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S. Tsujikawa, J. Ohashi, S.Kuroyanagi, A. De Felice, arXiv: 1305.3044
Tokyo University of Science, Shinji Tsujikawa

Proposed to resolve several cosmological problems plagued
in big bang cosmology, e.g.,
Horizon
problem
Flatness
problems
Responsible for the
origin of structure
Starobinsky , Guth , Sato , Kazanas ( early 1980’s)
・・・Cosmic acceleration around the energy scale

Old inflation (Sato, Kazanas, Guth)
1.
602 citations
3.
Inflationary universe: A possible solution to the horizon
and flatness problems
Alan H. Guth*
Stanford Linear Accelerator Center, Stanford University,
Stanford, California 94305
Received 11 August 1980
4414
citations
2.
168 citations

l “Old’’ inflation
Inflationary models
Many inflationary models have been proposed so far.
l Curvature inflation
The higher-­‐order curvature term leads to inflation.
Lagrangian:
Starobinsky (1980)
Inflation occurs due to the first-­‐order phase transition of a vacuum.
Sato (1981), Kazanas (1980), Guth (1981)
l Slow-­‐roll inflation
(first model of inflation)
Inflation is driven by the potential energy of a scalar field.
New, chaotic, power-­‐law, hybrid, natural, extra-­‐natural, eternal, D-­‐term,
F-­‐term, brane, oscillating, tachyon, hill-­‐top, KKLMMT, … (too many)
l K-­‐inflation
Inflation is driven by the kinetic energy of a scalar field.
Ghost condensate, DBI, Galileon,…
l Inflation in extended theories of gravity.
Brans-­‐Dicke, Higgs, Horndeski’s theory,….
There are also multi-­‐field models.

Standard slow-­‐roll inflation
S =
⇤
d 4 x ⇥
ä
a = 8 G
3
V ( )
V (⇥)
Inflation occurs under the condition
g
⇥
R
16 G + X V (⇥)
On the FLRW background, the scale factor obeys
˙2
˙⇥2 ⇥
X = 1 (⇥ )2
2
V ( )
Slow-­‐roll
The field evolves slowly along a
nearly flat potential (slow-­‐roll).
In this case
H
Cosmic acceleration
ȧ
a ⇥ const. a e H inf t

Perturbations
Inflaton
fluctuations
⇥
P R
Primordial
spectrum of
curvature
perturbations
CMB temperature anisotropies
from inflation
k
Larger scales
Scalar spectral index
k
Smaller scales
( 68% CL )
The primordial power spectrum is nearly scale-­‐invariant.

S =
⇤
d 4 x ⇥
g
⇥
R
16 G + X V (⇥)
Scalar perturbations
Tensor perturbations
ds 2 =
(1 + 2A)dt 2 +2 i Bdtdx i + a 2 (t) [(1 + 2R) ij + h ij ] dx i dx j
Scalar
S (2)
s = dtd 3 xa 3 Q s Ṙ 2 c 2 s
a 2 ( R)2
For standard potential-­‐driven inflation
Q s =
˙2
c 2
2H 2 , s =1
The amplitude of curvature perturbations
can be estimated by solving the equation
of motion which follows from the above
action.
The conditions for the avoidance
of ghosts and Laplacian instabilities
Q s > 0 c 2 s > 0
,

The CMB observations can discriminate between different potentials.
Scalar spectrum:
where
Tensor spectrum:
where
Tensor-­‐to-­‐scalar ratio:
The Planck constraints are
r

Discrimination between inflaton potentials
Lyth bound:
Smaller r
(i) Large-­‐field inflation (e.g., chaotic inflation, 1983)
During inflation the field evolves
over the large-­‐distance:
(ii) Small-­‐field inflation (e.g., natural inflation, 1990)
During inflation the field evolves
over the small-­‐distance:
A. Linde (young version)
K. Freese
(iii) Hybrid inflation (1994)
Inflation ends due to the waterfall transition
to the global minimum.
A. Linde (current version)

Potential-­‐driven slow-­‐roll inflation
Planck 2013 results. XXIII. Isotropy and Statistics of the CMB : arXiv:1303.5083

