscillate about its vacuum state at the minimum of the potential. At this point, we have an eectively
zero-temperature scalar in a state of coherent oscillation about the minimum of the potential, and the universe is a
huge Bose-Einstein condensate: hardly a hot Big Bang! In order to transition to a radiation-dominated hot Big Bang
cosmology, the energy in the in
aton eld must decay into Standard Model particles, a process generically termed
reheating. This process is model-dependent, but it typically happens very rapidly. Note that the eld  need not
be a fundamental eld like a Higgs boson (although it could in fact be fundamental). Any order parameter for a
phase transition will do, as long as it has the quantum numbers of vacuum, and the eective potential has the correct
properties. The in
aton  could well be a scalar composite of more fundamental degrees of freedom, the coordinate
of a brane in a higher-dimensional compactication from string theory, a supersymmetric modulus, or something
even more exotic. The simple single-eld picture we discuss here is therefore an eective representation of a large
variety of underlying fundamental theories. All of the physics important to in
ation is contained in the shape of the
potential V (). (The details of the underlying theory are important for understanding the epoch of reheating, since
the reheating process depends crucially on the specic couplings of the in
aton to the other degrees of freedom in the
theory.)
How long does in
ation need to go on in order to solve the
atness and horizon problems? We use a thermodynamic
argument, which rests on a simple fact about cosmological expansion: as long as there are no decays or annihilations of
massive particles, all other interactions conserve photon number, so that the number of photons in a comoving volume
is constant. Since the entropy of photons is proportional to the number density, that means the entropy per comoving
volume is also constant. Therefore, the total entropy in the Cosmic Microwave Background (or, equivalently, the total
number of photons) is a convenient measure of spatial volume in the universe. Since the entropy per photon s is (up
to a few constants) given by the cube of the temperature,
s  T 3; (101)
the total photon entropy S in our current horizon volume is of order
Shor  T
3
CMBd
3
H 

TCMB
H0
3
 1088; (102)
where we have taken the CMB temperature to be 2:7 K and the current Hubble parameter H0 to be 70 km=s=MpC.
(The interesting unit conversion from km/s/MpC to Kelvin is left as an exercise for the reader.)
Let us consider a highly over-simplied picture of the universe, in which no particle decays or annihilations occur
between the end of in
ation and today. In that case, the only time when the photon number (and therefore the
entropy) in the universe changes is during the reheating process itself, when the in
ation  decays into radiation and

25
A. Example: the 4 potential
We are now in a position to apply this to a specic case. We use the simple case of a quartic potential,
V () = 4: (112)
The slow roll equations (96, 94) imply that the eld evolves as:
_ =
V 0 ()
3H
=
r
m2Pl
24
V 0 ()
p
V ()
/ : (113)
Note that this potential does not much qualitatively resemble the schematic in Fig. 13: the \
atness" of the potential
arises because the energy density V () / 4 rises much more quickly than the kinetic energy, _2 / 2, so that if
the eld is far enough out on the potential, the slow roll approximation is self-consistent. The eld rolls down to the
potential toward the vacuum at the origin, and the equation of state is determined by the parameter ,
 () '
m2Pl
16

V 0 ()
V ()
2
=
1


mPl

2
: (114)
The eld value e at the end of in
ation is when  (e) = 1, or
e =
mPlp

: (115)
For  > e,  < 1 and the universe is in
ating, and for  < e,  > 1 and the expansion enters a decelerating phase.
Therefore, even this simple potential has the necessary characteristics to support a period of early-universe in
ation
followed by reheating and a hot Big Bang cosmology. What about the requirement that the universe in
ate for at
least 60 e-folds? Using Eq. (98), we can express the number of e-folds before the end of in
ation (100) as
N =
2
p

mPl
Z 
e
dx
p
 (x)
= 


mPl
2
1; (116)
where we integrate backward from e to  to be consistent with the sign convention (89). Therefore the eld value N
e-folds before the end of in
ation is
N = mPl
r
N + 1

; (117)
so that
60 = 4:4mPl: (118)
We obtain sucient in
ation, but at a price: the eld must be a long way (several times the Planck scale) out on
the potential. However, we do not necessarily have to invoke quantum gravity, since for small enough coupling , the
energy density in the eld can much less than the Planck density, and the energy density is the physically important
quantity.
In this section, we have seen that the basic picture of an early epoch in the universe dominated by vacuum-like
energy, leading to nearly exponential expansion, can be realized within the context of a simple scalar eld theory. The
equation of state for the eld approximates a cosmological constant p =  when the energy density is dominated
by the eld potential V (), and in
ation ends when the potential becomes steep enough that the kinetic energy
_2=2 dominates over the potential. To solve the horizon and
atness problems and create a universe consistent with
observation, we must have at least 60 or so e-folds of in
ation, although in principle in
ation could continue for much
longer than this minimum amount. This dynamical explanation for the
atness and homogeneity of the universe is an
interesting, but hardly compelling scenario. It could be that the universe started out homogeneous and
at because
of initial conditions, either through some symmetry we do not yet understand, or because there are many universes,
and we just happen to nd ourselves in a highly unlikely realization which is homogeneous and geometrically
at.
In the absence of any other observational handles on the physics of the very early universe, it is impossible to tell.
However,
atness and homogeneity are not the whole story: in
ation provides an elegant mechanism for explaining
the inhomogeneity of the universe as well, which we discuss in Section V.

26
V. PERTURBATIONS IN INFLATION
The universe we live in today is homogeneous, but only when averaged over very large scales. On small scales, the
size of people or solar systems or galaxies or even clusters of galaxies, the universe we see is highly inhomogeneous.
Our world is full of complex structure, created by gravitational instability acting on tiny \seed" perturbations in the
early universe. If we look as far back in time as the epoch of recombination, the universe on all scales was homogeneous
to a high degree of precision, a few parts in 105. Recent observational eorts such as the WMAP satellite have made
exquisitely precise measurements of the rst tiny inhomogeneities in the universe, which later collapsed to form the
structure we see today. (We discuss the WMAP observation in more detail in Section VI.) Therefore, another mystery
of Big Bang cosmology is: what created the primordial perturbations? This mystery is compounded by the fact that
the perturbations we observe in the CMB exhibit correlations on scales much larger than the horizon size at the
time of recombination, which corresponds to an angular multipole of ` ' 100, or about 1 as observed on the sky
today. This is another version of the horizon problem: not only is the universe homogeneous on scales larger than the
horizon, but whatever created the primordial perturbations must also have been capable of generating
uctuations
on scales larger than the horizon. In
ation provides just such a mechanism [7, 8, 9, 10, 11, 12, 13].
Consider a perturbation in the cosmological
uid with wavelength . Since the proper wavelength redshifts with
expansion, prop / a (t), the comoving wavelength of the perturbation is a constant, com = const: This is true of
photons or density perturbations or gravitational waves or any other wave propagating in the cosmological background.
Now consider this wavelength relative to the size of the horizon: We have seen that in general the horizon as measured
in comoving units is proportional to the conformal time, dH /  . Therefore, for matter- or radiation-dominated
expansion, the horizon size grows in comoving units, so that a comoving length which is larger than the horizon
at early times is smaller than the horizon at late times: modes \fall into" the horizon. The opposite is true during
in
ation, where the conformal time is negative and evolving toward zero: the comoving horizon size is still proportional
to  , but it now shrinks with cosmological expansion, and comoving perturbations which are initially smaller than
the horizon are \redshifted" to scales larger than the horizon at late times (Fig. 15).
If the universe is in
ating at early times, and radiation- or matter-dominated at late times, perturbations in the
density of the universe which are initially smaller than the horizon are redshifted during in
ation to superhorizon scales.
Later, as the horizon begins to grow in comoving coordinates, the perturbations fall back into the horizon, where they
act as a source for structure formation. In this way in
ation explains the observed properties of perturbations in the
universe, which exist at both super- and sub-horizon scales at the time of recombination. Furthermore, an important
consequence of this process is that the last perturbations to exit the horizon are the rst to fall back in. Therefore,
the shortest wavelength perturbations are the ones which exited the horizon just at the end of in
ation, N = 0, and
longer wavelength perturbations exited the horizon earlier. Perturbations about the same size as our horizon today
exited the horizon during in
ation at around N = 60. Perturbations which exited the horizon earlier than that,
N > 60, are still larger than our horizon today. Therefore, it is only possible to place observational constraints on
the end of in
ation, about the last 60 e-folds. Everything that happened before that, including anything that might
tell us about the initial conditions which led to in
ation, is most probably inaccessible to us.
This kinematic picture, however, does not itself explain the physical origin of the perturbations. In
ation driven by
a scalar eld provides a natural explanation for this as well. The in
aton eld  evolving on the potential V () will
not evolve completely classically, but will also be subject to small quantum
uctuations about its classical trajectory,
which will in general be inhomogeneous. Since the energy density of the universe during in
ation is dominated by
the in
aton eld, quantum
uctuations in  couple to the spacetime curvature and result in
uctuations in the
density of the universe. Therefore, in the same way that the classical behavior of the eld  provides a description
of the background evolution of the universe, the quantum behavior of  provides a description of the inhomogeneous
perturbations about that background. We defer a full treatment of in
aton perturbations to Appendix A, and in
the next section focus on the much simpler case of quantizing a decoupled scalar ' in an in
ationary spacetime. In
addition to its relative simplicity, this case has direct relevance to the generation of gravitational waves in in
ation.
A. The Klein-Gordon Equation in Curved Spacetime
Consider an arbitrary free scalar eld, which we denote ' to distinguish it from the in
aton eld . The Lagrangian
for the eld is
L =
1
2
g@'@'; (119)

27
FIG. 15: A conformal diagram of the horizon in an in
ationary universe. The comoving horizon shrinks during in
ation,
and grows during the radiation- and matter-dominated expansion, while the comoving wavelengths of perturbations remain
constant. This drives comoving perturbations to \superhorizon" scales.
and varying the action (119) gives the Euler-Lagrange equation of motion
1
p
g
@

g
p
g@'


