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