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
Vinícius Miranda – Final Presentation - Advanced CMB - University of Chicago – December 2012
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= −
Vinícius Miranda – Final Presentation - Advanced CMB - University of Chicago – December 2012
Ref: hep-th/9904176, astro-ph/0406496
Vinícius Miranda – Final Presentation - Advanced CMB - University of Chicago – December 2012
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
Vinícius Miranda – Final Presentation - Advanced CMB - University of Chicago – December 2012
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
=∂ ∂
=∂ ∂ + ∂ ∂
Vinícius Miranda – Final Presentation - Advanced CMB - University of Chicago – December 2012
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−− ≈ +
Vinícius Miranda – Final Presentation - Advanced CMB - University of Chicago – December 2012
Ref: astro-ph/1104.4500
Part II - Perturbations
Mukanov Equation
Vinícius Miranda – Final Presentation - Advanced CMB - University of Chicago – December 2012
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
Vinícius Miranda – Final Presentation - Advanced CMB - University of Chicago – December 2012
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
= − +
Vinícius Miranda – Final Presentation - Advanced CMB - University of Chicago – December 2012
Ref: hep-th/0310221
Dirac-Born-Infeld(DBI) Inflation
Vinícius Miranda – Final Presentation - Advanced CMB - University of Chicago – December 2012
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
Dirac-Born-Infeld(DBI) Inflation
Vinícius Miranda – Final Presentation - Advanced CMB - University of Chicago – December 2012
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
Vinícius Miranda – Final Presentation - Advanced CMB - University of Chicago – December 2012
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
Vinícius Miranda – Final Presentation - Advanced CMB - University of Chicago – December 2012
Ref: astro-ph/0910.2237 Ref: astro-ph/1207.2186
Potential Warp
DBI - Features
23 sH
N
V cHφη
φ
≡
+
1 N
TTφσ φ≡
Vinícius Miranda – Final Presentation - Advanced CMB - University of Chicago – December 2012
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= − + −
Ref: astro-ph/1207.2186
DBI - Features - Power Spectrum
Vinícius Miranda – Final Presentation - Advanced CMB - University of Chicago – December 2012
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
Vinícius Miranda – Final Presentation - Advanced CMB - University of Chicago – December 2012
Vinícius Miranda – Final Presentation - Advanced CMB - University of Chicago – December 2012
DBI - Features Analytical Solution
Analytical Numerical
NddNφφ ≡
Ref: astro-ph/1207.2186
0.01b = −1212 0d −×=
0 0.0506sc =
( ) 8.163 G c pSs φ =
DBI - Features Analytical Solution
Vinícius Miranda – Final Presentation - Advanced CMB - University of Chicago – December 2012
Analytical Numerical
Ref: astro-ph/1207.2186
0.01b = −1212 0d −×=
0 0.0506sc =
( ) 8.163 G c pSs φ =
21( )sXc
T φ−=
DBI - Features Analytical Solution
Vinícius Miranda – Final Presentation - Advanced CMB - University of Chicago – December 2012
Analytical Numerical
1ln sd cdN
σ ≡
Ref: astro-ph/1207.2186
0.01b = −1212 0d −×=
0 0.0506sc =
( ) 8.163 G c pSs φ =
DBI - Features Analytical Solution
Vinícius Miranda – Final Presentation - Advanced CMB - University of Chicago – December 2012
Analytical Numerical
12
ddNσσ ≡
Ref: astro-ph/1207.2186
0.01b = −1212 0d −×=
0 0.0506sc =
( ) 8.163 G c pSs φ =
DBI - Features Analytical Solution
Analytical Numerical
lnH N
Hdd
≡ −
Vinícius Miranda – Final Presentation - Advanced CMB - University of Chicago – December 2012
Ref: astro-ph/1207.2186
0.01b = −1212 0d −×=
0 0.0506sc =
( ) 8.163 G c pSs φ =
DBI - Features Analytical Solution
ln HH H
ddN
η ≡ −
Analytical Numerical
Vinícius Miranda – Final Presentation - Advanced CMB - University of Chicago – December 2012
Ref: astro-ph/1207.2186
0.01b = −1212 0d −×=
0 0.0506sc =
( ) 8.163 G c pSs φ =
Vinícius Miranda – Final Presentation - Advanced CMB - University of Chicago – December 2012
22
HH H H
ddNηδ η η −+≡
Analytical Numerical
DBI - Features Analytical Solution
Ref: astro-ph/1207.2186
0.01b = −1212 0d −×=
0 0.0506sc =
( ) 8.163 G c pSs φ =
DBI - Features Analytical Solution
Vinícius Miranda – Final Presentation - Advanced CMB - University of Chicago – December 2012
Analytical Numerical
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 +
≈ − + − − −
Ref: astro-ph/1207.2186
DBI - Features Analytical Solution
Vinícius Miranda – Final Presentation - Advanced CMB - University of Chicago – December 2012
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
− −=
Ref: astro-ph/1207.2186
' 0 )( ) (s ss G ss φ∆ = ≈=
DBI - Features Analytical Solution
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≈=
Ref: astro-ph/1207.2186
Vinícius Miranda – Final Presentation - Advanced CMB - University of Chicago – December 2012
DBI - Features Analytical Solution
Vinícius Miranda – Final Presentation - Advanced CMB - University of Chicago – December 2012
( ) 8.163 G c pSs φ =0.005b = − 122.44 10d −×=
Ref: astro-ph/1207.2186
DBI - Features Analytical Solution
Vinícius Miranda – Final Presentation - Advanced CMB - University of Chicago – December 2012
Ref: astro-ph/1207.2186
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
DBI - Features Analytical Solution
Vinícius Miranda – Final Presentation - Advanced CMB - University of Chicago – December 2012
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
= −
−=
−
−+ −
− −=
−
=
Ref: astro-ph/1207.2186
Vinícius Miranda – Final Presentation - Advanced CMB - University of Chicago – December 2012
DBI - Features Analytical Solution
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
Ref: astro-ph/1207.2186
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
Vinícius Miranda – Final Presentation - Advanced CMB - University of Chicago – December 2012
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!
Vinícius Miranda – Final Presentation - Advanced CMB - University of Chicago – December 2012
Bispectrum - Noncanonical
Vinícius Miranda – Final Presentation - Advanced CMB - University of Chicago – December 2012
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
Vinícius Miranda – Final Presentation - Advanced CMB - University of Chicago – December 2012
Bispectrum - GSR
Vinícius Miranda – Final Presentation - Advanced CMB - University of Chicago – December 2012
0 0.0506sc =
122.44 10d −×=
0.005b = −
GSR Numerical
Thank you