Towards more precise estimates of the primordial bispectrum - PowerPoint PPT Presentation

Towards more precise estimates of the primordial bispectrum Jinn-Ouk Gong APCTP , Pohang 790-784, Korea RESCEU Symposium on General Relativity and Gravitation (JGRG22) University of Tokyo, Tokyo, Japan 13th November, 2012 Celebrating 60th birthday of T. Futamase, H. Kodama and M. Sasaki Based on C. T. Byrnes and JG, arXiv:1210.1851 [astro-ph.CO] A. Achucarro, JG, G. A. Palma and S. P . Patil, to appear JG, K. Schalm and G. Shiu, to appear

Introduction Effects of non-trivial speed of sound Bispectrum in general slow-roll Running of f NL Summary Outline Introduction 1 Effects of non-trivial speed of sound 2 Bispectrum in general slow-roll 3 Running of f NL 4 Summary 5 Towards more precise estimates of the primordial bispectrum Jinn-Ouk Gong

Introduction Effects of non-trivial speed of sound Bispectrum in general slow-roll Running of f NL Summary General single field inflation � � � m 2 d 4 x �− g X ≡ − 1 Pl 2 g µν ∂ µ φ∂ ν φ S = 2 R + P ( X , φ ) with Originated from multi-field setup: light R and heavy F a ( t + ! ) + N a ( t + ! ) F ! a ( t , x ) = ! 0 Trajectory along the lightest direction Effects of heavy physics in curved traj a ( t ) ! 0 a Can we find universal features of N “heavy” physics? background a t + ! ( t , x ) ( ) trajectory ! 0 Write the action in terms of R (along traj) and F (off traj) 1 � Integrate out F : e S eff [ R ] = [ D F ] e S [ R , F ] 2 � � � � � ˙ � ˙ F = − 2˙ − � + M 2 = equiv to plugging linear sol: θ φ 0 / H R eff Effective single field action S eff [ R ] 3 Towards more precise estimates of the primordial bispectrum Jinn-Ouk Gong

Introduction Effects of non-trivial speed of sound Bispectrum in general slow-roll Running of f NL Summary Effects of heavy physics as non-trivial c s Effects of heavy physics in “speed of sound”   ≡ 1 + 4˙ θ 2 � ˙ � c − 2 θ : angular velocity of traj s M 2 eff Single field theory with non-trivial c 2 s : Footprint of heavy physics (Achucarro et al. 2012a) F borrows kinetic energy of R → propagation speed c s reduced EFT in � / M 2 eff : universal footprint of heavy physics Many scalar fields in BSM, e.g. moduli New observables poorly constrained → to be tested in next decades Towards more precise estimates of the primordial bispectrum Jinn-Ouk Gong

Introduction Effects of non-trivial speed of sound Bispectrum in general slow-roll Running of f NL Summary Splitting canonical action EFT = canonical ( c s = 1) + (occasional) departure from c s = 1 � ˙ � � R 2 − ( ∇ R ) 2 d 4 xa 3 ǫ m 2 S = + S 3 +··· Pl c 2 a 2 s � �� � = S 2 , “free” part � � � 1 d 4 xa 3 ǫ m 2 R 2 ˙ = S 2,canonical + − 1 + S 3 +··· Pl c 2 � �� � s � �� � c s = 1 part ≡ S 2,int Well known, accurate Green’s function (For example, JG & Stewart 2001, Choe, JG & Stewart 2004) Interaction valid for a limited interval (c.f. Chen & Wang 2010) � c.f. Using dy ≡ c s d τ = c s dt / a , q 2 ≡ a 2 ǫ / c s and v = 2 q R (Baumann, Senatore & Zaldarriaga 2011) � � � m 2 ( v ′ ) 2 − ( ∇ v ) 2 + q ′′ d 4 x Pl q v 2 S 2 = 2 But see later parts of this presentation Towards more precise estimates of the primordial bispectrum Jinn-Ouk Gong

