Prog. Theor. Exp. Phys. 2014, 06B103 (26 pages)
DOI: 10.1093/ptep/ptu081
CMB Cosmology
Inflation: 1980–201X
Jun’ichi Yokoyama1,2,∗
1
Research Center for the Early Universe (RESCEU), School of Science, The University of Tokyo, 7-3-1
Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
2
Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU), TODIAS, WPI,
The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8583, Japan
∗
E-mail: [email protected]
Received February 14, 2014; Revised April 30, 2014; Accepted May 1, 2014; Published June 11 , 2014
...............................................................................
This is an introductory review of theoretical developments and observational investigation of
inflationary cosmology with a particular emphasis on its role in the theoretical physics of the
Universe.
...............................................................................
Subject Index
1.
E80
Introduction
The classical Big Bang cosmology [1] was the first successful theory of the Universe based on physical science that succeeded in interpreting the three fundamental cosmological observations of the
time in a unified manner, namely, the homogeneous cosmic expansion originally discovered by Hubble [2] and Lemaitre [3], the cosmic microwave background radiation (CMB) discovered by Penzias
and Wilson [4], and the abundance of light elements [5,6]. On the other hand, it also suffered from
fundamental difficulties such as the horizon problem, the flatness problem [7], and the initial singularity problem. Since they are related with the initial condition of the Universe and there were no
reliable theories to describe the very early Universe, these problems had not been seriously studied
for a long time, even after the classical Big Bang cosmology was established by the discovery of
CMB.
In the late 1970s and early 1980s, however, the situation drastically changed in association with the
development of the grand unified theories (GUTs) of elementary interactions [8] which claim that
the three fundamental interactions are unified at an extremely high energy scale around MGUT =
1015–16 GeV. Since this energy scale is higher than the scale experimentally accessible through particle accelerators by more than ten digits, there emerged a trend to use the high temperature and
density regime in the early Universe as an arena for their verification. The most representative positive outcome was the study of baryogenesis initiated by Yoshimura [9]. We also note that since the
GUT scale was just a step off from the Planckian scale, MPl = (c5 /G)1/2 = 1.2 × 1019 GeV, where
quantum gravitational effects become important, cosmology based on GUTs stimulated serious study
of the birth of the Universe.
At the same time, however, GUTs also brought about a serious difficulty for the classical Big Bang
cosmology, namely, overproduction of magnetic monopoles [10,11] at the GUT phase transition [12]
which would over-close the Universe [13] with the density parameter monopole ∼ 1015 ! Fortunately,
© The Author(s) 2014. Published by Oxford University Press on behalf of the Physical Society of Japan.
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0/),
which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.
PTEP 2014, 06B103
J. Yokoyama
however, the resolution of this serious problem was already prepared in the GUT itself, as observed
independently by Sato [14–16] and Guth [17]. Inflationary cosmology thus came into being.
These pioneers observed that in the symmetric state of the Higgs field, which is realized thermally
4
, and
shortly after the Big Bang, the field has a large potential energy density of the order of MGUT
that if this state is metastable with a long enough lifetime, the Universe would be dominated by the
false vacuum energy density which would not decrease as long as the Higgs field remains in that state
no matter how much the Universe expands. As a result, an exponential cosmic expansion is realized
which was referred to as cosmic inflation. Since both the particle horizon and the curvature radius
are exponentially stretched during inflation, it can in principle provide solutions to the horizon and
the flatness problems.
In practice, however, this first model, now referred to as old inflation, was not successful because
inflation could not be terminated to realize a hot Friedmann Universe over the comoving scale corresponding to the current Hubble horizon. On the other hand, this model led a conceptual revolution
in cosmology that our Universe is not unique but there can be multiple universes that are causally
disconnected from each other [18,19].
The idea of solving various cosmological problems with an accelerated expansion was so attractive that a number of attempts to realize inflation followed by a transition to a radiation-dominated
regime were made immediately after the original proposal. Now the standard paradigm of inflation
is that the accelerated expansion is realized when a scalar field, dubbed the inflaton, slowly rolls
over its potential down to a global minimum in a time scale longer than the Hubble time [20–22].
The remarkable feature of the slow-roll inflation is that it can not only explain the global properties
of the observed Universe, but also provide seeds of density and curvature fluctuations that evolve
into large-scale structures [23–26]. The temperature (and polarization) anisotropies generated in the
CMB at the same time provide indirect cosmological tests of the inflation paradigm (see, e.g. [27–29]
among others).
In fact, the slow-roll inflation driven by a potential is not the only inflation scenario prevailing today,
and there are two alternatives. One is models realizing inflation without introducing any inflaton field
but modifying gravity from Einstein’s general relativity. Starobinsky was the first to show that quasiexponential expansion was realized by incorporating higher-order curvature terms in the action based
on quantum corrections [30]. Nowadays its simpler version, including a square scalar curvature term
besides the Einstein action, is referred to as the Starobinsky model. Since Starobinsky’s original
paper [30], which was written in an attempt to avoid the initial singularity, was earlier than the old
inflation models of Sato [14] and Guth [17], papers by these three authors are now regarded as the
original references of the inflationary cosmology [31].
The other alternative is models that make use of a scalar field with a higher-order kinetic function.
If we can realize a state with a constant canonical kinetic function, its energy density can have the
same equation of state as the cosmological constant to drive exponential inflation. Such models are
called k-inflation models [32].
All these models can be regarded as a subclass of the generalized G-inflation model [33,34] which
is the most general theory of single-field inflation with its field equations given by second-order
differential equations, as with most of the other theories of fundamental physics.
2.
Resolution of fundamental problems
Although the original inflation models of Sato and Guth realized exactly exponential cosmic expansion, it is not a necessary condition to resolve horizon and flatness problems. All we need is a
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sufficiently long period of accelerated expansion, when the particle horizon increases more rapidly
than t and the Hubble horizon, so the horizon problem can be solved. From the Einstein equations
for the cosmic scale factor, a(t),
2
K
8π G
ȧ
ρ,
(1)
+ 2 =
a
a
3
ä
4π G
=−
(ρ + 3P),
(2)
a
3
we see that such an expansion is possible if cosmic energy density, ρ, and pressure, P, satisfy ρ +
3P < 0. Then the energy density decreases less rapidly than a −2 (t) so that the curvature term in
the Friedmann equation (1), K /a 2 (t), becomes unimportant. As a result, the flatness problem is also
solved simultaneously. In this review we take c = = kB = 1 unless these quantities are explicitly
shown.
Of course, such a stage being dominated by a new energy component should not last forever but the
Universe should be converted to a state dominated by radiation to realize the initial state of the hot
Big Bang cosmology, since we do not wish to demolish its success. The accelerated or inflationary
expansion must be followed by the creation of radiation and entropy. In fact it is this epoch rather than
the inflationary expansion itself through which the monopole and other unwanted relic problems such
as domain walls, gravitinos, etc. are solved, since the current state of the Universe is characterized
by the CMB temperature T = 2.73 K and the corresponding entropy density, and we measure the
abundance of these relics in units of the entropy density.
Let us quantify how much inflationary expansion is required to solve the horizon and flatness
problems in terms of entropy consideration. First, the entropy contained in the current Hubble radius
H0−1 = 4.2 × 103 Mpc is given by that of the CMB photon and neutrino background with the effective temperature Tν = 1.95 K as S0 = 2.6 × 1088 . We consider the condition that the comoving
−3
Hubble volume at the beginning of inflation, 4π
3 Hinf at t = ti , contains more entropy than S0 after
the reheating, so that the currently observable region is well inside the initial Hubble radius at the
onset of inflation.
For simplicity, let us assume that during inflation the energy density takes a constant value ρinf with
the corresponding Hubble parameter Hinf , and that inflation continues from t = ti to t f to stretch the
scale factor by a f /ai ≡ e N , and that after inflation the Universe is dominated by a component with
equation-of-state P = wρ until radiation domination with the reheating temperature TR is achieved.
At this time the initial Hubble radius is stretched to
1
2 g T 4 − 3(1+w)
π
∗
−1 N
R
Hinf
e
≡ rH
(3)
30ρinf
which contains the entropy
1 1+3w −1+3w
1+w
4π 2 g∗ 3 4π 3
TR
16π 3 e3N 45g∗w 1+w Hinf − 1+w
S=
TR ×
rH =
,
(4)
90
3
270
4π 3
MPl
MPl
where g∗ represents the number of relativistic degrees of freedom at reheating. Requiring this to be
larger than S0 we find
g r 12.5 − 10.8w
w
1 + 3w
∗
N > 67.7 −
−
ln
ln
+
1+w
3 + 3w
106.75
6(1 + w)
0.01
2
TR
H
1 − 3w
ln
≡ Nmin , r ≡ 0.01
,
(5)
+
8
3 + 3w
10 GeV
2.4 × 1013 GeV
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J. Yokoyama
where r is the tensor-to-scalar ratio to be defined later in Eq. (99).
As mentioned above, this is the condition that the initial Hubble radius at the onset of inflation
is stretched to be larger than the current Hubble radius. However, while density perturbation on
this scale is observed to be 10−5 , we expect that the initial amplitude of fluctuation on this scale
can be close to unity just to initiate inflation [35]. Thus it is not sufficient to have N = Nmin but
1
we need more inflation by a factor of (10−5 )− 2 ∼ 500 to suppress the amplitude of fluctuations
to an acceptable level [36]. Thus the minimal condition to solve the horizon problem is actually
N > Nmin + ln 500 = Nmin + 6.2.
