Effective field theory approach. to quasi-single field inflation

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1 Effective field theory approach to quasi-single field inflation Toshifumi Noumi (The University of Tokyo, Komaba) based on arxiv: with Masahide Yamaguchi and Daisuke Yokoyama (Tokyo Institute of Technology) JGRG22 - RESCEU, 12th November 2012

2 Quasi-single field inflation 1/10 # classification of inflation models w.r.t. relevant dof relevant dof single field adiabatic mode ζ (massless) multiple field adiabatic + isocurvatures

3 Quasi-single field inflation 1/10 # classification of inflation models w.r.t. relevant dof relevant dof single field adiabatic mode ζ (massless) multiple field multi field quasi-single field adiabatic + isocurvatures light isocurvatures (m<<h) isocurvatures m H

4 Quasi-single field inflation 1/10 # classification of inflation models w.r.t. relevant dof relevant dof single field adiabatic mode ζ (massless) multiple field adiabatic + isocurvatures multi field quasi-single field light isocurvatures (m<<h) isocurvatures m H - well motivated by model building (string inspired, supergravity based,... )

5 Quasi-single field inflation 1/10 # classification of inflation models w.r.t. relevant dof relevant dof modulus in string theory single field (landscape) adiabatic mode ζ (massless) multiple field multi field quasi-single field adiabatic + isocurvatures light isocurvatures (m<<h) isocurvatures m H - well motivated by model building (string inspired, supergravity based,... )

6 Quasi-single field inflation 1/10 # classification of inflation models w.r.t. relevant dof relevant dof modulus in string theory single field (landscape) adiabatic mode ζ (massless) multiple field multi field quasi-single field adiabatic curvature + isocurvatures coupling in supergravity light isocurvatures (m<<h) isocurvatures m H - well motivated by model building (string inspired, supergravity based,... )

7 Quasi-single field inflation 1/10 # classification of inflation models w.r.t. relevant dof relevant dof modulus in string theory single field (landscape) adiabatic mode ζ (massless) multiple field multi field quasi-single field adiabatic curvature + isocurvatures coupling in supergravity light isocurvatures (m<<h) isocurvatures m H - well motivated by model building (string inspired, supergravity based,... ) - phenomenologically interesting (characteristic signatures in primordial perturbations)

8 QSI & non-gaussianities 2/10 # quasi-single field inflation [Chen-Wang 09] massless field δφ φ mixing σ massive field σ σ 3 coupling

9 QSI & non-gaussianities 2/10 # quasi-single field inflation [Chen-Wang 09] massless field δφ φ mixing σ massive field σ σ 3 coupling - can potentially give large non-gaussianities

10 QSI & non-gaussianities 2/10 # quasi-single field inflation [Chen-Wang 09] massless field δφ φ mixing σ massive field σ σ 3 coupling - can potentially give large non-gaussianities - intermediate shape between local and equilateral types

11 QSI & non-gaussianities 2/10 # quasi-single field inflation [Chen-Wang 09] massless field δφ φ mixing σ massive field σ σ 3 coupling - can potentially give large non-gaussianities k 3 k 1 0 k 2 k 3 k 2 = k 3 = k, k 1 = κk - intermediate shape between local and equilateral types - characteristic scaling in squeezed limit k1 k2 k3 κ 3/2 ν k 6 ν = 9/4 m 2 σ/h 2

12 QSI & non-gaussianities 2/10 # quasi-single field inflation [Chen-Wang 09] massless field δφ φ mixing σ massive field σ σ 3 coupling - can potentially give large non-gaussianities k 3 k 1 0 k 2 k 3 k 2 = k 3 = k, k 1 = κk - intermediate shape between local and equilateral types sensitive to the mass of σ - characteristic scaling in squeezed limit k1 k2 k3 κ 3/2 ν k 6 ν = 9/4 m 2 σ/h 2

13 Motivation 3/10 quasi-single field inflation: - naturally realized in supergravity - characteristic signatures in non-gaussianities (between local & equilateral, scaling in squeezed limit)

14 Motivation 3/10 quasi-single field inflation: - naturally realized in supergravity - characteristic signatures in non-gaussianities (between local & equilateral, scaling in squeezed limit) would like to discuss QSI - in general settings - in a systematic way

15 Motivation 3/10 quasi-single field inflation: - naturally realized in supergravity - characteristic signatures in non-gaussianities (between local & equilateral, scaling in squeezed limit) would like to discuss QSI - in general settings - in a systematic way effective field theory approach!

