Resonant non-Gaussianity based on Flauger & E.P. - - PowerPoint PPT Presentation

Resonant non-Gaussianity

based on Flauger & E.P. arXiv:1002.xxxx (hep-th) and Flauger, McAllister, E.P., Westphal & Xu arXiv:0907.2916 (hep-th) Enrico Pajer

Cornell University, Ithaca

Hamburg Feb 2010

Outline

1 Motivations 2 The model: inflation from axion monodromy 3 Non-Gaussianity in the bispectrum 4 Summary and conclusions

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 2 / 37

Motivations

Outline

1 Motivations 2 The model: inflation from axion monodromy 3 Non-Gaussianity in the bispectrum 4 Summary and conclusions

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 3 / 37

Motivations

Cosmological data

We are living in the golden age of

• bservational cosmology:

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 4 / 37

Motivations

Cosmological data

We are living in the golden age of

• bservational cosmology: COBE

goes to Stockholm,

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 4 / 37

Motivations

Cosmological data

We are living in the golden age of

• bservational cosmology: COBE

goes to Stockholm, WMAP has measured the CMB with percent

• accuracy. . .

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 4 / 37

Motivations

Cosmological data

We are living in the golden age of

• bservational cosmology: COBE

goes to Stockholm, WMAP has measured the CMB with percent

• accuracy. . .

and now Planck: the satellite, launched on May 2009, will have a full sky map by March! The situation on Dec 15, 2009:

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 4 / 37

Motivations

The picture emerging from the data

Inflation does not solve the horizon and flatness problem but can arguably alleviate them.

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 5 / 37

Motivations

The picture emerging from the data

Inflation does not solve the horizon and flatness problem but can arguably alleviate them. Neverless it is a spectacular model to generate cosmological perturbations.

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 5 / 37

Motivations

The picture emerging from the data

Inflation does not solve the horizon and flatness problem but can arguably alleviate them. Neverless it is a spectacular model to generate cosmological perturbations. So far the simplest models of inflation is compatible with the data, i.e. small, scale-invariant but slightly red tilted, Gaussian, adiabatic primordial curvature perturbations.

[see e.g. WMAP7]

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 5 / 37

Motivations

The picture emerging from the data

Inflation does not solve the horizon and flatness problem but can arguably alleviate them. Neverless it is a spectacular model to generate cosmological perturbations. So far the simplest models of inflation is compatible with the data, i.e. small, scale-invariant but slightly red tilted, Gaussian, adiabatic primordial curvature perturbations.

[see e.g. WMAP7]

Potential hints to go beyond a “vanilla” model of inflation? Can we distinguish the different models?

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 5 / 37

Motivations

Exciting signatures in the sky

Obervables that could deeply impact our picture of the early universe: Tensor modes:

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 6 / 37

Motivations

Exciting signatures in the sky

Obervables that could deeply impact our picture of the early universe: Tensor modes:

T anisotropies and polarization of the CMB. Bound: r < .20

[WMAP7+SN] .

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 6 / 37

Motivations

Exciting signatures in the sky

Obervables that could deeply impact our picture of the early universe: Tensor modes:

T anisotropies and polarization of the CMB. Bound: r < .20

[WMAP7+SN] .

A detection would support inflation and determine the high scale (order GUT) where it took place.

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 6 / 37

Motivations

Exciting signatures in the sky

Obervables that could deeply impact our picture of the early universe: Tensor modes:

T anisotropies and polarization of the CMB. Bound: r < .20

[WMAP7+SN] .

A detection would support inflation and determine the high scale (order GUT) where it took place.

non-Gaussianity:

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 6 / 37

Motivations

Exciting signatures in the sky

Obervables that could deeply impact our picture of the early universe: Tensor modes:

T anisotropies and polarization of the CMB. Bound: r < .20

[WMAP7+SN] .

A detection would support inflation and determine the high scale (order GUT) where it took place.

non-Gaussianity:

Three-point . Bounds: ∼ 1% (shape dependent).

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 6 / 37

Motivations

Exciting signatures in the sky

Obervables that could deeply impact our picture of the early universe: Tensor modes:

T anisotropies and polarization of the CMB. Bound: r < .20

[WMAP7+SN] .

A detection would support inflation and determine the high scale (order GUT) where it took place.

non-Gaussianity:

Three-point . Bounds: ∼ 1% (shape dependent). A detection would rule out the simplest class of models (a slowly rolling single canonically normalized field).

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 6 / 37

Motivations

Exciting signatures in the sky

Obervables that could deeply impact our picture of the early universe: Tensor modes:

T anisotropies and polarization of the CMB. Bound: r < .20

[WMAP7+SN] .

A detection would support inflation and determine the high scale (order GUT) where it took place.

non-Gaussianity:

Three-point . Bounds: ∼ 1% (shape dependent). A detection would rule out the simplest class of models (a slowly rolling single canonically normalized field).

Isocurvature modes, curvature, features in the spectrum, . . .

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 6 / 37

Motivations

Tensor modes and the Lyth bound

The detection of tensor modes, e.g. in the B-mode polarization, would fix the scale of inflation close to the GUT scale.