Taking this extra number of e-folds into account we find
TR
1 r 1
,
(6)
N > 55 + ln
+ ln
6
0.01
3
108 GeV
for w = 0, and
1 g∗ 1 r 1
TR
N > 67 − ln
,
+ ln
− ln
6
106.75
3
0.01
3
108 GeV
(7)
for w = 1.
Next let us consider the evolution of the total density parameter tot using
K
= H 2 (tot − 1).
a2
We find
tot (t0 ) − 1
=
tot (ti ) − 1
a(ti )Hinf
a0 H0
2
= e−2(N −Nmin ) < 500−2 = 4 × 10−6 .
(8)
(9)
Thus, if the horizon problem is solved by inflation, the flatness problem is also solved automatically
and such inflation models predict that the total density parameter today is equal to unity with four to
five digits’ accuracy.
3. Realization of inflation
3.1. Three mechanisms
As mentioned in the introduction one can classify inflation realization mechanisms into three broad
categories.
The most standard category is the slow-roll inflation models. Consider a scalar field φ, the inflaton,
with a canonical kinetic term X ≡ − 12 g μν ∂μ φ∂ν φ and a potential V [φ]. Taking the variation of the
action
√
√
4
S = L −gd x = (X − V [φ]) −gd4 x
(10)
with respect to the metric tensor, one finds the energy–momentum tensor
δS
2
= ∂μ φ∂ν φ + Lgμν .
Tμν = − √
−g δg μν
(11)
For homogeneous field configuration, the energy density, ρ, and the pressure, P, are given by
1
ρ = φ̇ 2 + V [φ],
2
1
P = φ̇ 2 − V [φ],
2
(12)
respectively. Hence if the potential is so flat that V [φ] > φ̇ 2 is satisfied, the equation-of-state
parameter w ≡ P/ρ is smaller than −1/3 and an accelerated expansion is realized.
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Next, in order to see the impact of a non-canonical kinetic term, let us rewrite the Lagrangian in
more general form as L = K (X, φ). Then calculating the energy–momentum tensor in the same way
as (11), we find
ρ = 2X K X − K ,
P = K,
(13)
in this case. Thus an accelerated expansion is possible, even without any potential, if X K X < −K
is satisfied. This is what is called k-inflation [32]. In particular, in the case when K X = ∂ K /∂ X = 0
holds we find de Sitter expansion. But of course, the solution must be found by solving the field
equation
K X (φ̈ + 3H φ̇) + 2K X X X φ̈ + 2K X φ X − K φ = 0,
(14)
with the Einstein equations.
One can find a simple model of k-inflation by expanding the Lagrangian as a power series of X ,
like
X2
.
(15)
K (X, φ) = −A(φ)X +
2M 4
Let us consider a regime where A(φ) can be approximated by unity, taking, say, A(φ) ≡ tanh[λ(φ f −
φ)/MG ] and assuming φ φ f initially with λφ f /MG being a positive constant well above unity.
Then we find a solution X ∼
= M 4 /6MG2 is real= M 4 =const., and de Sitter inflation with H 2 ∼
√
√
1
ized, where MG = MPl / 8π = 1/ 8π G = (c5 /8π G) 2 = 2.4 × 1018 GeV is the reduced Planck
scale.
The type of k-inflation in (15) is terminated when A(φ) flips its sign as it crosses φ f . Then X starts
to decrease rapidly and only the first term becomes relevant. Now the Universe is dominated by the
kinetic energy of a free scalar field whose energy density dissipates in proportion to a −6 (t) with the
equation-of-state parameter w = 1. This abrupt change from de Sitter to a power-law a(t) ∝ t 1/3
induces gravitational particle production to reheat the Universe in this model. We note that despite
the fact that the lowest-order kinetic term has the wrong sign during inflation, the theory is shown to
be perturbatively stable throughout the cosmic evolution [34,37].
So far we have implicitly assumed the gravity sector is expressed by the Einstein–Hilbert action
√
1
Sg = 2
R −gd4 x, κ 2 ≡ 8π G,
(16)
2κ
and considered a single scalar field as a matter ingredient. We may realize inflation even without any
scalar field matter by modifying gravity as
√
1
f (R) −gd4 x,
(17)
Sg = 2
2κ
where f (R) is a function of scalar curvature R. The trace of the field equation is given by
3 f (R) + f (R)R − 2 f (R) = 0.
(18)
One can therefore find a de Sitter solution, R = 12H 2 =const., if f (R) satisfies f (R) = 2 f (R),
namely, f (R) ∝ R 2 . A pure R 2 model cannot terminate inflation nor has a proper Einstein limit, so
we adopt
R2
.
(19)
f (R) = R +
6M 2
This theory should not be regarded as a mere perturbative extension of the Einstein gravity because
it contains an additional scalar degree of freedom called the scalaron.
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3.2.
J. Yokoyama
Inflation scenario
Let us describe cosmic evolution in a slow-roll inflation model as a prototype. Once inflationary
expansion sets in, the Universe rapidly becomes homogeneous, isotropic, and spatially flat practically
fairly soon, and the energy density other than the inflaton is soon diluted away. Hence we may consider
a spatially flat FLRW Universe from the beginning, except when we discuss the feasibility of inflation
in generic inhomogeneous and anisotropic spacetime. Thus the scalar field equation and the Einstein
equation read
φ̈ + 3H φ̇ + V [φ] = 0,
2
8πρφ
ρφ
ȧ
= H2 =
=
,
2
a
3MPl
3MG2
1
ρφ = φ̇ 2 + V [φ],
2
(20)
(21)
respectively.
To realize successful inflation the potential V [φ] must dominate ρφ for a sufficiently long time,
that is, the scalar field must remain practically constant in the cosmic expansion time scale. For this
purpose φ̈ must be negligibly small in the equation of motion. Then the field equations read
3H φ̇ + V [φ] = 0,
2
ȧ
8π V [φ]
V [φ]
= H2 =
=
,
2
a
3MPl
3MG2
which are called slow-roll equations of motion.
For these approximate equations to hold, we require the potential to satisfy
MG2 V [φ] 2
V [φ]
, |ηV | 1.
1, ηV ≡ MG2
V ≡
2
V [φ]
V [φ]
(22)
(23)
(24)
Here V and ηV are called the (potential) slow-roll parameters. Conversely, if these slow-roll conditions (24) are satisfied, inflationary expansion sets in soon. Inflation or accelerated expansion is
terminated when φ̇ 2 becomes larger than V [φ] or when | Ḣ | becomes larger than H 2 . Under the
slow-roll approximation, (22) and (23), the former occurs at V = 3/2 and the latter at V = 1. The
discrepancy is due to the invalidity of the slow-roll approximation then.
Using the slow-roll approximation, the number of e-folds of inflation after φ has crossed a value
φ N is given by
φf
dφ
1 φ f dφ ,
(25)
N=
H dt =
H
=
√
MG φ N
2V φ̇
φN
where φ f is the value at the end of inflation.
Figure 1 shows the typical shape of the inflaton’s potential. Near the local maximum at φ = 0 the
potential is so flat that inflation may be possible. Models of such a class are called small-field models.
On the other hand, if the potential increases at most with a power-law, one can see that the slow-roll
conditions (24) are satisfied for super-Planckian field values, so that inflation is realized there. Such
a model is called a large-field model.
In both cases the slow-roll conditions will no longer hold once the field approaches the global
minimum at φ = v, and it starts to oscillate around it. Thus the potential energy density is transferred
to field oscillation energy, which will eventually decay to particles interacting with φ. The Universe
will then be heated up to a radiation-dominated regime. This is the physical origin of the Big Bang
in the modern context.
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Fig. 1. Scalar field potential for single-field slow-roll inflation.
In fact, Starobinsky’s R 2 inflation can be regarded as a variant of a large-field model because the
modified gravity action can be converted to the Einstein gravity plus a scalar field with a potential.
Specifically, in terms of conformal transformation,
R
gμν ,
(26)
ḡμν ≡ 1 +
3M 2
we find the action is equivalent with
1
1 μν
4
ḡ ∂μ φ∂ν φ − V [φ]
R̄ −ḡd x +
Sg = 2
−ḡd4 x,
2κ
2
2
R
3
3 2 2
− 23 κφ
ln 1 +
, V [φ] ≡ MG M 1 − e
.
κφ ≡
2
3M 2
4
(27)
(28)
Thus the system is equivalent to scalar field matter in the Einstein gravity. For φ MG = κ −1 , the
potential has a plateau with a height 3MG2 M 2 /4 where inflation can occur. Inflation is followed by
scalar field oscillation around the origin and reheating proceeds through gravitational decay of the
inflaton φ.
4.
Slow-roll inflation models
Let us focus on specific models of slow-roll inflation with a canonical kinetic term.
4.1.
