features in dbi inflation - university of...

Features in DBI Inflation Vinícius Miranda

Outline

Vinícius Miranda – Final Presentation - Advanced CMB - University of Chicago – December 2012

Part I – Introduction (5 slides) • Definition canonical x noncanonical scalar fields

• Equations of motion

• Slow Roll Part-II Perturbations (3 slides)

• Dynamics of perturbations

• Generalize Slow Roll (GSR) formalism

Part III – DBI Inflation (4 slides) • Definition

• DBI equations of motion - Initial conditions

Outline


Part IV – Features in DBI Power Specturm (18 slides) • Definition

• Does GSR works?

• Analytical Solution

• How features affects CMB Cl’s

Extra - DBI Bispectrum (5 slides) • Definition

• GSR

• Comparison with Exact Solution

Part I - Introduction

Intro: Scalar Fields - Action

( )4 4116

,S d x gR d x g pG

Xφ φπ

= − +∫ ∫12

X g µνµ νφ φ= ∂ ∂

Special case: Canonical Field

( ), ( )p XX Vφ φ φ= −


Ref: hep-th/9904176, astro-ph/0406496


Intro: Scalar Fields - Canonical

Notation – Canonical Field 0

N

dNaH

η = ∫

end

ln aa

N

≡

0 1φ φ φ= +

Equation of Motion 2

0 02

0

3 0Hd d dVdt dt dφ φ

φ+ + =

Ref: http://background.uchicago.edu/~whu/Courses/Ast408_09/ast4PT_02.pdf

Intro: Cookbook of Inflation Canonical Field

“Slow Roll”

'H

VV

∝

0

0

3 0d dVdt

Hd

φφ

+ ≈

Field position specifies kinetic energy

Slow Roll parameters '''H

VV

η ∝

41 2s H Hn η− −≈ Power Spectrum tilt


Intro: Scalar Fields - Noncanonical

Notation – Noncanonical Field 0 sN

c da

NsH

= ∫

Effective sound speed (comoving gauge)

( ) ( )2

2 22/

//s

p pc

p XpX

X Xφ φ

φ φ φ

δδρ

=∂ ∂

=∂ ∂ + ∂ ∂


Ref: astro-ph/0406496

Intro: Scalar Fields - Noncanonical

No more direct connection between derivatives of the field and slow roll parameters

lnH N

Hdd

≡ −12

ln HH H

ddN

η = −

1ln sddN

cσ ≡

Power Spectrum tilt

14 21 s H Hn η σ−− ≈ +


Ref: astro-ph/1104.4500

Part II - Perturbations

Mukanov Equation


2

2 2 2

2 (ln ))

1) ((( )

d g sd ks ks

y ys

yk

+ = −

Noncanonical

2 sy ukc=1

01

ddN

u z φφ−

= −

( )1/2

s

pz

ac Hρ +

=

g zz

= −

Ref: Phys. Rept. 215, 203 (1992), astro-ph/0910.2237 Ref: hep-th/9904075, astro-ph/1104.4500

Generalize Slow Roll


Source depends on slow roll parameters => small

Green Funcion Method * *

0 0 0 00 02

( )( ) (ln ) ( ( ) ( ) ( )(2

))2 x

y x y u y xi dux u yy x yu

u y ui

g∞ −

+−

≈ ∫

Problem: curvature is not constant outside the horizon

GSR “guess” the higher order corrections to fix that issue

Ref: astro-ph/0110322 , astro-ph/0910.2237, astro-ph/1104.4500

Generalize Slow Roll

Noncanonical

min

2(GSR0) minln (ln ( ) '(ln )) dsG

ss W ks G s

η

∞= +∆ ∫

In slow roll ( )2' 1sG g n= −−

3 2

3sin(2 ) 3cos(2 ) 3sin(2 )( )2 2

x x xW xx x x

− −=

Vinícius Miranda – Final Presentation - Advanced CMB - University of Chicago – Dezember 2012

Ref: astro-ph/0910.2237, astro-ph/1104.4500

Part III – DBI

Dirac-Born-Infeld(DBI) Inflation

Phenomenological model motivated by string theory

21 1( )