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 7 / 37

Motivations

Tensor modes and the Lyth bound

The detection of tensor modes, e.g. in the B-mode polarization, would fix the scale of inflation close to the GUT scale. Measuring tensor modes puts a lower bound on the range of variation of the inflaton

[Lyth 98]

∆φ Mpl > r 0.01

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 7 / 37

Motivations

Tensor modes and the Lyth bound

The detection of tensor modes, e.g. in the B-mode polarization, would fix the scale of inflation close to the GUT scale. Measuring tensor modes puts a lower bound on the range of variation of the inflaton

[Lyth 98]

∆φ Mpl > r 0.01 In a fundamental theory a flat potential over a superplanckian distance is hard to control, e.g. η-problem.

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 7 / 37

Motivations

Tensor modes and the Lyth bound

The detection of tensor modes, e.g. in the B-mode polarization, would fix the scale of inflation close to the GUT scale. Measuring tensor modes puts a lower bound on the range of variation of the inflaton

[Lyth 98]

∆φ Mpl > r 0.01 In a fundamental theory a flat potential over a superplanckian distance is hard to control, e.g. η-problem. This is the main motivation to consider axion monodromy inflation

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 7 / 37

Motivations

Tensor modes and the Lyth bound

The detection of tensor modes, e.g. in the B-mode polarization, would fix the scale of inflation close to the GUT scale. Measuring tensor modes puts a lower bound on the range of variation of the inflaton

[Lyth 98]

∆φ Mpl > r 0.01 In a fundamental theory a flat potential over a superplanckian distance is hard to control, e.g. η-problem. This is the main motivation to consider axion monodromy inflation

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 7 / 37

Motivations

Tensor modes and the Lyth bound

The detection of tensor modes, e.g. in the B-mode polarization, would fix the scale of inflation close to the GUT scale. Measuring tensor modes puts a lower bound on the range of variation of the inflaton

[Lyth 98]

∆φ Mpl > r 0.01 In a fundamental theory a flat potential over a superplanckian distance is hard to control, e.g. η-problem. This is the main motivation to consider axion monodromy inflation Schematically Tensor modes ⇒ High scale ⇒ Large field ⇒ more UV-sensitive

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 7 / 37

Motivations

UV-sensitivity

EFT approach: learn about higher scales studying UV-sensitive

• bservables.

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 8 / 37

Motivations

UV-sensitivity

EFT approach: learn about higher scales studying UV-sensitive

• bservables.

Inflation is a UV-sensitive mechanism. Schematically V (φ) = 1 2m2φ2 +

• n

λn φn Mn−4

pl

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 8 / 37

Motivations

UV-sensitivity

EFT approach: learn about higher scales studying UV-sensitive

• bservables.

Inflation is a UV-sensitive mechanism. Schematically V (φ) = 1 2m2φ2 +

• n

λn φn Mn−4

pl

Within string theory and supergravity many models suffer from an η-problem.

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 8 / 37

Motivations

UV-sensitivity

EFT approach: learn about higher scales studying UV-sensitive

• bservables.

Inflation is a UV-sensitive mechanism. Schematically V (φ) = 1 2m2φ2 +

• n

λn φn Mn−4

pl

Within string theory and supergravity many models suffer from an η-problem. We need to invoke a symmetry, e.g. shift symmetry.

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 8 / 37

Motivations

UV-sensitivity

EFT approach: learn about higher scales studying UV-sensitive

• bservables.

Inflation is a UV-sensitive mechanism. Schematically V (φ) = 1 2m2φ2 +

• n

λn φn Mn−4

pl

Within string theory and supergravity many models suffer from an η-problem. We need to invoke a symmetry, e.g. shift symmetry. Then we need a fundamental theory (UV-finite) to ask if, how and where the symmetry is broken.

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 8 / 37

Motivations

Axion monodromy

The idea is to invoke shift symmetry to protect the flatness of the potential.

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 9 / 37

Motivations

Axion monodromy

The idea is to invoke shift symmetry to protect the flatness of the potential. Then the symmetry is broken in a controlled way inducing a monodromy.

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 9 / 37

Motivations

Axion monodromy

The idea is to invoke shift symmetry to protect the flatness of the potential. Then the symmetry is broken in a controlled way inducing a monodromy. This enlarges the field space and provides the potential for inflation.

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 9 / 37

Motivations

Axion monodromy

The idea is to invoke shift symmetry to protect the flatness of the potential. Then the symmetry is broken in a controlled way inducing a monodromy. This enlarges the field space and provides the potential for inflation. Generically non-perturbative effects generate modulations.

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 9 / 37

Motivations

Axion monodromy

The idea is to invoke shift symmetry to protect the flatness of the potential. Then the symmetry is broken in a controlled way inducing a monodromy. This enlarges the field space and provides the potential for inflation. Generically non-perturbative effects generate modulations.

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 9 / 37

Motivations

Axion monodromy

The idea is to invoke shift symmetry to protect the flatness of the potential. Then the symmetry is broken in a controlled way inducing a monodromy. This enlarges the field space and provides the potential for inflation. Generically non-perturbative effects generate modulations. The initial compact field space, e.g. the circle, is extended to a non-compact range, e.g. a spiral staircase.

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 9 / 37

The model

Outline

1 Motivations 2 The model: inflation from axion monodromy 3 Non-Gaussianity in the bispectrum 4 Summary and conclusions

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 10 / 37

The model

Axions in string and field theory

Axions are common in string and field theory. E.g. model dependent and independent axions from 10D forms.

