Inflationary non-Gaussianities in the most general second-order - PowerPoint PPT Presentation

1 Inflationary non-Gaussianities in the most general second-order scalar-tensor theories Antonio De Felice, IF, Naresuan U. Yukawa Institute – 4 March 2012 with Shinji Tsujikawa

2 Introduction • Accelerated expansion at early times

2 Introduction • Accelerated expansion at early times • Epoch: after QG and GUT (scale ?)

2 Introduction • Accelerated expansion at early times • Epoch: after QG and GUT (scale ?) • New physics: usually scalar field (?)

2 Introduction • Accelerated expansion at early times • Epoch: after QG and GUT (scale ?) • New physics: usually scalar field (?) • Or new gravity: 1st model Starobinsky model

3 General scalar-tensor theory d 4 x √− g � � � 1 2 M 2 S = P R + P ( φ, X ) − G 3 ( φ, X ) � φ + L 4 + L 5

3 General scalar-tensor theory d 4 x √− g � � � 1 2 M 2 S = P R + P ( φ, X ) − G 3 ( φ, X ) � φ + L 4 + L 5 G 4 ( φ, X ) R + G 4 ,X [( � φ ) 2 − ( ∇ µ ∇ ν φ ) ( ∇ µ ∇ ν φ )] , L 4 =

3 General scalar-tensor theory d 4 x √− g � � � 1 2 M 2 S = P R + P ( φ, X ) − G 3 ( φ, X ) � φ + L 4 + L 5 G 4 ( φ, X ) R + G 4 ,X [( � φ ) 2 − ( ∇ µ ∇ ν φ ) ( ∇ µ ∇ ν φ )] , L 4 = G 5 ( φ, X ) G µν ( ∇ µ ∇ ν φ ) − 1 6 G 5 ,X [( � φ ) 3 L 5 = − 3( � φ ) ( ∇ µ ∇ ν φ ) ( ∇ µ ∇ ν φ ) +2( ∇ µ ∇ α φ ) ( ∇ α ∇ β φ ) ( ∇ β ∇ µ φ )] . X ≡ − 1 2 ( ∂φ ) 2

4 Vacuum EOMs on flat FLRW • ds 2 = − dt 2 + a ( t ) 2 d x 2 , φ = φ ( t )

4 Vacuum EOMs on flat FLRW • ds 2 = − dt 2 + a ( t ) 2 d x 2 , φ = φ ( t ) 3 M 2 P H 2 F + P + 6 H G 4 ,φ ˙ φ + ( G 3 ,φ − 12 H 2 G 4 ,X + 9 H 2 G 5 ,φ − P ,X ) ˙ φ 2 0 = φ 3 + 3( G 5 ,φX − 2 G 4 , XX ) H 2 ˙ φ 4 − H 3 G 5 , XX ˙ φ 5 , +(6 G 4 ,φX − 3 G 3 ,X − 5 G 5 ,X H 2 ) H ˙ φ 2 − HG 5 ,X ˙ φ 3 + F M 2 H + 3 M 2 P H 2 F + P + 4 HG 4 ,φ ˙ 2[( G 5 ,φ − 2 G 4 ,X ) ˙ P ] ˙ 0 = φ φ 2 + 2 H ( G 5 ,φX − 2 G 4 , XX ) ˙ φ 3 − φ + (2 G 4 ,φX − G 3 ,X − 3 H 2 G 5 ,X ) ˙ +[2 G 4 ,φ + 4 H ( G 5 ,φ − G 4 ,X ) ˙ φ 2 + 2 H G 5 ,φφ − G 5 ,X H 2 − 2 G 4 ,φX � � � � 2 G 4 ,φφ + 3 H 2 G 5 ,φ − G 3 ,φ − 6 H 2 G 4 ,X φ 3 ˙ ˙ + φ 4 . − H 2 G 5 ,φX ˙

