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Investigation of
multi-field DBI inflation
Taichi Kidani
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This talk
- Motivation for the DBI inflation model
- Background dynamics
- Perturbation
- Summary
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Motivation for the DBI inflation model
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Inflation
Rapid expansion in the early universe solves;
-Horizon problem
-Flatness problem
-Monopole problem
FIG1. History of the UniverseNASA/WMAP Science Team(2008): http://map.gsfc.nasa.gov/media/060915/index.html
Fluctuations in CMB
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CMB anisotropies
NASA(2010) :http://map.gsfc.nasa.gov/media/101080/index.html
FIG2.CMB
Almost scale invariant
Curvature perturbation
PR 2109
ns 1 0.05
Almost Gaussian
10 fNL
local 74
214 fNL
equil 266
Inflation can predict! We can distinguish different models
Komatsu et. al., 2010
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Brane world scenario
In string theory, we have 10 dimensions.
Our Universe may be on a (mem)brane in the “bulk”.
Bulk:10 D
Brane:4D
y
ds2 h1/2 yK gdxdx h1/2 yK GIJ dy I dyJ
Brane Extra dimensionsFIG3.Brane in the bulk
h : warp factor
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DBI action
Lagrangian for a brane in the bulk: UTP det3
(In analogy with the Nambu-Goto action)
(Coming from interaction with the bulk or other branes)
P 1
f I D 1 V I
(γμν : induced metric on the 3-brane)
(DBI kinetic term)
(T3 : brane tension)
D det I
J 2 fX I
J
X IJ 1
2
I J
&where
f h
T3,
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Constraint on single field cases
Lidsey and Huston, 2007
Baumann and McAllister, 2006
ds2 d2 2dsX 5
5
Inflation is in: throatVV
max
FIG4.Calabi-Yau(courtesy of Jon Emery)
r 107
MP
r N 2
& WMAP data
UsingBoubekeur, Lyth,2005
&
1 ns 4 2 2s WMAP
Best-fit
(1-ns~0.013)
r 16cs
cs 106
fNL
equil 1
3cs
2
fNL
equil 1010
Too large!
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Multi-field case
Langlois, Renaux-Petel, Steer, Tanaka , 2008
R
S
1 TRS
0 TSS
R
S
*
R: Comoving curvature perturbation
S: Entropy perturbation
*: Horizon exit
φ
R
S
χ
FIG5.Curved trajectory
Sharp curve Large TRS Small cosΘ
cos 1
1 TRS
2
We can have both “small” cs and small fNLequil.
fNL
equil 35
108
1
cs
2cos2
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Multi-field potential
Chen, Gong, Koyama, Tasinato , 2010Potential:V
(Ouyang embedding case)
☆Angular mass mχ becomes:
Interaction with other branes and bulk.
FIG6.Potential
light: mχ2<1 tachyonic: mχ
2<0
The potential is naturally multi-field!!
(We have 5 angular directions + radial direction.)Complicated…
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Constant sound speed model
Copeland, Mizuno, Shaeri, 2010
V , 1
2 2 0
2 2
g
2
V0
4
f f06
&
cs 3
16 f0V0 3Inflationary attractor solution with
This potential has the essential feature of the potential
derived in string theory(transition in the angular direction)!!
We can analyse this model fully numerically! FIG7.Two- field potential
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Background dynamics
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Specific model with
1.2106
0 0.004
V0 51012
g 3109
FIG8.x field and sound speed
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FIG9.slow-roll parameters
Slow-roll background dynamics
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Perturbation
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Equations of motion
DecompositionAdiabatic perturbation vσ
Entropic perturbation vs
vk vsk
cs
2k2 z
z
vk
z
zvsk 0
vsk vk
cs
2k2
a2s
2
vsk
z
zvk 0
z a
d
dtHcx
3 / 2
a
cs
where &
numerically solve
vσk & vsk
PR k 3
2 2vk
2
z2
TRS PR
P*&
ξ : coupling
φ
σ
S
χ
FIG10.field decomposition
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Coupling and TRS
PR 1 TRS
2 P*Power spectrum for R:
where PR*=PS*=P*(* :around horizon exit)
ξ is non-negligible only during the curve.
TRS is 0 if ξ is 0 all the time.
(ξ quantifies how much PR is amplified)
PR is sourced by PS when the trajectory curves!
(Note: PR can be observed in CMB observation)
FIG11.coupling
dR
dt
aS (on super horizon scales)
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Delta-N formalism
n
n
aa
n
a
aaa
nN
nNxt ***2
21
21!
1,
This is true only if all the fields are slow-roll!
In this model, we have two fields φ
and χ, and consider the case where
N
N
2N
2
2N
2, ,…
t2,x N N
*
1
2
2N
2*
2(* : Initial hypersurface)
Sasaki, Stewart, 1995
Lyth, Malik, 1995
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Bispectrum
22
2
22
3
3
3
NNN
is called “equilateral type”. This vanishes
if δχ is Gaussian.
Is called “local type” and this has some value even
with Gaussian δχ
fNL
local
22
N,
N,2
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Numerical results
Full numerical result:
PR 2.3109
(all compatible with WMAP)
Results by delta-N:
TRS 103
fNL
equil 9.5
2.29109 PR 2.31109
(within 1% error)
fNL
local N,
N, 2 40
ns 0.972
FIG12.Curvature power spectrum in the long transition case
Kidani, Koyama, Mizuno, 2012
8107 r
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Summary
• DBI inflation is the most promising physical model to generate equilateral type non-G.
• However, single field model is strongly constrained in string theory.
• In string theory, multi-field models are natural due to the angular directions. Multi-field effects reduce the amount of equilateral type and suppress local type. Thus, we expect a tight connection between those types of non-G.
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Summary
• In toy models, we showed it is indeed possible to suppress equilateral type non-G and obtain large local type non-G so that they both satisfy the current observation.
• Measurements of both types of non-G by Plank will give us tight constraints on the form of potential in DBI inflation models.
fNL
local 5
fNL
equil 20
fNL
local 80
fNL
equil 500
WMAP PLANK
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Thank you for listening.