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 11 / 37

The model

Axions in string and field theory

Axions are common in string and field theory. E.g. model dependent and independent axions from 10D forms. Shift symmetry to all orders in perturbation theory φ(x) → φ(x) + constant

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 11 / 37

The model

Axions in string and field theory

Axions are common in string and field theory. E.g. model dependent and independent axions from 10D forms. Shift symmetry to all orders in perturbation theory φ(x) → φ(x) + constant Non-perturbative effects from e.g. brane or world-sheet instantons induce periodic potentials.

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 11 / 37

The model

Axions in string and field theory

Axions are common in string and field theory. E.g. model dependent and independent axions from 10D forms. Shift symmetry to all orders in perturbation theory φ(x) → φ(x) + constant Non-perturbative effects from e.g. brane or world-sheet instantons induce periodic potentials. The axion decay constant f determines the periodicity of the canonically normalized axion L ⊃ 1 2(∂φ)2 + Λ4 cos φ f

• ⇒ φ(x) → φ(x) + 2πf

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 11 / 37

The model

Axions in string and field theory

Axions are common in string and field theory. E.g. model dependent and independent axions from 10D forms. Shift symmetry to all orders in perturbation theory φ(x) → φ(x) + constant Non-perturbative effects from e.g. brane or world-sheet instantons induce periodic potentials. The axion decay constant f determines the periodicity of the canonically normalized axion L ⊃ 1 2(∂φ)2 + Λ4 cos φ f

• ⇒ φ(x) → φ(x) + 2πf

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 11 / 37

The model

Axions in string and field theory

Axions are common in string and field theory. E.g. model dependent and independent axions from 10D forms. Shift symmetry to all orders in perturbation theory φ(x) → φ(x) + constant Non-perturbative effects from e.g. brane or world-sheet instantons induce periodic potentials. The axion decay constant f determines the periodicity of the canonically normalized axion L ⊃ 1 2(∂φ)2 + Λ4 cos φ f

• ⇒ φ(x) → φ(x) + 2πf

Modulation are generic Flat potential ⇒ Shift symmetry ⇒ Axions ⇒ Non-perturbative modulations

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 11 / 37

The model

A cartoon of inflation from axion monodromy

We consider Type IIB (orientifolds) because moduli stabilization is more developed.

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 12 / 37

The model

A cartoon of inflation from axion monodromy

We consider Type IIB (orientifolds) because moduli stabilization is more developed. In the N = 1, 4D effective theory there is an axion c(x) coming from 10D C2 integrated over a two-cycle Σ2

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 12 / 37

The model

A cartoon of inflation from axion monodromy

We consider Type IIB (orientifolds) because moduli stabilization is more developed. In the N = 1, 4D effective theory there is an axion c(x) coming from 10D C2 integrated over a two-cycle Σ2 Wrapping a 5-brane over Σ2 induces a potential for c(x) (world-sheets with boundary).

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 12 / 37

The model

A cartoon of inflation from axion monodromy

We consider Type IIB (orientifolds) because moduli stabilization is more developed. In the N = 1, 4D effective theory there is an axion c(x) coming from 10D C2 integrated over a two-cycle Σ2 Wrapping a 5-brane over Σ2 induces a potential for c(x) (world-sheets with boundary). If the 5-brane is a warped region, the potential leads to viable inflation (COBE normalization)

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 12 / 37

The model

A cartoon of inflation from axion monodromy

We consider Type IIB (orientifolds) because moduli stabilization is more developed. In the N = 1, 4D effective theory there is an axion c(x) coming from 10D C2 integrated over a two-cycle Σ2 Wrapping a 5-brane over Σ2 induces a potential for c(x) (world-sheets with boundary). If the 5-brane is a warped region, the potential leads to viable inflation (COBE normalization) The moduli stabilization ´ a la KKLT does not spoil the shift symmetry.

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 12 / 37

The model

A cartoon of inflation from axion monodromy

We consider Type IIB (orientifolds) because moduli stabilization is more developed. In the N = 1, 4D effective theory there is an axion c(x) coming from 10D C2 integrated over a two-cycle Σ2 Wrapping a 5-brane over Σ2 induces a potential for c(x) (world-sheets with boundary). If the 5-brane is a warped region, the potential leads to viable inflation (COBE normalization) The moduli stabilization ´ a la KKLT does not spoil the shift symmetry. Non-perturbative corrections (e.g. to the K¨ ahler potential) induce small ripples

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 12 / 37

The model

Linear potential for the inflaton

The shift symmetry can be broken in the presence of boundaries. Consider a D5-brane wrapped on a two-cycle Σ. The DBI action −T5

• d5xe−Φ
• det (Gind + Bind)

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 13 / 37

The model

Linear potential for the inflaton

The shift symmetry can be broken in the presence of boundaries. Consider a D5-brane wrapped on a two-cycle Σ. The DBI action −T5

• d5xe−Φ
• det (Gind + Bind)

The shift b(x) → b(x) + const of b(x) =

• Σ B2

stores some potential energy. V (b) = T5

• L4 + b2 ∼ T5b

for large b This generates the linear inflaton potential (and break SUSY). COBE normalization and control require to red-shift T5

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 13 / 37

The model

The effective potential

Inflation is driven by a real scalar field with potential V (φ) = µ3φ + bµ3f cos φ f

The model

The effective potential

Inflation is driven by a real scalar field with potential V (φ) = µ3φ + bµ3f cos φ f

• b < 1 ⇒ monotonic potential

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 14 / 37

The model

The effective potential

Inflation is driven by a real scalar field with potential V (φ) = µ3φ + bµ3f cos φ f

• b < 1 ⇒ monotonic potential

φ ≫ Mpl gives large-field inflation. With µ = 6 · 10−4Mpl and φin ≃ 11Mpl one fits COBE. We will not discuss reheating.

