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Cosmology of the Lee-Wick Model

Cosmology of the Lee-Wick Model. Speaker: Yi-Fu Cai IHEP January 12, 2009 Work in collaboration with T. Qiu, R. Brandenberger and X. Zhang arXiv: 0810.4677. Outline. Introduction Model Background Cosmology Cosmological Fluctuations in the Lee-Wick Model Conclusions. Preliminaries

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Cosmology of the Lee-Wick Model

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  1. Cosmology of the Lee-Wick Model Speaker: Yi-Fu Cai IHEP January 12, 2009 Work in collaboration with T. Qiu, R. Brandenberger and X. Zhang arXiv: 0810.4677

  2. Outline • Introduction • Model • Background Cosmology • Cosmological Fluctuations in the Lee-Wick Model • Conclusions • Preliminaries • Formalism

  3. Introduction

  4. History I • 1969: T. D. Lee and G. C. Wick propose the Lee-Wick mechanism as a way of solving the hierarchy problem. • Higher derivative action →new degrees of freedom. • Equivalent to adding fields with opposite kinetic term. • → problem of ghosts. • 1970’s: debates on the consistency of the theory. • 1970’s: super-symmetry discovered → interest in Lee-Wick theory wanes.

  5. History II • 2007: Lee-Wick construction resurrected by Grinstein, O’Connell and Wise: Lee-Wick Standard Model. • No progress on conceptual issues related to ghosts. • Phenomenological studies (LHC).

  6. History II • 2007: Lee-Wick construction resurrected by Grinstein, O’Connell and Wise: Lee-Wick Standard Model. • No progress on conceptual issues related to ghosts. • Phenomenological studies (LHC). • Question: Can the Lee-Wick model be ruled out by cosmology?

  7. Results • Results: • Lee-Wick model leads to a non-singular cosmology. • Vacuum fluctuations in the contracting phase lead to a scale-invariant spectrum of cosmological perturbations.

  8. Results • Results: • Lee-Wick model leads to a non-singular cosmology. • Vacuum fluctuations in the contracting phase lead to a scale-invariant spectrum of cosmological perturbations. • Consequences: • Thus, Lee-Wick cosmology provides a possible alternative to inflationary cosmology. • Lee-Wick cosmology solves the singularity problem of inflationary cosmology. • Lee-Wick cosmology faces an anisotropy problem.

  9. Model

  10. Lee-Wick Theory: the model • Starting point: Higgs sector of Lee-Wick Standard Model: A high-derivative term is involved. Classically, a new degree of freedom is obtained. (so-called LW partner) To solve the quadratic divergences in particle physics ~ divergence

  11. ~ finite + Lee-Wick Theory: the model • Starting point: Higgs sector of Lee-Wick Standard Model: A high-derivative term is involved. Classically, a new degree of freedom is obtained. (so-called LW partner) To solve the quadratic divergences in particle physics ~ divergence

  12. Lee-Wick Theory: the model The Lagrangian: Equivalent language: LW partner Regular Higgs The mass terms can be diagonalized by rotating the field basis. Generically, there is a coupling between the two fields. Our choice:

  13. Background Cosmology

  14. Equations of Motion • Metric of space-time: • Einstein action coupled to Lee-Wick Model leads to the following equations for cosmological dynamics: • In addition, there are the coupled Klein-Gordon equations for and .

  15. Background dynamics • First, we see that a cosmological bounce is possible. • Cosmological bounce: • Next, we show that a cosmological bounce is inevitable (in the context of homogeneous and isotropic cosmology).

  16. Background dynamics I • 1. Initial conditions: Matter dominated contracting phase: • Regular matter dominates: • oscillating with amplitude • ~(x; t) oscillating with amplitude • F ()=( ~) = const >> 1 • A • 2. Deflationary contraction: • Onset of slow-rolling (up the hill) of when • Since • rapidly catches up, i.e. decreases towards 1.

  17. Background dynamics II • Note: period of deflation is generically short: • 3. Non-singular bounce: • Occurs when • Since but we have • Give rise to a nonsingular bounce.

  18. Background dynamics III • 4. Accelerated expansion (the time reverse of Period 2) • 5. Matter-dominated expansion (the time reverse of Period 1) • Note: We expect that the addition of radiation will not change the conclusion concerning the existence of the non-singular bounce.

