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Non-Gaussianities of Single Field Inflation with Non-minimal Coupling

Non-Gaussianities of Single Field Inflation with Non-minimal Coupling. Taotao Qiu 2010-12-28 Based on paper: arXiv: 1012.1697[Hep-th] (collaborated with Prof. K. C. Yang). Outline. Preliminary Non-Gaussianity in single field inflation with non-minimal coupling Summary. Preliminary.

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Non-Gaussianities of Single Field Inflation with Non-minimal Coupling

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  1. Non-Gaussianities of Single Field Inflation with Non-minimal Coupling Taotao Qiu 2010-12-28 Based on paper: arXiv: 1012.1697[Hep-th] (collaborated with Prof. K. C. Yang)

  2. Outline • Preliminary • Non-Gaussianity in single field inflation with non-minimal coupling • Summary

  3. Preliminary

  4. Why non-Gaussianities? • Observational development: • Data become more and more accurate to study the non-linear properties of the fluctuation in CMB and LSS. Y. Gong, X. Wang, Z. Zheng and X. Chen, Res. Astron. Astrophys. 10, 107 (2010) [arXiv:0904.4257 [astro-ph.CO]]. E. Komatsu et al., arXiv:1001.4538 [astro-ph.CO]; C. L. Bennett et al., arXiv:1001.4758 [astro-ph.CO]. [Planck Collaboration], arXiv: astro-ph/0604069. • Theoretical requirement: • The redundance of inflation models need to be distinguished.

  5. Observational constraints on non-Gaussianity • WMAP data: • WMAP 7yr (68% CL): • Planck data: E. Komatsu et al., Seven-year Wilkinson Microwave Anisotropy Probe (WMAP) observations: cosmological interpretation, arXiv:1001.4538 [SPIRES]. Planck collaboration, PLANCK the scientific programme, http://www.rssd.esa.int/SA/PLANCK/docs/Bluebook-ESA-SCI(2005)1.pdf [astro-ph/0604069] [SPIRES].

  6. Definition • Local non-Gaussianity: the non-Gaussianity at every space point has the form of the single random variable: • Nonlocal non-Gaussianity: the non-Gaussianity may be sourced by correlation functions of different space points. characterized by “shape” compared to the local case.

  7. Classification of NG shapes • Equilateral: • Squeezed • Folded

  8. Non-Gaussianity in single field inflation with non-minimal coupling

  9. Steps of non-Gaussianity Calculation • Get the constraint solution; • Expand the action w.r.t. the perturbations and the constraints; • Obtain the mode solution; • Calculate the 3-point correlation function with in-in formalism.

  10. Non-Gaussianities in single field inflation with non-minimal coupling Metric: ADM metric Action: where R is the Ricci Scalar and is the kinetic term of the inflaton field The equation for field: The Einstein Equations: where

  11. The equations Decomposite into 3+1 form: where and K is the trace of The constraint equations (varying the action w.r.t. and ): where

  12. A Theorem Theorem: To calculate n-th order perturbation, one only need to expand the constraints and to (n-2)-th order. Proof: Consider Lagrangian that contains constraints : The equation of motion: Expand to n-th order: Lagrangian becomes: Detailed analysis show that the coefficients before and are and respectively.

  13. A Theorem From the equation of motion: We can see that for 0th order: for 1st order: So the term of and will vanish in the expansion of , and we only need to consider up to (n-2)-th order.

  14. Solutions of constraint for linear coupling Consider a linear coupling case: The constraint equations: Comoving gauge: One may check that ->1, the result will return to GR! Constraint expansion: We calculate from the constraint equations: Define then and we have:

  15. Up to the 3rd Order Action: Decomposition to 3rd order: where , and are the 1st, 2nd and 3rd order term of , respectively.

  16. Up to the 3rd Order Action of 0th to 3rd order: (background action) (equation of motion) where a is the scale factor,

  17. Mode solution By varying the 2nd action w.r.t. , Using Fourier transformation: we can obtain the 2nd action in momentum space: Defining: where one have: where and thus

  18. Mode solution Define slow-roll parameters: The equation can be rewritten in the leading order of and where Solving the equation, we can get:

  19. Mode solution The above solution can be splitted into sub-horizon and super-horizon approximations: Sub-horizon: Super-horizon: The same is for : Sub-horizon: Super-horizon:

  20. Mode solution The power spectrum: The index: when : red spectrum; when : blue spectrum The constraint of nearly scale-invariance:

  21. Calculation of Non-Gaussianity Using the mode solutions, we can calculate the non- Gaussianity by in-in formalism. In-In Formalism: The 3-point correlation function is defined as: where is the vacuum in interaction picture. It is related to free vacuum through the interaction Hamiltonian with T being the time-ordered operator. So we have:

  22. Calculation of Non-Gaussianity For the 3rd order action , we have the interaction Hamiltonian: From which we can calculate the contributions of Non- Gaussianity from each part.

  23. Calculation of Non-Gaussianity Contribution from term:

  24. Calculation of Non-Gaussianity Contribution from term:

  25. Calculation of Non-Gaussianity Contribution from term:

  26. Calculation of Non-Gaussianity Contribution from term:

  27. Calculation of Non-Gaussianity Contribution from term: The results are very huge because it contains which we parameterized as , and it made the integral not the integer power law of , so different from the minimal coupling case, there are lots of integrals that cannot vanish.

  28. Calculation of Non-Gaussianity However, it can be obviously seen that many integrals have the same power-law of and can thus be combined. This will make things simpler. Define the shape of non-Gaussianity: we can have 20 shapes:

  29. Calculation of Non-Gaussianity

  30. Calculation of Non-Gaussianity

  31. Calculation of Non-Gaussianity

  32. Calculation of Non-Gaussianity

  33. Calculation of Non-Gaussianity

  34. Calculation of Non-Gaussianity

  35. Calculation of Non-Gaussianity The total shape: where SUMMARY: there are four classes of shapes: • Since we are assuming , they will definitely appear. • When (red spectrum), they will appear. • When (blue spectrum), they will not appear. • 4) When , they will appear and when , they will not.

  36. Calculation of Non-Gaussianity The estimator is defined as: and is also tedious. For example for the equilateral limit:

  37. Calculation of Non-Gaussianity

  38. Calculation of Non-Gaussianity

  39. Calculation of Non-Gaussianity

  40. Calculation of Non-Gaussianity

  41. Calculation of Non-Gaussianity Total : And have four classes as well as

  42. Summary

  43. Summary • Non-Gaussianities in single field inflation with non-minimal coupling • all the possible shapes of the 3-point correlation functions obtained; • different shapes will be involved in to give rise to non-Gaussianities for different tilt of power spectrum; • Possible to provide relation between 2- and 3-point correlation functions in order to constrain models. Surely, many more works remain to be done……

  44. Thank you! Happy New Year!

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