Large-field model
This model was originally proposed by Linde under the name chaotic inflation [22], since it makes
use of the chaotic initial condition of the Universe at the Planckian time tPl = (G/c5 )1/2 = 5.4 ×
10−44 s, when we expect large quantum fluctuations were dominant. Let us consider the following
simple Lagrangian as an example:
1
1
Lφ = − (∂φ)2 − V [φ], V [φ] = m 2 φ 2 .
(29)
2
2
By virtue of the uncertainty principle, at the Planckian time we expect both gradient energy and
potential energy densities were fluctuating with the Planckian magnitude
1
4
,
− (∂φ)2 MPl
2
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1 2 2
4
m φ MPl
2
(30)
PTEP 2014, 06B103
J. Yokoyama
2 /m M .
in each coherent domain. Taking m MPl we find φ can take a large value up to φ ∼ MPl
Pl
If the magnitude of φ is saturated, from the gradient energy constraint we find the coherent length of
4 /(Lm)2 M 4 , or L m −1 M −1 . In this case, φ is homogethe field, L, satisfies 12 (φ/L)2 ∼ MPl
Pl
Pl
neous over the Compton wavelength which is ∼ 105 times the horizon scale at the Planckian regime
for m ∼ 1013 GeV. To initiate inflation we do not need such an extremely homogeneous configuration,
but simply the inhomogeneity over the initial horizon scale should remain smaller than unity [35].
Then, over the horizon scale we may regard the field as homogeneous and apply the homogeneous
field equations which were derived in the previous section. Solving the field equation (20) and the
Einstein equation (21) we find
m MPl
φ(t) = φi − √ (t − ti ),
2 3π
4π m
m2
(t − ti )2
φi (t − ti ) −
a(t) = ai exp
3 MPl
6
1
φ 2 (t)
= a f exp
− 2π 2 .
2
MPl
(31)
(32)
√
Inflationary expansion is terminated around φ MPl / 4π when V becomes unity and |φ̇/φ|
becomes as large as H . Therefore, in order to solve the horizon problem one only needs the initial
amplitude φi 3MPl .
The most stringent constraint on this model comes from the amplitude of the density and curvature
perturbation generated during inflation, which sets the mass m ∼
= 1013 GeV and constrains the self4
−12
[38] (see Sect. 10.1).
coupling of the form λφ /4 as λ 10
4.2.
Small-field model
Soon after the old inflation model was proposed and its problems were elucidated, Linde proposed
a new inflation model based on the Coleman–Weinberg type potential of the GUT Higgs field,
in which inflation occurs as the scalar field induces a slow-roll over phase transition toward its
zero-temperature minimum after thermal correction of the potential has disappeared due to cosmic
expansion [20,21]. This was the first slow-roll inflation model and the first small-field inflation model
using a concave potential, which is sometimes called hill-top inflation [39]. Unfortunately, however,
the original new inflation predicted too short inflation and too large density fluctuations [24] so that
it could not serve as the right model of inflation. Furthermore, as realized by Linde himself, the Universe would have been far from a thermal state at the GUT era, hence one cannot expect thermal
symmetry restoration to set the appropriate initial condition for new inflation. These considerations
motivated chaotic inflation.
The required initial condition for small-field models can be achieved, however, without resorting
to thermal symmetry restoration [40,41]. Consider a simple Lagrangian
1
L = − (∂φ)2 − V [φ],
2
V [φ] =
λ 2
(φ − v 2 )2 ,
4
(33)
which has a local maximum at φ = 0. Since φ is a real scalar field this model admits a domain wall
solution connecting two vacuum states φ = ±v. Neglecting gravitational effects for the moment we
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find a solution
φ(x) = v tanh
λ
vz
2
(34)
that describes a domain wall on the x y-plane. Its thickness d0 can be estimated by equating (∇φ)2 ∼
−1
(v/d0 )2 and V [0] ≡ Vc as d0 ≈ vVc 2 . On the other hand, the Hubble horizon scale corresponding
to the vacuum energy density Vc is given by
− 1
1
2
2
3
8π G
−1
Vc
= MPl
,
(35)
Hc =
3
8π Vc
which is smaller than the thickness of the wall if v MPl . In this case one can find a region with a
large potential energy density V ∼ Vc whose dimension is larger than the Hubble scale near the center
of the domain wall. Such a field configuration provides a sufficient condition to initiate inflation there
without being affected by the configuration outside the Hubble horizon.
In this model, an initially random field configuration in the global space naturally creates domain
walls inside which inflation sets in near the central core. In this sense this model provides a natural
mechanism of small-field inflation on condition that the Universe continues to expand until the energy
density of the domain wall is locally dominant. We can also show that inflation ends after a finite time
except for the locus with φ = 0, and the reheating process proceeds just as in the chaotic inflation
model. Since the origin of inflation is provided by a topological defect, this model is called topological
inflation.
4.3.
Hybrid inflation
This is a model to induce inflation by a false vacuum energy density through a non-thermal symmetry
restoration by virtue of an extra scalar field [42]. The simplest model of hybrid inflation consists of
a real scalar field (φ) and a complex scalar field (χ ) with a Lagrangian
1
L = −(∂χ )† (∂χ ) − (∂φ)2 − V [χ , φ],
2
λ
1
V [χ , φ] = (|χ |2 − v 2 )2 + g 2 φ 2 |χ |2 + m 2 φ 2 .
2
2
(36)
From
∂V
= λ(|χ |2 − v 2 )χ + g 2 φ 2 χ
∂χ †
∂2V
≡ Mχ2 = λ(2|χ |2 − v 2 ) + g 2 φ 2
∂χ ∂χ †
(37)
we find that χ = 0 is a potential minimum if the inequality g 2 φ 2 > λv 2 is satisfied initially. Then
the potential reads
1
λ
(38)
V [χ = 0, φ] = v 4 + m 2 φ 2 .
2
2
If the√first term dominates the energy density of the Universe, inflation takes place. It is terminated at
φ < gλv when χ induces a phase transition, which typically occurs within one Hubble time unless
fine-tuning of parameters is applied. This model is attractive from the particle-physics viewpoint
because adequate inflation is possible even if φ is much smaller than the Planck scale [43], and
neither λ nor g requires fine-tuning. Instead the initial field configurations of the two scalar fields do
need tuning to realize inflation [44,45].
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In this model, since χ is a complex scalar field, a network of strings will be formed after the
phase transition. Were χ a real scalar field, a network of domain walls would have been formed and
over-dominated the energy density. Other mechanisms of formation of topological defects have been
proposed in [46–50] near or at the end of inflation. Hence there may well remain some relic defects
even if their energy scale is higher than the maximum temperature after inflation so that symmetry
restoration by thermal effects is impossible.
5.
Reheating
The entropy production process required after inflation to realize an appropriate initial state of the
hot Big Bang cosmology is called reheating, although in modern inflationary universe models it is
at this time that the Universe was dominated by radiation for the first time.
As is clear from the discussion in the previous section, the Universe is dominated by coherent
scalar field oscillation after potential-driven slow-roll inflation. Let us take the large-field massive
√
scalar model as an example. As the inflaton turns to satisfy |φ| MPl / 4π the scalar field starts to
oscillate around the origin with a period 2π/m. Such field oscillation is equivalent to a condensation
of the homogeneous zero-mode of the scalar field which decreases its amplitude through cosmic
expansion, to decay into radiation finally. In the initial oscillatory regime when its amplitude is large,
some non-perturbative particle production also takes place efficiently to some extent, which is called
preheating [51–53]. But the final reheating stage is always dominated by the perturbative decay of
the scalar field, which is understood by incorporating a decay term φ φ̇ in its equation of motion
[54–56]. Here φ is the decay rate of a φ particle which is much smaller than m since the inflaton
must be weakly coupled with other fields to suppress quantum fluctuations to an acceptable level, as
discussed in the following sections.
Multiplying (20) with φ̇ after introducing the abovementioned dissipation term, we find
d
dt
1 2 1 2 2
φ̇ + m φ = −(3H + φ )φ̇ 2 .
2
2
(39)
Since φ is rapidly oscillating in the cosmic expansion time scale, we can replace φ̇ 2 on the right-hand
2
side by an average over the oscillation period, φ̇ , which is identical to the total energy density of φ,
ρφ , thanks to the virial theorem [57]. Thus we find a Boltzmann equation
dρφ
= −(3H + φ )ρφ ,
dt
(40)
which is associated with that for radiation energy density ρr :
dρr
= −4Hρr + φ ρφ .
dt
(41)
These two equations can be solved to yield
a(t) −3
exp[−φ (t − t f )]
ρφ (t) = ρφ (t f )
a(t f )
t
a(t) −4
ρr (t) = φ
ρφ (τ )dτ.
t f a(τ )
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Thus the Universe turns to be dominated by radiation at t φ−1 , when the radiation temperature TR
is given by
1 1
2
8π
8π π 2 g∗ 4 2 ∼ 1 ∼ 1
TR
H=
ρ
=
(44)
= φ
=
2 r
2 30
2t
2
3MPl
3MPl
1
1
1 4 4
2
200
200
φ
11
TR ∼
MPl φ ∼
GeV,
(45)
= 0.1
= 10
g∗
g∗
105 GeV
where g∗ is the effective number of relativistic degrees of freedom.
6.
