Curvature Perturbation Spectrum in Two-field Inflation with a - PowerPoint PPT Presentation

Heavy Isocurvaton � � ) Curvature Perturbation Spectrum in Two-field Inflation with a Turning Trajectory Shi Pi( Physics Department, Peking University November 12th, 2012 Collaborate with Misao Sasaki, based on arXiv:1205.0161, JGRG 2012, RESCUE, University of Tokyo.

Heavy Isocurvaton Outline 1 Introduction 2 Quasi-single Field Inflation with Large Isocurvaton Mass 3 Non-Gaussianity of Equilateral Shape 4 Conclusion

Heavy Isocurvaton Introduction Primary Parameters Define the parameters Slow-roll parameter along the trajectory ǫ and η . Angular speed of rotation in field space ˙ θ ∼ V s . Effective mass perpendicular to the trajectory M eff = V ss + 3 ˙ θ .

Heavy Isocurvaton Introduction Classification The ordinary 2-field inflation can be classified by these parameters in the slow-roll region as ˙ θ ≪ H , M eff ≪ H : 2-field inflation with a negligible coupling 1 between adiabatic and curvature perturbations inside the horizon. Gordon 2001. ˙ θ ≪ H , M eff ∼ H : Quasi-single field inflation in the original 2 form. Chen 2010. ˙ θ ≪ H , M eff ≫ H : After integrating the heavy field out, one 3 can get an effective single field with a corrective speed of sound. Achucarro 2011,2012. Cespedes 2012.

Heavy Isocurvaton Introduction Classification The ordinary 2-field inflation can be classified by these parameters in the slow-roll region as ˙ θ ≪ H , M eff ≪ H : 2-field inflation with a negligible coupling 1 between adiabatic and curvature perturbations inside the horizon. Gordon 2001. ˙ θ ≪ H , M eff ∼ H : Quasi-single field inflation in the original 2 form. Chen 2010. ˙ θ ≪ H , M eff ≫ H : After integrating the heavy field out, one 3 can get an effective single field with a corrective speed of sound. Achucarro 2011,2012. Cespedes 2012. We are suppose to connect 2 and 3.

Heavy Isocurvaton Introduction “Massless” Slowball

Heavy Isocurvaton Introduction Quasi-single Panda

Heavy Isocurvaton Introduction Coaster with “Large Isocurvaton Mass”.

Heavy Isocurvaton Introduction EFT result In EFT, after integrating out the heavy field ( σ in our case), one have an effective single field inflation with an effective speed of sound c s which is � ˙ � 2 = 1 + 4 H 2 θ c − 2 , (1) s ˜ M 2 H eff Finally we got via EFT that � 2 � ˙ θ δ P R ∝ c − 1 − 1 ∼ 2 . s ˜ M eff Our main task is to verify this relation by in-in formulism.

Heavy Isocurvaton Quasi-single Field Inflation with Large Isocurvaton Mass Outline 1 Introduction 2 Quasi-single Field Inflation with Large Isocurvaton Mass 3 Non-Gaussianity of Equilateral Shape 4 Conclusion

Heavy Isocurvaton Quasi-single Field Inflation with Large Isocurvaton Mass Lagrangian The action for the fields can be decomposed into d 4 x √− g � � � − 1 R + σ ) 2 g µν ∂ µ θ∂ ν θ − 1 2( ˜ 2 g µν ∂ µ σ∂ ν σ − V sr ( θ ) − V ( σ ) S m = , where Rθ (tangent field) and σ (radial field), V sr ( θ ) is a slow-roll potential along the valley, V ( σ ) is a potential that forms the valley and traps the isocurvaton at σ = σ 0 , ˜ R denotes the radius of the minima valley, R = ˜ R + σ 0 is the constant radius where the trajectory is trapped with the centripetal force under consideration.