= 0: (120)
It will prove convenient to express the background FRW metric in conformal coordinates
g = a
2 ()  (121)
instead of the coordinate-time metric (83) we used in Section (IV). Here  is the conformal time and  =
diag: (1;1;1;1) is the Minkowski metric. In conformal coordinates, the free scalar equation of motion (120)
is
'00 + 2

a0
a

'0 r2' = 0; (122)
where 0 = d=d is a derivative with respect to conformal time. Note that unlike the case of the in
aton , we are
solving for perturbations and therefore retain the gradient term r2'. The eld ' is a decoupled spectator eld
evolving in a xed cosmological background, and does not eect the time evolution of the scale factor a (). An
example of such a eld is gravitational waves. If we express the spacetime metric as an FRW background gFRW plus
perturbation g , we can express the tensorial portion of the perturbation in general as a sum of two scalar degrees
of freedom
g0i = gi0 = 0
gij =
32
mPl

'+e^
+
ij + 'e^

ij


; (123)

28
where i; j = 1; 2; 3, and e^+;ij are longitudinal and transverse polarization tensors, respectively. It is left as an exercise
for the reader to show that the scalars '+; behave to linear order as free scalars, with equation of motion (122).
To solve the equation of motion (122), we rst Fourier expand the eld into momentum states 'k,
' (;x) =
Z
d3k
(2)3=2

'k () bke
ik
x + 'k () b

ke
ik
x

: (124)
Note that the coordinates x are comoving coordinates, and the wavevector k is a comoving wavevector, which does
not redshift with expansion. The proper wavevector is
kprop = k=a () : (125)
Therefore, the comoving wavenumber k is not itself dynamical, but is just a set of constants labeling a particular
Fourier component. The equation of motion for a single mode 'k is
'00k + 2

a0
a

'0k + k
2'k = 0: (126)
It is convenient to introduce a eld redenition
uk  a ()'k () ; (127)
and the mode function uk obeys a generalization of the Klein-Gordon equation to an expanding spacetime,
u00k +

k2
a00
a

uk = 0: (128)
(We have dropped the vector notation k on the subscript, since the Klein-Gordon equation depends only on the
magnitude of k.)
Any mode with a xed comoving wavenumber k redshifts with time, so that early time corresponds to short
wavelength (ultraviolet) and late time corresponds to long wavelength (infrared). The solutions to the mode equation
show qualitatively dierent behaviors in the ultraviolet and infrared limits:
 Short wavelength limit, k  a00=a. In this case, the equation of motion is that for a conformally Minkowski
Klein-Gordon eld,
u00k + k
2uk = 0; (129)
with solution
uk () =
1
p
2k

Ake
ik +Bke
ik


: (130)
Note that this is in terms of conformal time and comoving wavenumber, and can only be identied with an
exactly Minkowski spacetime in the ultraviolet limit.
 Long wavelength limit, k  a00=a. In the infrared limit, the mode equation becomes
a00uk = au
00
k ; (131)
with the trivial solution
uk / a ) 'k = const: (132)
This illustrates the phenomenon of mode freezing: eld modes 'k with wavelength longer than the horizon size
cease to be dynamical, and asymptote to a constant, nonzero amplitude.5 This is a quantitative expression of
our earlier qualitative notion of particle creation at the cosmological horizon. The amplitude of the eld at long
wavelength is determined by the boundary condition on the mode, i.e. the integration constants Ak and Bk.
Therefore, all of the physics boils down to the question of how we set the boundary condition on eld perturbations in
the ultraviolet limit. This is fortunate, since in that limit the eld theory describing the modes becomes approximately
Minkowskian, and we know how to quantize elds in Minkowski Space. Once the integration constants are xed, the
behavior of the mode function uk is completely determined, and the long-wavelength amplitude of the perturbation
can then be calculated without ambiguity. We next discuss quantization.
5 The second solution to this equation is a decaying mode, which is always subdominant in the infrared limit.

29
B. Quantization
We have seen that the equation of motion for eld perturbations approaches the usual Minkowski Space Klein-
Gordon equation in the ultraviolet limit, which corresponds to the limit of early time for a mode redshifting with
expansion. We determine the boundary conditions for the mode function via canonical quantization. To quantize
the eld 'k, we promote the Fourier coecients in the classical mode expansion (124) to annihilation and creation
operators
bk ! b^k; b

k ! b^
y
k; (133)
with commutation relation
h
b^k; b^
y
k0
i
 3 (k k0) : (134)
Note that the commutator in an FRW background is given in terms of comoving wavenumber, and holds whether we
are in the short wavelength limit or not. In the short wavelength limit, this becomes equivalent to a Minkowski Space
commutator. The quantum eld ' is then given by the usual expansion in operators b^k, b^
y
k
' (;x) =
Z
d3k
(2)3=2

'k () bke
ik
x + H:C:

(135)
The corresponding canonical momentum is
 (;x) 
L
 (@0')
= a2 ()
@'
@
: (136)
It is left as an exercise for the reader to show that the canonical commutation relation
[' (;x) ; (;x0)] = i3 (x x0) (137)
corresponds to a Wronskian condition on the mode uk,
uk
@uk
@
uk
@uk
@
= i; (138)
which for the ultraviolet mode function (130) results in a condition on the integration constants
jAkj
2 jBkj
2 = 1: (139)
This quantization condition corresponds to one of the two boundary conditions which are necessary to completely
determine the solution. The second boundary condition comes from vacuum selection, i.e. our denition of which
state corresponds to a zero-particle state for the system. In the next section, we discuss the issue of vacuum selection
in detail.
C. Vacuum Selection
Consider a quantum eld in Minkowski Space. The state space for a quantum eld theory is a set of states
jn(k1); : : : ; n(ki)i representing the number of particles with momenta k1; : : : ;ki. The creation and annihilation
operators a^yk and a^k act on these states by adding or subtracting a particle from the state:
a^yk jn(k)i =
p
n+ 1 jn(k) + 1i
a^k jn(k)i =
p
n jn(k) 1i : (140)
The ground state, or vacuum state of the space, is just the zero particle state:
a^k j0i = 0: (141)
Note in particular that the vacuum state j0i is not equivalent to zero. The vacuum is not nothing:
j0i 6= 0: (142)

30
To construct a quantum eld, we look at the familiar classical wave equation for a scalar eld,
@2
@t2
r2 = 0: (143)
To solve this equation, we decompose into Fourier modes uk,
 =
Z
d3k

akuk(t)e
ik
x + aku

k(t)e
ik
x

; (144)
where the mode functions uk(t) satisfy the ordinary dierential equation
uk + k
2uk = 0: (145)
This is a classical wave equation with a classical solution, and the Fourier coecients ak are just complex numbers.
The solution for the mode function is
uk / e
i!kt; (146)
where !k satises the dispersion relation
!2k k
2 = 0: (147)
To turn this into a quantum eld, we identify the Fourier coecients with creation and annihilation operators
ak ! a^k; a

k ! a^
y
k; (148)
and enforce the commutation relations
h
a^k; a^
y
k0
i
= 3 (k k0) : (149)
This is the standard quantization of a scalar eld in Minkowski Space, which should be familiar. But what probably
is not familiar is that this solution has an interesting symmetry. Suppose we dene a new mode function uk which is
a rotation of the solution (146):
uk = A(k)e
i!t+ik
x +B(k)ei!tik
x: (150)
This is also a perfectly valid solution to the original wave equation (143), since it is just a superposition of the Fourier
modes. But we can then re-write the quantum eld in terms of our original Fourier modes and new operators b^k and
b^yk and the original Fourier modes e
ik
x as:
 =
Z
d3k
h
b^ke
i!t+ik
x + b^yke
+i!tik
x
i
; (151)
where the new operators b^k are given in terms of the old operators a^k by
b^k = A(k)a^k +B
(k)a^yk: (152)
This is completely equivalent to our original solution (144) as long as the new operators satisfy the same commutation
relation as the original operators,
h
b^k; b^
y
k0
i
= 3 (k k0) : (153)
This can be shown to place a condition on the coecients A and B,
jAj2 jBj2 = 1: (154)
Otherwise, we are free to choose A and B as we please.
This is just a standard property of linear dierential equations: any linear combination of solutions is itself a
solution. But what does it mean physically? In one case, we have an annihilation operator a^k which gives zero when
acting on a particular state which we call the vacuum state:
a^k j0ai = 0: (155)

31
Similarly, our rotated operator b^k gives zero when acting on some state
b^k j0bi = 0: (156)
The point is that the two \vacuum" states are not the same
j0ai 6= j0bi : (157)
From this point of view, we can dene any state we wish to be the \vacuum" and build a completely consistent
quantum eld theory based on this assumption. From another equally valid point of view this state will contain
particles. How do we tell which is the physical vacuum state? To dene the real vacuum, we have to consider the
spacetime the eld is living in. For example, in regular special relativistic quantum eld theory, the \true" vacuum
is the zero-particle state as seen by an inertial observer. Another more formal way to state this is that we require the
vacuum to be Lorentz symmetric. This xes our choice of vacuum j0i and denes unambiguously our set of creation
and annihilation operators a^ and a^y. A consequence of this is that an accelerated observer in the Minkowski vacuum
will think that the space is full of particles, a phenomenon known as the Unruh eect [58]. The zero-particle state for
an accelerated observer is dierent than for an inertial observer. The case of an FRW spacetime is exactly analogous,
except that the FRW equivalent of an inertial observer is an observer at rest in comoving coordinates. Since an FRW
spacetime is asymptotically Minkowski in the ultraviolet limit, we choose the vacuum eld which corresponds to the
usual Minkowski vacuum in that limit,
uk () / e
ik ) Ak = 1; Bk = 0: (158)
This is known as the Bunch-Davies vacuum. This is not the only possible choice, although it is widely believed to
be the most natural. The issue of vacuum ambiguity of in
ationary perturbations is a subject which is extensively
discussed in the literature, and is still the subject of controversy. It is known that the choice of vacuum is potentially
sensitive to quantum-gravitational physics [59, 60, 61], a subject which is referred to as Trans-Planckian physics
[18, 62, 63]. For the remainder of our discussion, we will assume a Bunch-Davies vacuum.
The key point is that quantization and vacuum selection together completely specify the mode function, up to an
overall phase. This means that the amplitude of the mode once it has redshifted to long wavelength and frozen
out is similarly determined. In the next section, we solve the mode equation at long wavelength for an in
ationary
background.
D. Exact Solutions and the Primordial Power Spectrum
The exact form of the solution to Eq. (128) depends on the evolution of the background spacetime, as encoded in
a (), which in turn depends on the equation of state of the eld driving in
ation. We will consider the case where the
equation of state is constant, which will not be the case in general for scalar eld-driven in
ation, but will nonetheless
turn out to be a good approximation in the limit of a slowly rolling eld. Generalizing Eq. (77) to the case of arbitrary
equation of state parameter  = const:, the conformal time can be written
 =

