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

Motivations UV-sensitivity EFT approach: learn about higher scales studying UV-sensitive observables. Inflation is a UV-sensitive mechanism. Schematically φ n V ( φ ) = 1 2 m 2 φ 2 + � λ n M n − 4 pl n 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 observables. Inflation is a UV-sensitive mechanism. Schematically φ n V ( φ ) = 1 2 m 2 φ 2 + � λ n M n − 4 pl n 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 observables. Inflation is a UV-sensitive mechanism. Schematically φ n V ( φ ) = 1 2 m 2 φ 2 + � λ n M n − 4 pl n 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 � φ � L ⊃ 1 2( ∂φ ) 2 + Λ 4 cos ⇒ φ ( x ) → φ ( x ) + 2 πf f 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 � φ � L ⊃ 1 2( ∂φ ) 2 + Λ 4 cos ⇒ φ ( x ) → φ ( x ) + 2 πf f 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 � φ � L ⊃ 1 2( ∂φ ) 2 + Λ 4 cos ⇒ φ ( x ) → φ ( x ) + 2 πf f Modulation are generic Flat Shift Non-perturbative ⇒ ⇒ ⇒ Axions potential symmetry 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 C 2 integrated over a two-cycle Σ 2 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 C 2 integrated over a two-cycle Σ 2 Wrapping a 5-brane over Σ 2 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 C 2 integrated over a two-cycle Σ 2 Wrapping a 5-brane over Σ 2 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 C 2 integrated over a two-cycle Σ 2 Wrapping a 5-brane over Σ 2 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 C 2 integrated over a two-cycle Σ 2 Wrapping a 5-brane over Σ 2 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 � � det ( G ind + B ind ) d 5 xe − Φ − T 5 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 � � det ( G ind + B ind ) d 5 xe − Φ − T 5 � The shift b ( x ) → b ( x ) + const of b ( x ) = Σ B 2 stores some potential energy. � L 4 + b 2 ∼ T 5 b V ( b ) = T 5 for large b This generates the linear inflaton potential (and break SUSY). COBE normalization and control require to red-shift T 5 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 � φ � V ( φ ) = µ 3 φ + bµ 3 f cos f

The model The effective potential Inflation is driven by a real scalar field with potential � φ � V ( φ ) = µ 3 φ + bµ 3 f cos 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 � φ � V ( φ ) = µ 3 φ + bµ 3 f cos f b < 1 ⇒ monotonic potential φ ≫ M pl gives large-field inflation. With µ = 6 · 10 − 4 M pl and φ in ≃ 11 M pl 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 � φ � V ( φ ) = µ 3 φ + bµ 3 f cos f b < 1 ⇒ monotonic potential φ ≫ M pl gives large-field inflation. With µ = 6 · 10 − 4 M pl and φ in ≃ 11 M pl one fits COBE. We will not discuss reheating. f ≪ M pl 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 � φ � V ( φ ) = µ 3 φ + bµ 3 f cos f b < 1 ⇒ monotonic potential φ ≫ M pl gives large-field inflation. With µ = 6 · 10 − 4 M pl and φ in ≃ 11 M pl one fits COBE. We will not discuss reheating. f ≪ M pl 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 ≃ (10 f ) − 1 Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 14 / 37

The model Background oscillations The (Hubble) slow-roll parameters oscillate ˙ H � φ 0 � ǫ ≡ − H 2 ≃ ǫ 0 + ǫ osci cos f 1 + 3 bf cos( φ 0 ≃ f ) 2 φ 2 φ in 0 ǫ ˙ � φ 0 � η ≡ ǫH ≃ η 0 + η osci sin f � φ 0 � ≃ 0 + 6 b sin 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 ˙ H � φ 0 � ǫ ≡ − H 2 ≃ ǫ 0 + ǫ osci cos f 1 + 3 bf cos( φ 0 ≃ f ) 2 φ 2 φ in 0 ǫ ˙ � φ 0 � η ≡ ǫH ≃ η 0 + η osci sin f � φ 0 � ≃ 0 + 6 b sin 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. d 2 ζ k dx 2 − 2(1 + 2 ǫ + δ ) dζ k dx + ζ k = 0 , x 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. d 2 ζ k dx 2 − 2(1 + 2 ǫ + δ ) dζ k dx + ζ k = 0 , x 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. d 2 ζ k dx 2 − 2(1 + 2 ǫ + δ ) dζ k dx + ζ k = 0 , x 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 x res = 1 / (2 fφ ). 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. d 2 ζ k dx 2 − 2(1 + 2 ǫ + δ ) dζ k dx + ζ k = 0 , x 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 x res = 1 / (2 fφ ). Across the resonance a mode gets excited � π 2 x 3 / 2 H (1) ζ k ( x ) ≃ i 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. d 2 ζ k dx 2 − 2(1 + 2 ǫ + δ ) dζ k dx + ζ k = 0 , x 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 x res = 1 / (2 fφ ). Across the resonance a mode gets excited � π � π 2 x 3 / 2 H (1) 2 x 3 / 2 H (2) 3 / 2 ( x ) − c − ζ k ( x ) ≃ i k i 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. d 2 ζ k dx 2 − 2(1 + 2 ǫ + δ ) dζ k dx + ζ k = 0 , x 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 x res = 1 / (2 fφ ). Across the resonance a mode gets excited � π � π 2 x 3 / 2 H (1) 2 x 3 / 2 H (2) 3 / 2 ( x ) − c − ζ k ( x ) ≃ i k ( x ) i 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 � i � bfφ ∗ e − i log k f Γ c − k ( x ) ∼ 1 − , − 2 ix fφ ∗ 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 � i � bfφ ∗ e − i log k f Γ c − k ( x ) ∼ 1 − , − 2 ix fφ ∗ the resonance is seen numerically and anaytically 0.005 0.005 0.004 0.004 0.003 0.003 � � � � x �� � � � � x �� 0.002 0.002 Re � c k Re � c k 0.001 0.001 0 0 � 0.001 � 0.001 20 30 40 50 60 70 80 90 20 30 40 50 60 70 80 90 x x 0.005 0.005 0.004 0.004 0.003 0.003 � � � � x �� � � � � x �� 0.002 0.002 Re � c k Re � c k 0.001 0.001 0 0 � 0.001 � 0.001 20 30 40 50 60 70 80 90 20 30 40 50 60 70 80 90 x x Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 17 / 37

