Strong scale dependent bispectrum in the Starobinsky model of inflation

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

2. Outline • Introduction and Motivations • 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 • Summary and Conclusion

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! • Better observations are on the way, bounds will get tighter or we will have a detection. • The bounds are shape dependent, so it’s important to calculate the exact shape.

4. Conditions of Maldacena’s No-Go Theorem • Single field, canonical kinetic term, slow-roll and standard • initial conditions imply Simplest models Maldacena ’02 Seery et al.’05 • In this talk, I will violate one of the conditions to generate large NG! 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.

5. Motivations for features Why not? • 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. Covi et al. ’06 Joy et al. ’08 Introduction of new parameters (scales) that are fine-tuned to coincide with the CMB “glitches” at But there might be many features so the tuning can be alleviated • 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 ) Might be due to: - particle production - duality cascade during brane inflation - periodic features (instantons in axion monodromy inflation) - phase transitions - massive modes Chen ’10 • These are more realistic scenarios. One can learn about the microscopic • theory of inflation. • 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!

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

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

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

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

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

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

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

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.

14. The equilateral limit Small scales: Large scales: Fast decay Large enhancement Black – Full Green – Dirac fc.Red – Other 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.

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.

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

17. Comparison with previous works Valid on large scales Takamizu et al. ’10, 1004.1870 Obtained using the next-to-leading order gradient expansion method Agree with us Disagrees with us Computed using the In-In formalism, even some sub-leading order corrections Martin and Sriramkumar ’11, 1109.5838 They computed this Dirac delta function contribution is on: Large scales: Small scales: 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.

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: Linear growth – strong scale dependence • Interesting behavior on small scales: Large enhancement factor Angular frequency • It would be interesting to observationally constrain this type of models.