1
aH

1
1 

; (159)
and the Friedmann and Raychaudhuri Equations (14) give
a00
a
= a2H2 (2 ) ; (160)
where a prime denotes a derivative with respect to conformal time. The conformal time, as in the case of de Sitter
space, is negative and tending toward zero during in
ation. (Proof of these relations is left as an exercise for the
reader.) We can then write the mode equation (128) as
u00k +

k2 a2H2 (2 )

uk = 0: (161)
Using Eq. (159) to write aH in terms of the conformal time  , the equation of motion becomes
2 (1 )2 u00k +
h
(k)2 (1 )2 (2 )
i
uk = 0: (162)

32
This is a Bessel equation, with solution
uk /
p
k [J (k) iY (k)] ; (163)
where the index  is given by:
 =
3 
2 (1 )
: (164)
The quantity k has special physical signicance, since from Eq. (159) we can write
(k) (1 ) =
k
aH
; (165)
where the quantity (k=aH) expresses the wavenumber k in units of the comoving horizon size dH  (aH)1. Therefore,
the short wavelength limit is k ! 1, or (k=aH) 1. The long-wavelength limit is k ! 0, or (k=aH) 1.
The simple case of de Sitter space (p = ) corresponds to the limit  = 0, so that the Bessel index is  = 3=2 and
the mode function (163) simplies to
uk /

k i
k

eik : (166)
In the short wavelength limit, (k)! 1, the mode function is given, as expected, by
uk / e
ik : (167)
Selecting the Bunch-Davies vacuum gives uk / eik , and canonical quantization xes the normalization,
uk =
1
p
2k
eik : (168)
Therefore, the fully normalized exact solution is
uk =
1
p
2k

k i
k

eik : (169)
This solution has no free parameters aside from an overall phase, and is valid at all wavelengths, including after the
mode has been redshifted outside of the horizon and becomes non-dynamical, or \frozen". In the long wavelength
limit, k ! 0, the mode function (169) becomes
uk !
1
p
2k

i
(k)

=
i
2k

aH
k

/ a; (170)
consistent with the qualitative result (132). Therefore the eld amplitude 'k is given by
j'kj =



uk
a


!
H
p
2k3=2
= const: (171)
The quantum mode therefore displays the freezeout behavior we noted qualitatively above (Fig. 16). The amplitude
of quantum
uctuations is conventionally expressed in terms of the two-point correlation function of the eld '. It is
left as an exercise for the reader to show that the vacuum two-point correlation function is given by
h0 j' (;x)' (;x0)j 0i =
Z
d3k
(2)3



uk
a



2
eik
(xx
0)
=
Z
dk
k
P (k) eik
(xx
0); (172)
where the power spectrum P (k) is dened as
P (k) 

k3
22
 


uk
a



2
!

H
2
2
; k ! 0: (173)

33
FIG. 16: The normalized mode function in de Sitter space, showing oscillatory behavior on subhorizon scales k=aH > 1, and
mode freezing on superhorizon scales, k=aH < 1.
The power per logarithmic interval k in the eld
uctuation is then given in the long wavelength limit by the Hubble
parameter H = const: This property of scale invariance is exact in the de Sitter limit.
In a more general model, the spacetime is only approximately de Sitter, and we expect that the power spectrum of
eld
uctuations will only be approximately scale invariant. It is convenient to express this dynamics in terms of the
equation of state parameter ,
 =
1
H
dH
dN
: (174)
We must have  < 1 for in
ation, and for a slowly rolling eld jj  1 means that  will also be slowly varying,
 ' const: It is straightforward to show that for  = const: 6= 0 that:
 The Bunch-Davies vacuum corresponds to the positive mode of Eq. (163),
uk /
p
k [J (k) + iY (k)] : (175)
 Quantization xes the normalization as
uk =
1
2
r

k
p
k [J (k) + iY (k)] : (176)
 The power spectrum in the long-wavelength limit k=aH ! 0 is a power law in k:
[P (k)]1=2 ! 23=2
()
(3=2)
(1 )

H
2

k
aH (1 )
3=2
; (177)

34
where () is a gamma function, and
 =
3 
2 (1 )
: (178)
Proof is left as an exercise for the reader.6 Note that in the case  = const:, both the background and perturbation
equations are exactly solvable.
We can use these solutions as approximate solutions in the more general slow roll case, where   1 ' const:, so
that the dependence of the power spectrum on k is approximately a power-law,
P (k) / kn; (179)
with spectral index
n = 3 2 = 3
3 
1 
' 2: (180)
Equation (177) is curious, however, because it does not obviously exhibit complete mode freezing at long wavelength,
since a and H both depend on time. We can show that P (k) does in fact approach a time-dependent value at long
wavelength by evaluating
d
dN
"
H

k
aH
3=2
#
=
d
dN
"
H

k
aH
=(1)
#
= H

k
aH
=(1)


1 

k
aH
=(1)1 k
aH

k
aH

= 0; (181)
which can be easily shown using a / eN and H / eN . That is, the time-dependent quantities a and H in Eq. (177)
are combined in such a way as to form an exactly conserved quantity. Since it is conserved, we are free to evaluate it
at any time (or value of aH) that we wish. It is conventional to evaluate the power spectrum at horizon crossing, or
at aH = k, so that
P 1=2 (k) '

H
2

k=aH
; (182)
where we have approximated the -dependent multiplicative factor as order unity. 7
It is straightforward to calculate the spectral index (180) directly from the horizon crossing expression (182) by
using
a / eN ; H / eN ; (183)
so that we can write derivatives in k at horizon crossing as derivatives in the number of e-folds N ,
d ln kjk=aH = d ln (aH) =
1
aH
d (aH)
dN
dN = ( 1) dN: (184)
The spectral index is then, to lowest order in slow roll
n =
d lnP (k)
d ln k
=
k
H2
dH2
dk




k=aH
=
1
H2 ( 1)
dH2
dN
6 Note that the quantization condition (137) can be applied to the solution (163) exactly, resulting in the normalization condition (139),
without approximating the solution in the short-wavelength limit!
7 This is not the value of the scalar eld power spectrum at the moment the mode is physically crossing outside the horizon, as is often
stated in the literature: it is the value of the power spectrum in the asymptotic long-wavelength limit. It is easy to show from the exact
solution (176) that the mode function is still evolving with time as it crosses the horizon at k = aH, and the asymptotic amplitude
diers from the amplitude at horizon crossing by about a factor of two. See Ref. [64] for a more detailed discussion of this point.

35
=
2
( 1)
' 2; (185)
in agreement with (180). Note that we are rather freely changing variables from the wavenumber k to the comoving
horizon size (aH)1 to the number of e-folds N . As long as the cosmological evolution is monotonic, these are all
dierent ways of measuring time: the time when a mode with wavenumber k exits the horizon, the time at which the
horizon is a particular size, the number of e-folds N and the eld value  are all eectively just dierent choices of
a clock, and we can switch from one to another as is convenient. For example, in the slow roll approximation, the
Hubble parameter H is just a function of , H /
p
V (). Because of this, it is convenient to dene N (k) to be the
number of e-folds (100) when a mode with wavenumber k crosses outside the horizon, and N (k) to be the eld value
N (k) e-folds before the end of in
ation. Then the power spectrum can be written equivalently as either a function of
k or of :
P 1=2 (k) =

H
2

k=aH
=

H
2

=N (k)
'
s
2V (N )
3m2Pl
: (186)
Wavenumbers k are conventionally normalized in units of hMpc1 as measured in the current universe. We can
relate N to scales in the current universe by recalling that modes which are of order the horizon size in the universe
today, k  a0H0, exited the horizon during in
ation when N = [46; 60], so that we can calculate the amplitude of
perturbations at the scale of the CMB quadrupole today by evaluating the power spectrum for eld values between
46 and 60.
One example of a free scalar in in
ation is gravitational wave modes, where the transverse and longitudinal po-
larization states of the gravity waves evolve as independent scalar elds. Using Eq. (123), we can then calculate
the power spectrum in gravity waves (or tensors) as the sum of the two-point correlation functions for the separate
polarizations:
PT =


g2ij

= 2
32
m2Pl


'2

=
16H2
m2Pl
/ knT ; (187)
with spectral index
nT = 2: (188)
If the amplitude is large enough, such a spectrum of primordial gravity waves will be observable in the cosmic
microwave background anisotropy and polarization, or be directly detectable by proposed experiments such as Big
Bang Observer [65, 66].
The second type of perturbation generated during in
ation is perturbations in the density of the universe, which are
the dominant component of the CMB anisotropy T=T  =  105, and are responsible for structure formation.
Density, or scalar perturbations are more complicated than tensor perturbations because they are generated by
quantum
uctuations in the in
aton eld itself: since the background energy density is dominated by the in
aton,
uctuations of the in
aton up or down the potential generate perturbations in the density. The full calculation requires
self-consistent General Relativistic perturbation theory, and is presented in Appendix A. Here we simply state the
result: Perturbations in the in
aton eld  ' H=2 generate density perturbations with power spectrum
PR (k) =

N


2
=
H2
m2Pl




k=aH
/ knS1; (189)
where N is the number of e-folds. Scalar perturbations are therefore enhanced relative to tensor perturbations by a
factor of 1=. The scalar power spectrum is also an approximate power-law, with spectral index
nS 1 =

