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: The equation for field: The Einstein Equations: where where R is the Ricci Scalar and is the kinetic term of the inflaton field

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

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 0 th order: for 1 st 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 Comoving gauge: We calculate from the constraint equations: Definethenand One may check that ->1, the result will return to GR! Consider a linear coupling case: Constraint expansion: we have: The constraint equations:

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

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

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

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

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

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: where is the vacuum in interaction picture. It is related to free vacuum through the interaction Hamiltonian The 3-point correlation function is defined as: with T being the time-ordered operator. So we have:

22. Calculation of Non-Gaussianity For the 3 rd 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

35. 1) Since we are assuming, they will definitely appear. 2) When (red spectrum), they will appear. 3) When (blue spectrum), they will not appear. 4) When, they will appear and when, they will not. Calculation of Non-Gaussianity The total shape: where SUMMARY: there are four classes of shapes:

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

41. Total : And have four classes as well as

42. Summary

43. 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!