. . CMB Power Spectrum Formula in the Background-Field Method . . . . . Shoichi Ichinose ichinose@u-shizuoka-ken.ac.jp Laboratory of Physics, SFNS, University of Shizuoka Aug. 7, 2012 The 3rd UTQuest workshop ExDiP 2012 ”Superstring Cosmophysics” , Tokachi-Makubetsu Granvrio Hotel, Hokkaidou, Japan Aug. 7, 2012 The 3rd UTQuest workshop ExDiP Shoichi Ichinose (Univ. of Shizuoka) CMB Power Spectrum Formula in the Background-Field Method / 25
1. Introduction Sec 1. Introduction (1/2) History Cosmic Microwave Background Radiation Observation Data is accumulating . WMAP-5year . Dark Matter, Dark Energy ( ∼ Cosmological Term) ’Micro’ Theory of Gravity : Divergence Problem(Infra-red, Ultra-violet) Quntum Field Theory on dS 4 is not defined ’01 E. Witten, inf-dim Hilbert space ’03 J. Maldacena, Non-Gaussian ... ’06 S. Weinberg , in-in formalism Schwinger-Keldysh formalism in ’07 A.M. Polyakov ’09- T. Tanaka & Y. Urakawa ’11- H. Kitamoto & Y. Kitazawa Aug. 7, 2012 The 3rd UTQuest workshop ExDiP Shoichi Ichinose (Univ. of Shizuoka) CMB Power Spectrum Formula in the Background-Field Method / 25
1. Introduction Sec 1. Introduction (2/2) Recent Words and References A.M. Polyakov, ’09 Dark energy, like the black body radiation 150 years ago, hides secrets of fundamental physics E. Verlinde, ’10 Emergent Gravity A. Strominger et al, ’11 From Navier-Stokes to Einstein, arXiv:1101.2451 From Petrov-Einstein to Navier-Stokes, arXiv:1104.5502 Aug. 7, 2012 The 3rd UTQuest workshop ExDiP Shoichi Ichinose (Univ. of Shizuoka) CMB Power Spectrum Formula in the Background-Field Method / 25
Sec 2. Background Field Formalism Sec 2. Background Field Formalism (1/2) B.S. DeWitt, 1967; G. ’tHooft, 1973; I.Y. Aref’eva, A.A. Slavnov & L.D. Faddeev, 1974 Φ( x ) : Scalar Field , g µν ( x ) : Gravitational Field , V (Φ) = σ 4!Φ 4 , σ > 0 2 ∇ µ Φ ∇ µ Φ − m 2 ∫ d 4 x √ g ( − ( R − 2 λ ) − 1 ) 2 Φ 2 − V (Φ) S [Φ; g µν ] = 16 π G N (1) Background Expansion: Φ = Φ cl + φ , NOT expand g µν (2) Aug. 7, 2012 The 3rd UTQuest workshop ExDiP Shoichi Ichinose (Univ. of Shizuoka) CMB Power Spectrum Formula in the Background-Field Method / 25
Sec 2. Background Field Formalism Sec.2 Background Field Formalism (2/2) ∫ { S [Φ cl + φ ; g µν ] − δ S [Φ cl ; g µν ] } e i Γ[Φ cl ; g µν ] = D φ exp i φ Γ[Φ cl ; g µν ] ; δ Φ cl Φ cl is perturbatively solved, at the tree level, as D ( x − x ′ ) √ g δ V (Φ cl ) � ∫ � d 4 x ′ Φ cl ( x ) = Φ 0 ( x ) + , � δ Φ cl � x ′ √ g ( ∇ 2 − m 2 )Φ 0 = 0 √ g ( ∇ 2 − m 2 ) D ( x − x ′ ) = δ 4 ( x − x ′ ) , . (4) Φ 0 ( x ) : asymptotic fields for n-point function (see later part) Aug. 7, 2012 The 3rd UTQuest workshop ExDiP Shoichi Ichinose (Univ. of Shizuoka) CMB Power Spectrum Formula in the Background-Field Method / 25
