Primordial non-gaussianity from G-inflation

Transcription

1 PASCOS Cambridge, UK Primordial non-gaussianity from G-inflation Tsutomu Kobayashi Hakubi Center & Department of Physics Kyoto University Based on work with Masahide Yamaguchi (Tokyo Inst. Tech.) & Jun ichi Yokoyama (U of Tokyo) arxiv: ; [w/ Kohei Kamada (DESY)]; ;

2 What is G-inflation? G-inflation is inflation driven by a generic scalar field having second-order equations of motion G is motivated by the Galileon Nicolis, Rattazzi, Trincherini (2009) Cf. Burrage, de Rham, Seery, Tolley (2011)

3 What is G-inflation? G-inflation is inflation driven by a generic scalar field having second-order equations of motion G is motivated by the Galileon Nicolis, Rattazzi, Trincherini (2009) Cf. Burrage, de Rham, Seery, Tolley (2011) Galileon (in flat space) µ φ µ φ + b µ Second-order EOM L 2 =( φ) 2 L 3 =( φ) 2 φ L 4 =( φ) 2 (φ) 2 ( µ ν φ) 2 L 5 =( φ) 2 (φ) 3 3φ( µ ν φ) 2 + 2( µ ν φ) 3

4 Generalized Galileon The Galileon in flat space can be generalized to give the most general theory describing scalar + gravity system with second-order field equations

5 Generalized Galileon The Galileon in flat space can be generalized to give the most general theory describing scalar + gravity system with second-order field equations L 2 = K(φ,X) L 3 = G 3 (φ,x)φ Deffayet, Esposito-Farese, Vikman (2009); Deffayet, Pujolas, Sawicki, Vikman (2010); Deffayet, Gao, Steer, Zahariade (2011) L 4 = G 4 (φ,x)r + G 4X (φ) 2 ( µ ν φ) 2 L 5 = G 5 (φ,x)g µν µ ν φ 1 6 G 5X (φ) 3 3(φ)( µ ν φ) 2 +2( µ ν φ) 3

6 Generalized Galileon The Galileon in flat space can be generalized to give the most general theory describing scalar + gravity system with second-order field equations L 2 = K(φ,X) L 3 = G 3 (φ,x)φ Deffayet, Esposito-Farese, Vikman (2009); Deffayet, Pujolas, Sawicki, Vikman (2010); Deffayet, Gao, Steer, Zahariade (2011) L 4 = G 4 (φ,x)r + G 4X (φ) 2 ( µ ν φ) 2 L 5 = G 5 (φ,x)g µν µ ν φ 1 6 G 5X (φ) 3 3(φ)( µ ν φ) 2 +2( µ ν φ) 3 4 arbitrary functions of and φ X := ( φ) 2 /2

7 Generalized Galileon The Galileon in flat space can be generalized to give the most general theory describing scalar + gravity system with second-order field equations L 2 = K(φ,X) L 3 = G 3 (φ,x)φ Deffayet, Esposito-Farese, Vikman (2009); Deffayet, Pujolas, Sawicki, Vikman (2010); Deffayet, Gao, Steer, Zahariade (2011) L 4 = G 4 (φ,x)r + G 4X (φ) 2 ( µ ν φ) 2 L 5 = G 5 (φ,x)g µν µ ν φ 1 6 G 5X (φ) 3 3(φ)( µ ν φ) 2 +2( µ ν φ) 3 4 arbitrary functions of and φ X := ( φ) 2 /2 The most general single-field inflation model is obtained from the generalized Galileon

8 Special cases G 4 = M 2 Pl 2 + G 4 = f(φ)+ L 4 = M 2 Pl 2 R + L 4 = f(φ)r + G 5 = φ + L 5 = G µν µ φ ν φ + e.g., Germani, Kehagias (2010) K = 8ξ (4) X 2 (3 ln X), G 3 = 4ξ (3) X(7 3lnX), G 4 = 4ξ (2) X(2 ln X), G 5 = 4ξ (1) ln X ξ(φ)(gauss-bonnet)