The model Analytical result for the spectrum Oscillations in the spectrum: � k � n s − 1 � � φ k �� P s ( k ) = A s 1 + δn s cos k ∗ f � δn s ∼ 3 b 2 πfφ, Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 18 / 37

The model Analytical result for the spectrum Oscillations in the spectrum: � k � n s − 1 � � φ k �� P s ( k ) = A s 1 + δn s cos k ∗ f � δn s ∼ 3 b 2 πfφ, 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 0.00175 0.2 0.0015 0.00125 0.15 0.001 ∆ n s bf 0.1 0.00075 0.0005 0.05 0.00025 0 0 0 0.02 0.04 0.06 0.08 0.1 - 4 - 3.5 - 3 - 2.5 - 2 - 1.5 - 1 f log 10 f 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 8000 8000 6000 6000 4000 4000 � � � � 1 � C � � 2 Π � Μ K 2 � � � � � 1 � C � � 2 Π � Μ K 2 � 2000 2000 0 0 � 2000 � 2000 5 10 50 100 500 1000 5 10 50 100 500 1000 � � 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 8000 8000 6000 6000 4000 4000 � � � � 1 � C � � 2 Π � Μ K 2 � � � � � 1 � C � � 2 Π � Μ K 2 � 2000 2000 0 0 � 2000 � 2000 5 10 50 100 500 1000 5 10 50 100 500 1000 � � 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 Chaotic Inflation For b = 0 and using slow roll Linear Axion Inflation µ 3 � N = 50 N = 60 r ≃ 0 . 07 This is within Planck sensitivity! IIA Nil manifolds µ 10/3 � 2/3 N = 50 N = 60 0.92 0.94 0.96 0.98 1.0 1.02 n s 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 � ζ 2 n +1 � = 0 , � ζ 2 n � ∝ � ζ 2 � n 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 � ζ 2 n +1 � = 0 , � ζ 2 n � ∝ � ζ 2 � n Due torotational and translational invariance, the bispectrum � ζ 3 � depends on 3 variables � ζ 3 � ≡ (2 π ) 3 f NL F ( k 1 , k 2 , k 3 ) δ 3 ( k 1 + k 2 + k 3 ) f NL gives the size and the normalized F ( k 1 , k 2 , k 3 ) 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] : � t dt ′ < � ζ k 1 ( t ) ζ k 2 ( t ) ζ k 3 ( t ) , H I ( t ′ ) � < ζ k 1 ( t ) ζ k 2 ( t ) ζ k 3 ( t ) > = − i > t 0 where the intaraction Hamiltonian at order ζ 3 is obtained expanding around an inflationary solution � aǫ 2 ζζ ′ 2 + aǫ 2 ζ ( ∂ζ ) 2 − 2 ǫζ ′ ( ∂ζ )( ∂χ ) H I = + a ηζ 2 ζ ′ + ǫ 2 a ( ∂ζ )( ∂χ )( ∂ 2 χ ) + ǫ 4 a ( ∂ 2 ζ )( ∂χ ) 2 , 2 ǫ ˙ a 2 ǫ∂ − 2 ˙ χ ≡ 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] : � t dt ′ < � ζ k 1 ( t ) ζ k 2 ( t ) ζ k 3 ( t ) , H I ( t ′ ) � < ζ k 1 ( t ) ζ k 2 ( t ) ζ k 3 ( t ) > = − i > t 0 where the intaraction Hamiltonian at order ζ 3 is obtained expanding around an inflationary solution � aǫ 2 ζζ ′ 2 + aǫ 2 ζ ( ∂ζ ) 2 − 2 ǫζ ′ ( ∂ζ )( ∂χ ) H I = + a ηζ 2 ζ ′ + ǫ 2 a ( ∂ζ )( ∂χ )( ∂ 2 χ ) + ǫ 4 a ( ∂ 2 ζ )( ∂χ ) 2 , 2 ǫ ˙ a 2 ǫ∂ − 2 ˙ χ ≡ 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 ) = a lm Y lm (ˆ n ) . lm 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 ) = a