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 14 / 37

The model

The effective potential

Inflation is driven by a real scalar field with potential V (φ) = µ3φ + bµ3f cos φ f

• b < 1 ⇒ monotonic potential

φ ≫ Mpl gives large-field inflation. With µ = 6 · 10−4Mpl and φin ≃ 11Mpl one fits COBE. We will not discuss reheating. f ≪ Mpl many short ripples. Different from the superplanckian case that seems to be hard to achieve in string theory.

[Banks et al. 03]

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 14 / 37

The model

The effective potential

Inflation is driven by a real scalar field with potential V (φ) = µ3φ + bµ3f cos φ f

• b < 1 ⇒ monotonic potential

φ ≫ Mpl gives large-field inflation. With µ = 6 · 10−4Mpl and φin ≃ 11Mpl one fits COBE. We will not discuss reheating. f ≪ Mpl many short ripples. Different from the superplanckian case that seems to be hard to achieve in string theory.

[Banks et al. 03]

Oscillations in the CMB #osci ≃ (10f)−1

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 14 / 37

The model

Background oscillations

The (Hubble) slow-roll parameters oscillate ǫ ≡ − ˙ H H2 ≃ ǫ0 + ǫosci cos φ0 f

• ≃

1 2φ2 + 3bf φin cos(φ0 f ) η ≡ ˙ ǫ ǫH ≃ η0 + ηosci sin φ0 f

• ≃

0 + 6b sin φ0 f

• and can resonate with the perturbations ζ.

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 15 / 37

The model

Background oscillations

The (Hubble) slow-roll parameters oscillate ǫ ≡ − ˙ H H2 ≃ ǫ0 + ǫosci cos φ0 f

• ≃

1 2φ2 + 3bf φin cos(φ0 f ) η ≡ ˙ ǫ ǫH ≃ η0 + ηosci sin φ0 f

• ≃

0 + 6b sin φ0 f

• and can resonate with the perturbations ζ.

Notice that ˙ η ≫ ǫ so one can not use slow-roll formulae to compute the perturbations.

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 15 / 37

The model

The Mukhanov-Sasaki equation

Mukhanov-Sasaki equation leads to the power spectrum Slow roll is not enough because ǫ and δ are not approximatively constant, in fact oscillate fast. d2ζk dx2 − 2(1 + 2ǫ + δ) x dζk dx + ζk = 0 , where x ≡ −kτ and τ is the conformal time adτ ≡ dt

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 16 / 37

The model

The Mukhanov-Sasaki equation

Mukhanov-Sasaki equation leads to the power spectrum Slow roll is not enough because ǫ and δ are not approximatively constant, in fact oscillate fast. d2ζk dx2 − 2(1 + 2ǫ + δ) x dζk dx + ζk = 0 , where x ≡ −kτ and τ is the conformal time adτ ≡ dt We solve perturbatively in b.

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 16 / 37

The model

The Mukhanov-Sasaki equation

Mukhanov-Sasaki equation leads to the power spectrum Slow roll is not enough because ǫ and δ are not approximatively constant, in fact oscillate fast. d2ζk dx2 − 2(1 + 2ǫ + δ) x dζk dx + ζk = 0 , where x ≡ −kτ and τ is the conformal time adτ ≡ dt We solve perturbatively in b. There is a resonance between ζ and δ at xres = 1/(2fφ).

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 16 / 37

The model

The Mukhanov-Sasaki equation

Mukhanov-Sasaki equation leads to the power spectrum Slow roll is not enough because ǫ and δ are not approximatively constant, in fact oscillate fast. d2ζk dx2 − 2(1 + 2ǫ + δ) x dζk dx + ζk = 0 , where x ≡ −kτ and τ is the conformal time adτ ≡ dt We solve perturbatively in b. There is a resonance between ζ and δ at xres = 1/(2fφ). Across the resonance a mode gets excited ζk(x) ≃ i π 2 x3/2H(1)

3/2(x)

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 16 / 37

The model

The Mukhanov-Sasaki equation

Mukhanov-Sasaki equation leads to the power spectrum Slow roll is not enough because ǫ and δ are not approximatively constant, in fact oscillate fast. d2ζk dx2 − 2(1 + 2ǫ + δ) x dζk dx + ζk = 0 , where x ≡ −kτ and τ is the conformal time adτ ≡ dt We solve perturbatively in b. There is a resonance between ζ and δ at xres = 1/(2fφ). Across the resonance a mode gets excited ζk(x) ≃ i π 2 x3/2H(1)

3/2(x) − c− k i

π 2 x3/2H(2)

3/2(x) ,

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 16 / 37

The model

The Mukhanov-Sasaki equation

Mukhanov-Sasaki equation leads to the power spectrum Slow roll is not enough because ǫ and δ are not approximatively constant, in fact oscillate fast. d2ζk dx2 − 2(1 + 2ǫ + δ) x dζk dx + ζk = 0 , where x ≡ −kτ and τ is the conformal time adτ ≡ dt We solve perturbatively in b. There is a resonance between ζ and δ at xres = 1/(2fφ). Across the resonance a mode gets excited ζk(x) ≃ i π 2 x3/2H(1)