Generation of quantum fluctuations that eventually behave classically
Thanks to the smallness of the slow-roll parameters, the quantum-field-theoretic properties of the
inflaton during inflation are quite similar to those of a massless minimally coupled scalar field ϕ(x, t)
in de Sitter space:
ds 2 = −dt 2 + e2H t dx2 .
(46)
As is well known, such a scalar field shows anomalous growth of the square vacuum expectation
value as [58–60]
2
H
2
ϕ(x, t) =
H t.
(47)
2π
Decomposing the scalar field as
d3 k
†
(âk ϕk (t)eik·x + âk ϕk∗ (t)e−ik·x )
(48)
ϕ(x, t) =
(2π )3/2
d3 k
ϕ̂k (t)eik·x ,
≡
(2π )3/2
we find the momentum conjugate to ϕ(x, t) reads π(x, t) = a 3 (t)ϕ̇(x, t), hence the canonical
†
commutation relation [ϕ(x, t), π(x , t)] = iδ(x − x ) is equivalent to [âk , âk ] = δ (3) (k − k ) if we
impose the normalization condition
ϕk (t)ϕ̇k∗ (t) − ϕ̇k (t)ϕk∗ (t) =
i
a 3 (t)
.
(49)
†
Thus âk and âk satisfy the usual commutation relation and act as creation and annihilation operators
of the k-mode, respectively.
The mode function ϕk (t) satisfies
2
k2
d
d
+ 3H + 2H t ϕk (t) = 0,
(50)
dt 2
dt
e
as derived from the Klein–Gordon equation in de Sitter space, which is solved as
3
π
iH
(1)
H (−η) 2 H 3 (−kη) = √
(1 + ikη)e−ikη ,
(51)
ϕk (t) =
3
4
2
2k
under the condition (49). Here η is the conformal time defined by
t
t
dt
dt
1
=
=−
.
(52)
η≡
H
t
a(t)
e
H eH t
Note that since (50) is a second-order differential equation there are two independent solutions
(1)
(2)
proportional to H 3 (−kη) and H 3 (−kη). We have adopted only (51), which coincides with the
2
2
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positive-frequency mode in the Minkowski space at large wavenumber limit where the effect of
cosmic expansion is negligible.
k
is nothing but the ratio of physical wavenumber to the Hubble
Note also that −kη =
H a(t)
parameter. In the super-horizon limit k a(t)H , we find
k
ik
iH
iH
−ikη
ϕk = √
ei H a
1−
(1 + ikη)e
=√
3
3
Ha
2k
2k
2 iH
k
→√
1+O
,
(53)
3
H
a
2k
so that we find ϕk∗ (t) = −ϕk (t) and can regard that ϕ̂k (t) = ϕk (t)(âk − â−k ) holds when the wavelength corresponding to k is much longer than the Hubble radius. Then the momentum conjugate to
†
ϕ̂k reads π̂k (t) = a(t)3 ϕ̇k (t)(âk − â−k ), which means that both ϕ̂k and π̂k have the same operator
dependence in the super-horizon regime.
Thus in the super-horizon regime where the decaying mode is negligible, the quantum operators
ϕ̂k and π̂k commute with each other, so that quantum fluctuations behave as classical statistical
fluctuations [61]. Furthermore, the absolute square of mode function behaves as
†
|ϕk (t)|2 =
H2 H2
2
,
1
+
(kη)
→
2k 3
2k 3
when
k
−→ 0,
H a(t)
(54)
that is, it takes a constant value proportional to k −3 . Multiplying the phase space density over a loga4π k 3
d ln k, we find the dispersion takes a constant over each logarithmic
rithmic frequency interval, (2π
)3
frequency interval.
One can immediately see that (47) can be reproduced from this mode function by summing up
super-horizon fluctuations only as
ϕ(x, t) 2
H eH t
H
d3 k
|ϕk (t)|
=
(2π )3
2
H
2π
2
H t,
(55)
where the infrared cutoff is taken as the mode that left the Hubble radius at the beginning of inflation
(set to t = 0) [62].
The behavior that the square expectation value increases in proportion to time is the same as that
of Brownian motion with a step ±H/(2π ) at each time interval H −1 . Thus one can describe the
quantum fluctuations generated during inflation as follows:
H
, which is the Hawking
2π
temperature of de Sitter space, is continuously generated on the Hubble radius, and it is stretched
by subsequent cosmic expansion to form long-wave fluctuations.
For each Hubble time H −1 , quantum fluctuation with amplitude δϕ =
Finally, as a preparation for the calculations of curvature and tensor perturbations, we write
down the action of a massive scalar field ϕ using conformal time with the line element ds 2 =
a 2 (η)(−dη2 + dx2 ):
√
1
1 μν
1 2 2
4
dηd3 x a 2 φ 2 − (∇φ)2 − a 4 m 2 φ 2 , (56)
S=
−gd x − g ∂μ φ∂ν φ − m φ =
2
2
2
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where a prime denotes differentiation with respect to conformal time. Introducing a rescaled variable
χ ≡ aφ we find
a 1
3
2
2
2 2
(57)
dηd x χ − (∇χ ) − a m −
χ2 ,
S=
2
a
which is equivalent to the action of a scalar field with a time-dependent mass term in Minkowski
space.
In de Sitter space with a(η) = −1/(H η) the mode function is determined by
χk + k 2 χk −
2
χk = 0,
(−η)2
(58)
where we have taken m = 0. Its solution is given by
πη1
ϕk (t)
2
(1)
χk (η) = −
H 3 (−kη) =
4
a(t)
2
(59)
in agreement with (51) with the normalization condition χ χ ∗ − χ χ ∗ = i, equivalent to (49).
7.
Cosmological perturbation
In order to discuss quantum mechanical generation of cosmological perturbation, we incorporate
metric perturbation to the spatially flat homogeneous and isotropic background spacetime,
ds 2 = −dt 2 + a(t)2 dx2 ,
(60)
as
ds 2 = −(1 + 2A)dt 2 − 2a B j dtdx j + a 2 (δi j + 2HL δi j + 2HT i j )dx i dx j , Tr HT i j = 0,
(61)
where all new variables are functions of time and position [63–65]. As is well known, a spatial vector
B j can be decomposed to a rotation-free component and a divergence-free component as
Bj = ∂j B + Bj,
∂j B j = 0.
(62)
Similarly, the spatial tensor HT i j can be decomposed as
δi j 2
T j + ∂ j H
T i + HT T i j
∇ HT + ∂i H
HT i j = ∂i ∂ j −
3
T j = 0,
∂j H
∂ j HT T
k
j
= 0,
HT T
j
j
(63)
= 0.
HT T i j is the transverse-traceless tensor perturbation corresponding to gravitational waves. Thus
perturbation variables are classified to scalar, vector, and tensor variables according to the spatial
transformation law. Variables of different type are not mixed in linear perturbation theory and we
can treat them independently. Since vector perturbation has only a decaying mode under normal circumstances in the linear perturbation theory [63,64], we do not consider it hereafter and focus on
scalar and tensor perturbations.
First let us consider scalar perturbation, which is related to density and curvature fluctuations,
keeping only A, B, HL , and HT . In fact, not all of them are geometrical quantities, but include
gauge modes. They come from the arbitrariness of the choice of the background spacetime with
respect to which we define the perturbation variables. One should remember that the real physical
entity is an inhomogeneous and anisotropic spacetime, and that it may be regarded as homogeneous
and isotropic only after taking some average whose definition is not unique in general.
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To see the gauge dependence, let us consider two background spacetimes with coordinates x μ and
x μ , and assume that these two coordinates are related by a coordinate transformation
x 0 = x 0 + δx 0 ≡ x 0 + T
x i = x i + δx i ≡ x i + ∂ i L
(64)
where δx μ are small quantities whose relative magnitudes are of the order of the perturbation
variables. Here T and L are functions of space and time coordinates. By this coordinate transformation, we find that the scalar perturbation variables at the same coordinate values in two different
backgrounds are related as
A = A − Ṫ ,
B = B + a L̇ +
1
T,
a
∇2
L − H T,
H L = HL −
3
(65)
H T = HT + L ,
respectively. Here T and L represent gauge degrees of freedom on scalar perturbation. In order
to avoid their appearance, one may either find combinations of perturbation variables that remain
unchanged after gauge transformation, or fix the gauge completely. For example, if we set H T = 0,
L is fixed. If we further set B = 0, T is also fixed and there remains no gauge mode. In this case the
perturbed metric is given by
ds 2 = −(1 + 2)dt 2 + a 2 (1 + 2)dx2 ,
(66)
where we have rewritten A = and HL = to stress their gauge invariance. This is called the
longitudinal gauge.
In order to define the gauge, one may combine transformation properties of matter variables such
as a scalar field, which transforms as
φ(t, x) = φ(t − T, x j − ∂ j L) = φ(t − T ) + δφ = φ(t) − φ̇(t)T + δφ,
(67)
δφ = δφ − φ̇T.
(68)
that is,
This may be used to fix T .
8.
Generation of curvature fluctuations in inflationary cosmology
Here we consider the generation of curvature perturbations in single-field inflation models that cover
not only potential-driven slow-roll models but also k-inflation. For the most general single-field
inflation model with second-order field equations, see [34].
We start with an action
2
M
√
G
R + K (X, φ) .
