Brane SUSY Breaking and Inflation – implications for ... · Tree – level BSB SUSY broken at...

Brane SUSY Breaking and Inflation – implications for scalar

fields and CMB distorsion Augusto Sagnotti

Scuola Normale Superiore, Pisa

Rencontres de Moriond EW 2013 La Thuile, March 2 – 9 2013

E. Dudas, N. Kitazawa, AS, P.L. B 694 (2010) 80 [arXiv:1009.0874 [hep-th]]. E. Dudas, N. Kitazawa, S. Patil, AS, JCAP 1205 (2012) 012 [arXiv:1202.6630 [hep-th]] P. Fré, A.S., A.S. Sorin, to appear E. Dudas, N. Kitazawa, S. Patil, AS, in progress

Brane SUSY Breaking

2

(Sugimoto, 1999) (Antoniadis, Dudas, AS, 1999) (Aldazabal, Uranga, 1999) (Angelantonj, 1999)

• SUSY : D9 (T > 0, Q > 0) + O9- (T < 0, Q < 0) SO(32)

A. Sagnotti – Moriond EW 2013

• Dualities : link different strings • Orientifolds : link closed and open strings

Brane SUSY Breaking

3


• BSB : anti-D9(T > 0, Q < 0) + O9+ (T > 0, Q > 0) USp(32)


• Dualities : link different strings • Orientifolds : link closed and open strings

Brane SUSY Breaking

4


BSB: Tension unbalance exponential potential

S10 =1

2·210

Zd10x

p¡g

©e¡ 2Á

¡¡R + 4 (@Á)2

¢¡ T e¡Á + : : :

ª

Flat space : runaway behavior

String-scale breaking : early-Universe Cosmology ? A. Sagnotti – Moriond EW 2013

Tree – level BSB

SUSY broken at string scale in open sector, exact in closed Stable vacuum Goldstino in open sector

A climbing scalar in d dim’s

5

S =1

2·2

ZdDx

p¡g

·R ¡ 1

2(@Á)2 ¡ V (Á) + : : :

¸• Consider the action for gravity and a scalar φ :

• Look for cosmological solutions of the type

ds2 = ¡ e 2B(t) dt2 + e 2A(t) dx ¢ dx

• Make the convenient gauge choice V (Á) e2B = M2

(Halliwell, 1987) ……………… (Dudas,,Mourad, 2000) (Russo, 2004) ………………..

¯ =

rd¡ 1

d¡ 2; ¿ = M ¯ t ; ' =

¯ Áp2

; A = (d¡ 1) A

• In expanding phase : Ä' + _'p

1 + _' 2 +¡1 + _' 2

¢ 1

2V

@V

@'= 0

• Let :

• OUR CASE : V = exp( 2 ° ') ¡! 1

2 V

@V

@ '= °


A climbing scalar in d dim’s

6

• γ < 1 ? Both signs of speed a. “Climbing” solution (ϕ climbs, then descends):

b. “Descending” solution (ϕ only descends ):

_' =1

2

·r1¡ °

1 + °coth

³¿

2

p1¡ °2

´¡r

1 + °

1¡ °tanh

³¿

2

p1¡ °2

´¸

_' =1

2

·r1¡ °

1 + °tanh

³¿

2

p1¡ °2

´¡r

1 + °

1¡ °coth

³¿

2

p1¡ °2

´¸

vl = ¡ °p1¡ ° 2Limiting τ- speed (LM attractor):

γ 1 : LM attractor & descending solution disappear

_' =1

2 ¿¡ ¿

2• γ ≥ 1 ? Climbing ! E.g. for γ=1 :

CLIMBING : in ALL asymptotically exponential potentials with γ≥ 1 !


NOTE : only ϕo . Early speed singularity time !