( , ) ( ) ( )p X TT

VXφ φ φφ

= − − −

T(φ) is related to tension in branes (braneworld) theories

21( )sXc

T φ−= 1 (1 ) ( )

s

Tc

Vρ φ φ

= − +


Ref: hep-th/0310221



2

/ 1l

)im 1

( ( )sX T

X XT

cTφ φ

= − +

2

/ 1lim (

( ), ) ( )

X Tp X X V X

Tφ φ

φ

= − +

Limit to canonical field

Ref: hep-th/0310221

Limit to cs equals zero ( )2

TX φ=

( )X

T φ

Ratio critic to specify the dynamics of the system



0

0

3 sd dVc

dH

dtφ

φ≈

How flat the potential need to be?

In slow roll the following equation is true

Slow Roll

1''~'

( )H s H sV cV

cη σ+ '~H s Vc V

Because of the cs factor, potential does not to be very flat

DBI – Equations of Motion


22 3 2

2 2

1 1 (1 2 )3 (1 )2

Ns s s s

Hd d dV dTc cdN H dN H d d

c cφ φφ φ

= + + −

− − +

2

2N N

s

HH c

φ= −

Ref: hep-th/0310221, astro-ph/1207.2186

Part III – Features in DBI Power Spectrum

DBI - Features

Sharp transitions

Potential Warp

[ ]smooth( ) ( ) 1 ( )FT bT φ φ φ= +[ ]smooth( ) ( ) 1 ( )FV bV φ φ φ= +

( ) tanh 1s

dF φ φφ − = −

Dominant Interesting when

1sc ≈ 1sc


Ref: astro-ph/0910.2237 Ref: astro-ph/1207.2186

Potential Warp

DBI - Features

23 sH

N

V cHφη

φ

≡

+

1 N

TTφσ φ≡


Slow roll parameters 2

11 1

2 2 1ss

Hs

Hs

c c cc

η η σ+ −= −

+

211 (1 ) (1 )S s Hc cσ σ η= − + −


DBI - Features - Power Spectrum


What are the effects in the Power Spectrum caused by a

warp sharp feature? (1) Step of order b

(2) Oscillations that only damps away when

feature ~( )N s

dk ddk

Nφ φ

Can GSR reproduce these effects?

YES! Ref: astro-ph/1207.2186

DBI - Features Analytical Solution

Analytical Solution for small and sharp features

0 1φ φ φ= +

( )2200

0 00

3( )111

2( ) [ ( ) 2]

2ss Ns Nc

bF b F ecddφ φ φφ

−−−

= + +

( ) tanh 1s

dF φ φφ − = −

Because the feature is sharp, we can approximate cs0 by it’s value at φs Ref: astro-ph/1207.2186




Analytical Numerical

NddNφφ ≡


0.01b = −1212 0d −×=

0 0.0506sc =

( ) 8.163 G c pSs φ =





0.01b = −1212 0d −×=

0 0.0506sc =

( ) 8.163 G c pSs φ =

21( )sXc

T φ−=




1ln sd cdN

σ ≡


0.01b = −1212 0d −×=

0 0.0506sc =

( ) 8.163 G c pSs φ =




12

ddNσσ ≡


0.01b = −1212 0d −×=

0 0.0506sc =

( ) 8.163 G c pSs φ =



lnH N

Hdd

≡ −



0.01b = −1212 0d −×=

0 0.0506sc =

( ) 8.163 G c pSs φ =


ln HH H

ddN

η ≡ −




0.01b = −1212 0d −×=

0 0.0506sc =

( ) 8.163 G c pSs φ =


22

HH H H

ddNηδ η η −+≡




0.01b = −1212 0d −×=

0 0.0506sc =

( ) 8.163 G c pSs φ =




0.01b = −1212 0d −×=

0 0.0506sc =

( ) 8.163 G c pSs φ =

2 2 1' 22 5 83 3

13 3H

s

aHsGc

σ δ σ η +

≈ − + − − −




We have all the pieces to develop an analytical approximation of the power spectrum