lm Y lm (ˆ n ) . lm The three-point correlation function of the CMB d 3 � d 3 � d 3 � � k 1 k 2 k 3 l 1 m 1 (ˆ l 2 m 2 (ˆ l 3 m 3 (ˆ (2 π ) 3 Y ∗ k 1 ) Y ∗ k 2 ) Y ∗ � a l 1 m 1 a l 2 m 2 a l 3 m 3 � ∝ k 3 ) (2 π ) 3 (2 π ) 3 �� � × δ 3 F ( k 1 , k 2 , k 3 )∆ T l 1 ( k 1 )∆ T l 2 ( k 2 )∆ T l 3 ( k 3 ) . k i i 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 ) = a lm Y lm (ˆ n ) . lm The three-point correlation function of the CMB d 3 � d 3 � d 3 � � k 1 k 2 k 3 l 1 m 1 (ˆ l 2 m 2 (ˆ l 3 m 3 (ˆ (2 π ) 3 Y ∗ k 1 ) Y ∗ k 2 ) Y ∗ � a l 1 m 1 a l 2 m 2 a l 3 m 3 � ∝ k 3 ) (2 π ) 3 (2 π ) 3 �� � × δ 3 F ( k 1 , k 2 , k 3 )∆ T l 1 ( k 1 )∆ T l 2 ( k 2 )∆ T l 3 ( k 3 ) . k i i max ∼ 10 9 integrals with 10 10 operations Number of operations: l 3 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 ) = a lm Y lm (ˆ n ) . lm The three-point correlation function of the CMB d 3 � d 3 � d 3 � � k 1 k 2 k 3 l 1 m 1 (ˆ l 2 m 2 (ˆ l 3 m 3 (ˆ (2 π ) 3 Y ∗ k 1 ) Y ∗ k 2 ) Y ∗ � a l 1 m 1 a l 2 m 2 a l 3 m 3 � ∝ k 3 ) (2 π ) 3 (2 π ) 3 �� � × δ 3 F ( k 1 , k 2 , k 3 )∆ T l 1 ( k 1 )∆ T l 2 ( k 2 )∆ T l 3 ( k 3 ) . k i i max ∼ 10 9 integrals with 10 10 operations Number of operations: l 3 each. 10 19 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 one 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 one requires computational superpowers? This is an open problem. . . If F · F ′ is a scalar product of 3D shapes, define the cosine F · F ′ C ( 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 one requires computational superpowers? This is an open problem. . . If F · F ′ is a scalar product of 3D shapes, define the cosine F · F ′ C ( F, F ′ ) ≡ ( F · F ) ( F ′ · F ′ ) � When | C ( F, F ′ ) | ∼ 1, the two shapes are similar. The observational 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 Orthogonal Equilateral − 10 < f loc < 74 − 410 < f ort < 6 − 214 < f equi < 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 Orthogonal Equilateral − 10 < f loc < 74 − 410 < f ort < 6 − 214 < f equi < 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] (2 π ) 3 δ 3 ( k 1 + k 2 + k 3 ) ζ k 1 ζ k 2 ζ k 3 × � ζ k 1 ζ k 2 ζ k 3 � ≃ � � � dt ′ 2 a 3 ǫ ˙ k 2 ˙ ζ ∗ k 1 ζ ∗ ζ ∗ δ k 3 + 2 perm. + c.c. , x 3 / 2 H (1) ( x ) ∼ e ix ζ k ∼ � log x � ˙ δ ∼ sin 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] (2 π ) 3 δ 3 ( k 1 + k 2 + k 3 ) ζ k 1 ζ k 2 ζ k 3 × � ζ k 1 ζ k 2 ζ k 3 � ≃ � � � dt ′ 2 a 3 ǫ ˙ k 2 ˙ ζ ∗ k 1 ζ ∗ ζ ∗ δ k 3 + 2 perm. + c.c. , x 3 / 2 H (1) ( x ) ∼ e ix ζ k ∼ � log x � ˙ δ ∼ sin fφ The frequency of ζ k is stretched by the expansion from M pl to H when it exits the horizon. Enrico Pajer (Cornell) Resonant non-Gaussianity Hamburg Feb 2010 30 / 37