DBI inflation
• A natural inflation model that can generate large equilateral NG
• Inflaton is identified as a position modulus of a probe brane in extra-dimesnions
A small sound speed enhances NG
• Single field models in string theory are severely constrained
1
D3
anti-D3
inflaton
235 1108
equiNL
s
fc
= −
UVφ φ<716 10sr c ε −= <
1 4 0.04 0.013sn ε− ±: :
equil 300NLf >
Huston et.al ’07, Bean et.al. ’07
Silverstein & Tong ‘03
Multi-field DBI inflation • DBI inflation is naturally multi-field (i.e. 6 extra-dimensions = 6 fields)
• Multi-field effects could ameliorate the problem • Trispectrum is easier to detect
2
δσ
sδ
adiabatic
entropy 2116
1s RS
r cT
ε=+
2 235 1 1108 1
equiNL
s RS
fc T
= −+
,H HS ss
ζ δσ δσ
= =& &
* *RST Sζ ζ= +
2 2equi equiNL RS NLT fτ ∝
Renaux-Petel, et.al ’08, ‘09, Arroja, Mizuno Koyama ‘08
Arroja, Mizuno Koyama ‘Tanaka ’09, Mizuno Arroja Koyama ’09, Renaux-Petel ‘09
DBI Galileons • Single field extension including higher order derivative
interactions
• A probe brane in 5D A general brane action that gives the 2nd order e.o.m
3
( )21 2 1s
h gK
c
µν µν µ ν
µν µ ν
µ
φ φ
γ φ
γ φ−
= + ∂ ∂
= − ∂ ∂
= = + ∂
( )41 2 3 4[ ] KGBS d x h K R hα α α α= − + + +∫
(5) (5)
(5) (5)
[ ][ ]GB
R gR gGB
KK
[ ]R h
DBI Relativistic Galileon terms
de Rham & Tolley ’10, Burrage et.al. ’11, Goon et.al.’11, Mizuno Koyama ’10, …
Multi-field DBI Galileons
• In higher-codimensions n>1, it becomes multi-field model and for even n>3, the only possible term is the induced gravity term
• In non-relativistic limit
4
( )41 2 [ ]S d x h R hα α= − − +∫ , 1,2,..I
Ih Iµν µν µ νη φ φ= + ∂ ∂ =
2( ) 1φ∂ <<
41 21 ( )2
I I JI J I I JS d x µ µ λ ν µ ν
µ µ ν λα φ φ α φ φ φ φ φ φ⎛ ⎞= − ∂ ∂ + ∂ ∂ ∂ ∂ ∂ ∂ − ∂ ∂⎜ ⎟⎝ ⎠∫ W
Hinterbichler et.al. ’10
Padilla et.al. ’10 ‘11
Model • Embed a brane in 10-dimensions
• Induced metric
• Action
5
Renaux-Petel, Mizuno, Koyama ‘11
Equations of motion
6
Background
• Friedman equation
• In the relativistic regime, induced gravity gives a term that breaks the null energy condition
• Conditions for inflation
7
2
2 3 1/21, 1Dp D
Mc fVM c h
>> <<
Second-order action
Avoid instabilities
Sound speeds Entropic sound speeds are always smaller than adiabatic one and they become the same in the DBI inflation
Linear perturbations
8
adiabatic
entropy
2 2
2 1/2D
fH Mc h
α =
1 ( 1)9 Dcα < <<
1 5 , 11 9a D e Dc c c cα
α−
= = <<−
e ac c≤
δσ
sδ
Non-Gaussianity
9
2 2
2 1/2D
fH Mc h
α =
• Two parameters Mostly it gives equilateral NG but for some choice of
parameters, there appears orthogonal type NG
10
log Dc
α
0RST =
DBI
equilateral orthogonal
Single field
0equiNLf <
0equiNLf >
0orthogNLf <
11
log Dc
α
10RST =
DBI
equilateral orthogonal
Multi field
0equiNLf <
0equiNLf >
0orthogNLf <
Conclusion • Multi-field DBI galileon a probe brane action with induced gravity
• Induced gravity may break the energy condition need to check the ghost condition on non-slow roll inflation background
• Slow-roll inflation can be realised if
• Cosmological perturbations are controlled by two parmaeters
constraints on these parameters are obtained It is possible to create orthogonal type NG and positive
12
2
2 3 1/21, 1Dp D
Mc fVM c h
>> <<
2 2
2 1/2D
fH Mc h
α =
equiNLf