Non-Gaussianities of Single Field Inflation with Non-minimal Coupling

Non-Gaussianities of Single Field Inflation with Non-minimal

Coupling

Taotao Qiu2010-12-28

Based on paper: arXiv: 1012.1697[Hep-th]

(collaborated with Prof. K. C. Yang)

Outline

• Preliminary

• Non-Gaussianity in single field inflation with non-minimal coupling

• Summary

Preliminary

Why non-Gaussianities?• Observational development:

– Data become more and more accurate to study the non-linear properties of the fluctuation in CMB and LSS.

Y. Gong, X. Wang, Z. Zheng and X. Chen, Res. Astron. Astrophys. 10, 107 (2010) [arXiv:0904.4257 [astro-ph.CO]].

E. Komatsu et al., arXiv:1001.4538 [astro-ph.CO]; C. L. Bennett et al., arXiv:1001.4758 [astro-ph.CO].[Planck Collaboration], arXiv: astro-ph/0604069.

• Theoretical requirement: – The redundance of inflation models need to be distinguished.

Observational constraints on non-Gaussianity

• WMAP data:– WMAP 7yr (68% CL):

• Planck data:

E. Komatsu et al., Seven-year Wilkinson Microwave Anisotropy Probe (WMAP) observations: cosmological interpretation, arXiv:1001.4538 [SPIRES].

Planck collaboration, PLANCK the scientific programme,http://www.rssd.esa.int/SA/PLANCK/docs/Bluebook-ESA-SCI(2005)1.pdf[astro-ph/0604069] [SPIRES].

Definition• Local non-Gaussianity: the non-Gaussianity at every space point has the form of the

single random variable:

• Nonlocal non-Gaussianity: the non-Gaussianity may be sourced by correlation functions of

different space points.

characterized by “shape” compared to the local case.

Classification of NG shapes

• Equilateral:

• Squeezed

• Folded

Non-Gaussianity in single field inflation with non-

minimal coupling

Steps of non-Gaussianity Calculation• Get the constraint solution;

• Expand the action w.r.t. the perturbations and the constraints;

• Obtain the mode solution;

• Calculate the 3-point correlation function with in-in formalism.

Non-Gaussianities in single field inflation with non-minimal couplingMetric: ADM metric

Action:

The equation for field:

The Einstein Equations:

where

where R is the Ricci Scalar and is the kinetic term of the inflaton field

The equations

The constraint equations (varying the action w.r.t. and ):

where

Decomposite into 3+1 form:

where and K is the trace of

A TheoremTheorem: To calculate n-th order perturbation, one only need to expand the constraints and to (n-2)-th order.

Proof: Consider Lagrangian that contains constraints :

The equation of motion:

Expand to n-th order:

Lagrangian becomes:

Detailed analysis show that the coefficients before and are and respectively.

A TheoremFrom the equation of motion:

We can see that for 0th order:

for 1st order:

So the term of and will vanish in the expansion of , and we only need to consider up to (n-2)-th order.

Solutions of constraint for linear coupling

Comoving gauge:

We calculate from the constraint equations:

Define then and

One may check that

->1, the result

will return to GR!

Consider a linear coupling case:

Constraint expansion:

we have:

The constraint equations:

Up to the 3rd Order

Action:

Decomposition to 3rd order:

where , and are the 1st, 2nd and 3rd order term of , respectively.

Up to the 3rd OrderAction of 0th to 3rd order:

(background action)

(equation of motion)

where a is the scale factor,

Mode solutionBy varying the 2nd action w.r.t. , Using Fourier transformation:

we can obtain the 2nd action in momentum space:

Defining:

one have:

where

where

and thus

Mode solution

Solving the equation, we can get:

Define slow-roll parameters:

The equation can be rewritten in the leading order of and

where

Mode solution

Sub-horizon:

Super-horizon:

The same is for :

Sub-horizon:

Super-horizon:

The above solution can be splitted into sub-horizon and super-horizon approximations:

Mode solution

The power spectrum:

The index:

when : red spectrum; when : blue spectrum

The constraint of nearly scale-invariance:

Calculation of Non-Gaussianity

Using the mode solutions, we can calculate the non-Gaussianity by in-in formalism.

In-In Formalism:

where is the vacuum in interaction picture. It is related to free vacuum through the interaction Hamiltonian

The 3-point correlation function is defined as:

with T being the time-ordered operator.

So we have:


For the 3rd order action , we have the interaction Hamiltonian:

From which we can calculate the contributions of Non-Gaussianity from each part.

Calculation of Non-GaussianityContribution from term:


Contribution from term:







The results are very huge because it contains which we

parameterized as , and it made the integral not the

integer power law of , so different from the minimal coupling case,

there are lots of integrals that cannot vanish.


However, it can be obviously seen that many integrals have the same power-law of and can thus be combined. This will make things simpler.

Define the shape of non-Gaussianity:

we can have 20 shapes:







1) Since we are assuming , they will definitely appear.

2) When (red spectrum), they will appear.

3) When (blue spectrum), they will not appear.

4) When , they will appear and when , they will not.


The total shape:

where

SUMMARY: there are four classes of shapes:


The estimator is defined as:

and is also tedious. For example for the equilateral limit:






Total :

And have four classes as well as

Summary

Summary• Non-Gaussianities in single field inflation

with non-minimal coupling

all the possible shapes of the 3-point correlation functions obtained;

different shapes will be involved in to give rise to non-Gaussianities for different tilt of power spectrum;

Possible to provide relation between 2- and 3-point correlation functions in order to constrain models.

Surely, many more works remain to be done……

Thank you!Happy New

Year!

Documents

Non-Gaussianities of Single Field Inflation with Non-minimal Coupling