[ ]21 1 2 3( ) '( ) )l (n s

s s sd

ksC W ks C W ks C Y ksx

= +

∆

+

21 0

02

0

20 0

3

2(1 )

123 1

5 2 34

s

s

s

s s

C c b

cCc

c

b

C bc

= −

− = −

−

=

+−

2

3

6 cos(2 ) (4 3)sin(2 )( ) x x x xY xx

+ −=

3 2

3sin(2 ) 3cos(2 ) 3sin(2 )( )2 2

x x xW xx x x

− −=


' 0 )( ) (s ss G ss φ∆ = ≈=


Last Piece: Damping

0)im ( 1l

xW x

→=

Step Oscillation

0'( ) 0lim

xW x

→=

(lim ) 0x

W x→∞

= '( ) 3cos(2 )limx

W x x→∞

= −

Correction at x ~ 1

0)im ( 0l

xY x

→=

(lim ) 0x

Y x→∞

=

( )sinh( )

yyy

=s

d

y ksx

=1

lndNd

d sx

d dφφ

π π≈=





( ) 8.163 G c pSs φ =0.005b = − 122.44 10d −×=





0.25b = −122.44 10d −×=

0 0.0506sc =

( ) 8.163 G c pSs φ =

High b

Need non-linear rescaling (1) Amplitude of C1,C2,C3

(2) Damping



High b 20

1

02

0

20 0

3

1 21 2

123 1 1 2

5 2 34 1 2

n

l

1l

n s

s

s

s s

d

bcCb

c bCc b

c c bb

dd s

C

xd

φπ

= −

−=

−

−+ −

− −=

−

=




What are the effects of a feature on Cl space? 1

21Mpc0.05

sn

skA

−

−

=∆

0.218b = −

8163ss =

0.0507sc =

2 ln 12∆ ≈Improves likelihood


Extra – Bispectrum

Bispectrum - Introduction

• Bispectrum is zero if the field is gaussian

• Bispectrum probes field interaction - Free fields are gaussian

• Simpler to work with curvature perturbation

( )† 3( ), ( ) 2ˆ ˆ ' ( ')a a π δ + = q q q q q


21 3

3 31 1 33 22( (2 ) ( ) () ( )) ,( ,)t t t k k kπ δ∗ ∗ ∗⟨ ⟩ += +k k k k k k

* †ˆ ˆ( ) ( )( ) ( ) ( )q q qt t a at= −+q q

Bispectrum - Introduction

Curvature evolves according to the time evolution operator

0

0( , x )) e p (t

It

U t t t T i tH t d∗

= ∞ = −

= −

∫

Legendre transformation (third order in perturbation theory)

3( )I tH S= −

2 21 3 1 3( ) ( ) ( ) ( ) ( ) ( ) )2 (t

t t t t t t Hi d tt∗

∗ ∗ ∗ ∗ ∗ ∗−∞

⟨ ⟩ = ℜ − ⟨ ⟩

∫k k k k k k

We need the third order action!


Bispectrum - Noncanonical


The main contribution for equilateral triangle is

1322

1 11( )Is

dc

dH d

Ht

xη−

=

∫ k

2 3

2 3

23

2 2 2sH

i is

dd cac dt dt a

− ∂ = ∂

kkk k

Problems: (1) Divergent integral – needs regularization

(2) General Triangle – slow calculation

Bispectrum - GSR 7

2 2 2 221 2 3 1 2 3 1 1 33 2 1 2 3

1

, , ( ) (( ) ( ) , ,) ( ) ( ) ( )ii

k kk k k k k kk k k k k T k I k=

= ∆ ∆ ∆ + +∑

( )1 2 3 1 2 3( ) ( )ln (l )ni i iI k Wk k d sS s k kk s+ + +=+ ∫Advantages:

(1) Fast – Calculate integrals only once

(2) Integrals not divergent.

Problem: (1) Squeeze limit : tricky cancelation


Bispectrum - GSR


0 0.0506sc =

122.44 10d −×=

0.005b = −

GSR Numerical

Thank you

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