G-inflation. Tsutomu Kobayashi. RESCEU, Univ. of Tokyo. COSMO/CosPA The Univ. of Tokyo

Transcription

1 COSMO/CosPA The Univ. of Tokyo G-inflation Tsutomu Kobayashi RESCEU, Univ. of Tokyo Based on work with: Masahide Yamaguchi (Tokyo Inst. Tech.) Jun ichi Yokoyama (RESCEU & IPMU) arxiv:

2 G-inflation = Inflation driven by the Galileon field

3 The Galileon field L 1 = φ L 2 =( φ) 2 L 3 =( φ) 2 φ L 4 =( φ) 2 2(φ) 2 2( µ ν φ) 2 R 2 ( φ)2 Field equations are 2nd order Galilean shift symmetry in flat space µ φ µ φ + b µ L n 2(n 1) φ n L 5 =( φ) 2 (φ) 3 + Nicolis et al. 09; Deffayet et al. 09

4 Our inflaton Lagrangian L = R 2 + K(φ, X) F (φ, X)φ where X := 1 2 ( φ)2 Field equations are 2nd order Deffayet, Pujolas, Sawicki, Vikman ; TK, Yamaguchi, Yokoyama

5 Simple motivation The Galileon field has been used to explain current cosmic acceleration... Chow, Khoury 09; Silva, Koyama 09; TK, Tashiro, Suzuki 09; TK 10; Gannouji, Sami 10; De Felice, Tsujikawa 10; De Felice, Mukohyama, Tsujikawa 10;...

6 Simple motivation Why don t we use the Galileon field to drive inflation in the early universe?

7 Talk plan I. Introduction II. III. IV. G-inflation Primordial perturbations Summary

8 G-inflation

9 Standard picture of inflation One (or more) canonical scalar field(s) rolling slowly down a nearly flat potential L = X V (φ), X = 1 2 ( φ)2 3M 2 PlH 2 V (φ)

10 Kinematically driven inflation L = K(φ, X)

11 Kinematically driven inflation L = K(φ, X) d a 3 K φ X =0 dt K = K(X) k-inflation 3M 2 PlH 2 K Armendariz-Picon et al. 99; Ghost condensate Arkani-Hamed et al. 04 X K(X)

12 G-inflation: background L φ = K(φ, X) F (φ, X)φ 3H 2 = ρ 3H 2 2Ḣ = p + Scalar field EOM is automatically satisfied ρ = 2XK X K +3F X H φ 3 2F φ X p = K 2 F φ + F φ X X

13 de Sitter G-inflation K = K(X), F = fx, f = const

14 de Sitter G-inflation K = K(X), F = fx, f = const Look for exactly de Sitter solution: H = const φ = const satisfying: 3H 2 = K K X = 3fH φ

15 de Sitter G-inflation K = K(X), F = fx, f = const Look for exactly de Sitter solution: H = const φ = const satisfying: 3H 2 = K K X = 3fH φ K = 1 (K X ) 2 6f 2 X

16 de Sitter G-inflation K = K(X), F = fx, f = const Look for exactly de Sitter solution: H = const φ = const satisfying: φ >0 K(X) X 1 (K X ) 2 6f 2 X 3H 2 = K K X = 3fH φ K = 1 (K X ) 2 6f 2 X

17 Quasi-dS G-inflation K = K(X), F = f(φ)x Required to get n s 1 = 0

18 Quasi-dS G-inflation K = K(X), F = f(φ)x Required to get n s 1 = 0

19 Quasi-dS G-inflation K = K(X), F = f(φ)x Quasi-de Sitter solution: Required to get n s 1 = 0 H = H(t), φ = φ(t) Small rate of change satisfying: H 2 K(X) K X 3f(φ)H φ = Ḣ H 2 1 φ η = H φ 1

20 Graceful exit & Reheating Basic idea K = A(φ)X Inflation φ end Example: 0.5 φ φt A = tanh [λ(φ end φ)] 1.0

21 Graceful exit & Reheating Basic idea K = A(φ)X Inflation φ end Example: 0.5 A = tanh [λ(φ end φ)] 1.0 φ φt kination Reheating through gravitational particle production Ford 87 ρ p X a 6 ~ massless, canonical field (normal sign)