String Realizations

7

a. Two - derivative couplings : α΄ corrections ? b. [BUT: climbing weak string coupling]

Dimensional reduction of (critical) 10-dimensional low-energy EFT:

(Condeescu, Dudas, in progress)

ds2 = e¡(10¡d)(d¡2)

¾ g¹º dx¹ dxº + e¾ ±ij dxi dxj

SD =1

2·210

Zd 10x

p¡g

©e¡ 2Á

¡¡R + 4 (@Á)2

¢¡ T e¡Á + : : :

ª

Sd =1

2·2d

Zd dx

p¡g

½¡ R ¡ 1

2(@Á)2 ¡ 2(10¡ d)

(d¡ 2)(@¾)2 ¡ T e

32 Á¡ (10¡d)

(d¡2)¾ + : : :

¾

• Two scalar combinations (Φs and Φt). Focus on Φt :

Sd =1

2·2d

Zd dx

p¡g

½¡R ¡ 1

2(@©s)

2 ¡ 1

2(@©t)

2 ¡ T e¢ ©t

¾

¢ =

s2(d¡ 1)

(d¡ 2)

NOTE :

° = 1 8d < 10!


Climbing with a SUSY Axion

8

• No-scale reduction + 10D tadpole

(Kachru, Kallosh, Linde, Trivedi, 2003)

(Cremmer, Ferrara, Kounnas, Nanopoulos, 1983) (Witten, 1985)

S4 =1

2·24

Zd4x

p¡g

½R ¡ 1

2(@©t)

2 ¡ 1

2e

2p3

©t (@µ)2 ¡ V (©t; µ) + ¢ ¢ ¢¾

T = e¡ ©tp

3 + iµp3

V (©t; µ) =c

(T + ¹T )3+ V(non pert:)

KKLT uplift

©t =2p3

x ; µ =2p3

y

d2x

d¿2+

dx

d¿

s

1 +

µdx

d¿

¶2

+ e4x3

µdy

d¿

¶2

+1

2 V

@V

@x

"1 +

µdx

d¿

¶2#

+1

2 V

@V

@y

dx

d¿

dy

d¿¡ 2

3e

4x3

µdy

d¿

¶2

= 0 ;

d2y

d¿ 2+

dy

d¿

s

1 +

µdx

d¿

¶2

+ e4x3

µdy

d¿

¶2

+

µ1

2 V

@V

@x+

4

3

¶dx

d¿

dy

d¿

+1

2 V

@V

@y

"e¡

4x3 +

µdy

d¿

¶2#

= 0

AXION INITIALLY “FROZEN “ CLIMBING ! A. Sagnotti – Moriond EW 2013

Climbing can inject Inflation

9

a. “Hard” exponential of Brane SUSY Breaking b. “Soft” exponential (γ < 1/√3):

Non-BPS D3 brane gives γ = 1/2 [+ stabilization of Φs]

V (Á) = M4 ¡

e 2' + e 2 ° '¢

(Sen , 1998) (Dudas, Mourad, AS 2001)

BSB “Hard exponential“ makes initial climbing phase inevitable

“Soft exponential” drives inflation during subsequent descent

Would need :

° ¼ 1

12


ϕo : “hardness” of kick !

Mukhanov – Sasaki Equation

10

Schroedinger-like equation for scalar (or tensor) fluctuations : “MS Potential” : determined by the background

d 2vk(´)

d´2+

£k2 ¡ Ws(´)

¤vk(´) = 0

ds2 = a2(´)¡¡ d´2 + dx ¢ dx

¢

Scalar : z(´) = a2(´)Á 00(´)

a 0(´)

Tensor : z(´) = a

Ws =1

z

d 2 z

d´ 2

Initial Singularity : Ws g´!¡´0¡ 1

4

1

(´ + ´0)2

LM In°ation : Ws g!0

º2 ¡ 14

´2

·º =

3

2

1 ¡ ° 2

1 ¡ 3 ° 2

¸

P (k) » k3

¯¯v(¡²)

z(¡²)

¯¯2


Numerical Power Spectra

11

²Á ´ ¡_H

H2; ´Á ´

VÁÁV

nS ¡ 1 = 2( ´Á ¡ 3 ²Á) ;

nT ¡ 1 = ¡ 2 ²ÁPS;T »

Zdk

kknS;T¡1

Key features: 1. Harder “kicks” make ϕ reach later the attractor 2. Even with mild kicks the time scale is 103- 104 in t M ! 3. η re-equilibrates slowly


Analytic Power Spectra

12

WKB: vk(¡ ²) »1

4pjWs(¡²) ¡ k2j

exp

µZ ¡²

¡´?

pjWs(y) ¡ k2j dy

¶


Analytic Power Spectra

13

WKB: vk(¡ ²) »1

4pjWs(¡²) ¡ k2j

exp

µZ ¡²

¡´?

pjWs(y) ¡ k2j dy

¶

WIGGLES : cfr. Q.M. resonant transmission


An Observable Window ?