Introduction Effects of non-trivial speed of sound Bispectrum in general slow-roll Running of f NL Summary Features in the power spectrum Interaction Hamiltonian at quadratic order � � � � � � ∂ L (2) 1 H (2) int R − L (2) d 3 x ˙ d 3 xa 3 ǫ m 2 R 2 ˙ int ( t ) = = − 1 int Pl ∂ ˙ c 2 R s � �� � ≡− u ( t ) Features in the power spectrum � τ � �� � � a ( τ ′ ) d τ ′ � � � = (2 π ) 3 δ (3) ( k + q ) 2 π 2 � � R q ( τ ), H (2) int ( τ ′ ) R k ( τ ) � � R k ( τ ) � � R q ( τ ) =− i 0 � 0 � k 3 ∆ P R τ in � ∞ H 2 , t ≡ τ , κ ≡ k → ∆ P R = κ dtu ( t )sin(2 κ t ) with P R = 8 π 2 m 2 P R τ ⋆ k ⋆ Pl ǫ 0 Inverting this relation to write u in terms of observable ∆ P R / P R � ∞ � κ � u ( t ) = 2 i d κ ∆ P R e i κτ π κ P R 2 −∞ � Correlated bispectrum and power spectrum: B R = ( ··· ∆ P R / P R ) Towards more precise estimates of the primordial bispectrum Jinn-Ouk Gong

Introduction Effects of non-trivial speed of sound Bispectrum in general slow-roll Running of f NL Summary Leading bispectrum for varying c s Leading order action in terms of u ( t ) � � � � R 2 R − ( u + 2 s ) R ( ∇ R ) 2 � s ≡ ˙ c s d 4 xa 3 m 2 3 u ˙ S 3 ⊃ Pl ǫ Hc s Assumption: H , ǫ and η ∥ approximately constant ( K ≡ k 1 + k 2 + k 3 )   R ∗ R ∗ � 0 d � d �  m 2 k 1 ( τ ) k 2 ( τ ) d τ u Pl R ∗  2 i � R k 1 (0) � R k 2 (0) �  3 ǫ � B R ( k 1 , k 2 , k 3 ) = 2 ℜ R k 3 (0) k 3 ( τ ) + 2 perm H 2 τ 2 d τ d τ −∞ �� �� 0 m 2 � d τ u + 2 s Pl R ∗ R ∗ R ∗ � k 1 ( τ ) � k 2 ( τ ) � + ǫ k 1 · k 2 + 2 perm k 3 ( τ ) H 2 τ 2 −∞ � 1 � K � 1 � k ��� � 3 � � (2 π ) 4 P 2 d � ∆ P R ∆ P R R 2 ( k 1 k 2 ) 2 � = − k 3 + 2 perm � ( k 1 k 2 k 3 ) 3 K P R 2 k ⋆ dk k P R 2 k ⋆ k = K � � � � K � � 1 � k ��� K 2 − � + 1 � � k 1 k 2 + 2 perm ∆ P R d ∆ P R � k 1 · k 2 + 2 perm + k 1 k 2 k 3 � 2 K P R 2 k ⋆ dk k P R 2 k ⋆ k = K ��� �� � ∆ P R � k ��� � ∆ P R � k � d 2 � d � � � � − k 1 k 2 + 2 perm + k 1 k 2 k 3 � � dk 2 � k = K dk P R 2 k ⋆ P R 2 k ⋆ k = K (Achucarro, JG, Palma & Patil, to appear) Correlation between spectra is manifest! Towards more precise estimates of the primordial bispectrum Jinn-Ouk Gong

Introduction Effects of non-trivial speed of sound Bispectrum in general slow-roll Running of f NL Summary Modeling curvilinear trajectory A cosh turn in otherwise straight trajectory in 2-field system φ 2 η ⊥ > 0 φ 1 η ⊥ = 0 η ⊥ = 0 ˙ θ η ⊥ ,max η ⊥ = H = cosh 2 [2( N − N ⋆ )/ ∆ N ] (Equations of motion: see Achucarro et al. 2011) Towards more precise estimates of the primordial bispectrum Jinn-Ouk Gong