H2 ( 1)
d
dN

H2


' 4+ 2; (190)
where  is the second slow roll parameter (97). Therefore, for any particular choice of in
ationary potential, we have
four measurable quantities: the amplitudes PT and PR of the tensor and scalar power spectra, and their spectral
indices nT and nS . However, not all of these parameters are independent. In particular, the ratio r between the scalar
and tensor amplitudes is given by the parameter , as is the tensor spectral index nT :
r 
PT
PS
= 16 = 8nT : (191)

36
This relation is known as the consistency condition for single-eld slow roll in
ation, and is in principle testable by a
suciently accurate measurement of the primordial perturbation spectra.
In the next section, we apply these results to our example 4 potential and calculate the in
ationary power spectra.
E. Example: 4
For the case of our example model with V () = 4, it is now straightforward to calculate the scalar and tensor
perturbation spectra. We express the normalization of the power spectra as a function of the number of e-folds N by
P 1=2R =
H
mPl
p





=N
=
4
p
24
3m3Pl
[V (N )]
3=2
V 0 (N )
=
24
3

N + 1


1=2  105; (192)
where we have used the slow roll expressions for H (94) and  (98) and Eq. (117) for N . For perturbations about
the current size of our horizon, N = 60, and CMB normalization forces the self-coupling to be very small,
  1015: (193)
The presence of an extremely small parameter is not peculiar to the 4 model, but is generic, and is referred to as
the ne tuning problem for in
ation.
We can similarly calculate the tensor amplitude
P 1=2T =
4H
mPl
p

; (194)
which is usually expressed in terms of the tensor/scalar ratio
r = 16 (N ) =
mPl


V 0 (N )
V (N )
2
=
16


mPl
N
2
=
16
N + 1
' 0:26; (195)
where we have again taken N = 60. For this particular model, the power in gravitational waves is large, about
a quarter of the power in scalar perturbations. This is not generic, but is quite model-dependent. Some choices of
potential predict large tensor contributions (where \large" means of order 10%), and other choices of potential predict
very tiny tensor contributions, well below 1%.
The tensor spectral index nT is xed by the consistency condition (191), but the scalar spectral index nS is an
independent parameter because of its dependence on :
n = 1 4 (N ) + 2 (N ) ; (196)
where
 (N ) =
1
N + 1
; (197)
and
 (N ) =
m2Pl
8
"
V 00 (N )
V (N )

1
2

V 0 (N )
V (N )
2
#
=
m2Pl
8

12
2N

8
2N

=
1
2

mPl
N
2
=
1
2 (N + 1)
: (198)

38
3. Calculate the normalization of the scalar power spectrum by
P 1=2R =
H
mPl
p





=N
 105; (203)
where the CMB quadrupole corresponds to roughly N = 60. A more accurate calculation includes the uncer-
tainty in the reheat temperature, which gives a range N ' [46; 60], and a corresponding uncertainty in the
observable parameters.
4. Calculate the tensor/scalar ratio r and scalar spectral index nS at N = [46; 60] by
r = 16 (N ) ; (204)
and
ns = 1 4 (N ) + 2 (N ) ; (205)
where the second slow roll parameter  is given by:
 (N ) =
m2Pl
8
"
V 00 (N )
V (N )

1
2

V 0 (N )
V (N )
2
#
: (206)
The key point is that the scalar power spectrum PR and the tensor power spectrum PT are both completely determined
by the choice of potential V ().8 Therefore, if we measure the primordial perturbations in the universe accurately
enough, we can in principle constrain the form of the in
ationary potential. This is extremely exciting, because it
gives us a very rare window into physics at extremely high energy, perhaps as high as the GUT scale or higher, far
beyond the reach of accelerator experiments such as the Large Hadron Collider.
It is convenient to divide the set of possible single-eld potentials into a few basic types [69]:
 Large-eld potentials (Fig. 17). These are the simplest potentials one might imagine, with potentials of the
form V () = m22, or our example case, V () = 4. Another widely-noted example of this type of model
is in
ation on an exponential potential, V () = 4 exp (=), which has the useful property that both the
background evolution and the perturbation equations are exactly solvable. In the large-eld case, the eld is
displaced from the vacuum at the origin by an amount of order   mPl and rolls down the potential toward
the origin. Large-eld models are typically characterized by a \red" spectral index nS < 1, and a substantial
gravitational wave contribution, r  0:1.
 Small-eld potentials (Fig. 18). These are potentials characteristic of spontaneous symmetry breaking phase
transitions, where the eld rolls o an unstable equilibrium with V 0 () = 0 toward a displaced vacuum. Ex-
amples of small-eld in
ation include a simple quadratic potential, V () = 

2 2

2
, in
ation from a
pseudo-Nambu-Goldstone boson or a shift symmetry in string theory (called Natural In
ation) with a potential
typically of the form V () = 4 [1 + cos (=)], or Coleman-Weinberg potentials, V () = 4 ln (). Small-eld
models are characterized by a red spectral index n < 1, and a small tensor/scalar ratio, r  0:01.
 Hybrid potentials (Fig. 19). A third class of models are potentials for which there is a residual vacuum energy
when the eld is at the minimum of the potential, for example a potential like V () = 

2 + 2

2
. In this
case, in
ation will continue forever, so additional physics is required to end in
ation and initiate reheating. The
hybrid mechanism, introduced by Linde [70], solves this problem by adding a second eld coupled to the in
aton
which is stable for  large, but becomes unstable at a critical eld value c near the minimum of V () . During
in
ation, however, only  is dynamical, and these models are eectively single-eld. Typical models of this type
predict negligible tensor modes, r  0:01 and a \blue" spectrum, nS > 1, which is disfavored by the data, and
we will not discuss them in more detail here. (Ref. [48] contains a good discussion of current limits on general
hybrid models.) Note also that such potentials will also support large-eld in
ation if the eld is displaced far
enough from its minimum.
8 Strictly speaking, this is true only for scalar elds with a canonical kinetic term, where the speed of sound of perturbations is equal to
the speed of light. More complicated scenarios such as DBI in
ation [67] require specication of an extra free function, the speed of
sound cS (), to calculate the power spectra. For constraints on this more general class of models, see Ref. [68].

39
FIG. 17: A schematic of a large-eld potential.
FIG. 18: A schematic of a small-eld potential.
An important feature of all of these models is that each is characterized by two basic parameters, the \height"
of the potential 4, which governs the energy density during in
ation, and the \width" of the potential . (Hybrid
models have a third free parameter c which sets the end of in
ation.) In order to have a
at potential and a slowly
rolling eld, there must be a hierarchy of scales such that the width is larger than the height,   . As we saw in
the case of the 4 large-eld model, typical in
ationary potentials have widths of order the Planck scale   mPl
and heights of order the scale of Grand Unication  MGUT  1015 GeV; although models can be constructed for
which in
ation happens at a much lower scale [70, 71, 72].
The quantities we are interested in for constraining models of in
ation are the primordial power spectra PR and PT ,
which are the underlying source of the CMB temperature anisotropy and polarization. However, the observed CMB

40
FIG. 19: A schematic of a hybrid potential.
anisotropies depend on a handful of unrelated cosmological parameters, since the primordial
uctuations are processed
through the complicated physics of acoustic oscillations. This creates uncertainties due to parameter degeneracies:
our best-t values for r and nS will depend on what values we choose for the other cosmological parameters such as
the baryon density
b and the redshift of reionization zri. To accurately estimate the errors on r and nS , we must t
all the relevant parameters simultaneously, a process which is computationally intensive, and is typically approached
using Bayesian Monte Carlo Markov Chain techniques [73]. Here we simply show the results: Figure 20 shows the
regions of the r, nS parameter space allowed by the WMAP 5-year data set [74, 75]. We have t over the parameters

CDM,
b,
Lambda, H0, PR, zri, r, and ns, with a constraint that the universe must be
at, as predicted by in
ation,

b +
CDM +
Lambda = 1. We see that the data favor a red spectrum, nS < 1, although the scale-invariant limit
nS = 1 is still within the 95%-condence region. Our example in
ation model V () = 4 is convincingly ruled out
by WMAP, but the simple potential V () = m22 is nicely consistent with the data.9 Figure 21 shows the WMAP
constraint with r on a logarithmic scale, with the prediction of several small-eld models for reference. There is no
evidence in the WMAP data for a nonzero tensor/scalar ratio r, with a 95%-condence upper limit of r < 0:5. It is
possible to improve these constraints somewhat by adding other data sets, for example the ACBAR high-resolution
CMB anisotropy measurement [76] or the Sloan Digital Sky Survey [77, 78], which improve the upper limit on the
tensor/scalar ratio to r < 0:3 or so. Current data are completely consistent with Gaussianity and adiabaticity, as
expected from simple single-eld in
ation models. In the next section, we discuss the outlook for future observation.
VII. OUTLOOK AND CONCLUSION
The basic hot Big Bang scenario, in which the universe arises out of a hot, dense, smooth initial state and cools
through expansion, is now supported by a compelling set of observations, including the existence of the Cosmic
Microwave Background, the primordial abundances of the elements, and the evolution of structure in the universe,
all of which are being measured with unprecedented precision. However, this scenario leaves questions unanswered:
9 Liddle and Leach point out that 4 models are special because of their reheating properties, and should be more accurately evaluated
at N = 64 [56]. However, this assumes that the potential has no other terms which might become dominant during reheating, and in
any case is also ruled out by WMAP5.

41
FIG. 20: Constraints on the r, n plane from Cosmic Microwave Background measurements. Shaded regions are the regions
allowed by the WMAP5 measurement to 68% and 95% condence. Models plotted are \large-eld" potentials V () / 2 and
V () / 4.
FIG. 21: Constraints on the r, n plane from Cosmic Microwave Background measurements, with the tensor/scalar ratio
plotted on a log scale. In addition to the large-eld models shown in Fig. 20, three small-eld models are plotted against the
data: \Natural In
ation" from a pseudo-Nambu-Goldstone boson [79], with potential V () = 4 [1 cos (=)], a logarithmic
potential V () / ln () typical of supersymmetric models [80, 81, 82], and a Coleman-Weinberg potential V () / 4 ln ().