4 Vacuum EOMs on flat FLRW • ds 2 = − dt 2 + a ( t ) 2 d x 2 , φ = φ ( t ) 3 M 2 P H 2 F + P + 6 H G 4 ,φ ˙ φ + ( G 3 ,φ − 12 H 2 G 4 ,X + 9 H 2 G 5 ,φ − P ,X ) ˙ φ 2 0 = φ 3 + 3( G 5 ,φX − 2 G 4 , XX ) H 2 ˙ φ 4 − H 3 G 5 , XX ˙ φ 5 , +(6 G 4 ,φX − 3 G 3 ,X − 5 G 5 ,X H 2 ) H ˙ φ 2 − HG 5 ,X ˙ φ 3 + F M 2 H + 3 M 2 P H 2 F + P + 4 HG 4 ,φ ˙ 2[( G 5 ,φ − 2 G 4 ,X ) ˙ P ] ˙ 0 = φ φ 2 + 2 H ( G 5 ,φX − 2 G 4 , XX ) ˙ φ 3 − φ + (2 G 4 ,φX − G 3 ,X − 3 H 2 G 5 ,X ) ˙ +[2 G 4 ,φ + 4 H ( G 5 ,φ − G 4 ,X ) ˙ φ 2 + 2 H G 5 ,φφ − G 5 ,X H 2 − 2 G 4 ,φX � � � � 2 G 4 ,φφ + 3 H 2 G 5 ,φ − G 3 ,φ − 6 H 2 G 4 ,X φ 3 ˙ ˙ + φ 4 . − H 2 G 5 ,φX ˙ P , and ˙ φE 3 = ˙ • F = 1 + 2 G 4 /M 2 E 1 + 3 H ( E 1 + E 2 ) = 0 .

5 • Removing P from E 2 by E 1 (1 − 4 δ G 4 X − 2 δ G 5 X + 2 δ G 5 φ ) ǫ = δ P X + 3 δ G 3 X − 2 δ G 3 φ + 6 δ G 4 X − δ G 4 φ − 6 δ G 5 φ + 3 δ G 5 X +12 δ G 4 XX + 2 δ G 5 XX − 10 δ G 4 φX + 2 δ G 4 φφ − 8 δ G 5 φX + 2 δ G 5 φφ − δ φ ( δ G 3 X + 4 δ G 4 X − δ G 4 φ + 8 δ G 4 XX +3 δ G 5 X − 4 δ G 5 φ + 2 δ G 5 XX − 2 δ G 4 φX − 4 δ G 5 φX ) ,

5 • Removing P from E 2 by E 1 (1 − 4 δ G 4 X − 2 δ G 5 X + 2 δ G 5 φ ) ǫ = δ P X + 3 δ G 3 X − 2 δ G 3 φ + 6 δ G 4 X − δ G 4 φ − 6 δ G 5 φ + 3 δ G 5 X +12 δ G 4 XX + 2 δ G 5 XX − 10 δ G 4 φX + 2 δ G 4 φφ − 8 δ G 5 φX + 2 δ G 5 φφ − δ φ ( δ G 3 X + 4 δ G 4 X − δ G 4 φ + 8 δ G 4 XX +3 δ G 5 X − 4 δ G 5 φ + 2 δ G 5 XX − 2 δ G 4 φX − 4 δ G 5 φX ) , • Slow-roll parameters G 3 ,X ˙ ˙ ¨ P ,X X φX G 3 ,φ X G 4 ,X X H φ ǫ = − H 2 , δ φ = , δ P X = , δ G 3 X = , δ G 3 φ = , δ G 4 X = , H ˙ M 2 P H 2 F M 2 M 2 P H 2 F M 2 P HF P F φ G 4 ,XX X 2 G 4 ,φ ˙ G 4 ,φX ˙ φ φX G 4 ,φφ X G 5 ,φ X δ G 4 φ = , δ G 4 φX = , δ G 4 φφ = , δ G 4 XX = , δ G 5 φ = , M 2 M 2 M 2 M 2 M 2 P H 2 F P HF P HF P F P F G 5 ,X H ˙ G 5 ,XX H ˙ φX 2 G 5 ,φX X 2 G 5 ,φφ ˙ φX φX δ G 5 X = , δ G 5 XX = δ G 5 φX = , δ G 5 φφ = . M 2 M 2 M 2 M 2 P F P F P F P HF

6 • Perturbation theory / Uniform field gauge ds 2 = − [(1+ α ) 2 − a ( t ) − 2 e − 2 R ( ∂ψ ) 2 ] dt 2 +2 ∂ i ψ dt dx i + a ( t ) 2 e 2 R d x 2 ,

6 • Perturbation theory / Uniform field gauge ds 2 = − [(1+ α ) 2 − a ( t ) − 2 e − 2 R ( ∂ψ ) 2 ] dt 2 +2 ∂ i ψ dt dx i + a ( t ) 2 e 2 R d x 2 , • Action at second order � R 2 + 1 R − w 2 α ) ∂ 2 ψ − 2 w 1 R + 1 3 w 3 α 2 + w 4 � � dtd 3 x a 3 a 2 α∂ 2 R + 3 w 2 α ˙ a 2 ( ∂ R ) 2 − 3 w 1 ˙ a 2 (2 w 1 ˙ S 2 = ,

