21st of March of 2011

On the role of the boundary terms for the non-Gaussianity

Frederico Arroja

이화여자대학교EWHA WOMANS UNIVERSITY

FA and Takahiro Tanaka, arXiv:1103.1102 [astro-ph.CO]

Cosmological Perturbation and CMB

IntroductionPurpose 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.

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. ’10and scale dependence

Enqvist et al. ’10

- Parameterises the amplitude of the bispectrum

- Parameterises the amplitude of the trispectrum

Even some “detections”

• Discriminate between models Calculate 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

Silverstein and Tong’03DBI inflationExamples:

- Inflaton

Standard kinetic term inflation

Potential

For FLRW universes:

Some definitions:

Will determine the non-Gaussianity

The model: K-InflationAmendariz-Picon et al.’99

Sound speed

“Slow-roll” parameters

If it 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:

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:

Third order actionAfter 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.

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

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

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

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 terms affect 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

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 thebispectrum 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.