Climbing scalars and implications for Cosmology

Transcription

1 Climbing scalars and implications for Cosmology Emilian Dudas CPhT-Ecole Polytechnique E. D, N. Kitazawa, A.Sagnotti, P.L. B 694 (2010) 80 [arxiv: [hep-th]]. E. D, N. Kitazawa, S. Patil, A.Sagnotti, JCAP 1205 (2012) 012 [arxiv: [hep-th]] E. D, N. Kitazawa, S. Patil, A.Sagnotti, in progress C. Condeescu, E.D., in progress IFIN,Bucuresti, 23 aprilie 2013

2 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

3 Brane SUSY Breaking Dualities : link different strings Orientifolds : link closed and open strings Tree level BSB SUSY : D9 (T > 0, Q > 0) + O9 - (T < 0, Q < 0) SO(32) SUSY broken at string BSB : anti-d9(t > 0, Q < 0) + O9 scale + (T > in 0, open Q > 0) sector, USp(32) exact in closed Stable vacuum Goldstino in open sector (Sugimoto, 1999) (Antoniadis, E.D, Sagnotti, 1999) (Aldazabal, Uranga, 1999) (Angelantonj, 1999) S 10 = 1 BSB: Tension unbalance exponential potential Z d 10 x p g e 2 Á R + 4 (@Á) 2 T e Á + : : : ª Flat space : runaway behavior String-scale breaking : early-universe Cosmology? 3

4 A climbing scalar in d dim s Consider the action for gravity and a scalar φ : S = 1 Z d D x p g R (@Á)2 V (Á) + : : : Look for cosmological solutions of the type ds 2 = e 2B(t) dt 2 + e 2A(t) dx dx Make the convenient gauge choice V (Á) e 2B = M 2 (Halliwell, 1987) (E.D,Mourad, 2000) (Russo, 2004).. Let : = r d 1 d 2 ; = M t ; ' = Á p 2 ; A = (d 1) A In expanding phase : Ä' + _' p 1 + _' _' 2 1 2V OUR CASE : V = exp( 2 ')! 1 ' = 4

5 A climbing scalar in d dim s γ < 1? Both signs of speed a. Climbing solution (ϕ climbs, then descends): _' = 1 2 r 1 ³ 1 + coth p b. Descending solution (ϕ only descends ): _' = 1 2 r 1 ³ 1 + tanh p r 1 + ³ 1 tanh p r 1 + ³ 1 coth p NOTE : only ϕ o. Early speed singularity time! Limiting τ- speed (LM attractor): v l = p 1 2 γ 1 : LM attractor & descending solution disappear γ 1? Climbing! E.g. for γ=1 : _' = CLIMBING : in ALL asymptotically exponential potentials with γ 1! 5

6 String Realizations NOTE : a. Two - derivative couplings : α corrections? b. [BUT: climbing weak string coupling] (C.Condeescu, E.D, in progress) Dimensional reduction of (critical) 10-dimensional low-energy EFT: S D = Z d 10 x p g e 2 Á R + 4 (@Á) 2 T e Á + : : : ª ds 2 = e (10 d) (d 2) ¾ g ¹º dx ¹ dx º + e ¾ ± ij dx i dx j S d = 1 2 2d Z d d x p g ½ R 1 2 (@Á)2 2(10 d) (d 2) (@¾) 2 ¾ T e 3 2 Á (10 d) (d 2) ¾ + : : : Two scalar combinations (Φ s and Φ t ). Focus on Φ t : S d = 1 = 2 2d Z d d x p g s 2(d 1) (d 2) ½ R 1 2 (@ s) 2 1 ¾ 2 (@ t) 2 T e t = 1 8d < 10! 6