6 • Perturbation theory / Uniform field gauge ds 2 = − [(1+ α ) 2 − a ( t ) − 2 e − 2 R ( ∂ψ ) 2 ] dt 2 +2 ∂ i ψ dt dx i + a ( t ) 2 e 2 R d x 2 , • Action at second order � R 2 + 1 R − w 2 α ) ∂ 2 ψ − 2 w 1 R + 1 3 w 3 α 2 + w 4 � � dtd 3 x a 3 a 2 α∂ 2 R + 3 w 2 α ˙ a 2 ( ∂ R ) 2 − 3 w 1 ˙ a 2 (2 w 1 ˙ S 2 = , M 2 P F − 4 XG 4 ,X − 2 HX ˙ w 1 = φG 5 ,X + 2 XG 5 ,φ , 2 M 2 φG 3 ,X − 16 H ( XG 4 ,X + X 2 G 4 ,XX ) + 2 ˙ P HF − 2 X ˙ w 2 = φ ( G 4 ,φ + 2 XG 4 ,φX ) − 2 H 2 ˙ φ (5 XG 5 ,X + 2 X 2 G 5 ,XX ) + 4 HX (3 G 5 ,φ + 2 XG 5 ,φX ) , − 9 M 2 pl H 2 F + 3( XP ,X + 2 X 2 P ,XX ) + 18 H ˙ φ (2 XG 3 ,X + X 2 G 3 ,XX ) − 6 X ( G 3 ,φ + XG 3 ,φX ) w 3 = +18 H 2 (7 XG 4 ,X + 16 X 2 G 4 ,XX + 4 X 3 G 4 ,XXX ) − 18 H ˙ φ ( G 4 ,φ + 5 XG 4 ,φX + 2 X 2 G 4 ,φXX ) + 6 H 3 ˙ φ (15 XG 5 ,X + 13 X 2 G 5 ,XX + 2 X 3 G , 5 XXX ) − 18 H 2 X (6 G 5 ,φ + 9 XG 5 ,φX + 2 X 2 G 5 ,φXX ) , M 2 P F − 2 XG 5 ,φ − 2 XG 5 ,X ¨ w 4 = φ .

7 Second order Lagrangian • α = 2 w 1 ˙ R /w 2

7 Second order Lagrangian • α = 2 w 1 ˙ R /w 2 • Reduced action R 2 − c 2 � � � ˙ dtd 3 x a 3 Q a 2 ( ∂ R ) 2 s S 2 = ,

7 Second order Lagrangian • α = 2 w 1 ˙ R /w 2 • Reduced action R 2 − c 2 � � � ˙ dtd 3 x a 3 Q a 2 ( ∂ R ) 2 s S 2 = , w 1 (4 w 1 w 3 + 9 w 2 2 ) Q = , 3 w 2 2 3(2 w 2 1 w 2 H − w 2 w 1 w 2 − 2 w 2 2 w 4 + 4 w 1 ˙ 1 ˙ w 2 ) c 2 = . s w 1 (4 w 1 w 3 + 9 w 2 2 )

8 Three-point function • Look for � τf �R ( k 1 ) R ( k 2 ) R ( k 3 ) � = − i dτ a � 0 | [ R ( τ f , k 1 ) R ( τ f , k 2 ) R ( τ f , k 3 ) , H int ( τ )] | 0 � , τi