42
Why is the universe so big and so old? Why is the universe so close to geometrically
at? What created the initial
perturbations which later collapsed to form structure in the universe? The last of these questions is particularly
interesting, because recent observations of the CMB, in particular the all-sky anisotropy map made by the landmark
WMAP satellite, have directly measured the form of these primordial perturbations. A striking property of these
observed primordial perturbations is that they are correlated on scales larger than the cosmological horizon at the
time of last scattering. Such apparently acausal correlations can only be produced in a few ways [83]:
 In
ation.
 Extra dimensions [84].
 A universe much older than H10 [85, 86].
 A varying speed of light [87].
In addition, the WMAP data contain spectacular conrmation of the basic predictions of the in
ationary paradigm:
a geometrically
at universe with Gaussian, adiabatic, nearly scale-invariant perturbations. No other model explains
these properties of the universe with such simplicity and economy, and much attention has been devoted to the
implications of WMAP for in
ation [29, 48, 74, 75, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98]. In
ation also makes
predictions which have not been well tested by current data but can be by future experiments, most notably a deviation
from a scale-invariant spectrum and the production of primordial gravitational waves. A non-scale-invariant spectrum
is weakly favored by the existing data, but constraints on primordial gravity waves are still quite poor. The outlook
for improved data is promising: over the next ve to ten years, there will be a continuous stream of increasingly
high-precision data made available which will allow constraint of cosmological parameters relevant for understanding
the early universe. The most useful measurements for direct constraint of the in
ationary parameter space are
observations of the CMB, and current activity in this area is intense. The Planck satellite mission is scheduled to
launch in 2009 [99, 100], and will be complemented by ground- and balloon-based measurements using a variety of
technologies and strategies [43, 44, 101, 102, 103, 104, 105, 106].
At the same time, cosmological parameter estimation is a well-developed eld. A set of standard cosmological
parameters such as the baryon density
bh2, the matter density
mh2, the expansion rate H0  100h km=sec are being
measured with increasing accuracy. The observable quantities most meaningful for constraining models of in
ation
are the ratio r of tensor to scalar
uctuation amplitudes, and the spectral index nS of the scalar power spectrum.
This kind of simple parameterization is at the moment sucient to describe the highest-precision cosmological data
sets. Furthermore, the simplest slow-roll models of in
ation predict a nearly exact power-law perturbation spectrum.
In this sense, a simple concordance cosmology is well-supported by both data and by theoretical expectation. It
could be that the underlying universe really is that simple. However, the simplicity of concordance cosmology is at
present as much a statement about the data as about the universe itself. Only a handful of parameters are required
to explain existing cosmological data. Adding more parameters to the t does no good: any small improvement
in the t of the model to the data is oset by the statistical penalty one pays for introducing extra parameters
[107, 108, 109, 110, 111, 112, 113, 114, 115]. But the optimal parameter set is a moving target: as the data get better,
we will be able to probe more parameters. It may be that a \vanilla" universe [116] of a half-dozen or so parameters
will continue to be sucient to explain observation. But it is reasonable to expect that, as measurements improve in
accuracy, we will see evidence of deviation from such a lowest-order expectation. This is where the interplay between
theory and experiment gains the most leverage, because we must understand: (1) what deviations from a simple
universe are predicted by models, and (2) how to look for those deviations in the data. It is of course impossible to
predict which of the many possible signals (if any) will be realized in the universe in which we live. I discuss below
four of the best motivated possibilities, in order of the quality of current constraints. (For a more detailed treatment
of these issues, the reader is referred to the very comprehensive CMBPol Mission Concept Study [117].)
Features in the density power spectrum
Current data are consistent with a purely power-law spectrum of density perturbations, P (k) / knS1 with a \red"
spectrum (nS < 1) favored by the data at about a 90% condence level, a gure which depends on the choice of
parameter set and priors. Assuming it is supported by future data, the detection of a deviation from a scale-invariant
(nS = 1) spectrum is a signicant milestone, and represents a conrmation of one of the basic predictions of in
ation.
In slow-roll in
ation, this power-law scale dependence is nearly exact, and any additional scale dependence is strongly
suppressed. Therefore, detection of a nonzero \running"  = dnS=d ln k of the spectral index would be an indication
that slow roll is a poor approximation. There is currently no evidence for scale-dependence in the spectral index,
but constraints on the overall shape of the power spectrum are likely to improve dramatically through measurements
of the CMB anisotropy at small angular scales, improved polarization measurements, and better mapping of large-
scale structure. Planck is expected to measure the shape of the spectrum with 2 uncertainties of order
n  0:01

43
and
  0:01 [118, 119, 120, 121]. Over the longer term, measurements of 21cm radiation from neutral hydrogen
promises to be a precise probe of the primordial power spectrum, and would improve these constraints signicantly
[122].
Primordial Gravitational Waves
In addition to a spectrum PR of scalar perturbations, in
ation generically predicts a spectrum PT of tensor per-
turbations. The relative amplitude of the two is determined by the equation of state of the
uid driving in
ation,
r = 16 (207)
Since the scalar amplitude is known from the COBE normalization to be PR  H2=  1010, it follows that
measuring the tensor/scalar ratio r determines the in
ationary expansion rate H and the associated energy density
. Typical in
ation models take place with an energy density of around   (1015 GeV)4, which corresponds to a
tensor/scalar ratio of r  0:1, although this gure is highly model-dependent. Single-eld in
ation does not make a
denite prediction for the value of r: while many choices of potential generate a substantial tensor component, other
choices of potential result in an unobservably small tensor/scalar ratio, and there is no particular reason to favor one
scenario over another.
There is at present no observational evidence for primordial gravitational waves: the current upper limit on the
tensor/scalar ratio is around r  0:3. Detection of even a large primordial tensor signal requires extreme sensitivity.
The crucial observation is detection of the odd-parity, or B-mode, component of the CMB polarization signal, which
is suppressed relative to the temperature
uctuations, themselves at the 104 level, by at least another four orders
of magnitude. This signal is considerably below known foreground levels [123], severely complicating data analysis.
Despite the formidable challenges, the observational community has undertaken a broad-based eort to search for the
B-mode, and a detection would be a boon for in
ationary cosmology. Planck will be sensitive to a tensor/scalar ratio
of around r ' 0:1, and dedicated ground-based measurements can potentially reach limits of order r ' 0:01. The
proposed CMBPol polarization satellite would reach r of order 103 [117, 124], and direct detection experiments such
as BBO could in principle detect r of order 104 [65].
Primordial Non-Gaussianity
In addition to a power-law power spectrum, in
ation predicts that the primordial perturbations will be distributed
according to Gaussian statistics. Like running of the power spectrum, non-Gaussianity is suppressed in slow-roll in
a-
tion [125]. However, detection of even moderate non-Gaussianity is considerably more dicult. If the perturbations
are Gaussian, the two-point correlation function completely describes the perturbations. This is not the case for non-
Gaussian
uctuations: higher-order correlations contain additional information. However, higher-order correlations
require more statistics and are therefore more dicult to measure, especially at large angular scales where cosmic
variance errors are signicant. Current limits are extremely weak [88, 126], and future high angular resolution CMB
maps will still fall well short of being sensitive to a signal from slow-roll in
ation or even weakly non-slow-roll models
[127]. It will take a strong deviation from the slow-roll scenario to generate observable non-Gaussianity. However,
a measurement of non-Gaussianity would in one stroke rule out virtually all slow-roll in
ation models and force
consideration of more exotic scenarios such as DBI in
ation [67], Warm In
ation [128], or curvaton scenarios [129].
Isocurvature perturbations
In a universe where the matter consists of multiple components, there are two general classes of perturbation about
a homogeneous background: adiabatic, in which the perturbations in all of the
uid components are in phase, and
isocurvature, in which the perturbations have independent phases. Single-eld in
ation predicts purely adiabatic
primordial perturbations, for the simple reason that if there is a single eld  responsible for in
ation, then there
is a single order parameter governing the generation of density perturbations. This is a nontrivial prediction, and
the fact that current data are consistent with adiabatic perturbations is support for the idea of quantum generation
of perturbations in in
ation. However, current limits on the isocurvature fraction are quite weak [130, 131]. If
isocurvature modes are detected, it would rule out all single-eld models of in
ation. Multi-eld models, on the
other hand, naturally contain multiple order parameters and can generate isocurvature modes. Multi-eld models
are naturally motivated by the string \landscape", which is believed to contain an enormous number of degrees of
freedom. Another possible mechanism for the generation of isocurvature modes is the curvaton mechanism, in which
cosmological perturbations are generated by a eld other than the in
aton [132, 133].
The rich interplay between theory and observation that characterizes cosmology today is likely to continue for the
foreseeable future. As measurements improve, theory will need to become more precise and complete than the simple
picture of in
ation that we have outlined in these lectures, and single-eld in
ation models could yet prove to be a
poor t to the data. However, at the moment, such models provide an elegant, compelling, and (most importantly)
scientically useful picture of the very early universe.

44
FIG. 22: Foliations of an FRW spacetime. Comoving hypersurfaces (dashed lines) have constant density, but another choice of
gauge (solid lines) will have unphysical density
uctuations which are an artifact of the choice of gauge.
Acknowledgments
I would like to thank the organizers of the Theoretical Advanced Studies Institute (TASI) at Univ. of Colorado,
Boulder for giving me the opportunity to return to my alma mater to lecture. Various versions of these lectures
were also given at the Perimeter Institute Summer School on Particle Physics, Cosmology, and Strings in 2007, at
the Second Annual Dirac Lectures at Florida State University in 2008, and at the Research Training Group at the
University of Wurzburg in 2008. This research is supported in part by the National Science Foundation under grants
NSF-PHY-0456777 and NSF-PHY-0757693. I thank Dennis Bessada, Richard Easther, Hiranya Peiris, and Brian
Powell for comments on a draft version of the manuscript.
APPENDIX A: THE CURVATURE PERTURBATION IN SINGLE-FIELD INFLATION
In this section, we discuss the generation of perturbations in the density  (x)  = generated during in
ation.
The process is similar to the case of a free scalar eld discussed in Sec. V: the in
aton eld , like any other scalar,
will have quantum
uctuations which are stretched to superhorizon scales and subsequently freeze out as classical
perturbations. The dierence is that the energy density of the universe is dominated by the in
aton potential, so
that quantum
uctuations in  generate perturbations in the density . Dealing with such density perturbations
is complicated by the fact that in General Relativity, we are free to choose any coordinate system, or gauge, we
wish. To see why, consider the case of an FRW spacetime evolving with scale factor a (t) and uniform energy density
 (t;x) =  (t). What we mean here by \uniform" energy density, or homogeneity, is that the density is a constant
in comoving coordinates. But the physics is independent of coordinate system, so we could equally well work in
coordinates t0, x0 for which constant-time hypersurfaces do not have constant density (Fig. 22). Such a division of
spacetime into a time coordinate and a set of orthogonal spacelike hypersurfaces is called a foliation of the spacetime,
and is an arbitrary choice equivalent to a choice of coordinate system.
For an FRW spacetime, comoving coordinates correspond to a foliation of the spacetime into spatial hypersurfaces
with constant density: this is the most physically intuitive description of the spacetime. Any other choice of foliation
of the spacetime would result in density \perturbations" which are entirely due to the choice of coordinate system.

