1. A Galileon Bounce Model Based on JCAP 1110:036,2011 (arXiv:1108.0593) Collaborated with J. Evslin, Y. F. Cai, M. Z. Li, X. M. Zhang See also D. Easson et al, JCAP 1111:021,2011 (arXiv:1109.1047) Taotao Qiu LeCosPA Center, National Taiwan University 2012-04-19 1

2. Outline What is/Why bounce? A Galileon bounce model ◦ Background ◦ Perturbation Conclusion outlook 2

3. WHAT IS/WHY BOUNCE? 3

4. Problems/constraints from theoretical/observational aspects: (such as BBN,CMB(COBE), etc) Big Bang Standard Models of the Early Universe Big Bang Cosmology vs. Inflation Cosmology The age of galaxiesThe redshift of the galactic spectrum The He abundanceThe prediction of CMB temperature Flatness problemHorizon problem Singularity problem Monopole problem Structure formation problem Inflation 4

5. Problems/constraints from theoretical/observational aspects: (such as BBN,CMB(COBE), etc) Big Bang Standard Models of the Early Universe Big Bang Cosmology vs. Inflation Cosmology The age of galaxiesThe redshift of the galactic spectrum The He abundanceThe prediction of CMB temperature Flatness problemHorizon problem Singularity problem Monopole problem Structure formation problem Inflation 5

6. The Alternatives of Inflation Pre-big bang Scenario Ekpyrotic Scenario String gas/Hagedorn Scenario Non-local SFT Scenario Bouncing Scenario 6

7. (Non-singular) Bounce Cosmology Expansion Contraction IR size with Low energy scale Singularity problem avoided! Basic Picture: Y. Cai, T. Qiu, Y. Piao, M. Li and X. Zhang, JHEP 0710:071, 2007 7

8. How does Bounce solve other Big-Bang puzzles? Horizon problem: the horizon in the far past in contracting phase is very large; Flatness problem: (also provide mechanism for survival of quantum fluctuations, which Seeds for Large Scale Structure. See perturbation theory later on.) e. g. for radiation domination avoided if the spatial curva- ture in the contracting phase when the temperature is comparable to today is not larger than the current value. 8

9. If the energy density at the bounce point is given by the Grand Unification scale ( ), then and the wavelength of a perturbation mode is about Unwanted relics can also be avoided because of the low energy scale Trans-Planckian and Unwanted relics problem: Y. F. Cai, T. t. Qiu, R. Brandenberger and X. m. Zhang, Phys. Rev. D 80, 023511 (2009) How does Bounce solve other Big-Bang puzzles? 9

10. If the energy density at the bounce point is given by the Grand Unification scale ( ), then and the wavelength of a perturbation mode is about Unwanted relics can also be avoided because of the low energy scale Trans-Planckian and Unwanted relics problem: Y. F. Cai, T. t. Qiu, R. Brandenberger and X. m. Zhang, Phys. Rev. D 80, 023511 (2009) How does Bounce solve other Big-Bang puzzles? 10

11. Contraction: Expansion: Bouncing Point: Nearby: In order to connect to the observable universe (radiation dominant, matter dominant, etc), w goes to above -1 Y. Cai, T. Qiu, Y. Piao, M. Li and X. Zhang, JHEP 0710:071, 2007 So w crosses -1, similar to the behavior of Quintom! Note: If w>-1 at the beginning, w will cross twice. or From Friedmann Equation: Quintom: a model originally used as Dark Energy with w crossing -1. B. Feng et al., Phys. Lett. B 607, 35 (2005); Conditions for Bounce to Happen From the naïve picture, we can see: 11

12. Realization of a Crossing Behavior As for any kind of matter, which is (1) in 4D classical Einstein Gravity, (2)described by single simple component (either perfect fluid or single scalar field with lagrangian as ), and (3) coupled minimally to Gravity or other matter, its Equation of State can never cross the cosmological constant boundary (w=-1). No-Go theorem To realize crossing, one of the conditions should be violated i) Double field bounce: ii) Single field bounce with higher derivative term: Y. Cai, T. Qiu, R. Brandenberger, Y. Piao, X. Zhang, JCAP 0803:013,2008; Y. Cai, T. Qiu, J. Xia, X. Zhang, Phys.Rev.D79:021303,2009. (also known as Lee-Wick Bounce) Y. Cai, T. Qiu, R. Brandenberger, X. Zhang, Phys.Rev.D80:023511,2009; J. Karouby, T. Qiu, R. Brandenberger, Phys.Rev.D84:043505,2011. Bo Feng et al., Phys. Lett. B 607, 35 (2005); A. Vikman, Phys. Rev. D 71, 023515 (2005); Gong-Bo Zhao et al., Phys. Rev. D 72, 123515 (2005); J. Xia, Y. Cai, T. Qiu, G. Zhao and X. Zhang, Int.J.Mod.Phys.D17:1229-1243,2008. 12