Heavy Isocurvaton Quasi-single Field Inflation with Large Isocurvaton Mass EOM The Hubble equations and equations of motion are 1 2 R 2 ˙ 3 M 2 p H 2 θ 2 = 0 + V + V sr , R 2 ˙ − 2 M 2 p ˙ θ 2 H = 0 , R 2 ¨ θ 0 + 3 R 2 H ˙ θ 0 + V ′ 0 = sr , σ 0 + V ′ − R 2 ˙ θ 2 0 = σ 0 + 3 H ˙ ¨ 0 , We can see in the tangent direction of the trajectory, field Rθ obeys the ordinary equation of motion for single-field inflation.

Heavy Isocurvaton Quasi-single Field Inflation with Large Isocurvaton Mass Perturbative Hamiltonian Hamiltionian density in interaction picture (spatially flat gauge) 2 + R 2 � 1 � 2 a 2 ( ∂ i δθ ) 2 + 1 2 + 2 a 2 ( ∂ i δσ ) 2 + 1 1 2 R 2 ˙ ˙ a 3 2 M 2 eff δσ 2 H 0 = δθ δσ , 2 − c 2 a 3 δσ ˙ c 2 = 2 R ˙ H I = δθ, θ, 2 δθδσ 2 + aRδσ ( ∂ i δθ ) 2 + a 3 2 − a 3 ˙ − a 3 Rδσ ˙ θ ˙ H I 6 V ′′′ δσ 3 , = δθ 3 V ′′ + 3 ˙ M 2 θ 2 , = eff Our method is valid when � ˙ � 2 | V ′′′ | θ ≪ 1 , ≪ 1 . (2) H H

Heavy Isocurvaton Quasi-single Field Inflation with Large Isocurvaton Mass Perturbative Hamiltonian Hamiltionian density in interaction picture (spatially flat gauge) 2 + R 2 � 1 � 2 a 2 ( ∂ i δθ ) 2 + 1 2 + 2 a 2 ( ∂ i δσ ) 2 + 1 1 2 R 2 ˙ ˙ a 3 2 M 2 eff δσ 2 H 0 = δθ δσ , 2 − c 2 a 3 δσ ˙ c 2 = 2 R ˙ H I = δθ, θ = constant , 2 δθδσ 2 + aRδσ ( ∂ i δθ ) 2 + a 3 2 − a 3 ˙ − a 3 Rδσ ˙ θ ˙ H I 6 V ′′′ δσ 3 , = δθ 3 V ′′ + 3 ˙ θ 2 = constant , M 2 = eff In a constant turn case!

Heavy Isocurvaton Quasi-single Field Inflation with Large Isocurvaton Mass Illustrative Explanation Figure: The second order Figure: The 2-pt func with a interacting vertex heavy isocurvaton mediation. H 2 = − c 2 a 3 δσ ˙ θ , while c 2 = 2 R ˙ θ . And the curvature perturbation R is connected to θ via R = − H δθ. ˙ θ

Heavy Isocurvaton Quasi-single Field Inflation with Large Isocurvaton Mass Quantization Quantize the Fourier components − k a † δθ I u k a k + u ∗ = − k , k δσ I − k b † v k b k + v ∗ = − k . k The commutators [ a k , a † − k ′ ] = (2 π ) 3 δ 3 ( k + k ′ ) , [ b k , b † − k ′ ] = (2 π ) 3 δ 3 ( k + k ′ ) .

Heavy Isocurvaton Quasi-single Field Inflation with Large Isocurvaton Mass Quantization The equation for mode functions, k − 2 u ′′ τ u ′ k + k 2 u k = 0 , k + k 2 v k + M 2 k − 2 v ′′ τ v ′ eff H 2 τ 2 v k = 0 . Solve The EOMs by setting the initial conditions v k → i H τe − ikτ , Ru k , √ 2 k when k ≫ Ha .

