Thursday, 13th of September of 2012

Strong scale dependent bispectrum in the Starobinsky model of inflation

Frederico Arroja

이화여자대학교EWHA WOMANS UNIVERSITY

FA and M. Sasaki, JCAP 1208 (2012) 012 [arXiv:1204.6489 [astro-ph.CO]]

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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

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!

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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.

In this talk, I will violate one of the conditions to generate large NG!

Maldacena ’02 Seery 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

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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. ’06 Joy et al. ’08

Why not?

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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

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The background analytical solution

With the vacuum dominationassumption:

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.

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The slow-roll parameters

Analytical approximations:

- Transition scale

Plots from Martin and Sriramkumar ’11, 1109.5838

Always smallTemporarily 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 recovered8

Linear perturbationsIn 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:

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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

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The power spectrumPlot from Martin and Sriramkumar ’11,

1109.5838

Starobinsky ’92

Definition:

The analytical approximation is in good agreement with the numerical result.

Nearly scale invariant

Nearly scale invariant

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The bispectrumFA and Tanaka ’11, 1103.1102

The 3rd 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 vacuum Commutator

Some time after the modes of interest have left the horizon

Prescription

Boundary term

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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.

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The equilateral limitLarge 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

The bispectrum for any triangleContribution after the transition:

Closed analytical form for both the integrals before and after the transition for any triangle.

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The non-linearity function

Large scales

Small scales

Equilateral limit

If it is of order of one it may be observed

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Comparison with previous works

Martin and Sriramkumar ’11, 1109.5838

Takamizu et al. ’10, 1004.1870Obtained 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 resultBy 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

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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 frequencyLarge enhancement factor

It would be interesting to observationally constrain this type of models.

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