13. Realization of a Bounce Model For example: Lee-Wick Bounce Action: Y. F. Cai, T. t. Qiu, R. Brandenberger and X. m. Zhang, Phys. Rev. D 80, 023511 (2009) Picture: Bounce can happen, but… Or equivalent to: “WRONG” SIGN----GHOST MODE! 13

14. Galileon Theories Lagrangian with higher derivative operator/ 2 nd order EoM Only one dynamical degree of freedom w cross -1/ghost free Both of the two degrees of freedom are DYNAMICAL!!! The essence of the problem: A. Nicolis et al., Phys.Rev.D79:064036,2009; C. Deffayet et al., Phys.Rev.D79:084003,2009; C. Deffayet et al., arXiv:1103.3260 [hep-th]. Galileon Models: 14

15. Cosmological Applications of Galileon Theories Galileon as dark energy models:  R. Gannouji,M. Sami, Phys.Rev.D82:024011,2010.  A. De Felice, S. Tsujikawa, Phys.Rev.Lett.105:111301,2010.  C. Deffayet, O. Pujolas, I. Sawicki, A. Vikman, JCAP 1010:026,2010. Galileon as inflation and slow expansion models:  P. Creminelli, A. Nicolis, E. Trincherini, JCAP 1011:021,2010.  T. Kobayashi,M. Yamaguchi,J. Yokoyama, Phys.Rev.Lett.105:231302,2010.  C. Burrage,C. de Rham,D. Seery,A. Tolley, JCAP 1101:014,2011.  K. Kamada, T. Kobayashi, M. Yamaguchi, J. Yokoyama, Phys.Rev.D83:083515,2011.  Z. Liu, J. Zhang, Y. Piao, arXiv:1105.5713 [astro-ph.CO] Observational constraints on Galileon models:  S. Nesseris,A. De Felice, S. Tsujikawa, Phys.Rev.D82:124054,2010  A. Ali,R. Gannouji, M. Sami, Phys.Rev.D82:103015,2010. Galileon as spherically symmetric models:  D. Mota, M. Sandstad,T. Zlosnik, JHEP 1012:051,2010. … … … Can Galileon be used as bounce models??? 15

16. A GALILEON BOUNCE MODEL BACKGROUND PERTURBATION 16

17. Our New Bounce Model with Galileon The action: Stress energy tensor: From which we get energy density and pressure: which was also used in arXiv: 1007.0027 for “Galileon Genesis”. where 17

18. Considering, and thus is monotonic increasing, so the first term in H, is always larger than 0. Solution for Bounce to Happen So we get one property of the field: evolve as a monotonic function! In order to have bounce, H must reach 0, so negative branch is chosen. Reality of square root: From the Friedmann Equation we get the Hubble parameter: where 18

19. Asymptotic solution of Our Model with Equation of motion: Hubble parameter: In contracting phase: Analysis of the asymptotic behavior when Terms in EoM has different orders of t inconsistent ! EoM becomes: inconsistent ! consistent ! The only consistent solution has a radiation dominant behavior! iii) I.I. II. i) ii) 19

20. Numerical Plots of Our Model (1) Plots of Hubble parameter and scale factor in our model: Parameter choice: Bounce can happen naturally in our model around t=30. Reheating? 20

21. Plots of field and EoS w in our model: behaves as a monotonic function, and the equation of state is approximately 1/3 (radiation-dominant like) in contracting phase, and cross -1 before bounce in our model. Numerical Plots of Our Model (2) 21

22. QUINTOM BOUNCE WITH A GALILEON MODEL BACKGROUND PERTURBATION 22

23. Perturbation Theory  Theoretical aspects: stability must be guaranteed!  Observational aspects: should obtain a (nearly) scale- invariant power spectrum and small tensor-to-scalar ratio Why perturbations? Primordial perturbations provide seeds for structure formation and explains why our current universe is not complete isotropic. Two constraints for linear perturbations: 23

24. Perturbations of Our Bounce Model Perturbed metric in ADM form: Perturbed action: Gauge: uniform lapse function inverse shift vector Constraint equations: Solution: 24

25. is positive definite: no ghost instability! is model dependent: have to be checked numerically. Up to second order  Ghost instability:  Gradient instability: In our model, There are two kinds of instabilities at linear level: Stability of Perturbation of Our Model 25

