1. Open Inflation in the Landscape Misao Sasaki (YITP, Kyoto University) in collaboration with Daisuke Yamauchi (now at ICRR, U Tokyo) and Andrei Linde, Atsushi Naruko, Takahiro Tanaka Takehara workshop June 6-8, 2011 arXiv:1105.2674 [hep-th]

2. 2 1. String theory landscape  There are ~ 10 500 vacua in string theory vacuum energy  v may be positive or negative some of them have  v <<  P 4 typical energy scale ~  P 4 Lerche, Lust & Schellekens (’87), Bousso & Pochinski (’00), Susskind, Douglas, KKLT (’03),... which ?

3. 3  A universe jumps around in the landscape by quantum tunneling it can go up to a vacuum with larger  v if it tunnels to a vacuum with negative  v, it collapses within t ~ M P /|  v | 1/2. so we may focus on vacua with positive  v : dS vacua 0 vv Sato, MS, Kodama & Maeda (’81)

4. 4 2. Anthropic landscape Susskind (‘03)  Not all of dS vacua are habitable. “anthropic” landscape  v,f must not be larger than this value in order to account for the formation of stars and galaxies.  A universe jumps around in the landscape and settles down to a final vacuum with  v,f ~ M P 2 H 0 2 ~(10 -3 eV) 4.  Just before it has arrived the final vacuum, it must have gone through an era of (slow-roll) inflation and reheating, to create “matter and radiation.”

5. 5  Most plausible state of the universe before inflation is a de Sitter vacuum with  v ~ M P 4. false vacuum decay via O(4) symmetric (CDL) instanton inside bubble is an open universe Coleman & De Luccia (‘80)

6. 6  Anthropic principle suggests that # of e-folds of inflation inside the bubble should be ~ 50 – 60 : just enough to make the universe habitable. Garriga, Tanaka & Vilenkin (‘98), Freivogel et al. (‘04)  Nevertheless, the universe may be slightly open: revisit open inflation!  Natural outcome would be a universe with  0 <<1. “empty” universe: no matter, no life  Observational data excluded open universe with  0 <1. may be confirmed by Planck+BAO Colombo et al. (‘09)

7. 7 3. Open inflation in the landscape tunneling to a potential maximum ~ stochastic inflation  Simplest polynomial potential = Hawking-Moss model Hawking & Moss (‘82)  4 potential: Starobinsky (‘84) – constraints from scalar-type perturbations –  slow-roll inflation HM transition too large fluctuations of  unless # of e-folds >> 60 Linde (‘95)

8. 8 If inflation is short, too large perturbations from supercurvature mode of  H F : Hubble at false vacuum H R : Hubble after fv decay MS & Tanaka (‘96)  Two- (multi-)field model: “quasi-open inflation” a “heavy” field  undergoes false vacuum decay another “light” field  starts rolling after fv decay ~ perhaps naturally/easily realized in the landscape   Linde, Linde & Mezhlumian (‘95)

9. creation of open universe & supercurvature mode bubble wall open universe dS vacuum wavelength > curvature radius “supercurvature” mode

10. 10  To summarize: The models of the tunneling in the landscape with the simplest potentials such as or are ruled out by observations, assuming that inflation after the tunneling is short, N ~ 60. The same models are just fine if N >> 60. This means that we are testing the models of the landscape in combination with the probability measure(s), which may or may not predict that the last stage of inflation is short.

11. 11  Single field model if  fv ~ M P 4, the universe will most likely tunnel to a point where the energy scale is still very high unless potential is fine-tuned. FV decay rapid roll slow roll inflation rapid-roll stage will follow right after tunneling. perhaps no strong effect on scalar-type pert’s: suppressed by at rapid-roll phase need detailed analysis ∙∙∙ future issue Linde, MS & Tanaka (’99)

12. 12 but tensor perturbations may not be suppressed at all. Tensor perturbations and their effect on CMB Memory of H F (Hubble rate in the false vacuum) may remain in the perturbation on the curvature scale

13. 13 4. Single field model - evolution after tunneling - Right after tunneling, expansion is dominated by curvature:  rapid-roll phase : slow-roll parameter  curvature dominant phase kinetic energy grows until at

14. 14  rapid-roll phase  starts to decay at V dominates (curvature dominance ends) at no rapid-roll phase. slow-roll inflation starts at rapid-roll continues until - continued tracking is realized during rapid-roll phase

15. 15  exponential potential model Log 10 [r/H * 2 ] Log 10 [a(t) H * ] potential kinetic term curvature term added to realize slow-roll inflation

16. 16 5. Tensor perturbations  some technical details... CDL instanton bubble wall action

17. 17 analytic continuation to Lorentzian space (through r E =  /2) C-region: ~ outside the bubble  =const. r =const. Bubble wall r C =0 R-region: inside the bubble Euclidean vacuum C-region R-region  C =+ 8  C =- 8 R C time

18. 18 tensor mode function Euclidean vacuum Y ij : regular at r c =0 new variable w p : UTUT CC bubble wall

19. 19  analytic continuation from C-region to R-region(=open univ.) there will be time evolution of w p in R-region: effect of wall epoch of bubble nucleation: or final amplitude of X p depends both on the effect of wall & on the evolution after tunneling

20. 20 high freq continuum + low freq resonance wall fluctuation mode  Effect of tunneling/bubble wall on  s ~ ∞  s ~0.1  s ~0.02  s ~1  s ~0.7 scale-invariant wall tension

21. 21   rapid-roll phase (  * -)dependence of P T (p)  <<1: usual slow roll  ~1: small p modes remember H at false vacuum  >>1: No memory of H at false vacuum

22. 22 6. CMB anisotropy ℓ  ~1: small ℓ modes remember initial Hubble scales as at small ℓ, scale-invariant at large ℓ  <<1: the same as usual slow roll inflation  >>1: No memory of initial Hubble small ℓ modes enhanced for

23. 23 CMB anisotropy due to wall fluctuation (W-)mode affect only low ℓ scale-invariant part ℓ=2  s = 10 -2  s = 10 -5  s = 10 -4  s = 10 -3 W-mode dominates ℓ=2 MS, Tanaka & Yakushige (’97)

24. 24 7. Summary  Open inflation has attracted renewed interest in the context of string theory landscape  Landscape is already constrained by observations anthropic principle + landscape 1-  0 ~ 10 -2 – 10 -3 simple polynomial potentials a  2 – b  3 + c  4 lead to HM-transition, and are ruled out simple 2-field models, naturally realized in string theory, are ruled out due to large scalar-type perturbations on curvature scale If inflation after tunneling is short (N ~ 60):

25. 25  Tensor perturbations may also constrain the landscape “single-field models” it seems difficult to implement models with short slow-roll inflation right after tunneling in the string landscape. there will be a rapid-roll phase after tunneling. right after tunneling unless  >>1, the memory of pre-tunneling stage persists in the IR part of the tensor spectrum due to either wall fluctuation mode or evolution during rapid-roll phase We are already testing the landscape! large CMB anisotropy at small ℓ