＠基研研究会 August 21st 2013 Toshifumi Noumi (Math Phys Lab, RIKEN) Effective field theory approach to quasi-single field inflation and effects of heavy fields reference: ・JHEP 1306 (2013) 051 [arXiv:1211.1624] with Masahide Yamaguchi (TIT) and Daisuke Yokoyama (Seoul NU) ・[arXiv:1307.7110] with Masahide Yamaguchi (TIT)
＠基研研究会 August 21st 2013 Toshifumi Noumi (Math Phys Lab, RIKEN) Effective field theory approach to quasi-single field inflation and effects of heavy fields reference: ・JHEP 1306 (2013) 051 [arXiv:1211.1624] with Masahide Yamaguchi (TIT) and Daisuke Yokoyama (Seoul NU) ・[arXiv:1307.7110] with Masahide Yamaguchi (TIT) possible probe of high energy physics
＠基研研究会 August 21st 2013 Toshifumi Noumi (Math Phys Lab, RIKEN) Effective field theory approach to quasi-single field inflation and effects of heavy fields reference: ・JHEP 1306 (2013) 051 [arXiv:1211.1624] with Masahide Yamaguchi (TIT) and Daisuke Yokoyama (Seoul NU) ・[arXiv:1307.7110] with Masahide Yamaguchi (TIT) spontaneous breaking of time-diffeomorphism
inflation
inflation: accelerated expansion of the Universe - explains horizon problem, flatness problem, ... - generates primordial curvature fluctuations → seeds of structures of the Universe
# single-field slow-roll inflation - FRW spacetime a lot of models have been proposed and are being killed by experiments! observation horizon problem - Hubble parameter: approx. de-Sitter - introduce an inflaton field: L = − 1 2 ∂ µ φ∂ µ φ − V ( φ ) ds 2 = − dt 2 + a ( t ) 2 d � x 2 H ( t ) = ˙ a a ( reheating � a ( t f ) � ˙ ˙ H � ln � 60 � = � H � 1 � = � H 2 � 1 a ( t i )
how to distinguish models?
※ initial conditions in standard cosmology two gauges - , : fluctuation of scale factor - cosmic expansion cools Universe → CMB temp. fluctuations quantum fluctuations during inflation → sees of structures dof during inflation - unitary gauge others inflaton and graviton inflaton - spatially flat gauge ( ] φ φ γ ij metric g µ ν σ ds 2 = − dt 2 + a 2 e 2 ζ ( e γ ) ij dx i dx j scalar perturbation ζ tensor perturbation γ ij ζ γ ij
CMB as seen by Planck δ T T ∼ 10 − 5
as a probe of (very) high energy physics?
can affect primordial curvature perturbations!? can be used as a probe of high energy physics!? models based on high energy theory have been also discussed (ex. supergravity, superstring theory, ...) massive scalar fields other than inflaton one generic feature of such high energy based models: supergravity: generically extra dimensions: Kaluza-Klein modes superstring theory: moduli of compactification m scalar ∼ H
when heavy fields become relevant?
suppose that the potential has a massive direction in addition to the slow-roll direction slow-roll massive φ ⊥ φ sr
- single field approximation works well - don’t feel the massive potential if you roll along the bottom of potential...
- single field approximation works well - don’t feel the massive potential if you roll along the bottom of potential... no information about massive fields
two typical situation you feel massive potential turn and climb the potential potential itself is turning ※ in each case, you will feel centrifugal force during the turn conversion interaction from kinetic term： δθ r δ r θ r 2 ∂ µ θ∂ µ θ 3 δ r ˙ δθ
two typical situation you feel massive potential turn and climb the potential potential itself is turning ※ in each case, you will feel centrifugal force during the turn conversion interaction from kinetic term： can be a probe of high energy physics δθ r δ r θ r 2 ∂ µ θ∂ µ θ 3 δ r ˙ δθ
signatures from conversion interaction + heavy fields: # quasi-single field inflation [Chen-Wang ’09] - intermediate shape between local and equilateral types - potentially large non-Gaussianities # effective sound speed from heavy fields [ex. Achucarro et al ’11] - heavy fields can change dispersion relations of light fields # sudden turning trajectory [ex. Gao et al ’12] - a kind of resonances in primordial power spectra would like to discuss effects of heavy fields + conversion more systematically and in more general settings → effective field theory (EFT) approach to inflation
General action from EFT approach