26. Stability of Perturbation of Our Model Numeric plots for and Both and are positive all over the bouncing process, and we have which also behaves like radiation! 26

27. Spectrum of Perturbation of Our Model Blue spectrum inconsistent with observational data! Equation of motion: set In radiation dominant phase: Effective mass Solution: like a massless scalar field! 27

28. Mechanism of Getting Scale Invariant Power Spectrum D. Lyth and D. Wands, Phys.Lett.B524:5-14,2002. An alternative: Curvaton Mechanism Curvaton: a light scalar field other than inflaton to produce curvature perturbation. The simplest curvaton model: Curvature perturbation: For Gaussian part: where The equation of motion: with Solution: Power spectrum: 28

29. Curvaton Mechanism in Our Model Our curvaton action: The general solution: Our model: due to the background, in order to have scale invariant power spectrum, curvaton have to couple kinetically to the Galileon field The equation of motion: In radiation dominant phase: 29

30. Scale Invariant Power Spectrum from Curvaton growing constant decaying Superhorizon solution: There are two cases of getting scale invariant power spectrum: Subhorizon solution: From matching condition: and are independent of k!  q=2:  q=-4: 30

31. Back reaction of the Curvaton Field The energy density of : In contracting phase where the universe is radiation-like: From the equation of motion: In order for not to destroy the background evolution: one needs Question: will the growth of energy density of destroy the process of bounce? In our case which can produce scale invariant power spectrum:  q=-4: Safe from back reaction of  q=2: Needs severe fine-tuning. 31

32. Tensor Perturbation of Our Model Perturbed metric: Perturbed action (up to second order): Expand the tensor perturbation: Equation of motion: 32

33. Tensor Perturbation of Our Model In radiation dominant phase: Solution: Tensor spectrum: Spectrun index: In observable region we have, namely the spectrum is severely suppressed, so the tensor-to-scalar ratio like a massless scalar field! Blue spectrum! WMAP data predicts quite small r, so is consistent with our model! D. Larson et al. [WMAP collaboration], arXiv:1001.4635 [astro-ph.CO]. 33

34. Conclusion Bounce needs equation of state w cross -1, namely Quintom Bounce can be in form of galileon, where there are only two dynamical degrees of freedom and ghost can be eliminated. Quintom bounce in Galileon form: ◦ Background behavior: Radiation-dominant like. ◦ Perturbation1: free from instability but cannot provide scale invariant power spectrum ◦ Perturbation2: The way of providing scale-invariant power spectrum is curvaton. In our model there are two cases. ◦ Perturbation3: The back reaction is small in one case, but the other case needs fine tuning. ◦ Perturbation4: The tensor spectrum is blue and the tensor- to-scalar ratio is small. 34

35. 35 Outlook  Anisotropy  Reheating  Other issues…

36. Anisotropy 36 Has not been solved in this model! Problem: Solution: have the background evolution modified (w>1), or introduce some new suppression mechanism (in progress). Problem also for general bounce models to have scale invariant power spectrum! Attempts: V. Bozza and M. Bruni, JCAP 0910 (2009) 014; C. Lin, R. Brandenberger and L. Levasseur Perreault, JCAP 1104 (2011) 019. B. Xue and P. Steinhardt, Phys.Rev.Lett. 105 (2010) 261301 & Phys.Rev. D84 (2011) 083520 Bianchi IX metric will dominate over anything with w<1!

37. Reheating Through reheating, inflaton decay to matter and radiation after inflation  In galileon cosmology, reheating can help avoid divergence L. Levasseur, R. Brandenberger, A. Davis, arXiv:1105.5649  In bouncing cosmology, reheating is also important and maybe different from normal inflation T. Qiu, K. Yang, JCAP 1011:012,2010. Y. Cai, R. Brandenberger, X. Zhang, arXiv:1105.4286 What is the reheating process like of our Galileon bounce model??? (In progress) Reheating is important in inflationary scenario! Reheating mechanisms: The motivation of reheating in our bounce model:  Normal: L. Kofman, A. Linde, A. Starobinsky, Phys.Rev.Lett.73:3195-3198,1994.  Geometric: B. Bassett, S. Liberati, Phys.Rev.D58:021302,1998.  Curvaton: Bo Feng, Ming-zhe Li, Phys.Lett.B564:169-174,2003. …… Our bounce model “Galileon genesis”, P. Creminelli et al., 1007.0027 37

38. Other issues 38  Non-Gaussianites  Observational distinguishment  Connected with standard model physics  … … Not completed yet and still many things to do, but we take it deserved!

39. THANKS FOR ATTENTION! 39