16 EFT approach 4/10 # effective field theory approach [Cheung et al 07] time dependent background evolution during inflation φ 0 (t) (time-dependent vev:, ) R(t)

17 EFT approach 4/10 # effective field theory approach [Cheung et al 07] t time dependent background evolution during inflation spatial slicing φ 0 (t) (time-dependent vev:, ) R(t) x i

18 EFT approach 4/10 # effective field theory approach [Cheung et al 07] t time dependent background evolution during inflation spatial slicing φ 0 (t) (time-dependent vev:, ) R(t) spontaneous breakdown of time diffeo x i

19 EFT approach 4/10 # effective field theory approach [Cheung et al 07] t time dependent background evolution during inflation spatial slicing φ 0 (t) (time-dependent vev:, ) R(t) spontaneous breakdown of time diffeo x i unbroken time-dep. spatial diffeo δx i = i (t, x i )

20 EFT approach 4/10 # effective field theory approach [Cheung et al 07] t time dependent background evolution during inflation spatial slicing φ 0 (t) (time-dependent vev:, ) R(t) spontaneous breakdown of time diffeo x i unbroken time-dep. spatial diffeo δx i = i (t, x i ) assuming relevant dof. during inflation, we would be able to construct effective action for inflation based on the unbroken time-dependent diffeo

21 EFT approach 4/10 # effective field theory approach [Cheung et al 07] t time dependent background evolution during inflation spatial slicing φ 0 (t) (time-dependent vev:, ) R(t) spontaneous breakdown of time diffeo x i unbroken time-dep. spatial diffeo δx i = i (t, x i ) assuming relevant dof. during inflation, we would be able to construct effective action for inflation based on the unbroken time-dependent diffeo advantages: - systematic expansions in fluctuations and derivatives

22 EFT approach 4/10 # effective field theory approach [Cheung et al 07] t time dependent background evolution during inflation spatial slicing φ 0 (t) (time-dependent vev:, ) R(t) spontaneous breakdown of time diffeo x i unbroken time-dep. spatial diffeo δx i = i (t, x i ) assuming relevant dof. during inflation, we would be able to construct effective action for inflation based on the unbroken time-dependent diffeo advantages: - systematic expansions in fluctuations and derivatives - simplification in the dynamics of Goldstone boson

23 EFT approach 4/10 # effective field theory approach [Cheung et al 07] t time dependent background evolution during inflation spatial slicing φ 0 (t) (time-dependent vev:, ) R(t) spontaneous breakdown of time diffeo x i unbroken time-dep. spatial diffeo δx i = i (t, x i ) assuming relevant dof. during inflation, we would be able to construct effective action for inflation based on the unbroken time-dependent diffeo advantages: - systematic expansions in fluctuations and derivatives - simplification in the dynamics of Goldstone boson - relations between physics and non-gaussianities are clear!

24 General action 5/10 # general action of quasi-single field inflation relevant dof = three physical modes of graviton + additional massive scalar field σ

25 General action 5/10 # general action of quasi-single field inflation relevant dof = three physical modes of graviton + additional massive scalar field σ schematically written as S = S grav + S σ + S mix

26 General action 5/10 # general action of quasi-single field inflation relevant dof = three physical modes of graviton + additional massive scalar field σ ex. schematically written as S = S grav + S σ + S mix S mix = d 4 x g β 1 (t)δg 00 σ + β 2 (t)δg 00 0 σ +β 3 (t) 0 σ ( β 3 (t)+3hβ 3 (t))σ

27 General action 5/10 # general action of quasi-single field inflation ex. S mix = β 1 relevant dof = three physical modes of graviton + additional massive scalar field σ schematically written as S = S grav + S σ + S mix S mix = d 4 x g β 1 (t)δg 00 σ + β 2 (t)δg 00 0 σ +β 3 (t) 0 σ ( β 3 (t)+3hβ 3 (t))σ Stuckelberg method d 4 xa 3 2β 1 σ + (2β 2 β 3 ) σ + β 3 i i σ 2 ( i) 2 a 2 σ +3β 2 2 i i σ ( i ) 2 σ 2β 2 a 2 β 2 a 2 σ +... a 2

28 General action 5/10 # general action of quasi-single field inflation relevant dof = three physical modes of graviton + additional massive scalar field σ ex. schematically written as S = S grav + S σ + S mix S mix = d 4 x g β 1 (t)δg 00 σ + β 2 (t)δg 00 0 σ +β 3 (t) 0 σ ( β 3 (t)+3hβ 3 (t))σ S mix = β 1 d 4 xa 3 2β 1 σ + (2β 2 β 3 ) σ + β 3 i i σ 2 ( i) 2 a 2 σ +3β 2 2 i i σ ( i ) 2 σ 2β 2 a 2 β 2 a 2 σ +... a 2 large non-gaussianitis from mixings β 1 and β 2 1 ζ H