Sec 2. Background Field Formalism Sec.2 Background Field Formalism (2’/2) Aref’eva, Slavnov & Faddeev 1974 Harmonic Oscillator (Feynman’s text ’72) Density Matrix ( ˙ ∫ β x 2 2 + ω 2 [ − 1 ) ] ∫ 2 x 2 ρ ( x 2 , x 1 ; β ) = D x ( τ ) exp d τ . � 0 x (0)= x 1 , x ( β )= x 2 Background Field Expansion: x ( τ ) = x cl ( τ ) + y ( τ ) ( ˙ ∫ β √ x 2 2 + ω 2 1 [ − 1 ) ] cl 2 ρ ( x 2 , x 1 ; β ) = 2 π � β exp 2 x cl d τ . (6) � 0 Transition probability is given by δ δ δ x cl ( β ) ρ ( x 2 , x 1 ; β ) . (7) δ x cl (0) Aug. 7, 2012 The 3rd UTQuest workshop ExDiP Shoichi Ichinose (Univ. of Shizuoka) CMB Power Spectrum Formula in the Background-Field Method / 25
Sec 3. dS 4 Geometry, Conformal Time and Z 2 Symmetry Sec 3. dS 4 Geometry (1/3) background field g µν : dS 4 ds 2 = − dt 2 + e 2 H 0 t ( dx 2 + dy 2 + dz 2 ) ≡ g inf µν dx µ dx ν time variable: t → η (conformal time) ds 2 = ( H 0 η ) 2 ( − d η 2 + dx 2 + dy 2 + dz 2 ) 1 g µν ( χ ) d χ µ d χ ν , ( χ 0 , χ 1 , χ 2 , χ 3 ) = ( η, x , y , z ) = ˜ To regularize IR behavior, we introduce Z 2 Symmetry : t ↔ − t , Periodicity : t → t + 2 l , (8) Aug. 7, 2012 The 3rd UTQuest workshop ExDiP Shoichi Ichinose (Univ. of Shizuoka) CMB Power Spectrum Formula in the Background-Field Method / 25
Sec 3. dS 4 Geometry, Conformal Time and Z 2 Symmetry Sec 3. dS 4 Geometry (2/3) The perturbative solution Φ cl , (4), is given by � ∫ 1 δ V (Φ cl ) d 4 χ ′ , ˜ � Φ cl ( χ ) = Φ 0 ( χ ) + D ( χ, χ ′ ) � ( H 0 η ′ ) 4 δ Φ cl � χ ′ ∇ 2 − m 2 )Φ 0 = g ( ˜ √ − ˜ m 2 { 1 1 } ( H 0 η ) 2 ⃗ ∇ 2 − ∂ η ( H 0 η ) 2 ∂ η + ( H 0 η ) 4 − Φ 0 = 0 . (9) Switch to the spacially-Fourier-transformed expression: d 3 ⃗ d 3 ⃗ ∫ p ∫ p x ′ ) ˜ (2 π ) 3 e i ⃗ p · ⃗ p ( η ) , ˜ (2 π ) 3 e i ⃗ p · ( ⃗ x − ⃗ x ϕ ⃗ Φ 0 ( η,⃗ D ( χ, χ ′ ) = p ( η, η ′ ) , (10) x ) = D ⃗ ˜ D ⃗ p ( η, η ′ ): ’Momentum/Position propagator’ Aug. 7, 2012 The 3rd UTQuest workshop ExDiP Shoichi Ichinose (Univ. of Shizuoka) CMB Power Spectrum Formula in the Background-Field Method / 25
Sec 3. dS 4 Geometry, Conformal Time and Z 2 Symmetry Sec 3. dS 4 Geometry (3/3) ϕ ⃗ p ( η ) satisfies the following Bessel eigenvalue equation. m 2 { } 2 − 2 ( H 0 η ) 2 + M 2 ∂ η η∂ η + ϕ M ( η ) = { s ( η ) − 1 ˆ L η + M 2 } ϕ M ( η ) = 0 , m 2 1 M 2 ≡ ⃗ ( H 0 η ) 2 , ˆ p 2 , s ( η ) ≡ L η ≡ ∂ η s ( η ) ∂ η + . (11) ( H 0 η ) 4 Aug. 7, 2012 The 3rd UTQuest workshop ExDiP Shoichi Ichinose (Univ. of Shizuoka) CMB Power Spectrum Formula in the Background-Field Method / 25
Sec 4. Boundary Condition, Bunch-Davies Vacuum, Casimir Energy Sec 4. Bunch-Davies Vacuum (1/2) Boundary Condition for Free Wave Function Φ 0 = 0 Dirichlet for P = − (12) ∂ η Φ 0 = 0 Neumann for P = + Bunch-Davies Vacuum: the complete and orthonormal eigen functions ϕ n ( η ) of the operator s − 1 ˆ L η . Aug. 7, 2012 The 3rd UTQuest workshop ExDiP Shoichi Ichinose (Univ. of Shizuoka) CMB Power Spectrum Formula in the Background-Field Method / 25