S.T., J. Ohashi, S. Kuroyanagi, A De Felice, arXiv: 1305.3044
Potential−driven slow−roll inflation
10 0 Inclusion of high L data prefers
Planck+WP+BAO+high L
the smaller spectral index.
Planck+WP+BAO
4
power law inflation
10 −1
2
1
r 0.05
natural
2/3
10 −2
large−field
small−field
R 2 inflation
10 −3
very
small
field
hybrid2
hybrid1
0.94 0.95 0.96 0.97 0.98 0.99 1 1.01
n s

Inflation in extended theories of gravity
Most general single-­‐field scalar-­‐tensor theories with second-­‐order equations of motion
Horndeski (1974)
Deffayet et al (2011)
Charmousis et al (2011)
This action covers most of the dark energy models proposed in literature.
l Potential driven inflation:
l K-­‐inflation:
l f(R) gravity and scalar-­‐tensor gravity:
l Galileon:
l Gauss-­‐Bonnet coupling
:

Horndeski’s paper in 1973

Second-­‐order actions of scalar perturbations
We consider scalar metric perturbations
, ⇥, R
with the ADM metric
ds 2 = [(1 + ) 2 a(t) 2 e 2R (⇤⇥) 2 ] dt 2 +2⇤ i ⇥ dt dx i + a(t) 2 e 2R dx 2
We choose the uniform field gauge: =0
The second-­‐order action of scalar perturbations in the Horndeski’s theory reduces to
S 2 = dtd 3 xa 3 Q Ṙ2 c 2 s
a 2 ( R)2
Q>0 and c 2 s > 0 are required to avoid
ghosts and Laplacian instabilities.
where
Q = w 1(4w 1 w 3 +9w 2 2)
3w 2 2
, c 2 s = 3(2w2 1w 2 H w 2 2w 4 +4w 1 ẇ 1 w 2 2w 2 1ẇ 2 )
w 1 (4w 1 w 3 +9w 2 2 )
w 1 = MplF 2 4XG 4,X 2HX ˙G 5,X +2XG 5,
w 2 =2MplHF 2 2X ˙G 3,X 16H(XG 4,X + X 2 G 4,XX )+2˙(G 4, +2XG 4, X )
2H 2 ˙(5XG5,X +2X 2 G 5,XX )+4HX(3G 5, +2XG 5, X )
w 3 = 9MplH 2 2 F + 3(XP ,X +2X 2 P ,XX ) + 18H ˙(2XG 3,X + X 2 G 3,XX ) 6X(G 3, + XG 3, X )
+18H 2 (7XG 4,X + 16X 2 G 4,XX +4X 3 G 4,XXX ) 18H ˙(G 4, +5XG 4, X +2X 2 G 4, XX )
+6H 3 ˙(15XG5,X + 13X 2 G 5,XX +2X 3 G ,5XXX ) 18H 2 X(6G 5, +9XG 5, X +2X 2 G 5, XX )
w 4 = MplF 2 2XG 5, 2XG 5,X ¨

Power spectrum of curvature perturbations
where
Kobayashi et al. (2011)
De Felice and S.T. (2011)
where

Tensor power spectrum
Tensor perturbations:
The second-­‐order action is
where
The tensor power spectrum is
The tensor-­‐to-­‐scalar ratio is
These results can be readily used for concrete models of inflation.
e.g.,
where
These observables depend on the slope of the potential.

We can apply our general results to a variety of inflation models.
Standard potential-­‐driven inflation
P ( ,X)=X V ( )
f(R)
gravity
Brans-­‐Dicke theories
Non-­‐minimally coupled models
Galileon inflation
Field-­‐derivative couplings
Running kinetic couplings
k-­‐inflation
etc…
S = d 4 x g
M 2 pl
2 R + P ( ,X) G 3( ,X) + L 4 + L 5
L 4 = G 4 ( ,X) R + G 4,X [( ) 2 ( µ )(
µ
)]
L 5 = G 5 ( ,X) G µ (
µ
)
1
6 G 5,X[( ) 3 3( )( µ )(
µ
) + 2(
µ
)( )( µ )]

Inflation in f(R) gravity
The slow-­‐roll parameter is
Starobinsky (1980)
First model of inflation
Inflation occurs due to the presence
of the quadratic curvature term.
Inflation ends around
(quasi-­‐exponential expansion)

Planck constraints on Brans-Dicke gravity with the potential
Brans−Dicke theories
The observables are
10 0 Starobinsky’s f(R) model
10 −1
BD
=10 2
BD
=10 10
where N is the number of
e-­‐foldings from the end of
Inflation.
r 0.05
10 −2
BD
=1
BD
=0
10 −3
10 −4
BD
=−1.4
0.945 0.95 0.955 0.96 0.965 0.97 0.975 0.98
n s

Einstein frame
Jordan frame:
Confomral transformation
Einstein frame:
where
Einstein frame potential
Under observational
pressure
The potential is nearly flat.
Consistent with observations.