29 Power spectrum 6/10 # power spectrum S mix = d 4 xa 3 i i σ 2β 1 σ + (2β 2 β 3 ) σ + β 3 β 1 β2 β3 a 2

30 Power spectrum 6/10 # power spectrum S mix = d 4 xa 3 i i σ 2β 1 σ + (2β 2 β 3 ) σ + β 3 ζ k ζ k = H 2 k k β 1 β2 β3 a 2 βσ βσ

31 Power spectrum 6/10 # power spectrum S mix = d 4 xa 3 i i σ 2β 1 σ + (2β 2 β 3 ) σ + β 3 ζ k ζ k = H 2 k k β 1 β2 β3 a 2 βσ βσ ζ k ζ k =(2) 3 δ (3) (k + k ) 22 k 3 P ζ(k) deviation from single field P ζ (k) = H M 2 Pl c 1+ 4α 2 σ c 2 M 2 Pl ( Ḣ) β2 1 H 2 C 11 + β 2 2 C 22 + β 2 3 C 33 + β 1 H β 2 C 12 + β 1 H β 3 C 13 + β 2 β3 C 23

32 !'!" Power spectrum 7/10 C ij (m σ /H, c σ /c ) for fixed r s = c σ /c "!! &!'! C 11 "! "!'" C 22 "!'! &'! "'!!'&!'!"! " # $ % &! " # $ % & m σ /H m σ /H "!! "!! The dots are numerical results for r s =0.1 (red), 0.3 (orange), 1 (yellow), 3 (green), and 10 (blue). C 33 "! "!'" C 22 The curve is an analytic result "! for r s = 1. C 12!'" does not vanish even in the " heavy mass limit β 2 σ

33 3pt functions 8/10 # three point functions in the squeezed limit - original models of quasi-single field inflation k 3 k 1 0 k 2 k 3 k 2 = k 3 = k, k 1 = κk k1 k2 k3 κ 3/2 ν k 6 ν = 9/4 m 2 σ/h 2

34 3pt functions 8/10 # three point functions in the squeezed limit - original models of quasi-single field inflation k 3 k 1 0 k 2 k 3 k 2 = k 3 = k, k 1 = κk k1 k2 k3 κ 3/2 ν k 6 ν = 9/4 m 2 σ/h 2 - for general mixing and cubic couplings

35 3pt functions 8/10 # three point functions in the squeezed limit - original models of quasi-single field inflation k 3 k 1 0 k 2 k 3 k 2 = k 3 = k, k 1 = κk k1 k2 k3 κ 3/2 ν k 6 ν = 9/4 m 2 σ/h 2 - for general mixing and cubic couplings scaling does not depend on details of mixing

36 3pt functions 8/10 # three point functions in the squeezed limit - original models of quasi-single field inflation k 3 k 1 0 k 2 k 3 k 2 = k 3 = k, k 1 = κk k1 k2 k3 κ 3/2 ν k 6 ν = 9/4 m 2 σ/h 2 - for general mixing and cubic couplings scaling does not depend on details of mixing determined only from cubic interaction in the diagram

37 9/10 three point vertices 3, ( i) 2 a 2 2 σ, σ, σ 2, σ σ, σ 2, σ σ, σ 3, σ 2 σ, σ σ 2, σ ( iσ) 2 ( i ) 2 a 2 σ, ( i) 2 a 2 i i σ a 2 a 2, σ 3, σ ( iσ) 2 a 2 σ, i i σ a 2 ( iσ) 2 a 2 σ, i i σ a 2 σ momentum dependence κ 1 k 6 κ 3/2 ν k 6 for m σ < 3 2 H κ 3/2 k 6 sin[iν log κ + δ ν ] for m σ > 3 2 H κ 2 k 6 κ 3/2 ν k 6 for m σ < 2H κ 2 k 6 for m σ > 2H κ 1/2 ν k 6 for m σ < 2H κ 1 k 6 for m σ > 2H ( ) non-trivial scaling in the squeezed limit when mixing is relevant! ν = 9/4 m 2 σ/h 2 0 < ν < 3/2 or ν = pure imaginary

38 Summary and prospects 10/10 # summary applied EFT approach to QSI - systematic expansions in fluctuations and derivatives - simplification of action for in decoupling regime - relation between physics & non-gaussianities is clear calculated power spectrum for constant mixing discussed scaling of 3-pt functions in squeezed limit - sensitive to # of fields and their mass also discussed effects of heavy particles, sharp turning # prospects full non-gaussianities, detectability,... EFT for sugra based inflation, more on sharp turning,...

39 THANK YOU!!