Introduction Effects of non-trivial speed of sound Bispectrum in general slow-roll Running of f NL Summary Features from smooth curvilinear trajectory Η � max � 5, � N � 0.5, H 2 � M 2 � 1 � 300 Η � max � 5, � N � 0.5, H 2 � M 2 � 1 � 300 0.2 3 2 0.1 1 � P R � P R 0.0 0 � 1 � 0.1 � eq � f NL � folded � f NL � 2 � local � � 0.2 f NL � 3 0.1 0.2 0.5 1.0 2.0 5.0 10.0 20.0 0.1 0.2 0.5 1.0 2.0 5.0 10.0 20.0 k 1 � k � k 1 � k � 1.0 1.0 0.9 0.9 0.8 0.8 k 2 � k 1 k 2 � k 1 � 4 0.7 0.7 0.6 0.6 0.5 0.5 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 k 3 � k 1 k 3 � k 1 1.0 1.0 0.9 0.9 0.8 0.8 k 2 � k 1 k 2 � k 1 0.7 0.7 � � 2 0.6 0.6 0.5 0.5 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 k 3 � k 1 k 3 � k 1 Towards more precise estimates of the primordial bispectrum Jinn-Ouk Gong

Introduction Effects of non-trivial speed of sound Bispectrum in general slow-roll Running of f NL Summary General slow-roll approximation � R k ( τ ) = de Sitter piece + higher order corrections No guarantee for the hierarchy between slow-roll parameters Up to 1st order corrections in the standard SR known (Chen et al. 2007) Consistent account in more general contexts � Mode equation: z 2 ≡ 2 a 2 m 2 2 kz � Pl ǫ , y ≡ R k , dx =≡ − kc s dt / a , f ≡ 2 π xz / k � � � � � � f ′′ − 3 f ′ d 2 y 1 − 2 1 df 1 + i f ′ ≡ e ix dx 2 + y = y → y 0 ( x ) = x 2 x 2 f d log x x � �� � � �� � de Sitter solution ≡ g (log x ) � �� � departure from dS Green’s function solution (JG & Stewart 2001) � ∞ y ( x ) = y 0 ( x ) + i du � � y ∗ 0 ( u ) y 0 ( x ) − y ∗ u 2 g (log u ) 0 ( x ) y 0 ( u ) y ( u ) 2 x ≡ y 0 ( x ) + L ( x , u ) y ( u ) = y 0 ( x ) + L ( x , u ) y 0 ( u ) + L ( x , u ) L ( u , v ) y 0 ( v ) +··· Towards more precise estimates of the primordial bispectrum Jinn-Ouk Gong

Introduction Effects of non-trivial speed of sound Bispectrum in general slow-roll Running of f NL Summary 3rd order action reprocessed R 3 and ˙ ˙ R 2 R : cumbersome to compute with many derivatives � �� � 3 ∼ � R 3 ∼ ˙ y 0 + ˙ ˙ Ly 0 + L ˙ y 0 +··· Using partial int and linear eq to reduce the # of derivatives � � � � � c 2 � � ≡ a 3 ǫ a 2 ǫ � δ L d R − ∆ s ¨ ˙ � R + + H a 2 R � 1 c 2 a 2 ǫ c 2 δ R dt s s � �� � ≡ C = H ( 3 + η − 2 s ) � R 3 � � ¨ � � ˙ � � A − 3˙ AC − 2 A ˙ C + 2 AC 2 c 2 � d +···+ δ L A − 2 AC R 3 = s R 2 +··· A ˙ � � 1 a 3 ǫ 2 dt 3 δ R 2 � R 3 � � − ˙ � � c 2 � B + BC d +···+ δ L B B ˙ R 2 R = s 2 R 2 � � 1 a 3 ǫ 2 dt 3 δ R Field redefinition with more terms involved (JG, Schalm & Shiu, to appear) � � �� � m 2 a 2 ǫ � 2 + ǫ s + 9 us − 2 s 2 � � R 3 � 3 s + ǫη − 1 d η d d τ d 3 x Pl S 3 = − c s aH + (higher SR terms) c 2 3 c s 2 d τ d τ s � �� � ≡ C Towards more precise estimates of the primordial bispectrum Jinn-Ouk Gong