(69)
S = d4 x −g
2
In the spatially flat homogeneous and isotropic spacetime (60), the background equations read
3MG2 H 2 = ρ = 2X K X − K ,
2MG2 Ḣ + 3MG2 H 2 = −P = −K ,
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together with the scalar field equation of motion
φ̈ + 3H cs2 φ̇ +
Kφ 2
K Xφ 2 2
cs φ̇ −
c = 0,
KX
KX s
cs2 ≡
PX
KX
=
.
ρX
K X + 2X K X X
(71)
In order to derive second-order action with respect to the curvature perturbation in the comoving
gauge, R, which is conserved outside the Hubble horizon if there is only adiabatic fluctuations, it is
convenient to use (3 + 1) formalism:
ds 2 = −N 2 dt 2 + h i j (dx i + N i dt)(dx j + N j dt),
h i j ≡ a 2 (t)e2R δi j ,
(72)
in which constraint equations are easily obtained for perturbation variables [66]. Since we are concerned with scalar-type fluctuations we put N = 1 + α and Ni = ∂i ψ, where α and ψ are also
perturbation variables of the same order of R. Correspondence with the general perturbed metric,
(61) through (63), is given as
A = α, B = −aψ, HL = R, and HT = 0,
(73)
with all the other vector and tensor modes set to zero. From (65) we immediately find that the gauge L
is fixed in the above expression, and equation (68) tells us that T can also be fixed by taking δφ = 0.
Then, indeed, R coincides with the comoving curvature perturbation, and there is no more gauge
degree of freedom.
Writing the action (69) in this gauge, we find
√
√
MG2
1
4
2 (3)
d x h N (MG R + 2K ) +
d4 x h N −1 (E i j E i j − E 2 ) + · · · ,
S=
(74)
2
2
1
E i j ≡ (ḣ i j − Ni| j − N j|i ), E ≡ TrE,
(75)
2
where terms replaced by · · · are all total derivative terms, so they do not affect the dynamics of the
system. Here R (3) is scalar curvature of 3-space and | denotes covariant derivative with respect to h i j .
Differentiating the action (74) with respect to N we find the Hamiltonian constraint
R (3) + 2
K
KX
1
− 4X 2 − 2 (E i j E i j − E 2 ) = 0,
2
N
MG
MG
(76)
which yields
H 2
1
∇ ψ = − 2 ∇ 2 R + α, MG2 ≡ X K X + 2X 2 K X X .
2
a
a
The momentum constraint obtained by differentiation with respect to Ni reads
1
j
j
(E − Eδi ) = 2H α,i − 2Ṙ,i = 0,
N i
|j
(77)
(78)
which gives α = Ṙ/H up to a term that does not depend on spatial coordinates. Inserting these
relations in the action we obtain an action for R only as
Ḣ
(∂ R)2
2
3
3 2
, H ≡ − 2 .
Ṙ − H
(79)
S2 = MG dtd xa
2
2
H
a
H
1
1
Introducing new variables z ≡ a(2) 2 /H = a(2 H ) 2 /cs , v ≡ MG z R, and using the conformal
time, we find
1
z 2
3
2
2
2
dηd x v − cs (∇v) + v ,
(80)
S2 =
2
z
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which, like (57), is an action of a free scalar field with a time-dependent mass term
η̇ H
z ηH ηH ṡ
2
= (a H )
+
2 − H − s +
1−s+
−
z
2
2
H
2H
˙ H
ċs
≡ (a H )2 (2 + q), η H ≡
, s≡
,
H H
H cs
(81)
with the important difference that the “sound velocity” cs may not be identical to unity. Still, one can
quantize v in the same way as a free scalar field if the time variation of cs is negligible. q is a small
number consisting of slow-roll parameters that can be read off from (81).
Using the de Sitter scale factor a = −1/(H η) one can obtain the mode function in the same way
as (59),
1
πη1
i
4 2 ∼3
1
3
2
(1)
−ikcs η
∼
1−
e
1+ q
vk = −
Hν (−kcs η) = √
, ν=
= , (82)
4
kcs η
2
9
2
2kcs
which approaches a constant in the long wave limit.
Thus the power spectrum of curvature perturbations multiplied by the phase space density reads
4π k 3
4π k 3 vk 2
H2
2
PR (k) ≡
|
R
|
=
=
.
(83)
k
(2π )3
(2π )3 MG z 8π 2 MG2 cs H
Here each quantity has weak time dependence and it should be evaluated at the time k-mode left the
sound horizon when |kcs η| = 1 was first satisfied.
Then the spectral index of curvature perturbation is defined and given by
d ln PR (k)
= −2 H − η H − s.
(84)
d ln k
In the case when the scalar field has a canonical kinetic term, as in the standard theory of particle
physics, we find cs = 1 and H = φ̇ 2 /(2MG2 H 2 ). Thus we find
2 2 H
δϕ 2
PR (k) =
= H
.
(85)
2π φ̇
φ̇
ns − 1 ≡
Here δϕ = H/(2π ) is the amplitude of scalar field fluctuation, as discussed in Sect. 6. This equality
is intuitively understandable since it represents the relative fluctuation of local scale factor
δa
δϕ
= δ N = H δt = H ,
a
φ̇
(86)
where N ≡ ln a. Taking HL = 0 and identifying δφ with δϕ through an appropriate gauge transformation by T to the flat gauge, we find the above expression is indeed equivalent with the comoving
curvature perturbation. This δ N method [67,68] is often used in the literature.
For slow-roll inflation models with a canonical kinetic term, from (22) and (23) we find H = V
and η H = −2ηV + 4V , so the spectral index is given by
n s − 1 = −6V + 2ηV .
(87)
Furthermore, the scale dependence of the spectral index which is called the running spectral index
is given by
V (φ)V (φ)
dn s
= 16V ηV − 24V2 − 2ξV , ξV ≡ MG4
.
(88)
d ln k
V 2 (φ)
These are fundamental observables of CMB anisotropy.
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9.
J. Yokoyama
Tensor perturbation
Next we consider tensor perturbations generated during inflation as quantum gravitational waves
[69–71]. First we incorporate a small metric perturbation h μν in Minkowski space as gμν = ημν +
h μν . The indices of h μν are raised by the Minkowski tensor. We again need to calculate the action
up to second order in the perturbed variables, so we should use g μν = ημν − h μν + h μα h να . Taking
,j
the longitudinal and traceless gauge, we find h 00 = h 0i = 0, h αα = h ii = 0, h i j = 0, with which the
Ricci scalar is given by
3
1
,μ
(89)
R = h i j h i j,μ + h i j,μ h i j,μ − h i j,l h jl,i .
4
2
Here Greek indices run from 0 to 3 while Latin indices from 1 to 3. Of course, what we need is a
perturbed action in cosmological background spacetime with the metric
ds 2 = a 2 (η)[−dη2 + (δi j + h i j )dx i dx j ] ≡ g̃μν dx μ dx ν ,
(90)
which is obtained from (61) through (63) by identifying HT T i j = h i j and putting all other perturbation variables to zero.
To calculate the desired quantity we make use of a conformal transformation g̃μν = 2 gμν ,
through which the Ricci tensor transforms as
R̃μν = Rμν − 2(ln );μν − gμν g σ τ (ln );τ σ + 2(ln );μ (ln );ν − gμν g σ τ (ln );σ (ln );τ .
(91)
Taking = a(η) we find the Ricci scalar in the metric (90) is given by
a i j a −2
R + 6 − 3 h hi j .
(92)
R̃ = a
a
a
Since we wish to calculate properties of quantum gravitational waves generated during inflation,
let us assume inflation is driven by a constant vacuum energy density equivalent to a cosmological
constant and calculate the second-order action, to yield
MG2
S2T =
( R̃ − 2) −g̃d4 x 2
2nd order
2
M
j
j,l
dηd3 xa 2 (h ij h i − h ij,l h i ),
= G
(93)
8
where we have used a 4 = 2aa − a 2 .
This action has the same form as a free scalar field again, so we can quantize it in the same way as
before. Indeed, taking new variables as z T ≡ a/2, u i j ≡ MG z T h i j , we find
1
a 2
3
2
2
S2T =
dηd x u i j − (∇u i j ) + u i j ,
(94)
2
a
which is indeed of the same form as (57).
For exact de Sitter space with H = 0, we have a = −1/(H η), whereas in the case H = 0, one
can express the scale factor around an arbitrary time η∗ using the Hubble parameter at that time H∗ as
1
1
−η 2 −νT
3 1 − H /3
1
, νT =
.
(95)
a=−
H∗ η∗ 1 − H −η∗
2 1 − H
Therefore we find the normalized mode function
πη1
2
u iAj = −
Hν(1)
(−kη)eiAj (k), A = +, ×,
(96)
T
4
where eiAj (k) is a polarization tensor that satisfies eiAj (k)e∗i j B (k) = δ AB .
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Thus the power spectrum of long-wave tensor perturbation is given by
PT (k) ≡
A ∗i j A
4π k 3
4π k 3 u i j u
2H 2
∗i j
h
h
=
=
,
ij
(2π )3
(2π )3 MG2 z 2T
π 2 MG2
(97)
with the spectral index
d ln PT (k)
= −2 H .
d ln k
Finally, the ratio of this power spectrum to the curvature perturbation,
nt ≡
r≡
PT (k)
= 16cs H = −8cs n t ,
PR (k)
(98)
(99)
is called the tensor-to-scalar ratio. It is one of the most important quantities subject to ongoing and
future observational tests of inflation. See Ref. [34] for the relation between r and n t in more generic
theories.
