1. 21 st of March of 2011 On the role of the boundary terms for the non-Gaussianity Frederico Arroja 이화여자대학교 EWHA WOMANS UNIVERSITY FA and Takahiro Tanaka, arXiv:1103.1102 [astro-ph.CO] Cosmological Perturbation and CMB

2. Introduction Purpose of the talk:  To clarify (if clarification was needed) the role of the boundary terms in the calculation of the bispectrum for general models of k-inflation. These boundary terms appear from many integrations by parts to simplify the third order action  We work in the comoving gauge; Many integrations by parts are needed to show that the action has the right slow-roll suppression, this generates many boundary terms. Maldacena ’02 Recognized their importance and used field redefinitions to take them into account. Seery and Lidsey ’06 In the UC gauge and for the standard kinetic term model showed that one doesn’t need to do field redefinitions but then one should include the boundary interactions. Collins ’11 Repeats part of Maldacena’s calculation but keeps these terms in the action.

3. Motivations for higher-order statistics and non-linear perturbations The observational constraints on the non-linearity parameters and will improve significantly in the near future. NVSS: Xia et al. ’10 Clusters: Hoyle et al. ’10 and scale dependence Enqvist et al. ’10 - Parameterises the amplitude of the bispectrum - Parameterises the amplitude of the trispectrum Even some “detections”

4. Discriminate between modelsCalculate more observables  Why general k-inflation? Single field, canonical kinetic term, slow-roll and with standard initial conditions implies Maldacena ’02 Generates large non-Gaussianity through the non-standard kinetic term e.g. the bispectrum Motivations for higher-order statistics The bispectrum is a function of three momentum vectors, it contains much more information about the dynamics than the power spectrum. The trispectrum contains even more information. Because it includes the standard kinetic term model, DBI-inflation as particular cases Seery et al. ’06 ‘08

5. Silverstein and Tong’03 DBI inflation Examples: - Inflaton Standard kinetic term inflation Potential

6. For FLRW universes: Some definitions: Will determine the non-Gaussianity The model: K-Inflation Amendariz-Picon et al.’99 Sound speed

7. “Slow-roll” parameters Ifit does not imply that the inflaton is rolling slowly. The can be arbitrary, only its rate of change has to be small. Adiabatic sound speed  A “perfect” scalar field is dual to an irrotational barotropic perfect fluid. FA and M. Sasaki ’10 In general these two speeds are different: Note:

8. Comoving gauge: Perturbations Solution (leading order in slow-roll): - Comoving curvature perturbation Neglect tensor perturbations because they don’t contribute to the tree-level scalar bispectrum. Chen et al.’06 Usual quantisation:

9. Third order action After really many integrations by parts and excluding the boundary terms: Maldacena ’02 Chen et al.’06 The field redefinition: First order eom No slow-roll approximation was made.  Eliminates from the action the terms proportional to the eom.

10. The bispectrum from the field redefinition LOCAL SHAPE Chen et al.’06 The field redefinition introduces extra terms in the bispectrum as In Fourier space, the second line is: The first line is calculated as (tree-level): The omitted terms vanish when evaluated outside the horizon because

11. The boundary terms Total spatial derivative terms were omitted because they do not contribute to the three point function, Their contribution is proportional to which vanishes due to momentum conservation. Agrees with the previous results and shows that the boundary terms are important. The total action contains boundary terms previously omitted Non-zero bispectrum at leading order and in the limit

12. The field redefinition in the action  At the same time it eliminate the terms proportional to the eom and the previous boundary term. We neglected total spatial gradients and time derivatives terms that do not contribute to the leading order

13. What determines ? Discussion The boundary terms are necessary to erase from the action terms with two time derivatives on These interactions are not present in the original action and are not produced when we insert the solution of the constraints back in the action. They are generated by the integrations by parts. The boundary terms appear as a way of keeping track of these integrations.  The boundary termsaffect the bispectrum result. Terms containing cannot remain in the action because they were not present initially, after integrations by parts they should disappear. Terms containing only do not change the result because their Hamiltonian commutes

14. Conclusions  We obtained explicitly all total time derivative interactions in the comoving-gauge third order action for a general k-inflation model. but the leading order tree-level bispectrum produced from the time boundary interactions is non-zero and equal to the bispectrum coming from the field redefinition. We showed that total spatial gradients can be safely ignored These boundary terms are necessary to erase terms from the action that contain higher-derivatives in time that were generated by integrations by parts. One can ignore all the boundary terms that appear when one simplifies the action but then one has to perform a field redefinition to eliminate terms in the action that are proportional to the first order equation of motion. On the other hand, one might choose to keep all the boundary terms and calculate the bispectrum using the usual method without the need to do the field redefinition. We have shown that in the end the bispectrum of the curvature perturbation is the same in both procedures.