3/2(x) − c− k (x)i

π 2 x3/2H(2)

3/2(x) ,

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 16 / 37

The model

The effect of the resonance on the modes

The equation for c−

k (x) can be solved exactly

c−

k (x)

∼ bfφ∗e−i log k

f Γ

• 1 −

i fφ∗ , −2ix

• Enrico Pajer

(Cornell) Resonant non-Gaussianity Hamburg Feb 2010 17 / 37

The model

The effect of the resonance on the modes

The equation for c−

k (x) can be solved exactly

c−

k (x)

∼ bfφ∗e−i log k

f Γ

• 1 −

i fφ∗ , −2ix

• the resonance is seen numerically and anaytically

20 30 40 50 60 70 80 90 x 0.001 0.001 0.002 0.003 0.004 0.005 Reck

x

20 30 40 50 60 70 80 90 x 0.001 0.001 0.002 0.003 0.004 0.005 Reck

x

20 30 40 50 60 70 80 90 x 0.001 0.001 0.002 0.003 0.004 0.005 Reck

x

20 30 40 50 60 70 80 90 x 0.001 0.001 0.002 0.003 0.004 0.005 Reck

x

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 17 / 37

The model

Analytical result for the spectrum

Oscillations in the spectrum: Ps(k) = As k k∗ ns−1 1 + δns cos φk f

• δns

∼ 3b

• 2πfφ,

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 18 / 37

The model

Analytical result for the spectrum

Oscillations in the spectrum: Ps(k) = As k k∗ ns−1 1 + δns cos φk f

• δns

∼ 3b

• 2πfφ,

Agreement with the numerics.

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 18 / 37

The model

Observational constraints on the spectrum

We have evolved this signlat with CAMB and compared it with

• WMAP5. The one- and two-sigma exclusion contours are

0.02 0.04 0.06 0.08 0.1 f 0.05 0.1 0.15 0.2 ∆ns

• 4
• 3.5
• 3
• 2.5
• 2
• 1.5
• 1

log10f 0.00025 0.0005 0.00075 0.001 0.00125 0.0015 0.00175 bf Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 19 / 37

The model

Observational constraints on the spectrum

The best fit next to the unbinned WMAP5 data looks like

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 20 / 37

The model

Observational constraints on the spectrum

The best fit next to the unbinned WMAP5 data looks like

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 20 / 37

The model

Observational constraints on the spectrum

The best fit next to the unbinned WMAP5 data looks like

5 10 50 100 500 1000

• 2000

2000 4000 6000 8000 1C 2Π ΜK2 5 10 50 100 500 1000

• 2000

2000 4000 6000 8000 1C 2Π ΜK2

The improvement of the fit is not statistically significant.

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 20 / 37

The model

Observational constraints on the spectrum

The best fit next to the unbinned WMAP5 data looks like

5 10 50 100 500 1000

• 2000

2000 4000 6000 8000 1C 2Π ΜK2 5 10 50 100 500 1000

• 2000

2000 4000 6000 8000 1C 2Π ΜK2

The improvement of the fit is not statistically significant. The bound of WMAP5 data on the parameters of the model is roughly fb < 10−4

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 20 / 37

The model

Tensor modes

A string theory model of large-field inflation and detectable tensor modes For b = 0 and using slow roll r ≃ 0.07 This is within Planck sensitivity!

Chaotic Inflation

0.92 0.94 0.96 0.98 1.0 1.02

ns

IIA Nil manifolds µ10/32/3 N = 50 N = 60 Linear Axion Inflation µ3 N = 50 N = 60

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 21 / 37

Non-Gaussianity in the bispectrum

Outline

1 Motivations 2 The model: inflation from axion monodromy 3 Non-Gaussianity in the bispectrum 4 Summary and conclusions

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 22 / 37

Non-Gaussianity in the bispectrum

n-point functions

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 23 / 37

Non-Gaussianity in the bispectrum

n-point functions

For a Gaussian distributed varible ζ ζ2n+1 = 0 , ζ2n ∝ ζ2n

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 23 / 37

Non-Gaussianity in the bispectrum

n-point functions

For a Gaussian distributed varible ζ ζ2n+1 = 0 , ζ2n ∝ ζ2n Due torotational and translational invariance, the bispectrum ζ3 depends on 3 variables ζ3 ≡ (2π)3fNLF(k1, k2, k3)δ3(k1 + k2 + k3) fNL gives the size and the normalized F(k1, k2, k3) gives the shape.