45
Such unphysical perturbations are referred to as gauge modes. Another way to think of this is that the division
between what we call \background" and what we call \perturbation" is itself gauge-dependent. For perturbations
with wavelength smaller than the horizon, it is possible to dene background and perturbation without ambiguity,
since all observers can agree on a denition of time coordinate t and on an average density  (t). Not so for superhorizon
modes: if we consider a perturbation mode with wavelength much larger than the horizon size, observers in dierent
horizons will see themselves in independently evolving, homogeneous patches of the universe: a \perturbation" can
be dened only by comparing causally disconnected observers, and there is an inherent gauge ambiguity in how we
do this. The canonical paper on gauge issues in General Relativistic perturbation theory is by Bardeen [134]. A good
pedagogical treatment with a focus on in
ationary perturbations can be found in Ref. [135].
In practice, instead of the density perturbation , the quantity most directly relevant to CMB physics is the
Newtonian potential  on the surface of last scattering. For example, this is the quantity that directly appears in
Eq. (50) for the Sachs-Wolfe Eect. The Newtonian potential is related to the density perturbation  through the
Poisson Equation:
r2 = 4Ga2; (A1)
where the factor of a2 comes from dening the gradient r relative to comoving coordinates. Like , the Newtonian
potential  is a gauge-dependent quantity: its value depends on how we foliate the spacetime. For example, we
are free to choose spatial hypersurfaces such that the density is constant, and the Newtonian potential vanishes
everywhere:  (t;x) = 0. This foliation of the spacetime is equivalent to the qualitative picture above of dierent
horizon volumes as independently evolving homogeneous universes. Observers in dierent horizons use the density 
to synchronize their clocks with one another. Such a foliation is not very useful for computing the Sachs-Wolfe eect,
however! Instead, we need to dene a gauge which corresponds to the Newtonian limit in the present universe. To
accomplish this, we describe the evolution of a scalar eld dominated cosmology using the useful
uid
ow approach
[136, 137, 138, 139, 140]. (An alternate strategy involves the construction of gauge-invariant variables: see Refs.
[141, 142] for reviews.)
Consider a scalar eld  in an arbitrary background g . The stress-energy tensor of the scalar eld may be written
T = ;; g

1
2
g;; V ()

: (A2)
Note that we have not yet made any assumptions about the metric g or about the scalar eld . Equation (A2) is
a completely general expression. We can dene a
uid four-velocity for the scalar eld by
u 
;
p
g;;
: (A3)
It is not immediately obvious why this should be considered a four-velocity. Consider any perfect
uid lling spacetime.
Each element of the
uid has four-velocity u (x) at every point in spacetime which is everywhere timelike,
u (x)u (x) = 1 8x: (A4)
Such a collection of four-vectors is called a timelike congruence. We can draw the congruence dened by the
uid
four-velocity as a set of
ow lines in spacetime (Fig. 23). Each event P in spacetime has one and only one
ow line
passing through it. The
uid four-velocity is then a set of unit-normalized tangent vectors to the
ow lines, uu = 1.
For a scalar eld, we construct a timelike congruence by Eq. (A3), which is by construction unit normalized:
uu =
g;;
g;;
= 1: (A5)
We then dene the \time" derivative of any scalar quantity f(x) by the projection of the derivative along the
uid
four-velocity:
_f  uf;: (A6)
In particular, the time derivative of the scalar eld itself is
_  u; =
q
g;; : (A7)
Note that in the homogeneous case, we recover the usual time derivative,
r = 0) _ =
q
g00;0;0 =
d
dt
: (A8)

46
FIG. 23: A timelike congruence in spacetime. Each event P is intersected by exactly one world line in the congruence.
The stress-energy tensor (A2) in terms of _ takes the form
T =

1
2
_2 + V ()

uu +

1
2
_2 V ()

(uu g) : (A9)
We can then dene a generalized density  and and pressure p by
 
1
2
_2 + V () ;
p 
1
2
_2 V () : (A10)
Note that despite the familiar form of these expressions, they are dened without any assumption of homogeneity of
the scalar eld or even the imposition of a particular metric.
In terms of the generalized density and pressure, the stress-energy (A2) is
T = uu + ph ; (A11)
where the tensor h is dened as:
h  uu g : (A12)
The tensor h can be easily seen to be a projection operator onto hypersurfaces orthogonal to the four-velocity u.
For any vector eld A, the product hA is identically orthogonal to the four-velocity:
(hA
)u = A (hu
) = 0: (A13)
Therefore, as in the case of the time derivative, we can dene gradients by projecting the derivative onto surfaces
orthogonal to the four-velocity
(rf)  hf; : (A14)

47
FIG. 24: A comoving foliation of spacetime. Spatial hypersurfaces are everywhere orthogonal to the
uid four-velocity u.
In the case of a scalar eld
uid with four-velocity given by Eq. (A3), the gradient of the eld identically vanishes,
(r) = 0: (A15)
Note that despite its relation to a \spatial" gradient, rf is a covariant quantity, i.e. a four-vector.
Our fully covariant denitions of \time" derivatives and \spatial" gradients suggest a natural foliation of the
spacetime into spacelike hypersurfaces, with time coordinate orthogonal to those hypersurfaces. We can dene spatial
hypersurfaces to be everywhere orthogonal to the
uid
ow (Fig. 24). This is equivalent to choosing a coordinate
system for which ui = 0 everywhere. Such a gauge choice is called comoving gauge. In the case of a scalar eld, we
can equivalently dene comoving gauge as a coordinate system in which spatial gradients of the scalar eld ;i are
dened to vanish. Therefore the time derivative (A6) is just the derivative with respect to the coordinate time in
comoving gauge
_ =

@
@t

c
: (A16)
Similarly, the generalized density and pressure (A10) are just dened to be those quantities as measured in comoving
gauge.
The equations of motion for the
uid can be derived from stress-energy conservation,
T; = 0 = _u
 + (rp) + (+ p) ( _u + u) ; (A17)
where the quantity  is dened as the divergence of the four-velocity,
  u;: (A18)
We can group the terms multiplied by u separately, resulting in familiar-looking equations for the generalized density
and pressure
_+  (+ p) = 0;
(rp) + (+ p) _u = 0: (A19)
The rst of these equations, similar to the usual continuity equation in the homogeneous case, can be rewritten using
the denitions of the generalized density and pressure (A10) in terms of the eld as
+  _+ V 0 () = 0: (A20)
This suggests identifying the divergence  as a generalization of the Hubble parameter H in the homogeneous case. In
fact, if we take g to be a
at Friedmann-Robertson-Walker (FRW) metric and take comoving gauge, u = (1; 0; 0; 0),
we have
u; = 3H; (A21)

48
and the generalized equation of motion (A20) becomes the familiar equation of motion for a homogeneous scalar,
+ 3H _+ V 0 () = 0: (A22)
Now consider perturbations g about a
at FRW metric,
g = a
2 () [ + g ] ; (A23)
where  is the conformal time and  is the Minkowski metric  = diag (1;1;1;1). A general metric perturbation
g can be separated into components which transform independently under coordinate transformations [134],
g = g
scalar
 + g
vector
 + g
tensor
 : (A24)
The tensor component is just the transverse-traceless gravitational wave perturbation, discussed in Section V, and
vector perturbations are not sourced by single-eld in
ation. We therefore specialize to the case of scalar perturbations,
for which the metric perturbations can be written generally in terms of four scalar functions of space and time A, B,
R, and HT :
g00 = 2A
g0i = @iB
gij = 2 [Rij + @i@jHT ] : (A25)
We are interested in calculatingR. Recall that in the Newtonian limit of General Relativity, we can write perturbations
about the Minkowski metric in terms of the Newtonian potential  as:
ds2 = (1 + 2) dt2 (1 2) ijdx
idxj : (A26)
Similarly, we can write Newtonian perturbations about a
at FRW metric as
ds2 = a2 ()

(1 + 2) d2 (1 2) ijdx
idxj

: (A27)
We therefore expect  / R in the Newtonian limit. A careful calculation [138, 141] gives
 =
3 (1 + w)
5 + 3w
R; (A28)
so that in a matter-dominated universe,
 =
3
5
R: (A29)
In these expressions, R is the curvature perturbation measured on comoving hypersurfaces. To see qualitatively why
comoving gauge corresponds correctly to the Newtonian limit in the current universe, consider the end of in
ation.
Since in
ation ends at a particular eld value  = e, comoving gauge corresponds to a foliation for which in
ation
ends at constant time at all points in space: all observers synchronize their clocks to  = 0 at the end of in
ation.
This means that the background, or unperturbed universe is exactly the homogeneous case diagrammed in Fig. 13,
and the comoving curvature perturbation R is the Newtonian potential measured relative to that background.
To calculate R, we start by calculating the four-velocity u in terms of the perturbed metric.10 If we specialize to
comoving gauge, ui  0, the norm of the four-velocity can be written
uu = a
2 (1 + 2A)

u0

2
= 1; (A30)
and the timelike component of the four-velocity is, to linear order,
u0 =
1
a
(1A)
u0 = a (1 +A) : (A31)
10 This treatment closely follows that of Sasaki and Stewart [139], except that we use the opposite sign convention for N .