9 The generalized Galileon in 4D was already formulated in 1973! International Journal of Theoretical Physics, Vol. 10, No. 6 (1974), pp S econd-o rder S calar-t ensor Field Equations in a Four-Dimensional Space GREGORY WALTER HORNDESKI Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada Received: 10 July It~'~['7 8cde~ Ih R /to 4 " ede h l/ lk L = L(φ, φ, 2 φ, 3 φ, ; g µν, g µν, 2 g µν, 3 g µν, ) cde Ih ik _ 4 ~.. 6cde h / k + "~/(g)(~-+ 2~g/~)8~Rca fh + 2 x/(g)(2,y,('3-2,3("1 + 4pX3)6~OIcI@IJ h -3 %/(g)(2o ~'' + 4"///`' + po~a)~lc Ic + 2x/(g)af86~le~l'f~lalh + N/(g){4Jf9 - p(2,~'" + 4'~"" + p_o~ + 20/{'9)} (4.21)

10 The generalized Galileon in 4D was already formulated in 1973! International Journal of Theoretical Physics, Vol. 10, No. 6 (1974), pp Revisited by Charmousis, Copeland, Padilla, Saffini [ ] S econd-o rder S calar-t ensor Field Equations in a Four-Dimensional Space GREGORY WALTER HORNDESKI Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada Received: 10 July It~'~['7 8cde~ Ih R /to 4 " ede h l/ lk L = L(φ, φ, 2 φ, 3 φ, ; g µν, g µν, 2 g µν, 3 g µν, ) cde Ih ik _ 4 ~.. 6cde h / k + "~/(g)(~-+ 2~g/~)8~Rca fh + 2 x/(g)(2,y,('3-2,3("1 + 4pX3)6~OIcI@IJ h -3 %/(g)(2o ~'' + 4"///`' + po~a)~lc Ic + 2x/(g)af86~le~l'f~lalh + N/(g){4Jf9 - p(2,~'" + 4'~"" + p_o~ + 20/{'9)} (4.21)

11 The generalized Galileon in 4D was already formulated in 1973! International Journal of Theoretical Physics, Vol. 10, No. 6 (1974), pp Revisited by Charmousis, Copeland, Padilla, Saffini [ ] S econd-o rder S calar-t ensor Field Equations in a Four-Dimensional Space GREGORY WALTER HORNDESKI Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada Received: 10 July It~'~['7 8cde~ Ih R /to 4 " ede h l/ lk L = L(φ, φ, 2 φ, 3 φ, ; g µν, g µν, 2 g µν, 3 g µν, ) cde Ih ik _ 4 ~.. 6cde h / k This is equivalent to the generalized Galileon + "~/(g)(~-+ 2~g/~)8~Rca fh + 2 x/(g)(2,y,('3-2,3("1 + 4pX3)6~OIcI@IJ h -3 %/(g)(2o ~'' + 4"///`' + po~a)~lc Ic + 2x/(g)af86~le~l'f~lalh + N/(g){4Jf9 - p(2,~'" + 4'~"" + p_o~ + 20/{'9)} (4.21)

12 Background example 1 K(φ,X) = V (φ)+k(φ)x +, G i (φ,x) = g i (φ)+h i (φ)x +. Slowly-rolling φ Potential-dominated inflation H V (φ) g 4 (φ) L 4 = g 4 (φ)r +

13 Background example 1 K(φ,X) = V (φ)+k(φ)x +, G i (φ,x) = g i (φ)+h i (φ)x +. Slowly-rolling φ Potential-dominated inflation H V (φ) g 4 (φ) L 4 = g 4 (φ)r + Modified friction term in scalar-field EOM 3H K φ +6 Hh 3 X + H 2 h 4 φ + H 3 h 5 X V φ + 12H 2 g 4φ can enhance friction Germani, Kehagias (2010); Kamada et al. (2011)

14 Background example 2 Shift symmetry: φ φ + c, i.e., K = K(X), G i = G i (X) Inflation can be driven by constant kinetic energy