14

WMAP9 power spectrum :

NOTE : ¯¯¢C`

C`

¯¯ =

r2

2` + 1

(Dudas, Kitazawa, Patil, AS, 2013, in progress) A. Sagnotti – Moriond EW 2013

An Observable Window ?

15

WMAP9 power spectrum :

But with a harder “kick”…

(Dudas, Kitazawa, Patil, AS, 2013, in progress) A. Sagnotti – Moriond EW 2013

Qualitatively the low-k tail

16

Thank you for your attention

Another way of presenting the results in slide 15 2 parameters to adjust : “hardness” of kick & time of horizon exit


Summary & Outlook

17

• BRANE SUSY BREAKING (d ≤ 10) : “critical” exponential potentials

• “HARD” exponential of BSB + “MILD” exponential (for inflation) :

WIDE IR depression of scalar spectrum (~ 6 e-folds)

[MILDER IR enhancement of tensor spectrum]

LARGE quadrupole depression & qualitatively next few multipoles !

[ LARGE CLASS of integrable potentials with climbing (to appear) ]


WITH “short” inflation (~ 60 e-folds) :

BISPECTRUM ?

18

Extra slides


19

Scales


• BSB potential: T10 =1

(® 0)5! T4 =

1

(® 0)2

µRp® 0

¶6

=¡M¢4

• Attractor Power spectra:

PS(k) =1

16¼GN ²

µH?

2¼

¶2 µk

a H?

¶nS¡1

PT (k) =1

¼GN ²

µH?

2¼

¶2 µk

a H?

¶ns¡1

nS = 1¡ 6² + 2´ nT = ¡ 2 ²

² = 8¼GN

µV 0

V

¶2

; ´ = 16¼GN

µV 00

V

¶2

• COBE normalization & bounds on ∈:

H? ¼ 1015 £ (²)12 GeV

¹M ¼ 6:5 1016 £ (²)14 GeV

10¡4 <PT

PS< 1:28 ! 10¡5 < ² < 0:08

3:5 1015 GeV < M < 3 1016 GeV

3 1012 GeV < H? < 3:4 1014 GeV

Integrable Scalar Potentials, I (climbing+barrier)

20

A number of exactly solvable (exp-like) potentials can be identified with techniques drawn from the theory of integrable systems. For instance, if V (Á) = ¸

³e

2° ' + e 2 ° '

´

(Fré, A.S., Sorin, to appear)

e' = e'0

·sinh(!°¿)

cosh !(¿ ¡ ¿0)

¸ °

1¡°2

eA = eA0

"cosh°

2

!(¿ ¡ ¿0)

sinh(!°¿)

# 1

1¡°2

!2 =¸

°

p1¡ °2 e 2A0

p1¡°2

Qualitatively : as in the figure

Letting :

ds2 = ¡ e 2A d¿ 2 + e2Ad¡1 dx ¢ dx

21 A. Sagnotti – Moriond EW 2013

Integrable Scalar Potentials, II (graceful exit)

An integrable potential where climbing is followed by a graceful exit.

(Fré, A.S., Sorin, to appear)

ds2 = ¡ e¡ 2A d¿2 + e2Ad¡1 dx ¢ dx

V(') = arctan¡e¡ 2'

¢

A = log¡x y¢

; ' = log

µx

y

¶

z = x + i y

L = 2 Imh¡ _z2 ¡ 8 C log z ¡ 8 D

i

Imh_z2 ¡ 8 C log z ¡ 8 D

i= 0 (H: C:)

System solved by a contour integral, and the shape of the contour is determined so that t is REAL

t =

Z z

a

dzrlog

³zz0

´

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