Heavy Isocurvaton Quasi-single Field Inflation with Large Isocurvaton Mass Solution The solution is H 2 k 3 (1 + ikτ ) e − ikτ , √ u k = R and √ π eff /H 2 ≤ 9 / 4 , v k = − ie i ( ν + 1 2 ) π 2 H ( − τ ) 3 / 2 H (1) for M 2 ν ( − kτ ) , 2 � 9 / 4 − M 2 eff /H 2 , or where ν = √ π eff /H 2 > 9 / 4 , 2 H ( − τ ) 3 / 2 H (1) v k = − ie − π 2 µ + i π for M 2 iµ ( − kτ ) , 4 � eff /H 2 − 9 / 4 . M 2 where µ =

Heavy Isocurvaton Quasi-single Field Inflation with Large Isocurvaton Mass 2-point function We use in-in formulism to calculate the 2-point function of δθ 2 � t � t � � �� � � �� ¯ � δθ 2 � dt ′ H I ( t ′ ) δθ 2 dt ′ H I ( t ′ ) ≡ � 0 | T exp i I ( t ) T exp − i | 0 � t 0 t 0 (0) + δ P R ∼ P R H 4 � � 1 + δ P R = . 4 π 2 R 2 ˙ (0) θ 2 P R

Heavy Isocurvaton Quasi-single Field Inflation with Large Isocurvaton Mass Feynman Rules u(k) v(k) u*(k) v*(k) u(k)v’(k)

Heavy Isocurvaton Quasi-single Field Inflation with Large Isocurvaton Mass Correction to Power Spectrum t=∞ t=∞ + + t=∞ t=∞ - -

Heavy Isocurvaton Quasi-single Field Inflation with Large Isocurvaton Mass Calculating α t=∞ t=∞ + +

Heavy Isocurvaton Quasi-single Field Inflation with Large Isocurvaton Mass Interchange the momenta t=∞ t=∞ + +

Heavy Isocurvaton Quasi-single Field Inflation with Large Isocurvaton Mass “Split” the integral t=∞ t=∞ t=∞ 2 + + = The Cut-in-the-Middle integral α is � ∞ 2 � � dx x − 1 / 2 H (1) iµ ( x ) e ix � � α = . � � � 0 �

Heavy Isocurvaton Quasi-single Field Inflation with Large Isocurvaton Mass Calculating β t=∞ t=∞ - -

Heavy Isocurvaton Quasi-single Field Inflation with Large Isocurvaton Mass Take the Conjugate t=∞ t=∞ * - -

Heavy Isocurvaton Quasi-single Field Inflation with Large Isocurvaton Mass Sum the Integral t=∞ t=∞ t=∞ * - - -2Re = The Cut-in-the-Side integral β is � ∞ � ∞ dx 1 x − 1 / 2 H (1) dx 2 x − 1 / 2 ( H (1) iµ ( x 1 ) e − ix 1 iµ ( x 2 )) ∗ e − ix 2 . β = 2Re 1 2 0 x 1

Heavy Isocurvaton Quasi-single Field Inflation with Large Isocurvaton Mass The Correction to Power Spectrum � ˙ � 2 δ P R θ e − µπ ( α − β ) , = π P (0) H R α − β = t=∞ t=∞ 2 -2Re

Heavy Isocurvaton Quasi-single Field Inflation with Large Isocurvaton Mass The Correction to Power Spectrum � ˙ � 2 δ P R θ e − µπ ( α − β ) , = π P (0) H R α − β = t=∞ t=∞ 2 -2Re

Heavy Isocurvaton Quasi-single Field Inflation with Large Isocurvaton Mass Calculating α α can be directly integrated, √ 2 � �� e µπ/ 2 � e − µπ √ 1 2 � � α = − sinh µπ + i + 2 coth µπ � � π 2 2 � � � � → 1 , when µ → ∞ . CIM is exponentially suppressed!

Heavy Isocurvaton Quasi-single Field Inflation with Large Isocurvaton Mass Calculating β Use the asymptotic formula of Hankel function when x ≪ µ : − x 2 � 2 e πµ 1 � � � x � iµ H (1) 4 µe − i π iµ → µ exp . 4 e iµ (ln µ − 1) 2 The main contribution to β comes from infrared x ≪ 1 . The result is � 1 β = − 2 e µπ � �� 1 + O . πµ 2 µ 2