15 Background example 2 Shift symmetry: φ φ + c, i.e., K = K(X), G i = G i (X) Inflation can be driven by constant kinetic energy Scalar-field EOM J +3HJ =0 where J a 3 0 de Sitter attractor J := φk X +6HXG 3X +6H 2 φ (G4X +2XG 4XX ) +2H 3 X (3G 5X +2XG 5XX ) H = const. X = const. Use for dark energy, see Deffayet, Pujolas, Sawicki, Vikman (2010)

16 ζ h ij Perturbations

17 Tensor perturbations g ij = a 2 (t)(δ ij + h ij )

18 Tensor perturbations g ij = a 2 (t)(δ ij + h ij ) Quadratic action for tensor perturbations S (2) T = 1 8 dtd 3 xa 3 G T ḣ 2 ij F T a 2 ( h ij ) 2 F T := 2 G T := 2 G 4 X φg5x + G 5φ, H G 4 2XG 4X X φg 5X G 5φ

19 Tensor perturbations g ij = a 2 (t)(δ ij + h ij ) Quadratic action for tensor perturbations S (2) T = 1 8 dtd 3 xa 3 G T ḣ 2 ij F T a 2 ( h ij ) 2 F T > := 0 2 G T > := 0 2 G 4 X φg5x + G 5φ, H G 4 2XG 4X X φg 5X G 5φ Stability conditions

20 Scalar perturbations g ij = a 2 (t)e 2ζ δ ij

21 Scalar perturbations g ij = a 2 (t)e 2ζ δ ij Quadratic action for scalar perturbations S (2) S = dtd 3 xa 3 G ζ2 S F S a 2 ( ζ) 2 F S := 1 a d dt a Θ G2 T F T, Σ := XK X +2X 2 K XX + 12H φxg 3X +6H φx 2 G 3XX 2XG 3φ 2X 2 G 3φX 6H 2 G 4 +6 H 2 7XG 4X + 16X 2 G 4XX +4X 3 G 4XXX H φ G 4φ +5XG 4φX +2X 2 G 4φXX +30H 3 φxg5x + 26H 3 φx 2 G 5XX G S := Σ Θ 2 G2 T +3G T +4H 3 φx 3 G 5XXX 6H 2 X 6G 5φ +9XG 5φX +2X 2 G 5φXX, Θ := φxg 3X +2HG 4 8HXG 4X 8HX 2 G 4XX + φg 4φ +2X φg 4φX H 2 φ 5XG5X +2X 2 G 5XX +2HX (3G 5φ +2XG 5φX )

22 Scalar perturbations g ij = a 2 (t)e 2ζ δ ij Quadratic action for scalar perturbations S (2) S = dtd 3 xa 3 G ζ2 S F S a 2 ( ζ) 2 F S := > 0 1 a d a Θ G2 T dt G S := > 0 Σ Θ 2 G2 T +3G T F T, Stability conditions Σ := XK X +2X 2 K XX + 12H φxg 3X +6H φx 2 G 3XX 2XG 3φ 2X 2 G 3φX 6H 2 G 4 +6 H 2 7XG 4X + 16X 2 G 4XX +4X 3 G 4XXX H φ G 4φ +5XG 4φX +2X 2 G 4φXX +30H 3 φxg5x + 26H 3 φx 2 G 5XX +4H 3 φx 3 G 5XXX 6H 2 X 6G 5φ +9XG 5φX +2X 2 G 5φXX, Θ := φxg 3X +2HG 4 8HXG 4X 8HX 2 G 4XX + φg 4φ +2X φg 4φX H 2 φ 5XG5X +2X 2 G 5XX +2HX (3G 5φ +2XG 5φX )

23 Primordial power spectrum Tensor P T =8 G1/2 T F 3/2 T Scalar P ζ = 1 2 G 1/2 S F 3/2 S H 2 4π 2 H 2 4π 2 ct kη= 1 cs kη= 1 c 2 T := F T /G T Evaluated at sound horizon crossing c 2 s := F S /G S Generic formulas which can be used for any single-field models