8 Three-point function • Look for � τf �R ( k 1 ) R ( k 2 ) R ( k 3 ) � = − i dτ a � 0 | [ R ( τ f , k 1 ) R ( τ f , k 2 ) R ( τ f , k 3 ) , H int ( τ )] | 0 � , τi • 3rd order action � dt d 3 x a 3 { a 1 α 3 + α 2 ( a 2 R + a 3 ˙ R + a 4 ∂ 2 R /a 2 + a 5 ∂ 2 ψ/a 2 ) S 3 = + α [ a 6 ∂ i R ∂ i ψ/a 2 + a 7 ˙ R ∂ 2 R /a 2 + a 9 ( ∂ i ∂ j ψ∂ i ∂ j ψ − ∂ 2 ψ∂ 2 ψ ) /a 4 RR + a 8 ˙ + a 10 ( ∂ i ∂ j ψ∂ i ∂ j R − ∂ 2 ψ∂ 2 R ) /a 4 + a 11 R ∂ 2 ψ/a 2 + a 12 ˙ R ∂ 2 ψ/a 2 + a 13 R ∂ 2 R /a 2 + a 14 ( ∂ R ) 2 /a 2 + a 15 ˙ R 2 ] R 3 + b 2 R ( ∂ R ) 2 /a 2 + b 3 ˙ R 2 R + c 1 ˙ R ∂ i R ∂ i ψ/a 2 + c 2 ˙ R 2 ∂ 2 ψ/a 2 + c 3 ˙ R R ∂ 2 ψ/a 2 + b 1 ˙ R + d 2 R ) ( ∂ i ∂ j ψ∂ i ∂ j ψ − ∂ 2 ψ∂ 2 ψ ) /a 4 + d 3 ∂ i R ∂ i ψ ∂ 2 ψ/a 4 } , + ( d 1 ˙

9 Reducing the action • S 3 : � dt d 3 x a 3 f 1 + af 2 + f 3 /a � � S 3 = ,

9 Reducing the action • S 3 : � dt d 3 x a 3 f 1 + af 2 + f 3 /a � � S 3 = , � Q 2 � � Q 2 � Q Q R 3 + R 2 + A 9 ˙ R ˙ ˙ f 1 ≡ A 1 + A 3 − A 5 A 4 − A 6 R ∂ i R ∂ i X w 2 w 2 w 2 w 1 1 1 1 + 1 � � A 5 ˙ R + A 6 R ( ∂ i ∂ j X )( ∂ i ∂ j X ) , w 2 1 � 2 L 1 Q Q � 2 L 1 Q L 1 Q R ∂ 2 R + A 8 R ( ∂ R ) 2 − A 9 R 2 ∂ 2 R + A 6 R ( ∂ R ) 2 ˙ R ˙ ˙ f 2 ≡ A 2 − A 3 L 1 + A 5 − A 7 w 1 w 1 w 1 w 1 + A 7 − 2 A 5 L 1 R ( ∂ i ∂ j R )( ∂ i ∂ j X ) − 2 A 6 L 1 R ( ∂ i ∂ j R )( ∂ i ∂ j X ) − A 9 L 1 ∂ 2 R ∂ i R ∂ i X , ˙ w 1 w 1 w 1 � A 5 L 2 � R [( ∂ i ∂ j R )( ∂ i ∂ j R ) − ( ∂ 2 R ) 2 ] + A 6 L 2 1 R [( ∂ i ∂ j R )( ∂ i ∂ j R ) − ( ∂ 2 R ) 2 ] ˙ f 3 ≡ 1 − A 7 L 1 + A 9 L 2 1 ( ∂ R ) 2 ∂ 2 R .

10 Reducing the action — 2 � • Finally S 3 = dt L 3 � � R 2 + a C 2 M 2 P R ( ∂ R ) 2 + a 3 C 3 M P ˙ R 3 + a 3 C 4 ˙ d 3 x a 3 C 1 M 2 P R ˙ L 3 = R ( ∂ i R )( ∂ i X ) P ) ∂ 2 R ( ∂ X ) 2 + a C 6 ˙ ∂ 2 R ( ∂ R ) 2 − R ∂ i ∂ j ( ∂ i R )( ∂ j R ) + a 3 ( C 5 /M 2 R 2 ∂ 2 R + C 7 � � /a δ L 2 � � � � ∂ 2 R ∂ i R ∂ i X − R ∂ i ∂ j ( ∂ i R )( ∂ j X ) � + a ( C 8 /M P ) + F 1 , � δ R � 1

11 Three-point function • Looking for � τf �R ( k 1 ) R ( k 2 ) R ( k 3 ) � = − i dτ a � 0 | [ R ( τ f , k 1 ) R ( τ f , k 2 ) R ( τ f , k 3 ) , H int ( τ )] | 0 � , τi

11 Three-point function • Looking for � τf �R ( k 1 ) R ( k 2 ) R ( k 3 ) � = − i dτ a � 0 | [ R ( τ f , k 1 ) R ( τ f , k 2 ) R ( τ f , k 3 ) , H int ( τ )] | 0 � , τi �R ( k 1 ) R ( k 2 ) R ( k 3 ) � = (2 π ) 3 δ (3) ( k 1 + k 2 + k 3 )( P R ) 2 B R ( k 1 , k 2 , k 3 ) ,