Climbing scalars and implications for Cosmology

Emilian Dudas CPhT-Ecole Polytechnique

IFIN,Bucuresti, 23 aprilie 2013

E. D, N. Kitazawa, A.Sagnotti, P.L. B 694 (2010) 80 [arXiv:1009.0874 [hep-th]]. E. D, N. Kitazawa, S. Patil, A.Sagnotti, JCAP 1205 (2012) 012 [arXiv:1202.6630 [hep-th]] E. D, N. Kitazawa, S. Patil, A.Sagnotti, in progress C. Condeescu, E.D., in progress

Outline

• Brane SUSY breaking • A climbing scalar in D dimensions • Climbing with a SUSY axion (KKLT) • Climbing and inflation, power spectrum • Kasner approach: higher-derivative

corrections, models with no big-bang • Outlook

2

Brane SUSY Breaking

3

(Sugimoto, 1999) (Antoniadis, E.D, Sagnotti, 1999) (Aldazabal, Uranga, 1999) (Angelantonj, 1999)

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

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

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 ?

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

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

4

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) ……………… (E.D,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

5

• 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

6

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

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

(C.Condeescu, E.D, 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

7

• 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 !

Climbing and Inflation

8

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) (E.D .J.Mourad, A.Sagnotti 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

9

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

10

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

11

WKB: vk(¡ ²) »1

4pjWs(¡²) ¡ k2j

exp

µZ ¡²

¡´?

pjWs(y) ¡ k2j dy

¶

WIGGLES : cfr. Q.M. resonant transmission

12

An Observable Window ?

13

WMAP9/Planck power spectrum :

NOTE : ¯¯¢C`

C`

¯¯ =

r2

2` + 1 But with a harder “kick”…

(E.D, Kitazawa, Patil, Sagnotti, in progress) Qualitatively the low-k tail

14

Another way of presenting the results in slide 11 2 parameters to adjust : “hardness” of kick & time of horizon exit

Kasner approach

15

Search for approximate Kasner-like solutions near big-bang (t=0)

The leading order e.o.m. close to big-bang reduce to

whereas for the exponential potential V = α exp (∆ φ) the descending solution exists if ∆ p > - 2 . Then we find: • for asymmetric metric there is always a descending solution • for the symmetric (FRW) case a_i = a, the descending solution exists if

,in agreement with the exact solution

16

The method can be used to analyze the climbing behaviour of any lagrangian (and any potential). Some results (FRW case):

• Higher-derivative corrections typically spoil the climbing behaviour. Specific operators preserve it. Quartic order:

• Most other higher-derivatives spoils it. Ex: DBI

The scalar close to big-bang is force to slow-down

The scalar potential V = α exp (∆ φ) is now regular for both descending and climbing solution, for any ∆ .

Examples with no big-bang

17

Consider the potentials with asymptotic behaviour

For: • Kasner/FRW solutions starting on either

side of the minimum

scalar starts near big-bang necessarily on the flat side ¡1

•

Moreover, for the scalar is exponentially damped to the minimum, whereas for there is damping

plus oscillations. For , no singular solutions anymore. Scalar forced to stay close to minimum. No big-bang !

Summary & Outlook

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• 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 (Fre,Sagnotti,Sorin, to appear) ]

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

BISPECTRUM ?

19

Multumesc pentru atentie

Kasner approach used to analyze climbing for various Models, confirms and extend previous analysis.

20

Extra slides

21

More analytical spectra

22

23

Scales • BSB potential: T10 =

1

(® 0)5! T4 =

1

(® 0)2

µRp® 0

¶6

=¡M¢4

• Attractor Power spectra:

• 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