COSMO/CosPA 2010 @ 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:1008.0603
G-inflation = Inflation driven by the Galileon field
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
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 1008.0048; TK, Yamaguchi, Yokoyama 1008.0603
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;...
Simple motivation Why don t we use the Galileon field to drive inflation in the early universe?
Talk plan I. Introduction II. III. IV. G-inflation Primordial perturbations Summary
G-inflation
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)
Kinematically driven inflation L = K(φ, X)
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)
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
de Sitter G-inflation K = K(X), F = fx, f = const
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
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
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
Quasi-dS G-inflation K = K(X), F = f(φ)x Required to get n s 1 = 0
Quasi-dS G-inflation K = K(X), F = f(φ)x Required to get n s 1 = 0
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
Graceful exit & Reheating Basic idea K = A(φ)X + 1.0 0.5 0.0 Inflation φ end Example: 0.5 φ φt A = tanh [λ(φ end φ)] 1.0
Graceful exit & Reheating Basic idea K = A(φ)X + 1.0 0.5 0.0 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)
Phase diagram
Phase diagram Stable violation of null energy condition Creminelli, Luty, Nicolis, Senatore 06 Creminelli, Nicolis, Trincherini 10
Primordial perturbations
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
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
Deffayet, Pujolas, Sawicki, Vikman 1008.0048; TK, Yamaguchi, Yokoyama 1008.0603 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
Deffayet, Pujolas, Sawicki, Vikman 1008.0048; TK, Yamaguchi, Yokoyama 1008.0603 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 2 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
Stable example K = A(φ)X + X2 2M 3 µ, F = X M 3 1.5 1.0 Inflation Reheating c 2 s 0.5 c 2 s 3 6 µ M Pl > 0 G 0.0 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 log a
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
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
Tensor-to-scalar ratio K = X + X2 2M 3 µ, F = fx H 2 µm 3 M 2 Pl r 16 6 3 3µ M Pl 3/2 Conventional consistency relation is violated r = 8c s n T f,φ M =0.00435 M Pl,µ=0.032 M Pl P R =2.4 10 9,r=0.17 r can be large!
Summary
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!
Thank you!
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
Numerical Example K = X + X2 2M 3 µ, F = eαφ M 3 X 0.15 0.006 0.005 0.10 0.004 φ = const H H 2 0.05 = Ḣ H 2 = 0 Φ 0.003 0.002 0.001 0.00 55 50 45 40 35 log a 0.000 55 50 45 40 35 log a