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Eliminating Ghost and Anisotropy in Bouncing Cosmology

交叉学科理论研究中心. Interdisciplinary Center for Theoretical Study. Eliminating Ghost and Anisotropy in Bouncing Cosmology. Taotao Qiu Institute of Astrophysics, Central China Normal University 201 4 -03-06. Based on Phys.Rev. D88 (2013) 043525 (TQ, Xian Gao, Emmanuel Saridakis)

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Eliminating Ghost and Anisotropy in Bouncing Cosmology

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  1. 交叉学科理论研究中心 Interdisciplinary Center for Theoretical Study Eliminating Ghost and Anisotropy in Bouncing Cosmology Taotao Qiu Institute of Astrophysics, Central China Normal University 2014-03-06 Based on Phys.Rev. D88 (2013) 043525(TQ, Xian Gao, Emmanuel Saridakis) Related works: JCAP 1110: 036, 2011, JHEP 0710 (2007) 071, Phys.Rev. D80 (2009) 023511 See Also: Mingzhe Li, Phys.Lett. B724 (2013) 192-197 Collaborated with R. Brandenberger, Y. F. Cai, J. Evslin, Xian Gao, M. Z. Li, Y. S. Piao, E. N. Saridakis, X. M. Zhang

  2. Outline • Preliminary of non-singular universe • Problems of bouncing cosmology 1. Ghost; 2. Anisotropy. • Model to eliminate these problems • Background • Perturbation • Conclusion

  3. Preliminary of non-singular universe——Why do we need bounce in the early universe? Problems of bouncing cosmology 1. Ghost; 2. Anisotropy. Model to eliminate these problems Background Perturbation Conclusions

  4. Standard Models of the Early Universe 1. Big Bang Cosmology(Alpher/Bethe/Gamow) Pros and Cons: • The age of galaxies • The redshift of the galactic spectrum • The He abundance • The prediction of CMB temperature • Flatness problem • Horizon problem • Unwanted relics problem • Singularity problem The sketch plot of Big Bang

  5. Standard Models of the Early Universe 2. Inflation Cosmology(Guth/Sato/Linde/Starobinsky/Fang…) Pros and Cons: • Flatness problem • Horizon problem • Unwanted relics problem • Singularity problem • TransPlanckian problem • (Martin & Brandenberger, 2000) Inflation

  6. The Singularity Problem The universe will meet a singularity when (1) it is described by General Relativity; (2) it satisfies Null Energy Condition; Where at finite time point S. Hawking R. Penrose for any null vector : S.W. Hawking, G.F.R. Ellis, Cambridge University Press, Cambridge, 1973; Borde and Vilenkin, Phys.Rev.Lett.72,3305 (1994).(Khoury et al., 01)

  7. TransPlanckian problem If the period of inflation is only slightly longer than the minimal value required for inflation to address the problems of standard big bang cosmology, then the wavelengths of all fluctuation modes which are currently inside the Hubble radius were smaller than the Planck length at the beginning of the period of inflation, the physics of which we do not understand. J. Martin, and R. Brandenberger, Phys.Rev.D63:123501 (2001).

  8. The Alternatives of Inflation Phenomenologically possible scenarios: 1. Inflation scenario: reaches 0 (singularity!) 2. Emergent scenario: 3. Bounce scenario: doesn’t reach 0 at its minimum doesn’t reach 0 at infinite past

  9. How does Bounce solve other Big-Bang puzzles? 1. Horizon problem: In Big-Bang scenario: the horizon in the far past is smaller than the length-scale of fluctuations, so it is strange why the observed fluctuations are nearly homogeneous and isotropic. To have horizon larger than fluctuations requires w<-1/3 in expanding phase or w>-1/3 in contracting phase, where w is the equation of state of the universe.

  10. How does Bounce solve other Big-Bang puzzles? 2. Flatness problem: In Big-Bang scenario: e. g. for radiation domination The small observed today requires even smaller in the early universe, causing initially fine-tuning! In bounce scenario: This problem can be avoided if the spatial curva-ture in the contracting phase when the temperature is comparable to today is not larger than the current value.

  11. How does Bounce solve other Big-Bang puzzles? 3. Trans-Planckian and Unwanted relics problem: In Big-Bang scenario: Quantum effects will be robust at scale larger than Planck scale, which is out of control and makes the effective theory invalid. Unwanted relics may also appear! J. Martin, and R. Brandenberger, Phys.Rev.D63:123501 (2001). In bounce scenario: sketch of bounce • 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. sketch of inflation Y. F. Cai, T. t. Qiu, R. Brandenberger and X. m. Zhang, Phys. Rev. D 80, 023511 (2009)

  12. Preliminary of non-singular universe • Problems of bouncing cosmology • Model to eliminate these problems • Background • Perturbation • Conclusions

  13. Preliminary of non-singular universe • Problems of bouncing cosmology 1. Ghost; 2. Anisotropy. • Model to eliminate these problems • Background • Perturbation • Conclusions