10.
Inflationary cosmology and observations
Nowadays much effort is being made to compare various predictions of inflationary cosmology to
observations of CMB anisotropy and large-scale structures. Since the first-year WMAP result was
disclosed [28,72,73], the cosmological parameters of a homogeneous and isotropic universe, as well
as properties of fluctuations, have been determined quite accurately with some error bars. Indeed,
until the WMAP papers were published, although we had the concordance cosmology, we did not
have any sensible error bars because the concordance values of the cosmological parameters had
been obtained by combining a number of different observational results and we did not have a means
to calculate error bars by making a sensible weighted average.
Contemporary precision cosmology makes use of the Markov Chain Monte Carlo (MCMC) method
to constrain values of cosmological parameters as well as the amplitude and spectrum of primordial
fluctuations using the temperature and E-mode polarization data of the CMB [74]. Note, however, that
the final values of these parameters, as well as the magnitude of errors obtained by these procedures,
change considerably depending on which observational data sets one uses, as well as what model
parameters one wishes to fit.
Current cosmological data as a whole show good agreement with a spatially flat universe with nonvanishing cosmological constant () and cold dark matter (CDM) based on the simple single-field
inflation paradigm that predicts almost scale-invariant adiabatic fluctuations. Thus it makes sense to
estimate various parameters of the CDM model using the MCMC technique.
Below we quote results from the seven- and nine-year WMAP reports [75,76], as well as the Planck
2013 results [77], to get an idea of what constraints will survive in future independent of the subtleties
of each experiment. The abbreviations in parentheses after each quoted number refer to the dataset
used, whose meaning should be self-explanatory.
First, as the most important test of the global spacetime, the spatial curvature is constrained as
−0.0133 < K 0 ≡ −
K
a02 H02
< 0.0084 (W7HSTBAO 95%CL),
−0.09 < K 0 < 0.001 (P13WPhighL 95%CL),
(100)
−0.0075 < K 0 < 0.0052 (P13lensWPhighLBAO 95%CL),
which is consistent with (9). Hence, from now on we fix K 0 = 0, since hte inflationary cosmology
we discuss here predicts so with much higher accuracy than (100), as discussed in Sect. 2.
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The amplitude and the spectral index of curvature perturbations are given by
PR (k0 ) = (2.43 ± 0.11) × 10−9 ,
PR (k0 ) = (2.430 ± 0.091) × 10
−9
n s = 0.967 ± 0.014 (W7 68%CL),
, n s = 0.968 ± 0.012 (W7HSTBAO 68%CL),
PR (k0 ) = (2.41 ± 0.10) × 10−9 , n s = 0.972 ± 0.013 (W9 68%CL),
−9
PR (k0 ) = (2.427+0.078
−0.079 ) × 10 , n s = 0.971 ± 0.010 (W9BAOH0 68%CL),
(101)
(102)
where the pivot scale is taken as k0 = 0.002 Mpc−1 . Planck takes the pivot scale at k0 = 0.05 Mpc−1
and finds
PR (k0 ) = (2.23 ± 0.16) × 10−9 , n s = 0.9616 ± 0.0094 (P13 68%CL)
−9
PR (k0 ) = (2.200+0.056
−0.054 ) × 10 , n s = 0.9608 ± 0.0054 (P13WPhighLBAO 68%CL).
(103)
If we allow scale dependence of the spectral index, the values change considerably as
dn s
n s = 1.027+0.050
= −0.034 ± 0.026, (W7 68%CL),
−0.051 ,
d ln k
(104)
dn s
n s = 1.008 ± 0.042,
= −0.022 ± 0.020, (W7HSTBAO 68%CL),
d ln k
dn s
= −0.019 ± 0.025, (W9 68%CL),
n s = 1.009 ± 0.049,
d ln k
(105)
dn s
n s = 1.020 ± 0.029,
= −0.023 ± 0.011, (W9eCMBBAOH0 68%CL).
d ln k
These numbers refer to the values at k0 = 0.002 Mpc−1 . Planck, on the other hand, finds
dn s
= −0.013 ± 0.009 (P13WP 68%CL),
d ln k
(106)
dn s
= −0.011 ± 0.008 (P13lensWPhighL 68%CL),
d ln k
at k0 = 0.05 Mpc−1 .
As for the amplitude of tensor perturbations, as of February 2014, only upper bounds have been
obtained. If we incorporate tensor perturbations in the power-law CDM model the constraints are
r < 0.36 (W7 95%; no running),
r < 0.24 (W7BAOH0 95%; no running),
r < 0.38 (W9 95%; no running),
(107)
r < 0.13 (W9eCMBBAOH0 95%; no running),
r < 0.11 (P13WPhighL 95%; no running),
whereas if we further incorporate a running spectral index they read
r < 0.49 (W7 95%; with running),
r < 0.49 (W7BAOH0 95%; with running),
r < 0.50 (W9 95%; with running),
(108)
r < 0.47 (W9eCMBBAOH0 95%; with running),
r < 0.26 (P13WPhighL 95%; with running).
Thus the constraint changes significantly depending on whether we allow a running spectral index
of curvature perturbation or not.
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J. Yokoyama
These constraints are often graphically displayed in an (n s , r ) plane [78] with one- and two-sigma
contours and compared with various inflation models. We do not do the same here for two reasons.
First, constraints vary appreciably depending on what dataset one uses and what underlying model
one adopts, as seen above. Second, occupying the center of the likelihood contour at the current level
of observations does not necessarily mean that such a model is the right one, and we should be open
minded for further development of observations.
Instead, below we calculate these observables in each class of models discussed in Sect. 3.1.
10.1.
Large-field models
Consider a massive scalar model as an example of large-field models. The number of e-folds of
inflation after φ crossed φ N is given from (32) as
1
N∼
=
4
φN
MG
2
1
− .
2
(109)
Hence the slow-roll parameters read
V = ηV = 2
MG
φN
2
=
1
, ξV = 0,
2N + 1
(110)
which gives the amplitude of curvature perturbation, (83),
(2N + 1)2
PR (k) =
24π 2
m
MG
2
.
(111)
Setting N = 55 on the pivot scale tentatively, we find m = 1.6 × 1013 GeV and the quartic coupling
is constrained as λ 8 × 10−13 in the case that such a term coexists as V [φ] = 12 m 2 φ 2 + λ4 φ 4 and
the first term dominates the observable regime of inflation, namely, the last ∼ 60 e-folds.
The spectral index and its running are given by
ns = 1 −
2
= 0.964,
2N + 1
dn s
8
=−
= −6.5 × 10−4 ,
d ln k
(2N + 1)2
(112)
in agreement with observations. The amplitude of the tensor perturbation is large enough to be
observable soon, r = 16V = 0.15.
Let us also consider R 2 inflation in the Einstein frame as a kind of large-field model. From (24),
(25), and (28) we find
3
N∼
= e
4
2
3 κφ N
,
4
V =
3
(e
1
2
3 κφ N
− 1)2
3
∼
,
=
4N 2
2
κφ
4 e 3 N −2 ∼ 1
ηV = −
=− ,
2
3
N
κφ
N
(e 3
− 1)2
(113)
which yield
ns = 1 −
2
= 0.964,
N
r=
for N = 55.
20/26
12
= 4.0 × 10−3 ,
N2
(114)
PTEP 2014, 06B103
10.2.
J. Yokoyama
Small-field model
Next we consider a small-field model (33). From V [φ] = λ4 (φ 2 − v 2 )2 , the slow-roll parameters are
given by
8MG2 φ 2
4MG2 (3φ 2 − v 2 )
96MG4 φ 2
,
η
=
,
ξ
=
.
V
V
(φ 2 − v 2 )2
(φ 2 − v 2 )2
(φ 2 − v 2 )3
V =
(115)
Since inflation ends when H = V = 1, the field value, φ f , at that time is given by
φ 2f
=v
2
+ 4MG2
√
− 16MG2 + 8MG2 v 2 ∼
= (β 2 − 2 2β)MG2 ,
(116)
where we have set v ≡ β MG and the last approximate equality holds when β 10. The number for
e-folds from φ = φ N to φ f reads
N=
φf
φN
H
φf
dφ
1
β2
1
β2 φ f
2
2
ln
ln
.
−
(φ
−
φ
)
−
=
f
N
4
φN
4
φN
2
φ̇
8MG2
(117)
The last approximation is valid when β 10–20. The amplitude of the curvature perturbation is
given by
PR (k0 ) =
λ(φ 2N − v 2 )4
768π 2 MG6 φ 2N
λβ 8
768π 2
MG
φN
2
8N
exp
+1 ,
√
β2
768π 2 (β − 2)2
λβ 8
where the second and third approximations are based on φ N v and φ f (β −
tively. The spectral index and its running are given by
ns − 1 = −
dn s
=
d ln k
8(3φ 2N + v 2 )MG2
−
8
β2
(φ 2N − v 2 )2
(320v 2 φ 2N + 192φ 4N )MG4
−
(φ 2N − v 2 )4
,
−
(118)
√
2)MG , respec-
(119)
320
β6
φN
MG
2
.