7 Climbing with a SUSY Axion (Kachru, Kallosh, Linde, Trivedi, 2003) No-scale reduction + 10D tadpole KKLT uplift T = e t p 3 + i µ p 3 (Cremmer, Ferrara, Kounnas, Nanopoulos, 1983) (Witten, 1985) S 4 = Z d 4 x p g ½ R 1 2 (@ t) 2 1 ¾ 2 e p 2 3 t (@µ) 2 V ( t ; µ) + V ( t ; µ) = d 2 x d 2 + dx d c (T + ¹ T ) 3 + V (non pert:) s 1 + µ 2 dx + e 4x 3 d µ 2 dy + 1 d 2 V t = 2 p 3 x ; µ = 2 p " 1 + µ # 2 dx d V d 2 y d 2 + dy d d s µ dx 1 + dy d 2 3 e 4x 3 e 4x e 4x 3 + µ dy d µ 2 dy = 0 ; d µ 2 dy + d # 2 = 0 µ dx 3 d dy d AXION INITIALLY FROZEN CLIMBING! 7

8 Climbing and Inflation a. Hard exponential of Brane SUSY Breaking b. Soft exponential (γ < 1/ 3): Would need : ¼ 1 12 V (Á) = M 4 e 2 ' + e 2 ' Non-BPS D3 brane gives γ = 1/2 [+ stabilization of Φ s ] (Sen, 1998) (E.D.J.Mourad, A.Sagnotti 2001) BSB Hard exponential makes initial climbing phase inevitable Soft exponential drives inflation during subsequent descent ϕ o : hardness of kick! 8

9 Mukhanov Sasaki Equation Schroedinger-like equation for scalar (or tensor) fluctuations : d 2 v k ( ) + k 2 W s ( ) v k ( ) = 0 d 2 MS Potential : determined by the background Initial Singularity : W s! 0 g 1 4 LM In ation : W s ǵ!0 º º = ( + 0) 2 ds 2 = a 2 ( ) d 2 + dx dx Scalar : z( ) = a 2 ( ) Á 0 0( ) a 0 ( ) T ensor : z( ) = a W s = 1 d 2 z z d 2 P (k)» k 3 v( ²) z( ²) 2 9

10 Numerical Power Spectra Key features: 1. Harder kicks make ϕ reach later the attractor 2. Even with mild kicks the time scale is in t M! 3. η re-equilibrates slowly ² Á _ H H 2 ; Á VÁÁ V P S;T» Z dk k kn S;T 1 n S 1 = 2( Á 3 ² Á ) ; n T 1 = 2 ² Á 10

11 Analytic Power Spectra WKB: v k ( ²)» 1 p 4 jws ( ²) k 2 j exp µz ²? p jws (y) k 2 j dy WIGGLES : cfr. Q.M. resonant transmission 11

12 An Observable Window? 12

13 WMAP9/Planck power spectrum : NOTE : C` = C` r 2 2` + 1 But with a harder kick Qualitatively the low-k tail (E.D, Kitazawa, Patil, Sagnotti, in progress) 13

14 Another way of presenting the results in slide 11 2 parameters to adjust : hardness of kick & time of horizon exit 14

15 Kasner approach 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 15

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. 16

17 Examples with no big-bang Consider the potentials with asymptotic behaviour For: Kasner/FRW solutions starting on either side of the minimum Moreover, for scalar starts near big-bang necessarily on the flat side 1 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! 17

18 Summary & Outlook BRANE SUSY BREAKING (d 10) : critical exponential potentials HARD exponential of BSB + MILD exponential (for inflation) : WITH short inflation (~ 60 e-folds) : 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) ] BISPECTRUM? 18

19 Kasner approach used to analyze climbing for various Models, confirms and extend previous analysis. Multumesc pentru atentie 19

20 Extra slides 20

21 More analytical spectra 21

22 22

23 Scales BSB potential: T 10 = 1 ( 0 ) 5! T 4 = 1 ( 0 ) 2 µ 6 R p = M 4 0 Attractor Power spectra: COBE normalization & bounds on : H? ¼ (²) 1 2 GeV ¹M ¼ 6: (²) 1 4 GeV 10 4 < P T P S < 1:28! 10 5 < ² < 0:08 3: GeV < M < GeV GeV < H? < 3: GeV 23