Non-Gaussianities of Single Field Inflation with Non-minimal Coupling
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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)
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Outline
• Preliminary
• Non-Gaussianity in single field inflation with non-minimal coupling
• Summary
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Preliminary
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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.
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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].
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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.
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Classification of NG shapes
• Equilateral:
• Squeezed
• Folded
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Non-Gaussianity in single field inflation with non-
minimal coupling
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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.
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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
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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
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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.
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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.
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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:
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Up to the 3rd Order
Action:
Decomposition to 3rd order:
where , and are the 1st, 2nd and 3rd order term of , respectively.
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Up to the 3rd OrderAction of 0th to 3rd order:
(background action)
(equation of motion)
where a is the scale factor,
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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
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Mode solution
Solving the equation, we can get:
Define slow-roll parameters:
The equation can be rewritten in the leading order of and
where
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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:
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Mode solution
The power spectrum:
The index:
when : red spectrum; when : blue spectrum
The constraint of nearly scale-invariance:
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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:
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Calculation of Non-Gaussianity
For the 3rd order action , we have the interaction Hamiltonian:
From which we can calculate the contributions of Non-Gaussianity from each part.
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Calculation of Non-GaussianityContribution from term:
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Calculation of Non-Gaussianity
Contribution from term:
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Calculation of Non-Gaussianity
Contribution from term:
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Calculation of Non-Gaussianity
Contribution from term:
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Calculation of Non-Gaussianity
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.
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Calculation of Non-Gaussianity
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:
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Calculation of Non-Gaussianity
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Calculation of Non-Gaussianity
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Calculation of Non-Gaussianity
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Calculation of Non-Gaussianity
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Calculation of Non-Gaussianity
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Calculation of Non-Gaussianity
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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.
Calculation of Non-Gaussianity
The total shape:
where
SUMMARY: there are four classes of shapes:
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Calculation of Non-Gaussianity
The estimator is defined as:
and is also tedious. For example for the equilateral limit:
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Calculation of Non-Gaussianity
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Calculation of Non-Gaussianity
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Calculation of Non-Gaussianity
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Calculation of Non-Gaussianity
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Calculation of Non-Gaussianity
Total :
And have four classes as well as
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Summary
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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……
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Thank you!Happy New
Year!