(120)
For β = 15 and N = 55, for example, we find n s = 0.964 and negligibly small running, and the self
coupling is given by λ 7 × 10−14 .
10.3.
Hybrid inflation model
Here we consider the case where vacuum energy drives inflation, which is distinct from the above
two classes. The potential in this regime is approximated as
V [φ] = V0 +
m2 2
φ ,
2
(121)
and so the slow-roll parameters are given by
1 m 4
V =
18 H
φ
MG
2
,
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ηV =
m2
,
3H 2
ξV = 0.
(122)
PTEP 2014, 06B103
J. Yokoyama
In this case one cannot determine the energy scale of inflation from the amplitude of curvature
fluctuations alone. Unless φ is significantly larger than MG , the spectral index is determined by
ηV as
2m 2
> 1,
(123)
n s − 1 2ηV =
3H 2
which is disfavored by current observations. The running spectral index takes a small value,
2 3 dn s
φ 2
φ 2
2m
3
16V ηV =
(n s − 1)
,
(124)
d ln k
3H 2
MG
MG
as does the tensor-to-scalar ratio,
2 2 φ 2
φ 2
2m
2
r =2
(n s − 1)
.
3H 2
MG
MG
(125)
One should note that if the second term dominates over the vacuum energy in (121), the prediction
becomes closer to a large-field model and the spectral index can be smaller than unity.
10.4.
Non-canonical models and multi-field models
So far we have focused on the amplitude and spectral index of primordial curvature and tensor perturbations. Indeed, in single-field slow-roll inflation there is only one fluctuating degree, so only
adiabatic fluctuations are generated with no isocurvature counterparts. Furthermore, the curvature
perturbation generated in slow-roll inflation behaves as a practically free massless scalar field during
inflation, and so its vacuum fluctuation is Gaussian distributed. This is why the power spectrum fully
quantifies the properties of fluctuations.
In some classes of non-canonical inflation models, including DBI inflation [79,80], ghost inflation [81], k-inflation [32], and G-inflation [33], the inflaton may change rapidly and its interaction
may not be negligible. In such circumstances, deviation from Gaussian statistics may be significant
[66,82–85]. Furthermore, in models with multiple fluctuating fields besides the inflaton, one may
also expect a large deviation from Gaussian [86–88].
Much work has been done in recent years to quantify the skewness or the bispectrum of the temperature anisotropy, and a number of non-standard models were proposed that predicted observationally
falsifiable magnitudes of the deviation from Gaussian using the expected Planck data [89]. As a result,
these models were indeed basically falsified by Planck, which confirmed Gaussian distribution with
a relatively high accuracy [90]. So we do not go into further detail.
Another possible signature of multi-field models would be the presence of isocurvature modes.
But, again, CMB observations do not need any isocurvature components, at least on relevant
scales [76].
11.
Conclusion
The fact that we live in a large and old Universe with a large amount of entropy can be regarded as a
primary consequence of inflation in the early Universe. Any attempt to avoid inflation by providing
alternative explanations to the other predictions of inflationary cosmology, namely, generation of
adiabatic density perturbation with nearly scale-invariant spectrum as well as tensor perturbation
should also have a sound explanation for the aforementioned global properties of the spacetime.
Some people, however, try to refute this viewpoint by claiming that the global properties of the Universe, namely, homogeneity, isotropy, and longevity, had been known long before the first inflation
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models [14,17,30] were proposed so that they cannot be regarded as the predictions of inflationary
cosmology. This is true indeed. One should recognize, however, that theories in physics have double roles, namely, one to give a new prediction which may be justified or falsified by experiments
and observations, and the other to provide sensible explanations to known mysteries of the nature.
Inflationary cosmology certainly provides a simple mechanism to explain fundamental properties of
the global structure of space, playing the latter role of theoretical physics. Thus the abovementioned
refutation is a long way off the mark.
Furthermore, the simplest class of slow-roll inflation models predicted nearly scale-invariant, adiabatic, and Gaussian curvature and density fluctuations, all of which are now being proven to be true
by dedicated CMB experiments. In particular, the observed angular correlation between temperature
anisotropy and E-mode polarization clearly indicates that the observed fluctuations were generated
as super-Hubble fluctuations, as predicted by inflation [74].
We are tempted to interpret the fact that Planck did not confirm any sizable non-Gaussianity [90]
despite the recent efforts of many theorists to make models of non-Gaussian fluctuations as indicating that nature preferred the simplest inflation models. This, on the other hand, highlights the
difficulty in singling out the right model responsible for our Universe from the small numbers of
observational clues available. For example, among the models occupying the central regions of the
likelihood contours in the (n s , r ) plane at the moment, it is very difficult to distinguish between the
R 2 model [30,91,92] and a Higgs inflation model with a large and negative non-minimal coupling
[93–95] from the observations of perturbation variables alone. We should also consider the reheating
processes of these models for more detailed comparison, which may be observationally probed by
future space-based laser interferometers such as DECIGO [96,97].
Our final goal is to identify the inflaton in a particle physics context. As argued above, the more
observations agree with the standard prediction, the harder it is to single out the inflaton from purely
observational grounds, so particle-physics considerations are also very important. Currently we know
only one scalar field in nature experimentally. In this sense, the case where the standard Higgs field
is responsible for inflation is interesting and deserves serious consideration. Since its self coupling
is too large to produce an appropriate amplitude of curvature perturbation, several remedies have
been proposed besides the non-minimally coupled model mentioned above [93–95], namely new
Higgs inflation [98], Higgs G-inflation [99], and running kinetic inflation [100], all of which can be
treated on an equal footing as a variant of generalized G-inflation [101]. Since the predictions of each
model are different, if particle physics identity is specified, observations may show the form of these
additional interactions.
Further open questions are how inflation started at the birth of the universe, and how it ended, that
is, how the Universe was reheated. Observationally, as mentioned above, future space-based laser
interferometers such as DECIGO [96,97] can provide us with useful information on the reheating
history after inflation [102–104]. Hence we hope these satellites will be put into action as soon as
possible.
Note Added
After the original version of this manuscript was submitted on February 14, 2014, on March 17,
2014 the BICEP2 collaboration announced detection of B-mode polarization of the CMB [105] and
claimed that it originates from tensor perturbations generated during inflation corresponding to r 0.2 (or r 0.16 after subtracting foregrounds based on the best foreground model), disfavoring
synchrotron or dust at 2.3 σ and 2.2 σ , respectively. While their discovery would have profound
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J. Yokoyama
implications [106], further confirmation is certainly desired to determine the actual value of r through
multi-frequency observations in a wider area of the sky. Hence I decided not to change Sect. 10 of
the manuscript and left “X” in the title as it is. We hope the forthcoming Planck data will achieve
the above desire, so that we should be able to put X = 4 to refer to the year of the discovery of the
primordial tensor mode.
Acknowledgements
This manuscript was written in association with the UT-Quest project under the support of Grant-in-Aid for
Scientific Research on Innovative Areas No. 21111006. It is partially based on a Japanese text book written by
the present author and edited by Katsuhiko Sato [107].
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
[28]
[29]
[30]
[31]
[32]
[33]
[34]
[35]
[36]
[37]
[38]
[39]
G. Gamow, Phys. Rev., 70, 572 (1946).
E. Hubble, Proc. Nat. Acad. Sci., 15, 168 (1929).
G. Lemaître, Annales de la Societe Scientifique de Bruxelles, 47, 49 (1927).
A. A. Penzias and R. W. Wilson, Astrophys. J., 142, 419 (1965).
R. A. Alpher, H. Bethe, and G. Gamow, Phys. Rev., 73, 803 (1948).
C. Hayashi, Prog. Theor. Phys., 5, 224 (1950).
R. H. Dicke and P. J. E. Peebles, The big bang cosmology – enigmas and nostrums. In General
Relativity, An Einstein Centenary Survey, eds. S. W. Hawking and W. Israel (Cambridge University
Press, Cambridge, 1979).
H. Georgi and S. L. Glashow, Phys. Rev. Lett., 32, 438 (1974).
M. Yoshimura, Phys. Rev. Lett., 41, 281 (1978).
G. ’t Hooft, Nucl. Phys., B 79, 276 (1974).
A. M. Polyakov, JETP Lett., 20, 194 (1974).
T. W. B. Kibble, J. Phys., A 9, 1387 (1976).
J. Preskill, Phys. Rev. Lett., 43, 1365 (1979).
K. Sato, Mon. Not. Roy. Astron. Soc., 195, 467 (1981).
K. Sato, Phys. Lett., B 99, 66 (1981).
M. B. Einhorn and K. Sato, Nucl. Phys., B 180, 385 (1981).
A. H. Guth, Phys. Rev., D 23, 347 (1981).
K. Sato, M. Sasaki, H. Kodama, and K.-i. Maeda, Prog. Theor. Phys., 65, 1443 (1981).
K. Sato, H. Kodama, M. Sasaki, and K.-i. Maeda, Phys. Lett., B 108, 103 (1982).
A. D. Linde, Phys. Lett., B 108, 389 (1982).
A. Albrecht and P. J. Steinhardt, Phys. Rev. Lett., 48, 1220 (1982).
A. D. Linde, Phys. Lett., B 129, 177 (1983).
V. F. Mukhanov and G. V. Chibisov, JETP Lett., 33, 532 (1981).
A. A. Starobinsky, Phys. Lett., B 117, 175 (1982).
S. W. Hawking, Phys. Lett., B 115, 295 (1982).
A. H. Guth and S. Y. Pi, Phys. Rev. Lett., 49, 1110 (1982).
G. F. Smoot et al., Astrophys. J., 396, L1 (1992).
H. V. Peiris et al., Astrophys. J. Suppl., 148, 213 (2003).
P. A. R. Ade et al., arXiv:1303.5082.
A. A. Starobinsky, Phys. Lett., B 91, 99 (1980).
Class for Physics of the Royal Swedish Academy of Sciences, AAPPS Bulletin, 20–21, 11 (2011).