  19. dominates freezes still oscillates dominates again Sketch A heavier field is much stabler than a lighter one at low energy densities and curvatures. Contracting: Near the bounce: A bounce happens: when the contribution of LW scalar to the energy density catches up to that of the normal scalar Expanding:

  20. Numerical Results The plots of the equation-of-state, hubble parameter, and scale factor in Lee-Wick Bounce:

  21. Cosmological Fluctuations in the Lee-Wick Model Preliminaries Formalism

  22. Comparison with inflationary cosmology Space-time sketch of inflationary cosmology:

  23. Comparison with inflationary cosmology Crucial facts: • Fluctuations originate on sub-Hubble scales • Fluctuations propagate for a long time on super-Hubble scales

  24. Space-time sketch for the Lee-Wick model Space-time sketch for the Lee-Wick model

  25. Space-time sketch for the Lee-Wick model Note: Fluctuations emerge inside the Hubble radius and propagate on super-Hubble scales as in inflationary cosmology.

  26. Trans-Planckian Problem for Inflationary Cosmology R.B., hep-ph/9910410

  27. Trans-Planckian Problem for Inflationary Cosmology R.B., hep-ph/9910410 • Success of inflation: At early times scales are inside the Hubble radius → causal generation mechanism is possible. • Problem: If time period of inflation is more than 70H−1, then λp(t) < lplat the beginning of inflation • → new physics MUST enter into the calculation of the fluctuations.

  28. Trans-Planckian Window of Opportunity • If evolution in Period I is non-adiabatic, then scale-invariance of the power spectrum will be lost [J. Martin and RB, 2000] • → Planck scale physics testable with cosmological observations!

  29. No Trans-Planckian problem for Lee-Wick Cosmology • Note: The physical length of fluctuation modes is always greater than 1mm → fluctuations are in the far IR regime throughout.

  30. Singularity Problem of Inflationary Cosmology • Standard cosmology: Penrose-Hawking theorems → initial singularity → incompleteness of the theory. • Inflationary cosmology: In scalar field-driven inflationary models the initial singularity persists → incompleteness of the theory. [Borde and Vilenkin]

  31. Singularity Problem Resolved in Lee-Wick Cosmology • The Lee-Wick bounce is non-singular. All curvature invariants are bounded and space-time is geodesically complete.

  32. Cosmological Fluctuations in the Lee-Wick Model Preliminaries Formalism

  33. Lee-Wick Bounce: Perturbations Perturbed scalars: Perturbed metric: Pert. Equation: Status： Contracting phase The Bounce Expanding phase

  34. Contracting Expanding irrelevant irrelevant Lee-Wick Bounce: Perturbations I One approach: gravitational potential The solution: Initial conditions: Bunch-Davies vacuum Matching relations:

  35. Contracting Expanding Output: Scale-invariant Spectrum!! irrelevant irrelevant Lee-Wick Bounce: Perturbations I One approach: gravitational potential The solution: Initial conditions: Bunch-Davies vacuum Matching relations:

  36. Lee-Wick Bounce: Perturbations II Another approach: curvature perturbation on uniform density Contracting: Expanding: Scale-invariant and constant Comments: 1, zeta is no longer a conserved quantity outside hubble radius when the universe is contracting, but it is continuous through the bounce; 2, there exists entropic perturbations, but less important; 3, the spectrum for Ekpyrotic bounce is not scale-invariant,

  37. Lee-Wick Bounce: Numerical Results The plots of the power spectrum and spectral index

  38. Lee-Wick Bounce: Non-linear Perts. Non-gaussianity parameter: WMAP5 data: • The contribution from redefinition: So it gives • Also other contributions…

  39. Lee-Wick Bounce: Non-linear Perts. Summary of main results for Non-gaussianity: • No slow roll —— a sizable amplitude; • No slow roll —— new shapes; • No conservation (zeta) —— new origins; • No one has done it —— To be appeared ^_^

  40. Conclusions

  41. Summary • An alternative to inflation • Avoiding initial singularity • Scale-invariant spectrum • Large non-gaussianity • A new window between particle physics & cosmology

  42. Problems remained • Entropic perturbations; • Excess of tensor to scalar ratio; • Anisotropic instability; • … These issues deserve future studies

  43. Lee-Wick Cosmology: Other phenomena & Outlook • Lee-Wick Dark Energy S. Lee . arXiv:arXiv:0810.1145 • A candidate of DM? • Related to fundamental theories, (SUSY, String, …?)

  44. THANK YOU&Happy Spring Festival !

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