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 23 / 37

Non-Gaussianity in the bispectrum

Computing the bispectrum

Primordial perturbations from inflation are computed using the in-in formalism

[Maldacena 03] :

< ζk1(t)ζk2(t)ζk3(t) >= −i t

t0

dt′ <

• ζk1(t)ζk2(t)ζk3(t), HI(t′)
• >

where the intaraction Hamiltonian at order ζ3 is obtained expanding around an inflationary solution HI =

• aǫ2ζζ′2 + aǫ2ζ(∂ζ)2 − 2ǫζ′(∂ζ)(∂χ)

+a 2ǫ ˙ ηζ2ζ′ + ǫ 2a(∂ζ)(∂χ)(∂2χ) + ǫ 4a(∂2ζ)(∂χ)2 , χ ≡ a2ǫ∂−2 ˙ ζ

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 24 / 37

Non-Gaussianity in the bispectrum

Computing the bispectrum

Primordial perturbations from inflation are computed using the in-in formalism

[Maldacena 03] :

< ζk1(t)ζk2(t)ζk3(t) >= −i t

t0

dt′ <

• ζk1(t)ζk2(t)ζk3(t), HI(t′)
• >

where the intaraction Hamiltonian at order ζ3 is obtained expanding around an inflationary solution HI =

• aǫ2ζζ′2 + aǫ2ζ(∂ζ)2 − 2ǫζ′(∂ζ)(∂χ)

+a 2ǫ ˙ ηζ2ζ′ + ǫ 2a(∂ζ)(∂χ)(∂2χ) + ǫ 4a(∂2ζ)(∂χ)2 , χ ≡ a2ǫ∂−2 ˙ ζ

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 24 / 37

Non-Gaussianity in the bispectrum

From primordial perturbations to the CMB

The 2D temperature fluctuations decomposed in spherical harmonics ∆T T (ˆ n) =

• lm

almYlm(ˆ n) .

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 25 / 37

Non-Gaussianity in the bispectrum

From primordial perturbations to the CMB

The 2D temperature fluctuations decomposed in spherical harmonics ∆T T (ˆ n) =

• lm

almYlm(ˆ n) . The three-point correlation function of the CMB al1m1al2m2al3m3 ∝

• d3

k1 (2π)3 d3 k2 (2π)3 d3 k3 (2π)3 Y ∗

l1m1(ˆ

k1)Y ∗

l2m2(ˆ

k2)Y ∗

l3m3(ˆ

k3) ×δ3

• i

ki

• F(k1, k2, k3)∆T

l1(k1)∆T l2(k2)∆T l3(k3) .

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 25 / 37

Non-Gaussianity in the bispectrum

From primordial perturbations to the CMB

The 2D temperature fluctuations decomposed in spherical harmonics ∆T T (ˆ n) =

• lm

almYlm(ˆ n) . The three-point correlation function of the CMB al1m1al2m2al3m3 ∝

• d3

k1 (2π)3 d3 k2 (2π)3 d3 k3 (2π)3 Y ∗

l1m1(ˆ

k1)Y ∗

l2m2(ˆ

k2)Y ∗

l3m3(ˆ

k3) ×δ3

• i

ki

• F(k1, k2, k3)∆T

l1(k1)∆T l2(k2)∆T l3(k3) .

Number of operations: l3

max ∼ 109 integrals with 1010 operations

each.

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 25 / 37

Non-Gaussianity in the bispectrum

From primordial perturbations to the CMB

The 2D temperature fluctuations decomposed in spherical harmonics ∆T T (ˆ n) =

• lm

almYlm(ˆ n) . The three-point correlation function of the CMB al1m1al2m2al3m3 ∝

• d3

k1 (2π)3 d3 k2 (2π)3 d3 k3 (2π)3 Y ∗

l1m1(ˆ

k1)Y ∗

l2m2(ˆ

k2)Y ∗

l3m3(ˆ

k3) ×δ3

• i

ki

• F(k1, k2, k3)∆T

l1(k1)∆T l2(k2)∆T l3(k3) .

Number of operations: l3

max ∼ 109 integrals with 1010 operations

each. 1019 operatons, numerically very challenging!

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 25 / 37

Non-Gaussianity in the bispectrum

The cosine of shapes

How can we look for any non-Gaussian signal when even a single

• ne requires computational superpowers? This is an open
• problem. . .

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 26 / 37

Non-Gaussianity in the bispectrum

The cosine of shapes

How can we look for any non-Gaussian signal when even a single

• ne requires computational superpowers? This is an open
• problem. . .

If F · F ′ is a scalar product of 3D shapes, define the cosine C(F, F ′) ≡ F · F ′

• (F · F) (F ′ · F ′)

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 26 / 37

Non-Gaussianity in the bispectrum

The cosine of shapes

How can we look for any non-Gaussian signal when even a single

• ne requires computational superpowers? This is an open
• problem. . .

If F · F ′ is a scalar product of 3D shapes, define the cosine C(F, F ′) ≡ F · F ′

• (F · F) (F ′ · F ′)

When |C(F, F ′)| ∼ 1, the two shapes are similar. The

• bservational constraints on one apply to the other as well

[Babich et al. 04] .

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 26 / 37

Non-Gaussianity in the bispectrum

The state of the art

Ruling out the ”simplest“ model of inflation? Single-field slow-roll inflation with canonical kinetic term, i.e. vanilla inflation, leads to undetectable non-Gaussianity.

