1. Thursday, 13 th of September of 2012 Strong scale dependent bispectrum in the Starobinsky model of inflation Frederico Arroja 이화여자대학교 EWHA WOMANS UNIVERSITY FA and M. Sasaki, JCAP 1208 (2012) 012 [arXiv:1204.6489 [astro-ph.CO]] 1

2. 2 Outline  Introduction and Motivations  Summary and Conclusion  The model  The background approximate analytical solution  Linear perturbations Analytical approximation to the mode function The power spectrum  The bispectrum The equilateral limit Appendix: For any triangle  The non-linearity function  Comparison with previous works

3. Introduction  Scalar (CMB temperature) perturbations have been observed. The non-Gaussianity (nG) (bispectrum, trispectrum, …) are other observables in addition to the power spectrum. There are many inflation models that give similar predictions for the power spectrum, which one (if any) is the correct one? We need to discriminate between models by using other observables, e.g. nG.  CMB is Gaussian to ~0.1%! However a detection of such small primordial nG would have profound implications! 3  The bounds are shape dependent, so it’s important to calculate the exact shape.  Better observations are on the way, bounds will get tighter or we will have a detection.

4.  In this talk, I will violate one of the conditions to generate large NG! Maldacena ’02Seery et al.’05 Conditions of Maldacena’s No-Go Theorem Single field, canonical kinetic term, slow-roll and standard initial conditions imply The Starobinsky model (‘92) breaks temporarily slow-roll but inflation never stops. - Was proposed to explain the correlation fc. of galaxies which was requiring more power on large scales in a EdS universe paradigm. - Also, it allows approximate analytical treatment of perturbations and was used to study superhorizon nonconservation of the curvature perturbation. Interesting bispectrum signatures. Simplest models 4

5. Motivations for features Might be due to: - particle production - duality cascade during brane inflation - periodic features (instantons in axion monodromy inflation) - phase transitions - massive modes  The inflaton’s potential might not be a smooth function.  Models with features in the potential (Lagrangian) have been shown to provide better fits to the power spectrum of the CMB. Introduction of new parameters (scales) that are fine-tuned to coincide with the CMB “glitches” at  Once we fix these parameters to get a better fit to the power spectrum the bispectrum signal is completely fixed: predictable  Interesting bispectrum signatures: scale-dependence (e.g. “ringing” and localization of ) PLANCK is out there taking data, its precision is higher so the current constraints on nG will improve considerably. It’s time for theorists to get the predictions in! But there might be many features so the tuning can be alleviated  These are more realistic scenarios. One can learn about the microscopic theory of inflation. Chen ’10 Covi et al. ’06Joy et al. ’08 Why not? 5

6. The model Vacuum domination assumption: Parameters of the model: -> Transition value In following plots used: Starobinsky ’92 Einstein gravity + canonical scalar field with the potential: to satisfy COBE normalization 6

7. The background analytical solution With the vacuum domination assumption: Equations of motion: Definitions of the slow-roll parameters: - Cosmic time - conformal time Will always be small, inflation doesn’t stop. Allows to solve the Klein-Gordon eq. after the transition analytically. 7

8. The slow-roll parameters Analytical approximations: - Transition scale Plots from Martin and Sriramkumar ’11, 1109.5838 Always small Temporarily large Starobinsky ’92 Temporarily large Subscripts: 0 transition quantities + before transition – after transition The analytical approximations are in good agreement with the numerical results. continuous discontinuous at late time SR is recovered 8

9. Linear perturbations In the co-moving gauge, the 3-metric is perturbed as: In Fourier space the eom is: Usual quantization: - gauge invariant, co-moving curvature perturbation wavenumber annihilation operator creation operator Using the Mukhanov-Sasaki variable, it becomes: 9

10. Analytical approximation to the mode function Bogoliubov coefficients are: Martin and Sriramkumar ’11, 1109.5838 Before the transition: Usual SR mode function with standard Bunch-Davies vacuum initial conditions Even after the transition one has Negative frequency modes like in the slow-roll case, so the general solution is: Starobinsky ’92 10

11. The power spectrum Plot from Martin and Sriramkumar ’11, 1109.5838 Starobinsky ’92 Definition: The analytical approximation is in good agreement with the numerical result. Nearly scale invariant 11

12. The bispectrum FA and Tanaka ’11, 1103.1102 The 3 rd order action: Leading terms After one integration by parts, takes the convenient form: In the In-In formalism the tree-level bispectrum is: Interacting vacuum Free vacuumCommutator Some time after the modes of interest have left the horizon Prescription Boundary term 12

13. The equilateral limit The contribution before the transition to the integral is small compared with the contribution from after the transition, the later is: Closed analytical form for both the integrals before and after the transition. 13

14. The equilateral limit Large scales: Small scales: Black – Full Green – Dirac fc. Red – Other Large enhancementFast decay For a smooth transition of width: Number of e-foldings to cross: The simple scaling: implies the range of scales affected as:  This gives the cut-off scale for the small scales linear growth. For smaller scales the amplitude should go quickly to zero. 14

15. The bispectrum for any triangle Contribution after the transition: Closed analytical form for both the integrals before and after the transition for any triangle. 15

16. The non-linearity function Large scales Small scales Equilateral limit If it is of order of one it may be observed 16

17. Comparison with previous works Martin and Sriramkumar ’11, 1109.5838 Takamizu et al. ’10, 1004.1870 Obtained using the next-to-leading order gradient expansion method Disagrees with us Large scales: Small scales: They computed this Dirac delta function contribution is on: Same results with opposite sign Becomes the leading result By adding the Dirac delta fc. contribution to their result we recover our previous answer.  One cannot neglect the Dirac delta fc. contribution. Valid on large scales Agree with us Computed using the In-In formalism, even some sub-leading order corrections 17

18. Summary and Conclusions  Computed the tree-level leading-order bispectrum in one of the Starobinsky models of inflation. It’s a canonical scalar field with a vacuum dominated potential. The linear term has an abrupt slope change. After this transition, the slow-roll approximation breaks down for some time. and become large.  Despite this, the mode admits approximate analytical solutions for background, linear perturbations and we now computed analytically the bispectrum. In the equilateral limit and on large scales, the non-linearity function is: Interesting behavior on small scales: Linear growth – strong scale dependence Angular frequency Large enhancement factor  It would be interesting to observationally constrain this type of models. 18