  14. Conditions for Bounce to Happen From the naïve picture, we can see: Contraction: Bouncing Point: Nearby: Expansion: From Friedmann Equation: Or in other words: NEC is violated! Y. Cai, T. Qiu, Y. Piao, M. Li and X. Zhang, JHEP 0710:071, 2007, cite 100+

  15. Conditions for Bounce to Happen Moreover, in order to connect to the observable universe (radiation dominant, matter dominant, etc), w goes to above -1, so w must crosses -1! w crossing -1 can never be reached for matter which is • in 4D classical Einstein Gravity, • described by single simple component: ---- perfect fluid ---- single scalar field with lagrangian as (3) coupled minimally to Gravity or other matter. How to realize crossing: No-Go theorem To realize crossing, one of the conditions should be violated J. Xia, Y. Cai, T. Qiu, G. Zhao and X. Zhang, Int.J.Mod.Phys.D17:1229-1243,2008.

  16. Ghost due to NEC violation! Null Energy Condition will generally cause ghost mode! • Example: “Phantom” Energy Lagrangian: Ghost mode! Quantization: Hamiltonian (density) & particle number (density) unbounded energy Quantization recipe 1: Non-unitarity Quantization recipe 2: S. Carroll, M. Hoffman, M. Trodden, Phys.Rev. D68 (2003) 023509; J. Cline, S. Jeon, G. Moore, Phys.Rev. D70 (2004) 043543.  

  17. Preliminary of non-singular universe • Problems of bouncing cosmology 1. Ghost; 2. Anisotropy. • Model to eliminate these problems • Background • Perturbation • Conclusions (maybe better dubbed as “Inconsistency between isotropy & scale invariance”!)

  18. Inconsistency between isotropy & scale invariance On one hand… Starting with the metric that contains anisotropy: Energy density Friedmann Equation: Anisotropy term Matter in the universe Equation of motion for anisotropy: Scale factor anisotropy Any matter in contracting phase with w<1 (scaling slower than a^{-6}) will be overwhelmed by the anisotropy term, which will cause the collapse of universe into a totally anisotopic one (no bounce)!!! J. Erickson, D. Wesley, P. Steinhardt, N. Turok, Phys.Rev. D69 (2004) 063514.

  19. Inconsistency between isotropy & scale invariance On the other hand… In contracting phase, scale invariant spectrum can only be generated when w=0! (Fabio Finelli, Robert Brandenberger, Phys.Rev. D65 (2002) 103522. ) • Theoretical aspects: stability must be guaranteed! (will talk later!) • Observational aspects: should obtain a (nearly) scale-invariant power spectrum and small tensor-to-scalar ratio Digression: perturbation theory We need primordial perturbations to provide seeds for structure formation and explains why our current universe is not complete isotropic. Two constraints for linear perturbations: Planck Collaboration (P.A.R. Ade et al.), arXiv:1303.5082 

  20. Inconsistency between isotropy & scale invariance In contracting phase, scale invariant spectrum can only be generated when w=0! (Fabio Finelli, Robert Brandenberger, Phys.Rev. D65 (2002) 103522.) Assume: viable for expanding phase with or contracting phase with The perturbation equation: Solution: grow for decay for Power spectrum: growing mode dominant (matter contraction) constant mode dominant (de sitter expantion)

  21. Inconsistency between isotropy & scale invariance! Summarize: Anisotropy Free Obtaining scale invariant perturbations Inconsistency!

  22. Preliminary of non-singular universe • Problems of bouncing cosmology • Model to eliminate these problems • Background • Perturbation • Conclusions

  23. Galileon Theories Galileon theory (Nicolis/Deffayet … ,2008): Lagrangian with higher derivative operator and 2nd order EoM The properties of Galileon Models: • Multi-degrees of freedom but only one is dynamical • w cross -1/ghost free e. g. E. o. M.: Ghost Free! Earlier idea: see G. Horndeski, Int. J. Theor. Phys. 10: 363, 1974.

  24. “Old” Galileon-Bounce Model The action (conformal Galileon model): the Galileon field F/M: free parameters T.Qiu, J. Evslin, Y. Cai, M. z. Li, X. Zhang, JCAP 1110:036,2011 See also P. Creminelli, A. Nicolis, E. Trincherini, JCAP 1011 (2010) 021for “Galileon Genesis”

  25. Preliminary of non-singular universe • Problems of bouncing cosmology • Model to eliminate these problems • Background • Perturbation • Conclusions

  26. Our New Galileon Bounce Model The action: Energy density and pressure: G-term Equation of motion: where

  27. Our New Galileon Bounce Model We take the potential to be of the form: Earlier application: Ekpyrotic models (J. Khoury et al., Phys.Rev. D64 (2001) 123522 ) I. Solutions before the bounce: (slow roll region) G-term is less important. Eolve like common single field. Slow roll region Fast roll region for large c. (can solve anisotropy problem as in Ekpyrotic scenarios.) Bounce region II. Solutions during the bounce: (bounce region) G-term cannot be neglected. Complicated but roughly III. Solutions after the bounce: (fast roll region) G-term is very important. induces

  28. Our New Galileon Bounce Model Numerical solutions in our model: So the university will not be dominated by its anisotropy!

