Investigation of multi-field DBI...
Transcript of Investigation of multi-field DBI...
Investigation of
multi-field DBI inflation
Taichi Kidani
This talk
- Motivation for the DBI inflation model
- Background dynamics
- Perturbation
- Summary
Motivation for the DBI inflation model
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
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
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
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,
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!
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
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…
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
Background dynamics
Specific model with
1.2106
0 0.004
V0 51012
g 3109
FIG8.x field and sound speed
FIG9.slow-roll parameters
Slow-roll background dynamics
Perturbation
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
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)
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
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
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
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.
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
Thank you for listening.