Sec 4. Boundary Condition, Bunch-Davies Vacuum, Casimir Energy Sec 4. Bunch-Davies Vacuum (2/2) { s ( η ) − 1 ˆ 2 } ϕ n ( η ) = 0 ϕ n ( η ) ≡ ( n | η ) = ( η | n ) , L η + M n , ∫ 1 / H 0 (∫ − 1 /ω ) ∫ − 1 /ω d η d η + ( H 0 η ) 2 ( n | η )( η | k ) = 2 ( H 0 η ) 2 ( n | η )( η | k ) − 1 / H 0 1 /ω − 1 / H 0 = ( n | k ) = δ n , k , ( H 0 η ) 2 ϵ ( η ) ϵ ( η ′ )ˆ { δ ( | η | − | η ′ | ) for P = − ( η | η ′ ) = ( H 0 η ) 2 δ ( | η | − | η ′ | ) for P = + ∫ 1 / H 0 (∫ − 1 /ω ) ∫ − 1 /ω d η d η + ( H 0 η ) 2 | η )( η | = 2 ( H 0 η ) 2 | η )( η | = 1 , − 1 / H 0 1 /ω − 1 / H 0 ∑ | n )( n | = 1 , (13) Aug. 7, 2012 The 3rd UTQuest workshop ExDiP Shoichi Ichinose (Univ. of Shizuoka) CMB Power Spectrum Formula in the Background-Field Method / 25
Sec 4. Boundary Condition, Bunch-Davies Vacuum, Casimir Energy Sec 5. Casimir Energy (1/2) Casimir energy: free part of the effective action in (3) 2 ∇ µ φ ∇ µ φ − m 2 d 4 x √ g ( − 1 ) ∫ ∫ exp {− H − 3 0 E dS 4 2 φ 2 Cas } = D φ exp i [∫ ∫ − 1 /ω ] d 3 ⃗ d η {− 1 p 2 ln( − s ( η ) − 1 ˆ p 2 ) } L η − ⃗ = exp (2 π ) 3 2 (14) − 1 / H 0 ∫ ∞ 0 ( e − t − e − tM ) / t dt = ln M , det M > 0, From the formula: ∫ ∞ d τ 1 − H − 3 0 E dS 4 Cas = 2 Tr H ⃗ p ( η, η ′ ; τ ) , (15) τ 0 Aug. 7, 2012 The 3rd UTQuest workshop ExDiP Shoichi Ichinose (Univ. of Shizuoka) CMB Power Spectrum Formula in the Background-Field Method / 25
Sec 4. Boundary Condition, Bunch-Davies Vacuum, Casimir Energy Sec 5. Casimir Energy (2/2) where H ⃗ p ( η, η ′ ; τ ) is the Heat-Kernel: { ∂ } ∂τ − ( s − 1 ˆ p 2 ) p ( η, η ′ ; τ ) = 0 L η + ⃗ H ⃗ , p ( η, η ′ ; τ ) = ( η | e ( s − 1 ˆ p 2 ) τ | η ′ ) L η + ⃗ H ⃗ p 2 τ ∑ e − M n 2 τ ϕ n ( η ) ϕ n ( η ′ ) → ( η | η ′ ) as τ → +0 = e ⃗ . (16) n Aug. 7, 2012 The 3rd UTQuest workshop ExDiP Shoichi Ichinose (Univ. of Shizuoka) CMB Power Spectrum Formula in the Background-Field Method / 25
Sec 5. Spatial Wick Rotation Sec 6. Spatial Wick Rotation ∫ − 1 /ω ∫ ∞ d 3 ⃗ ∫ p d η { 1 d τ τ ( η | e τ ( s ( η ) − 1 ˆ p 2 ) | η ) } − H − 3 0 E dS 4 L η + ⃗ Cas = (2 π ) 3 2 , (17) 2 − 1 / H 0 0 Diverges very badly ! To regularize it, we do Wick rotation for space-components of momentum p x , p y , p z − → ip x , ip y , ip z (18) The regularized expression is Casimir energy for AdS 4 . The finiteness (both for IR and for UV) is shown. S.I. arXiv:0812.1263, 0801.3064 Aug. 7, 2012 The 3rd UTQuest workshop ExDiP Shoichi Ichinose (Univ. of Shizuoka) CMB Power Spectrum Formula in the Background-Field Method / 25
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