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 27 / 37

Non-Gaussianity in the bispectrum

The state of the art

Ruling out the ”simplest“ model of inflation? Single-field slow-roll inflation with canonical kinetic term, i.e. vanilla inflation, leads to undetectable non-Gaussianity. On the other hand Inflation with large non-Gaussianity:

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 27 / 37

Non-Gaussianity in the bispectrum

The state of the art

Ruling out the ”simplest“ model of inflation? Single-field slow-roll inflation with canonical kinetic term, i.e. vanilla inflation, leads to undetectable non-Gaussianity. On the other hand Inflation with large non-Gaussianity:

non-canonical kinetic terms (e.g. DBI)

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 27 / 37

Non-Gaussianity in the bispectrum

The state of the art

Ruling out the ”simplest“ model of inflation? Single-field slow-roll inflation with canonical kinetic term, i.e. vanilla inflation, leads to undetectable non-Gaussianity. On the other hand Inflation with large non-Gaussianity:

non-canonical kinetic terms (e.g. DBI) extra light spectator fields (e.g. curvaton)

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 27 / 37

Non-Gaussianity in the bispectrum

The state of the art

Ruling out the ”simplest“ model of inflation? Single-field slow-roll inflation with canonical kinetic term, i.e. vanilla inflation, leads to undetectable non-Gaussianity. On the other hand Inflation with large non-Gaussianity:

non-canonical kinetic terms (e.g. DBI) extra light spectator fields (e.g. curvaton) multifields

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 27 / 37

Non-Gaussianity in the bispectrum

The state of the art

Ruling out the ”simplest“ model of inflation? Single-field slow-roll inflation with canonical kinetic term, i.e. vanilla inflation, leads to undetectable non-Gaussianity. On the other hand Inflation with large non-Gaussianity:

non-canonical kinetic terms (e.g. DBI) extra light spectator fields (e.g. curvaton) multifields violations of slow roll.

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 27 / 37

Non-Gaussianity in the bispectrum

The state of the art

Ruling out the ”simplest“ model of inflation? Single-field slow-roll inflation with canonical kinetic term, i.e. vanilla inflation, leads to undetectable non-Gaussianity. On the other hand Inflation with large non-Gaussianity:

non-canonical kinetic terms (e.g. DBI) extra light spectator fields (e.g. curvaton) multifields violations of slow roll.

Only a handful of shapes have been constrained by observations

[WMAP7, Smith et al. 09] . They are all scale invariant.

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 27 / 37

Non-Gaussianity in the bispectrum

Observational constraints

Observational constraints

[WMAP7] on three different shapes:

Local −10 < floc < 74 Orthogonal −410 < fort < 6 Equilateral −214 < fequi < 266

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 28 / 37

Non-Gaussianity in the bispectrum

Observational constraints

Observational constraints

[WMAP7] on three different shapes:

Local −10 < floc < 74 Orthogonal −410 < fort < 6 Equilateral −214 < fequi < 266 There is no evidence of non-Gaussianity so far...

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 28 / 37

Non-Gaussianity in the bispectrum

Resonant non-Gaussianity

Large non-Gaussianity from modulations Modulations on the potential violate slow roll and can induce large non-Gaussianity. We now present resonant non-Gaussianity.

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 29 / 37

Non-Gaussianity in the bispectrum

Resonant non-Gaussianity

Large non-Gaussianity from modulations Modulations on the potential violate slow roll and can induce large non-Gaussianity. We now present resonant non-Gaussianity. They are very large and are not scale invariant

[Chen et al. 08] .

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 29 / 37

Non-Gaussianity in the bispectrum

Resonant non-Gaussianity

Large non-Gaussianity from modulations Modulations on the potential violate slow roll and can induce large non-Gaussianity. We now present resonant non-Gaussianity. They are very large and are not scale invariant

[Chen et al. 08] .

Resonant non-Gaussianity is orthogonal to any other known

• shape. Hence there are almost no constraints on it.

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 29 / 37

Non-Gaussianity in the bispectrum

Calculation of resonant of non-Gaussianity

Schematically

[Chen et al 08, Flauger & E.P. 10]

ζk1ζk2ζk3 ≃ (2π)3δ3(k1 + k2 + k3)ζk1ζk2ζk3 ×

• dt′2a3ǫ ˙

δ

• ζ∗

k1ζ∗ k2 ˙

ζ∗

k3 + 2 perm.

• + c.c. ,

ζk ∼ x3/2H(1)(x) ∼ eix ˙ δ ∼ sin log x fφ

• Enrico Pajer

(Cornell) Resonant non-Gaussianity Hamburg Feb 2010 30 / 37

Non-Gaussianity in the bispectrum

Calculation of resonant of non-Gaussianity

Schematically

[Chen et al 08, Flauger & E.P. 10]

ζk1ζk2ζk3 ≃ (2π)3δ3(k1 + k2 + k3)ζk1ζk2ζk3 ×

• dt′2a3ǫ ˙

δ

• ζ∗

k1ζ∗ k2 ˙

ζ∗

k3 + 2 perm.

• + c.c. ,

ζk ∼ x3/2H(1)(x) ∼ eix ˙ δ ∼ sin log x fφ

• The frequency of ζk is stretched by the expansion from Mpl to H when

it exits the horizon.

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 30 / 37

Non-Gaussianity in the bispectrum

Calculation of resonant of non-Gaussianity

Schematically

[Chen et al 08, Flauger & E.P. 10]

ζk1ζk2ζk3 ≃ (2π)3δ3(k1 + k2 + k3)ζk1ζk2ζk3 ×

• dt′2a3ǫ ˙

δ

• ζ∗

k1ζ∗ k2 ˙

ζ∗

k3 + 2 perm.

• + c.c. ,

ζk ∼ x3/2H(1)(x) ∼ eix ˙ δ ∼ sin log x fφ

• The frequency of ζk is stretched by the expansion from Mpl to H when

it exits the horizon. Necessary condition H < ω < Mpl ⇒ fφ ≪ M2

pl and f ≪ Mpl

where ω is the frequency of background oscillations.