  29. Preliminary of non-singular universe • Problems of bouncing cosmology • Model to eliminate these problems • Background • Perturbation • Conclusions

  30. Perturbation Theory Perturb our universe Space-time matter The perturbed action: (vanishing) eq. of motion Nonlinear corrections (non-Gaussianities) background Linear perturbation (power spectrum) Gauge fixing: uniform-φ gauge

  31. Stability of Perturbation of Our Model The second order perturbed action of our model: is the comoving curvature perturbation. There are two kinds of instabilities at linear level: • Ghost instability: • Gradient instability: (quantum instability) (Runaway behavior of classical fluctuations) In our model, which is model dependent and have to be checked numerically.

  32. Stability of Perturbation of Our Model Numeric plots for and Both and are positive all over the bouncing process! So we solved the ghost problem!

  33. Spectrum of Perturbation of Our Model The perturbation equation: Solution: Power spectrum: in contracting phase blue spectrum, inconsistent with observational data! How should we do?

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

  35. Scale Invariant Power Spectrum via Curvaton Our curvaton action: and are arbitrary functions of φ, is the potential of Background equation of motion: Energy density and pressure:

  36. Scale Invariant Power Spectrum via Curvaton Perturb the field: Perturbation equation of motion: with Power spectrum: The condition for spectrum to be scale-invariant (proof omitted): which can be realized by requiring specific forms of G, W and F. Here two examples are given.

  37. Scale Invariant Power Spectrum via Curvaton Example I: “faking” From the conditions above, we have: which split into two possibilities: Possibility a) (constant mode) Possibility b) (increasing mode)

  38. Scale Invariant Power Spectrum via Curvaton Backreaction of Example I For background energy density: in contracting phase (this is true for any constant w). In case where there is no potential term in the curvaton: EOM Energy density (safe) For For (unsafe)

  39. Scale Invariant Power Spectrum via Curvaton Example II: (no “faking”, ) One has , and thus the condition becomes: Furtherly assuming the potential of the curvaton: One has: growing decaying and the power spectrum is:

  40. Scale Invariant Power Spectrum via Curvaton Backreaction of Example II The equation of motion for the curvaton: where or The solution is: or So as long as , our background is safe from backreaction of curvaton, which can be guaranteed by ( ) provided . So we also get a (stable) scale-invariant power spectrum!

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

  42. Tensor Perturbation of Our Model The solution of (in contracting phase): Constant mode Growing mode for Decaying mode for In our case where , we have: Blue spectrum! In observable region , we have ,, namely the spectrum is severely suppressed, so the tensor-to-scalar ratio: consistent with the newest PLANCK data!

  43. Preliminary of non-singular universe • Problems of bouncing cosmology 1. Ghost; 2. Anisotropy. • Model to eliminate these problems • Background • Perturbation • Conclusion

  44. Conclusion • Bounce can solve Big-Bang problems, including singularity as well as others. • The two problems of Bounce cosmology: I. Ghost appearance due to NEC violation; II. Inconsistency between anisotropy-free and scale invariance of perturbations. • Solution to these problems: • Make use of Galileon theory to get rid of ghosts; • Using Ekpyrotic-like potential to be free of anisotropy; • Using curvaton mechanism to get scale invariant perturbations out of anisotropy-free background models. (two examples are given.) • Additional remarks: • Backreaction: will not affect our conclusion in most of the cases; • Tensor perturbation:blue which causes small tensor-to-scalar ratio.

  45. THANKS FOR ATTENTION!

  46. Brief mention: small Non-Gaussianities German scientists’ work that based on our model: Planck data: for local non-Gaussianity Our model: Final (local) non-Gaussianity: Cited from: A. Fertig, J.-L. Lehners, E. Mallwitz, arXiv:1310.8133 [hep-th].

  47. Standard Models of the Early Universe 2. Inflation Cosmology The simplest model Inflation The main success of inflation: • Flatness problem • Horizon problem • Unwanted relics problem if then Guth 1981; Sato 1981; Linde 1982; Albrecht & Steinhardt, 1982; Starobinsky 1980; Fang 1980; … However, the singularity problem remains unsolved!

  48. Galileon in Cosmology 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] Galileon as bounce models: • T. Qiu, J. Evslin, Y. F. Cai, M. Li, X. Zhang, JCAP 1110 (2011) 036; • D. Easson, I. Sawicki, A. Vikman, JCAP 1111 (2011) 021 • Y. F. Cai, D. Easson, R. Brandenberger, JCAP 1208 (2012) 020; • Y. F. Cai, R. Brandenberger, P. Peter, Class.Quant.Grav. 30 (2013) 075019 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; • …………

  49. Avoidance of Big-Rip The full action containing Galileon field and curvaton: • Reconstruction approach Equations: With these equations and ansatzed evolution of the universe, we can probably reconstruct the form of functions F, G and W!

  50. Avoidance of Big-Rip The “input” behaviors of scale factor and Hubble parameter: • Reconstruction approach The numerical results of reconstruction of field evolutions: (assume )

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