24 Let s focus on the simplest non-trivial example L = R 2 + K(φ,X) G(φ,X)φ, and evaluate non-gaussianity

25 Key quantities Recall the quadratic action S = dtd 3 x σ 1 c 2 s ζ 2 1 a 2 ( ζ)2

26 Key quantities Recall the quadratic action S = dtd 3 x σ 1 c 2 s ζ 2 1 a 2 ( ζ)2 c 2 s := F S G S Sound speed

27 Key quantities Recall the quadratic action S = dtd 3 x σ 1 c 2 s ζ 2 1 a 2 ( ζ)2 σ := XF S c 2 (H φxg X ) 2 s := F S G S Sound speed

28 Key quantities Recall the quadratic action S = dtd 3 x σ 1 c 2 s ζ 2 1 a 2 ( ζ)2 σ := XF S c 2 (H φxg X ) 2 s := F S G S Sound speed reduces to (= Ḣ/H2 ) in standard (k-)inflation σ =, not necessarily slow-roll suppressed in (kinetically driven) G-inflation

29 Key quantities Recall the quadratic action S = dtd 3 x σ 1 c 2 s ζ 2 1 a 2 ( ζ)2 σ := reduces to XF S c 2 (H φxg X ) 2 s := F S G S (= Ḣ/H2 ) in standard (k-)inflation Tensor-to-scalar ratio r = 16σc s Sound speed σ =, not necessarily slow-roll suppressed in (kinetically driven) G-inflation

30 Cubic action S 3 = χ := a2 σ c 2 s 2 ζ dtd 3 xa 3 C1 H ζ 3 + C 2 ζ ζ 2 + C 3 a 4 H 2 2 ζ( ζ) 2 + C 4 a 2 H 2 ζ 2 2 ζ + C 5 Hζ 2 ζ + C 6 a 4 H 2 ζ( ζ χ)+ C 7 a 4 2 ζ( χ) 2 + C 8 a 2 ζ( ζ)2 + C 9 a ζ( ζ 2 χ)+ 2 δl f(ζ) a3 δζ 1 C 1 = H σ Θ c I 2 s G φx (G X + XG XX ) Hσ c 2 sθ 2 + H2 σ c 4 sθ 2, C 2 = σ c 2 3 H2 s c 2 sθ Θ, HΘ C 3 = H2 φxgx Θ 3, C 4 = 2H2 φx (GX + XG XX ) Θ 3, σ d H 2 δ C 5 = 2c 2 sh dt c 2 sθ 2, C 6 = 2H φxg X Θ 2, C 7 = σ 4 φxg X Θ, C 8 = σ + H2 σ Θ 2 c 2 1 2s 2 Θ, s HΘ C 9 = σ c 2 2H s Θ + σ, 2

31 Cubic action S 3 = χ := a2 σ c 2 s 2 ζ dtd 3 xa 3 C1 H ζ 3 + C 2 ζ ζ 2 + C 3 a 4 H 2 2 ζ( ζ) 2 + C 4 a 2 H 2 ζ 2 2 ζ + C 5 Hζ 2 ζ + C 6 a 4 H 2 ζ( ζ χ)+ C 7 a 4 2 ζ( χ) 2 + C 8 a 2 ζ( ζ)2 + C 9 a ζ( ζ 2 χ)+ 2 δl f(ζ) a3 δζ 1 C 1 = H σ Θ c I 2 s G φx (G X + XG XX ) Hσ c 2 sθ 2 + H2 σ c 4 sθ 2, C 2 = σ c 2 3 H2 s c 2 sθ Θ, HΘ C 3 = H2 φxgx Θ 3, New terms from G(φ,X) C 4 = 2H2 φx (GX + XG XX ) Θ 3, σ d H 2 δ C 5 = 2c 2 sh dt c 2 sθ 2, C 6 = 2H φxg X Θ 2, C 7 = σ 4 φxg X Θ, C 8 = σ + H2 σ Θ 2 c 2 1 2s 2 Θ, s HΘ C 9 = σ c 2 2H s Θ + σ, 2