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 30 / 37

Non-Gaussianity in the bispectrum

Calculation of resonant of non-Gaussianity

The result for resonant non-Gaussianity is [Flauger & E.P.] ζk1ζk2ζk3 = (2π)7∆4

ζ

δ3(k1 + k2 + k3) k2

1k2 2k2 3

fres ×  sin log K/k∗ fφ∗

• + fφ∗
• i,j

ki kj cos log K/k∗ fφ∗   , fres ≡ 3 √ 2πb 8(fφ∗)3/2 .

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 31 / 37

Non-Gaussianity in the bispectrum

Calculation of resonant of non-Gaussianity

The result for resonant non-Gaussianity is [Flauger & E.P.] ζk1ζk2ζk3 = (2π)7∆4

ζ

δ3(k1 + k2 + k3) k2

1k2 2k2 3

fres ×  sin log K/k∗ fφ∗

• + fφ∗
• i,j

ki kj cos log K/k∗ fφ∗   , fres ≡ 3 √ 2πb 8(fφ∗)3/2 . One can try to use this explicit shape to compare resonant non-Gaussianity with the data.

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 31 / 37

Non-Gaussianity in the bispectrum

Resonant enhancement of non-Gaussianity

The size of the resonant non-Gaussianity is fres ≃ 3 √ 2π 8 b (fφ)3/2

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 32 / 37

Non-Gaussianity in the bispectrum

Resonant enhancement of non-Gaussianity

The size of the resonant non-Gaussianity is fres ≃ 3 √ 2π 8 b (fφ)3/2 Large resonant non-Gaussianity The non Gaussian signal is Liner in b as for the spectrum

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 32 / 37

Non-Gaussianity in the bispectrum

Resonant enhancement of non-Gaussianity

The size of the resonant non-Gaussianity is fres ≃ 3 √ 2π 8 b (fφ)3/2 Large resonant non-Gaussianity The non Gaussian signal is Liner in b as for the spectrum Fixing b, one finds fres ∝ f−3/2. Remember that δns ∝ f1/2. Both spectrum and bispectrum are potential observables.

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 32 / 37

Non-Gaussianity in the bispectrum

Resonant enhancement of non-Gaussianity

The size of the resonant non-Gaussianity is fres ≃ 3 √ 2π 8 b (fφ)3/2 Large resonant non-Gaussianity The non Gaussian signal is Liner in b as for the spectrum Fixing b, one finds fres ∝ f−3/2. Remember that δns ∝ f1/2. Both spectrum and bispectrum are potential observables. Non-scale-invariant due to the sinusoidal oscillation. It is scale invariant if avareged over the period.

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 32 / 37

Non-Gaussianity in the bispectrum

Oscillations in the spectrum and bispectrum

These are the observational constraints from the spectrum together with a contour plot of fres as functions of f and b.

• 4
• 3
• 2
• 1

log10f

• 4
• 3
• 2
• 1

log10b

TT TTT

For small f the bispectrum is the most relevant observable. The f range is restricted in the explicit string theory construction.

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 33 / 37

Non-Gaussianity in the bispectrum

Correlation with other shapes

The cosine as function of f is fast oscillating

0.0001 0.0002 0.0005 0.001 0.002 0.005 f 0.1 0.05 0.05 0.1 CSres,Sortho 0.0001 0.0002 0.0005 0.001 0.002 0.005 f 0.75 0.5 0.25 0.25 0.5 0.75 1 CSres,S 0.0001 0.0002 0.0005 0.001 0.002 0.005 f 0.1 0.05 0.05 0.1 CSres,Sequil 0.0001 0.0002 0.0005 0.001 0.002 0.005 f 0.1 0.05 0.05 0.1 CSres,Sloc

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 34 / 37

Non-Gaussianity in the bispectrum

Correlation with other shapes

The cosine as function of f is fast oscillating

0.0001 0.0002 0.0005 0.001 0.002 0.005 f 0.1 0.05 0.05 0.1 CSres,Sortho 0.0001 0.0002 0.0005 0.001 0.002 0.005 f 0.75 0.5 0.25 0.25 0.5 0.75 1 CSres,S 0.0001 0.0002 0.0005 0.001 0.002 0.005 f 0.1 0.05 0.05 0.1 CSres,Sequil 0.0001 0.0002 0.0005 0.001 0.002 0.005 f 0.1 0.05 0.05 0.1 CSres,Sloc

Notice that the correlation is always less than 10% and goes to zero for small f.

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 34 / 37

Non-Gaussianity in the bispectrum

Correlation with other shapes: heuristic

The resonant shape oscillates as F res(k1, k2, k3) ∝ cos log K fφ

• Enrico Pajer

(Cornell) Resonant non-Gaussianity Hamburg Feb 2010 35 / 37

Non-Gaussianity in the bispectrum

Correlation with other shapes: heuristic

The resonant shape oscillates as F res(k1, k2, k3) ∝ cos log K fφ

• Upon ntegration, at most half a period can contribute, hence for

any slowly varying shape |C(F res, F)| πfφ ≪ 1

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 35 / 37

Summary and conclusions

Outline

1 Motivations 2 The model: inflation from axion monodromy 3 Non-Gaussianity in the bispectrum 4 Summary and conclusions

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 36 / 37

Summary and conclusions

Summary

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 37 / 37

Summary and conclusions

Summary

Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 37 / 37