Higgs inflation
There are several ways to accommodate Higgs for inflation.
Bezrukov and Shaposhnikov (2008)
Nakayama and Takahashi (2010),
De Felice, S.T., Elliston, Tavakol (2011)
Germani and Keghagias (2010)
Kamada, Kobayashi,
Yamaguchi, Yokoyama (2010)

Planck constraints on non-minimally coupled chaotic inflation
Non−minimally coupled models
r 0.05
10 −1
10 −2
=0
(a)
=−6.0
×10 −3
10 0 Favored observationally.
(b)
=−2.0×10 −3
=−0.1
=−10 4
=0
(a) n=2
(b) n=4
10 −3
0.94 0.95 0.96 0.97 0.98
n s
We also have
/⇠ 2 ' 4.3 ⇥ 10 10

Running kinetic couplings
0.3
0.25
Running kinetic coupling models
µ=0
(b)
(a) n=2
(b) n=4
The dilatonic running coupling
leads to the decrease of r.
r 0.05
0.2
0.15
µ=0.1
µ=−0.03
µ=0
However both the quadratic
and quartic potentials are
outside the 68 % CL boundary.
0.1
0.05
(a)
µ=0.1
µ=10
µ=10
This comes from the fact that
Planck data give tight upper
bounds on the scalar spectral
index than that of WMAP.
0
0.945 0.95 0.955 0.96 0.965 0.97 0.975 0.98
n s

Field derivative couplings to the Einstein tensor
0.25
0.2
Field−derivative coupling models
(b)
(a) n=2
(b) n=4
This coupling leads to the slow-­‐down
of the velocity of inflaton, which
leads to the decrease of r.
For smaller M, the scalar spectral
index gets larger.
r 0.05
0.15
0.1
smaller M
(a)
Both the quadratic and quartic potentials
are outside the 68 % CLboundary.
0.05
A similar conclusion was reached for
the Galileon couplings.
0
0.945 0.95 0.955 0.96 0.965 0.97 0.975 0.98
n s

f NL
B R f NL
・・・nonlinear estimator of primordial scalar non-Gaussianity (NG)
R(k 1 )R(k 2 )R(k 3 )⇥ = (2⇥) 3 (3) (k 1 + k 2 + k 3 )B R (k 1 ,k 2 ,k 3 )
Deviation from the Gaussian distribution
Equilateral
Local
k 1 k 2
Observations
WMAP
f equil
NL
= 51 ± 272
Planck
k 1 k 2
k 3 k 3
f equil
NL
= 42 ± 75
Theory
Standard slow-­‐roll inflation
L = X V ( )
Observations
Theory
WMAP
f local
NL = 37 ± 40
Planck
f local
NL =2.7 ± 5.8

Non-Gaussianities in the Horndeski’s theory
l Local shape:
Under the slow-­‐variation approximation, the non-­‐linear estimator is
Same as that derived by
Maldacena in standard
slow-­‐roll inflation (2002).
Gao and Steer (2011)
A. De Felice and S.T. (2011)
The small local non-­‐Gaussianity in the squeezed limit is a generic
feature in the single-­‐field slow-­‐variation inflation.
l Equilateral shape:
The leading-­‐order non-­‐linear estimator is

Observational constraints on power-law k-inflation
The power-­‐law inflation can be realized by the Lagrangian
(dilatonic ghost condensate)
In this case the observables are
c s
>0.079
0.02 0.04 0.06 0.08 0.1
c s
Generally the k-­‐inflation models
are tightly constrained from the
equilateral type NG.

Summary
Overall, the single-­‐field small-­‐field models with
are favored.

Potentials favored from observations
(Einstein frame)
Natural inflation