Measuring local-type might not rule out single-field inflation.Jonathan GancPhysics Dept., Univ. of Texas

July 6, 2011PASCOS 2011, Univ. of Cambridge

based on arXiv: 1104.0244

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Overview of presentation

I will demonstrate that single-field, slow-roll, canonical kinetic-term inflation with a non Bunch-Davies (BD) initial state has an enhanced local-limit bispectrumI will discuss the observed from this scenario in the CMB and whether it could affect the interpretation of a Planck detection of .

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Conventional wisdom

Single field inflation produces ≈ (5/12) (1 – ns)≈0.01, regardless of potential, kinetic term, or initial vacuum state

Creminelli & Zaldarriaga, 2004

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What is… ?… the bispectrum?

ζ(x1)

ζ(x2)ζ(x3)

the Fourier transform of the three-point function of the curvature perturbation ζ

… the squeezed or local limit?k1 when one of the wavenumbers

is much smaller than the other two, e.g. k3 ≪ k1 ≈ k2.k2

k3

… local-type or ?the best-fit parameter to a target bispectrum by the family of local bispectra, i.e. those generated from a Gaussian field by .

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Consider slow-roll inflationCanonical action:

Assume slow-roll,

Write action in terms of (Maldacena 2003):

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To quantize ζ…… we promote it to an operator :

The mode functions , are independent solutions of the classical equation of motion for ζ.The vacuum, slow-roll mode function is

We can represent a non-vacuum state by performing a Bogoliubov transformation:

new state has occupation number .

vacuum or Bunch-Davies state⇒ ,

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To calculate the bispectrum…

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… we use the in-in formalism:

We find

Bunch-Davies vs. non-Bunch-DaviesBunch-Davies

(, )

in the squeezed limit (

non-Bunch-Davies

𝐵𝜁∝12 𝑖𝑘1

𝐵𝜁∝1𝑖𝑘3

enhancement of in squeezed limit vs. BD!

This effect noticed only recently (Agullo & Parker 2011).Why was it missed earlier? People expected signal only in folded limit.

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What is the observable signal in the CMB from this enhancement?

is calculated from the CMB by fitting the observed angular bispectrum (using transfer functions and projecting onto a sphere) to that predicted by the local bispectrum

The angular bispectrum must be calculated numerically.

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What we findIf we suppose that across the wavenumbers visible today,

Thus, , even for very large .Such a signal is not distinguishable in the CMB.However, it’s larger than predicted by the consistency relation (c.f. ).

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But…… we’ve glossed over , the phase angle between the Bogoliubov parameters and .Why is this usually OK?

Expect to set , at early time by matching mode functions to some non-slow-roll equations. Relative phase will be dominated by exponential factors in mode functions . Thus, we expect . is very large, so oscillates quickly and averages out.

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What happens if ?Depending on choice of , we can get large positive or negative :

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Can achieve for

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What about the consistency relation?

The consistency relation predicts for single-field models.Here we can have .Is this a counterexample?

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Is this a counterexample to the consistency relation?

Initial conditions are set at some time , when is inside the horizon, i.e. .Non-BD terms are multiplied by. On average, the rapidly oscillating term so the above expression ; thus, we get contributions from non-BD terms.In exact local limit, , and we get zero contribution from non-BD terms.

Consistency relation does hold in exact local limit

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The takeawaySlow-roll single-field inflation with an excited initial state can produce an larger than expected.In the more probable case, and still not detectable in the CMB.If we allow the phase angle to be constant, we can get large, detectable .

The consistency relation is a useful guideline but it holds precisely only in the exact squeezed limit.

(A similar conclusion is reached in Ganc & Komatsu 2010).

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