C. Armendariz-Picon, T. Damour, and V. F. Mukhanov, Phys. Lett., B 458, 209 (1999).
T. Kobayashi, M. Yamaguchi, and J. Yokoyama, Phys. Rev. Lett., 105, 231302 (2010).
T. Kobayashi, M. Yamaguchi, and J. Yokoyama, Prog. Theor. Phys., 126, 511 (2011).
D. S. Goldwirth and T. Piran, Phys. Rept., 214, 223 (1992).
A. Kashlinsky, I. I. Tkachev, and J. Frieman, Phys. Rev. Lett., 73, 1582 (1994).
J. Garriga and V. F. Mukhanov, Phys. Lett., B 458, 219 (1999).
D. S. Salopek, Phys. Rev. Lett., 69, 3602 (1992).
L. Boubekeur and D. H. Lyth, JCAP, 0507, 010 (2005).
24/26
PTEP 2014, 06B103
[40]
[41]
[42]
[43]
[44]
[45]
[46]
[47]
[48]
[49]
[50]
[51]
[52]
[53]
[54]
[55]
[56]
[57]
[58]
[59]
[60]
[61]
[62]
[63]
[64]
[65]
[66]
[67]
[68]
[69]
[70]
[71]
[72]
[73]
[74]
[75]
[76]
[77]
[78]
[79]
[80]
[81]
[82]
[83]
[84]
[85]
[86]
[87]
[88]
[89]
[90]
[91]
[92]
[93]
[94]
[95]
J. Yokoyama
A. Vilenkin, Phys. Rev. Lett., 72, 3137 (1994).
A. D. Linde, Phys. Lett., B 327, 208 (1994).
A. D. Linde, Phys. Rev., D 49, 748 (1994).
D. H. Lyth, Phys. Rev. Lett., 78, 1861 (1997).
N. Tetradis, Phys. Rev., D 57, 5997 (1998).
L. E. Mendes and A. R. Liddle, Phys. Rev., D 62, 103511 (2000).
J. Yokoyama, Phys. Lett., B 212, 273 (1988).
J. Yokoyama, Phys. Lett., B 231, 49 (1989).
J. Yokoyama, Phys. Rev. Lett., 63, 712 (1989).
K. Kamada, K. Nakayama, and J. Yokoyama, Phys. Rev., D 85, 043503 (2012).
K. Kamada, Y. Miyamoto, and J. Yokoyama, JCAP, 1210, 023 (2012).
J. H. Traschen and R. H. Brandenberger, Phys. Rev., D 42, 2491 (1990).
L. Kofman, A. D. Linde, and A. A. Starobinsky, Phys. Rev. Lett., 73, 3195 (1994).
L. Kofman, A. D. Linde, and A. A. Starobinsky, Phys. Rev., D 56, 3258 (1997).
L. F. Abbott, E. Farhi, and M. B. Wise, Phys. Lett., B 117, 29 (1982).
A. Albrecht, P. J. Steinhardt, M. S. Turner, and F. Wilczek, Phys. Rev. Lett., 48, 1437 (1982).
J. Yokoyama, Phys. Rev., D 70, 103511 (2004).
M. S. Turner, Phys. Rev., D 28, 1243 (1983).
T. S. Bunch and P. C. W. Davies, Proc. Roy. Soc. Lond., A 360, 117 (1978).
A. Vilenkin and L. H. Ford, Phys. Rev., D 26, 1231 (1982).
A. D. Linde, Phys. Lett., B 116, 335 (1982).
D. Polarski and A. A. Starobinsky, Class. Quant. Grav., 13, 377 (1996).
A. Vilenkin, Nucl. Phys., B 226, 527 (1983).
J. M. Bardeen, Phys. Rev., D 22, 1882 (1980).
H. Kodama and M. Sasaki, Prog. Theor. Phys. Suppl., 78, 1 (1984).
V. F. Mukhanov, H. A. Feldman, and R. H. Brandenberger, Phys. Rept., 215, 203 (1992).
D. Seery and J. E. Lidsey, JCAP, 506, 003 (2005).
A. A. Starobinsky, JETP Lett., 42, 152 (1985).
M. Sasaki and E. D. Stewart, Prog. Theor. Phys., 95, 71 (1996).
A. A. Starobinsky, JETP Lett., 30, 682 (1979).
V. A. Rubakov, M. V. Sazhin, and A. V. Veryaskin, Phys. Lett., B 115, 189 (1982).
L. F. Abbott and M. B. Wise, Nucl. Phys., B 244, 541 (1984).
C. L. Bennett et al., Astrophys. J. Suppl., 148, 1 (2003).
D. N. Spergel et al., Astrophys. J. Suppl., 148, 175 (2003).
C. L. Bennett et al., Astrophys. J. Suppl., 148, 1 (2003).
E. Komatsu et al., Astrophys. J. Suppl., 192, 18 (2011).
G. Hinshaw et al., Astrophys. J. Suppl., 208, 19 (2013).
P. A. R. Ade et al., arXiv:1303.5076.
S. Dodelson, W. H. Kinney, and E. W. Kolb, Phys. Rev., D 56, 3207 (1997).
M. Alishahiha, E. Silverstein, and D. Tong, Phys. Rev., D 70, 123505 (2004).
E. Silverstein and D. Tong, Phys. Rev., D 70, 103505 (2004).
N. Arkani-Hamed, P. Creminelli, S. Mukohyama, and M. Zaldarriaga, JCAP, 404, 001 (2004).
D. Langlois, S. Renaux-Petel, D. A. Steer, and T. Tanaka, Phys. Rev. Lett., 101, 061301 (2008).
D. Babich, P. Creminelli, and M. Zaldarriaga, JCAP, 408, 009 (2004).
T. Kobayashi, M. Yamaguchi, and J. Yokoyama, Phys. Rev., D 83, 103524 (2011).
A De Felice and S. Tsujikawa, Phys. Rev., D 84, 083504 (2011).
N. Bartolo, S. Matarrese, and A. Riotto, Phys. Rev., D 69, 043503 (2004).
T. Suyama and M. Yamaguchi, Phys. Rev., D 77, 023505 (2008).
D. H. Lyth and Y. Rodriguez, Phys. Rev. Lett., 95, 121302 (2005).
N. Bartolo, E. Komatsu, S. Matarrese, and A. Riotto, Phys. Rept., 402, 103 (2004).
P. A. R. Ade et al., arXiv:1303.5084.
A. Vilenkin, Phys. Rev., D 32, 2511 (1985).
M. B. Mijic, M. S. Morris, and W.-M. Suen, Phys. Rev., D 34, 2934 (1986).
J. L. Cervantes-Cota and H. Dehnen, Nucl. Phys., B 442, 391 (1995).
F. L. Bezrukov and M. Shaposhnikov, Phys. Lett., B 659, 703 (2008).
A. O. Barvinsky, A. Yu. Kamenshchik, and A. A. Starobinsky, JCAP, 811, 021 (2008).
25/26
PTEP 2014, 06B103
[96]
[97]
[98]
[99]
[100]
[101]
[102]
[103]
[104]
[105]
[106]
[107]
J. Yokoyama
N. Seto, S. Kawamura, and T. Nakamura, Phys. Rev. Lett., 87, 221103 (2001).
S. Kawamura et al., Class. Quant. Grav., 23, S125 (2006).
C. Germani and A. Kehagias, Phys. Rev. Lett., 105, 011302 (2010).
K. Kamada, T. Kobayashi, M. Yamaguchi, and J. Yokoyama, Phys. Rev., D 83, 083515 (2011).
K. Nakayama and F. Takahashi, JCAP, 1011, 009 (2010).
K. Kamada, T. Kobayashi, T. Takahashi, M. Yamaguchi, and J. Yokoyama, Phys. Rev., D 86, 023504
(2012).
N. Seto and J. Yokoyama, J. Phys. Soc. Jap., 72, 3082 (2003).
K. Nakayama, S. Saito, Y. Suwa, and J. Yokoyama, Phys. Rev., D 77, 124001 (2008).
K. Nakayama, S. Saito, Y. Suwa, and J. Yokoyama, JCAP, 806, 20 (2008).
P. A. R. Ade et al., arXiv:1403.3985.
J. Yokoyama, AAPPS Bulletin, 24, 27 (2014).
J. Yokoyama, Utyuron I (Nihon Hyoron-sha, Tokyo, 2012), 2nd ed.
26/26