49
The velocity divergence  is then
 = u; = u
0
;0 +

0u
0
= 3H

1A
1
aH

@R
@
+
1
3
@i@i
@HT
@

; (A32)
where the unperturbed Hubble parameter is dened as
H 
1
a2
@a
@
: (A33)
Fourier expanding HT ,
@i@iHT = k
2HT ; (A34)
we see that for long-wavelength modes k  aH, the last term in Eq. (A32) can be ignored, and the velocity divergence
is
 ' 3H

1A
1
aH
@R
@

: (A35)
Remembering the denition of the number of e-folds in the unperturbed case,
N 
Z
Hdt: (A36)
we can dene a generalized number of e-folds as the integral of the velocity divergence along comoving world lines:
N 
1
3
Z
ds =
1
3
Z
 [a (1 +A) d ]: (A37)
Using Eq. (A35) for  and evaluating to linear order in the metric perturbation results in
N = R
Z
Hdt; (A38)
and we have a simple expression for the curvature perturbation,
R = N N: (A39)
This requires a little physical interpretation: we dened comoving hypersurfaces such that the eld has no spatial
variation,
(r) = 0 )  = const: (A40)
Then N is the number of e-folds measured on comoving hypersurfaces. But we can equivalently foliate the spacetime
such that spatial hypersurfaces are
at, and the eld exhibits spatial
uctuations:
A = R = 0 )  6= const: (A41)
On
at hypersurfaces, the eld varies, but the curvature does not, so that the metric on these hypersurfaces is exactly
of the FRW form (11) with k = 0. We then see immediately that
N =
Z
Hdt = const: (A42)
is the number of e-folds measured on
at hypersurfaces, and has no spatial variation. The curvature perturbation R
is the dierence in the number of e-folds between the two sets of hypersurfaces (Fig. 25). This can be expressed to
linear order in terms of the eld variation  on
at hypersurfaces as
R = N N =
N

 (A43)
where R is measured on comoving hypersurfaces, and N= and  are measured on
at hypersurfaces. We can

50
FIG. 25: Flat and comoving hypersurfaces.
express N as a function of the eld :
N =
Z
Hdt =
Z
H
_
d: (A44)
For monotonic eld evolution, we can express _ as a function of , so that
N

=
H
_
; (A45)
and the curvature perturbation is given by
R = N N =
N

 =
H
_
: (A46)
Note that this is an expression for the metric perturbation R on comoving hypersurfaces, calculated in terms of quan-
tities dened on
at hypersurfaces. For  produced by quantum
uctuations in in
ation, the two-point correlation
function is
p
h2i =
H
2
; (A47)
and the two-point correlation function for curvature perturbations is
p
hR2i =
H2
2 _
=
H
mPl
p

; (A48)
which is the needed result.
[1] A. H. Guth, Phys. Rev. D 23, 347 (1981).
[2] A. D. Linde, Phys. Lett. B 108, 389 (1982).
[3] A. Albrecht and P. J. Steinhardt, Phys. Rev. Lett. 48, 1220 (1982).
[4] E. B. Gliner, Sov. Phys.-JETP 22, 378 (1966).
[5] E. B. Gliner and I. G. Dymnikova, Sov. Astron. Lett. 1, 93 (1975).
[6] A. Linde, Lect. Notes Phys. 738, 1 (2008) [arXiv:0705.0164 [hep-th]].
[7] A. A. Starobinsky, JETP Lett. 30 (1979) 682 [Pisma Zh. Eksp. Teor. Fiz. 30 (1979) 719].
[8] V. F. Mukhanov and G. V. Chibisov, JETP Lett. 33 (1981) 532 [Pisma Zh. Eksp. Teor. Fiz. 33 (1981) 549].
[9] S. W. Hawking, Phys. Lett. B 115, 295 (1982).

51
[10] S. W. Hawking and I. G. Moss, Nucl. Phys. B 224, 180 (1983).
[11] A. A. Starobinsky, Phys. Lett. B 117 (1982) 175.
[12] A. H. Guth and S. Y. Pi, Phys. Rev. Lett. 49, 1110 (1982).
[13] J. M. Bardeen, P. J. Steinhardt and M. S. Turner, Phys. Rev. D 28, 679 (1983).
[14] D. H. Lyth and A. Riotto, Phys. Rept. 314, 1 (1999) [arXiv:hep-ph/9807278].
[15] G. S. Watson, arXiv:astro-ph/0005003.
[16] A. Riotto, arXiv:hep-ph/0210162.
[17] C. H. Lineweaver, arXiv:astro-ph/0305179.
[18] W. H. Kinney, arXiv:astro-ph/0301448.
[19] M. Trodden and S. M. Carroll, arXiv:astro-ph/0401547.
[20] D. Baumann and H. V. Peiris, arXiv:0810.3022 [astro-ph].
[21] S. Weinberg, \Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity," John Wiley
& Sons (1972) Ch. 13.
[22] W. L. Freedman et al. [HST Collaboration], Astrophys. J. 553, 47 (2001) [arXiv:astro-ph/0012376].
[23] M. J. White, D. Scott and J. Silk, Ann. Rev. Astron. Astrophys. 32, 319 (1994).
[24] W. Hu and S. Dodelson, Ann. Rev. Astron. Astrophys. 40, 171 (2002) [arXiv:astro-ph/0110414].
[25] A. Kosowsky, arXiv:astro-ph/0102402.
[26] D. Samtleben, S. Staggs and B. Winstein, Ann. Rev. Nucl. Part. Sci. 57, 245 (2007) [arXiv:0803.0834 [astro-ph]].
[27] W. Hu, arXiv:0802.3688 [astro-ph].
[28] E. W. Kolb and M. S. Turner, \The Early Universe," Addison-Wesley (1990), Ch. 3.
[29] J. Dunkley et al. [WMAP Collaboration], arXiv:0803.0586 [astro-ph].
[30] P. S. Henry, Nature, 231, 516 (1971).
[31] B. E. Corey and D. T. Wilkinson, Bull. Amer. Astron. Soc,. 8, 351 (1976).
[32] G. F. Smoot, M. V. Gorenstein, and R. A. Muller, Phys. Rev. Lett 39, 898 (1977).
[33] C. L. Bennett et al., Astrophys. J. 464, L1 (1996) [arXiv:astro-ph/9601067].
[34] G. Hinshaw et al. [WMAP Collaboration], arXiv:0803.0732 [astro-ph].
[35] R. K. Sachs and A. M. Wolfe, Astrophys. J. 147, 73 (1967).
[36] A. D. Sakharov, JETP 49, 345 (1965).
[37] Y. B. Zeldovich and R. A. Sunyaev, Astrophys. Space Sci. 4, 301 (1969).
[38] R. A. Sunyaev and Y. B. Zeldovich, Astrophys. Space Sci. 7, 3 (1970).
[39] P. J. E. Peebles and J. T. Yu, Astrophys. J. 162, 815 (1970).
[40] C. P. Ma and E. Bertschinger, Astrophys. J. 455, 7 (1995) [arXiv:astro-ph/9506072].
[41] A. Kosowsky, New Astron. Rev. 43, 157 (1999) [arXiv:astro-ph/9904102].
[42] M. Zaldarriaga, arXiv:astro-ph/0305272.
[43] E. M. Leitch, J. M. Kovac, N. W. Halverson, J. E. Carlstrom, C. Pryke and M. W. E. Smith, Astrophys. J. 624, 10
(2005) [arXiv:astro-ph/0409357].
[44] J. L. Sievers et al., arXiv:astro-ph/0509203.
[45] T. E. Montroy et al., Astrophys. J. 647, 813 (2006) [arXiv:astro-ph/0507514].
[46] J. H. Wu et al., arXiv:astro-ph/0611392.
[47] M. R. Nolta et al. [WMAP Collaboration], arXiv:0803.0593 [astro-ph].
[48] E. Komatsu et al. [WMAP Collaboration], arXiv:0803.0547 [astro-ph].
[49] M. Kamionkowski, Science 280, 1397 (1998) [arXiv:astro-ph/9806347].
[50] J. J. Levin, Phys. Rept. 365, 251 (2002) [arXiv:gr-qc/0108043].
[51] A. A. Starobinsky, Phys. Lett. B 91 (1980) 99.
[52] A. D. Linde, Mod. Phys. Lett. A 1, 81 (1986).
[53] A. H. Guth, Phys. Rept. 333, 555 (2000) [arXiv:astro-ph/0002156].
[54] A. Aguirre, arXiv:0712.0571 [hep-th].
[55] S. Winitzki, \Eternal in
ation," Hackensack, USA: World Scientic (2008).
[56] A. R. Liddle and S. M. Leach, Phys. Rev. D 68, 103503 (2003) [arXiv:astro-ph/0305263].
[57] W. H. Kinney and A. Riotto, JCAP 0603, 011 (2006) [arXiv:astro-ph/0511127].
[58] W. G. Unruh, Phys. Rev. D 14, 870 (1976).
[59] L. Hui and W. H. Kinney, Phys. Rev. D 65, 103507 (2002) [arXiv:astro-ph/0109107].
[60] U. H. Danielsson, Phys. Rev. D 66, 023511 (2002) [arXiv:hep-th/0203198].
[61] R. Easther, B. R. Greene, W. H. Kinney and G. Shiu, Phys. Rev. D 66, 023518 (2002) [arXiv:hep-th/0204129].
[62] J. Martin and R. H. Brandenberger, Phys. Rev. D 63, 123501 (2001) [arXiv:hep-th/0005209].
[63] J. C. Niemeyer, Phys. Rev. D 63, 123502 (2001) [arXiv:astro-ph/0005533].
[64] W. H. Kinney, Phys. Rev. D 72, 023515 (2005) [arXiv:gr-qc/0503017].
[65] T. L. Smith, M. Kamionkowski and A. Cooray, Phys. Rev. D 73, 023504 (2006) [arXiv:astro-ph/0506422].
[66] B. C. Friedman, A. Cooray and A. Melchiorri, Phys. Rev. D 74, 123509 (2006) [arXiv:astro-ph/0610220].
[67] E. Silverstein and D. Tong, Phys. Rev. D 70, 103505 (2004) [arXiv:hep-th/0310221].
[68] N. Agarwal and R. Bean, Phys. Rev. D 79, 023503 (2009) [arXiv:0809.2798 [astro-ph]].
[69] S. Dodelson, W. H. Kinney and E. W. Kolb, Phys. Rev. D 56, 3207 (1997) [arXiv:astro-ph/9702166].
[70] A. D. Linde, Phys. Rev. D 49, 748 (1994) [arXiv:astro-ph/9307002].
[71] L. Knox and M. S. Turner, Phys. Rev. Lett. 70, 371 (1993) [arXiv:astro-ph/9209006].