32 Cubic action S 3 = χ := a2 σ c 2 s σ 2 ζ dtd 3 xa 3 C1 H ζ 3 + C 2 ζ ζ 2 + C 3 a 4 H 2 2 ζ( ζ) 2 + C 4 a 2 H 2 ζ 2 2 ζ + C 5 Hζ 2 ζ + C 6 a 4 H 2 ζ( ζ χ)+ C 7 a 4 2 ζ( χ) 2 + C 8 a 2 ζ( ζ)2 + C 9 a ζ( ζ 2 χ)+ 2 δl f(ζ) a3 δζ 1 C 1 = H σ Θ c I 2 s G φx (G X + XG XX ) Hσ c 2 sθ 2 + H2 σ c 4 sθ 2, σ C 2 = c 2 3 H2 s c 2 sθ Θ, HΘ C 3 = H2 φxgx Θ 3, New terms from G(φ,X) C 4 = 2H2 φx (GX + XG XX ) Θ 3, σ d H 2 δ C 5 = 2c 2 sh dt c 2 sθ 2, σ instead of C 6 = 2H φxg X Θ 2, σ C 7 = σ 4 φxg X Θ, C 8 = σ + H2 Θ 2 c 2 1 2s 2 Θ σ σ, s HΘ σ C 9 = c 2 2H σ s Θ +, 2

33 Non-Gaussianity f equi NL = O σ 2 c 2 s + O σ 2 XG XX G X + O σ IGS, σ := max{1, σ} C.f. r = 16σc s I := XK XX + 2X2 3 K XXX + H φg X +

34 Non-Gaussianity f equi NL = O σ 2 c 2 s + O σ 2 XG XX G X + O σ IGS, σ := max{1, σ} C.f. r = 16σc s I := XK XX + 2X2 3 K XXX + H φg X + k-inflation σ = f NL 1 c 2 s Large fnl (r) r = 16c s Small r (fnl) Seery, Lidsey (2005); Chen et al. (2007)

35 Non-Gaussianity f equi NL = O σ 2 c 2 s + O σ 2 XG XX G X + O σ IGS, σ := max{1, σ} C.f. r = 16σc s I := XK XX + 2X2 3 K XXX + H φg X + k-inflation σ = G-inflation σ = f NL 1 c 2 s r = 16c s Both fnl and r can be large Large fnl (r) Small r (fnl) Seery, Lidsey (2005); Chen et al. (2007) Say, f NL = 210 with r =0.17 is possible in kinetically driven models

36 Summary Generalized G-inflation is the most general single-field inflation model, obtained from the generalized Galileon Large r and large f NL are compatible in G-inflation

37 Thank you!

38 Backup

39 Horndeski & the Galileon L H = δµνσ αβγ κ 1 µ α φr +κ 3 α φ µ φr νσ βγ +δ αβ µν βγ νσ (F +2W )R µν αβ κ 1X µ α φ ν β φ σ γ φ +2κ 3X α φ µ φ ν β φ σ γ φ +2F X µ α φ ν β φ +2κ 8 α φ µ φ ν β φ 6(F φ +2W φ Xκ 8 ) φ + κ 9 Horndeski (1974) K = κ 9 +4X X dx (κ 8φ 2κ 3φφ ), G 3 = 6F φ 2Xκ 8 8Xκ 3φ +2 X dx (κ 8 2κ 3φ ), G 4 = 2F 4Xκ 3, G 5 = 4κ 1, Generalized Galileon

40 Ḣ and stability k-inflation: F S = M 2 Pl Garriga, Mukhanov (1999) = Ḣ H 2 > 0 Stable In more general cases, the sign of Ḣ and the stability criteria are not correlated Stable cosmology with Ḣ>0 is possible Interesting scenarios with null energy condition violation: Creminelli et al. (2006); Creminelli, Nicolis, Trincherini (2010)