22 Phase diagram

23 Phase diagram Stable violation of null energy condition Creminelli, Luty, Nicolis, Senatore 06 Creminelli, Nicolis, Trincherini 10

24 Primordial perturbations

25 Cosmological perturbations ds 2 = (1 + 2α)dt 2 +2a 2 β,i dtdx i + a 2 (1 + 2R)δ ij dx i dx j φ = φ(t) Unitary gauge: δφ =0 1. Expand the action to 2nd order 2. Eliminate α and β using constraint eqs 3. Quadratic action for R

26 Cosmological perturbations ds 2 = (1 + 2α)dt 2 +2a 2 β,i dtdx i + a 2 (1 + 2R)δ ij dx i dx j φ = φ(t) Unitary gauge: δφ =0 1. Expand the action to 2nd order 2. Eliminate α and β using constraint eqs 3. Quadratic action for R δt 0 i = F X φ3 α,i Uniform φ hypersurfaces comoving hypersurfaces

27 Deffayet, Pujolas, Sawicki, Vikman ; TK, Yamaguchi, Yokoyama Quadratic action S (2) = 1 2 dτd 3 xz 2 G(R ) 2 F( R) 2 where a φ z = H F φ3 X /2 F = K X +2F φ X +2H φ 2F 2 XX 2 +2F XX X φ 2(F φ XF φx ) G = K X +2XK XX +6F X H φ +6F 2 XX 2 2(F φ + XF φx )+6F XX HX φ

28 Deffayet, Pujolas, Sawicki, Vikman ; TK, Yamaguchi, Yokoyama Quadratic action S (2) = 1 2 dτd 3 xz 2 G(R ) 2 F( R) 2 where No ghost and gradient instabilities if a φ z = G > 0, c 2 H F φ3 X /2 s = F/G > 0 F = K X +2F φ X +2H φ 2FXX F XX X φ 2(F φ XF φx ) G = K X +2XK XX +6F X H φ +6F 2 XX 2 2(F φ + XF φx )+6F XX HX φ

29 Stable example K = A(φ)X + X2 2M 3 µ, F = X M Inflation Reheating c 2 s 0.5 c 2 s 3 6 µ M Pl > 0 G log a

30 Primordial spectrum Consider G-inflation with: Sasaki-Mukhanov equation K = K(X), F = f(φ)x New variables: d 2 u dy 2 + k 2 z,yy z u =0 dy = c s dτ z = (FG) 1/4 z u = zr z,yy z 1 [2 + 3C(X)] ( y) 2 C(X) = K K X Q X Q Q(X) = (K XK X) 2 18Xc 2 s FG

31 Primordial spectrum Normalized mode: u = π 2 yh (1) 3/2+C ( ky) P R = Q 4π 2 cs k=1/( τ), n s 1= 2C f,φ R can be generated even from exact de Sitter where Q(X) = (K XK X) 2 18M 4 Pl Xc2 s FG * Tensor mode dynamics: unchanged

32 Tensor-to-scalar ratio K = X + X2 2M 3 µ, F = fx H 2 µm 3 M 2 Pl r µ M Pl 3/2 Conventional consistency relation is violated r = 8c s n T f,φ M = M Pl,µ=0.032 M Pl P R = ,r=0.17 r can be large!

33 Summary

34 Summary G-inflation: A general class of single field inflation L φ = K(φ, X) F (φ, X)φ Large n s 1 0 r Consistency relation G-inflation would make gravitational wave people happy!

35 Thank you!

36 Example K = X + X2 2M 3 µ, F = X M 3 H 2 M 2 Pl 1 6 µ M Pl X 1 M 3 MPl 3 µ M Pl 3µ µm 3 M Pl

37 Numerical Example K = X + X2 2M 3 µ, F = eαφ M 3 X φ = const H H = Ḣ H 2 = 0 Φ log a log a