＃ EFT approach to inflation [Cheung-Creminelli-Fitzpatrick-Kaplan-Senatore ’07] - systematic expansions in perturbations and derivatives - interactions at different orders are related by symmetry advantages: spontaneous breakdown of time diffeo ※ unbroken time-dep. spatial diffeo (time-dep. vev of inflaton) spatial slicing relevant dof + symmetry breaking structure → effective action ※ write down all possible interactions preserving unbroken symmetry! inflation：approx. de-Sitter expansion → time-dep. scalar curvature R ( t ) t � x i = � i ( t, x i ) x i
relevant dof = metric only - simplest case (⇔ single field inflation) ※ time-diffeo breaking → 2 transverse and 1 longitudinal physical modes we would like to construct the most general action - constructed from the metric - invariant under unbroken time-dependent spatial diffeo - around given FRW background (background field satisfy the eom) : extrinsic curvature on constant-time spatial slices (inflaton is eaten by graviton) free indices: upper 0’s ＃ EFT approach to inflation [Cheung-Creminelli-Fitzpatrick-Kaplan-Senatore ’07] g µ ν g µ ν d 4 x �� g F ( R µ νρσ , g 00 , K µ ν , � µ , t ) � S = K µ ν
＃ Generic action with heavy fields [Noumi-Yamaguchi-Yokoyama ’12] (background field satisfy the eom) (inflaton is eaten by graviton) ※ time-diffeo breaking → 2 transverse and 1 longitudinal physical modes : extrinsic curvature on constant-time spatial slices - in our case... - around given FRW background - invariant under unbroken time-dependent spatial diffeo - constructed from the metric and we would like to construct the most general action relevant dof = metric g µ ν + additional massive scalar field σ g µ ν σ d 4 x p� g F ( R µ νρσ , g 00 , K µ ν , r µ , t, σ ) Z S = K µ ν
with - Goldstone boson non-linearly realizes time diffeo - at the linear order 2. introduced the Goldstone boson π via Stuckelberg method 1. expand the action around a given FRW background ＃ Generic action with heavy fields [Noumi-Yamaguchi-Yokoyama ’12] π x ) = π ( x ) − ξ 0 ( x ) t → ˜ t = t + ξ 0 ( x ) π ( x ) → ˜ π (˜ ζ = − H π
＃ Generic action with heavy fields [Noumi-Yamaguchi-Yokoyama ’12] ※ interactions at different orders are related by symmetry : typically, ※ nontrivial cubic interaction from conversion ※ model is specified by time-dep. parameters such as and : kinetic term of σ, self-interaction of σ, .... 3. write the action schematically as : no σ (⇔ single field) S = S π + S σ + S mix S π S σ S mix : conversion of π and σ, .... π 2 � ( ∂ i π ) 2 ⌘� Z ⇣ Pl ˙ d 4 x a 3 � M 2 ˙ S π 3 H a 2 π 2 � ( ∂ i π ) 2 � Z ⇣ ⌘ d 4 x a 3 � 2 β ˙ ˙ S mix 3 πσ � β σ a 2 f NL = O (1 ∼ 10) H ( t ) β
Example: effects of heavy field oscillations [Noumi-Yamaguchi ’13]
1. turning potential 2. phase transition (of massive direction) heavy field oscillations can occur in the case of
two effects of heavy field oscillations: 1. deformations of Hubble parameter 2. conversion interactions
bottom of the potential → π-π interaction if background trajectory oscillates... - deformed Hubble parameter oscillating (slow-roll direction) # Deformations of Hubble parameter φ ⊥ φ sr Pl ˙ H = ˙ sr + ˙ Friedman equation: − 2 M 2 φ 2 φ 2 ⊥ H = ˙ ˙ H sr + δ ˙ H π 2 − ( ∂ i π ) 2 � Z Pl δ ˙ dtd 3 x a 3 ( − M 2 H ) ˙ a 2
# conversion interaction × → oscillating π-σ interaction: - coupling β oscillates with frequency m - for turning background trajectory... → π-σ conversion appears during the turn Goldstone heavy scalar two dof of scalar perturbations: conversion π σ β ( t ) ˙ πσ σ π Z dtd 3 x a 3 β ( t ) ˙ πσ
two effects of heavy field oscillations: ※ and is oscillating ① Hubble deformation → π-π interaction ② π-σ conversion interaction π 2 − ( ∂ i π ) 2 � Z Pl δ ˙ dtd 3 x a 3 ( − M 2 H ) ˙ a 2 Z dtd 3 x a 3 β ( t ) ˙ πσ δ H ( t ) β ( t )
# effects on primordial power spectrum if there are no oscillations... single slow-roll → almost scale invariant power spectrum H 2 sr P ζ ( k ) = 8 ⇡ 2 M 2 sr ✏ sr P ζ ( k ) H 2 sr 8 ⇡ 2 M 2 sr ✏ sr k 0
# effects on primordial power spectrum if there are no oscillations... heavy field oscillation → deviations single slow-roll → almost scale invariant power spectrum H 2 sr P ζ ( k ) = (1 + C δ H + C conv ) 8 ⇡ 2 M 2 sr ✏ sr P ζ ( k ) H 2 sr 8 ⇡ 2 M 2 sr ✏ sr k 0
cf. swing wavy features around : can be understood as resonances # effects on primordial power spectrum C δ H ( k ) for m = 20 H sr α 2 10 k ∼ 20 k ∗ 8 6 4 2 k 0 10 20 30 40 50 k ∗ - 2 k ∗ : scale of turning/transition
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