52
[72] W. H. Kinney and K. T. Mahanthappa, Phys. Rev. D 53, 5455 (1996) [arXiv:hep-ph/9512241].
[73] A. Lewis and S. Bridle, Phys. Rev. D 66, 103511 (2002) [arXiv:astro-ph/0205436].
[74] W. H. Kinney, E. W. Kolb, A. Melchiorri and A. Riotto, Phys. Rev. D 74, 023502 (2006) [arXiv:astro-ph/0605338].
[75] W. H. Kinney, E. W. Kolb, A. Melchiorri and A. Riotto, Phys. Rev. D 78, 087302 (2008) [arXiv:0805.2966 [astro-ph]].
[76] C. L. Reichardt et al., arXiv:0801.1491 [astro-ph].
[77] J. Loveday [the SDSS Collaboration], arXiv:astro-ph/0207189.
[78] K. N. Abazajian et al. [SDSS Collaboration], arXiv:0812.0649 [astro-ph].
[79] K. Freese, J. A. Frieman and A. V. Olinto, Phys. Rev. Lett. 65, 3233 (1990).
[80] G. R. Dvali, Q. Sha and R. K. Schaefer, Phys. Rev. Lett. 73, 1886 (1994) [arXiv:hep-ph/9406319].
[81] E. D. Stewart, Phys. Rev. D 51, 6847 (1995) [arXiv:hep-ph/9405389].
[82] J. D. Barrow and P. Parsons, Phys. Rev. D 52, 5576 (1995) [arXiv:astro-ph/9506049].
[83] D. N. Spergel and M. Zaldarriaga, Phys. Rev. Lett. 79, 2180 (1997) [arXiv:astro-ph/9705182].
[84] J. Khoury, B. A. Ovrut, P. J. Steinhardt and N. Turok, Phys. Rev. D 64, 123522 (2001) [arXiv:hep-th/0103239].
[85] J. Khoury, P. J. Steinhardt and N. Turok, Phys. Rev. Lett. 92, 031302 (2004) [arXiv:hep-th/0307132].
[86] R. Brandenberger and N. Shuhmaher, JHEP 0601, 074 (2006) [arXiv:hep-th/0511299].
[87] A. Albrecht and J. Magueijo, Phys. Rev. D 59, 043516 (1999) [arXiv:astro-ph/9811018].
[88] D. N. Spergel et al. [WMAP Collaboration], Astrophys. J. Suppl. 170, 377 (2007) [arXiv:astro-ph/0603449].
[89] L. Alabidi and D. H. Lyth, JCAP 0608, 013 (2006) [arXiv:astro-ph/0603539].
[90] U. Seljak, A. Slosar and P. McDonald, JCAP 0610, 014 (2006) [arXiv:astro-ph/0604335].
[91] J. Martin and C. Ringeval, JCAP 0608, 009 (2006) [arXiv:astro-ph/0605367].
[92] J. Lesgourgues, A. A. Starobinsky and W. Valkenburg, JCAP 0801, 010 (2008) [arXiv:0710.1630 [astro-ph]].
[93] H. V. Peiris and R. Easther, JCAP 0807, 024 (2008) [arXiv:0805.2154 [astro-ph]].
[94] L. Alabidi and J. E. Lidsey, arXiv:0807.2181 [astro-ph].
[95] J. Q. Xia, H. Li, G. B. Zhao and X. Zhang, Phys. Rev. D 78, 083524 (2008) [arXiv:0807.3878 [astro-ph]].
[96] J. Hamann, J. Lesgourgues and W. Valkenburg, JCAP 0804, 016 (2008) [arXiv:0802.0505 [astro-ph]].
[97] T. L. Smith, M. Kamionkowski and A. Cooray, Phys. Rev. D 78, 083525 (2008) [arXiv:0802.1530 [astro-ph]].
[98] H. Li et al., arXiv:0812.1672 [astro-ph].
[99] [Planck Collaboration], arXiv:astro-ph/0604069.
[100] F. R. Bouchet [Planck Collaboration], Mod. Phys. Lett. A 22, 1857 (2007).
[101] C. L. Kuo et al., arXiv:astro-ph/0611198.
[102] J. E. Ruhl et al. [The SPT Collaboration], arXiv:astro-ph/0411122.
[103] K. W. Yoon et al., large angular scale CMB polarimeter," arXiv:astro-ph/0606278.
[104] A. C. Taylor [the Clover Collaboration], New Astron. Rev. 50, 993 (2006) [arXiv:astro-ph/0610716].
[105] D. Samtleben and f. t. Q. collaboration, arXiv:0806.4334 [astro-ph].
[106] B. P. Crill et al., arXiv:0807.1548 [astro-ph].
[107] R. Trotta, Mon. Not. Roy. Astron. Soc. 378, 72 (2007) [arXiv:astro-ph/0504022].
[108] J. Magueijo and R. D. Sorkin, Mon. Not. Roy. Astron. Soc. Lett. 377, L39 (2007) [arXiv:astro-ph/0604410].
[109] D. Parkinson, P. Mukherjee and A. R. Liddle, Phys. Rev. D 73, 123523 (2006) [arXiv:astro-ph/0605003].
[110] A. R. Liddle, P. Mukherjee and D. Parkinson, Astron. Geophys. 47, 4.30-4.33 (2006) [arXiv:astro-ph/0608184].
[111] A. R. Liddle, Mon. Not. Roy. Astron. Soc. Lett. 377, L74 (2007) [arXiv:astro-ph/0701113].
[112] C. Pahud, A. R. Liddle, P. Mukherjee and D. Parkinson, Mon. Not. Roy. Astron. Soc. 381, 489 (2007) [arXiv:astro-
ph/0701481].
[113] E. V. Linder and R. Miquel, arXiv:astro-ph/0702542.
[114] A. R. Liddle, P. S. Corasaniti, M. Kunz, P. Mukherjee, D. Parkinson and R. Trotta, arXiv:astro-ph/0703285.
[115] G. Efstathiou, arXiv:0802.3185 [astro-ph].
[116] R. Easther, AIP Conf. Proc. 698, 64 (2004) [arXiv:astro-ph/0308160].
[117] D. Baumann et al., arXiv:0811.3919 [astro-ph].
[118] W. H. Kinney, Phys. Rev. D 58, 123506 (1998) [arXiv:astro-ph/9806259].
[119] E. J. Copeland, I. J. Grivell and A. R. Liddle, Mon. Not. Roy. Astron. Soc. 298, 1233 (1998) [arXiv:astro-ph/9712028].
[120] L. P. L. Colombo, E. Pierpaoli and J. R. Pritchard, arXiv:0811.2622 [astro-ph].
[121] P. Adshead and R. Easther, JCAP 0810, 047 (2008) [arXiv:0802.3898 [astro-ph]].
[122] V. Barger, Y. Gao, Y. Mao and D. Marfatia, arXiv:0810.3337 [astro-ph].
[123] A. Kogut et al., arXiv:0704.3991 [astro-ph].
[124] J. Dunkley et al., arXiv:0811.3915 [astro-ph].
[125] J. M. Maldacena, JHEP 0305, 013 (2003) [arXiv:astro-ph/0210603].
[126] P. Creminelli, A. Nicolis, L. Senatore, M. Tegmark and M. Zaldarriaga, JCAP 0605, 004 (2006) [arXiv:astro-ph/0509029].
[127] M. Liguori, A. Yadav, F. K. Hansen, E. Komatsu, S. Matarrese and B. Wandelt, Phys. Rev. D 76, 105016 (2007)
[Erratum-ibid. D 77, 029902 (2008)] [arXiv:0708.3786 [astro-ph]].
[128] I. G. Moss and C. Xiong, JCAP 0704, 007 (2007) [arXiv:astro-ph/0701302].
[129] M. Sasaki, J. Valiviita and D. Wands, Phys. Rev. D 74, 103003 (2006) [arXiv:astro-ph/0607627].
[130] K. Moodley, M. Bucher, J. Dunkley, P. G. Ferreira and C. Skordis, [arXiv:astro-ph/0407304].
[131] R. Bean, J. Dunkley and E. Pierpaoli, Phys. Rev. D 74, 063503 (2006) [arXiv:astro-ph/0606685].
[132] D. H. Lyth, C. Ungarelli and D. Wands, Phys. Rev. D 67, 023503 (2003) [arXiv:astro-ph/0208055].
[133] D. H. Lyth and D. Wands, Phys. Rev. D 68, 103516 (2003) [arXiv:astro-ph/0306500].

53
[134] J. M. Bardeen, Phys. Rev. D 22 (1980) 1882.
[135] E. Komatsu, arXiv:astro-ph/0206039.
[136] S. W. Hawking, Astrophys. J. 145, 544 (1966).
[137] G. F. R. Ellis and M. Bruni, Phys. Rev. D 40, 1804 (1989).
[138] A. R. Liddle and D. H. Lyth, Phys. Rept. 231, 1 (1993) [arXiv:astro-ph/9303019].
[139] M. Sasaki and E. D. Stewart, Prog. Theor. Phys. 95, 71 (1996) [arXiv:astro-ph/9507001].
[140] A. Challinor and A. Lasenby, Astrophys. J. 513, 1 (1999) [arXiv:astro-ph/9804301].
[141] H. Kodama and M. Sasaki, Prog. Theor. Phys. Suppl. 78, 1 (1984).
[142] V. F. Mukhanov, H. A. Feldman and R. H. Brandenberger, Phys. Rept. 215, 203 (1992).