Non-minimal k-inflation Wade Naylor High Energy Theory Group - PowerPoint PPT Presentation

Non-minimal k-inflation Wade Naylor High Energy Theory Group Osaka University (HETOU) APS2012, Kyoto, 2nd March 2012 In collaboration with T. Kubota, M. Nobuhiko and N. Okuda

Nonminimal k-inflation Nomination for shortest title at APS2012 APS2012, Kyoto, 2nd March 2012 In collaboration with T. Kubota, M. Nobuhiko and N. Okuda

/~20 Contents 3 • Comment (my non-Gaussianity) • Introduction • Model • Motivation • Jordan ≣ Einstein • Details on non-minimal k-inflation • Inflationary attractors even for K(φ)<0 • Power spectrum and tilt • Final comments (classical stability + non-Gaussianity)

/~20 My non-Gaussianity spectrum ⇒ I should know everything they know need students like Misumi and Okuda 4 • Prof. Moss (Ph.D.) • Prof. Sasaki (Post Doc) • Prof. Kubota (HETOU) • Kubota-Moss-Sasaki scale invariant • Non-Gaussianity ⇒ I don’t know and

/~20 Het Camp 2011 5 • Misumi and Okuda are nowhere to be seen?

/~20 Introduction between two frames including non-Gaussianity 084022. al., Phys. Rev. D 80 (2009); ibid. 81 (2010) ] 6 • k-inflation can be motivated from effective field theory + curvature coupling terms, ξφ 2 R • Jordan or Einstein frame? • Single field models give full agreement • Qiu and Yang, Non-Gaussianities of single field inflation with non-minimal coupling , Phys. Rev. D 83 (2011) • non-minimal coupled DBI (ξ=0, 1/6) [ see Easson et

/~20 Non-minimal k-inflation Model term 7 � d 4 x √− g � � S = f ( ϕ ) R + 2 P ( ϕ , X ) X = − 1 P ( ϕ , X ) = K ( ϕ ) X + L ( ϕ ) X 2 + · · · 2 g µ ν ∂ µ ϕ∂ ν ϕ • For example in a two field Lagrangian: L = 1 2( ∂φ ) 2 + 1 M ( ∂φ ) 2 + 1 2 M 2 ρ 2 + V ( φ ) − 1 2( ∂ρ ) 2 + ρ 2 ξ R ( φ 2 + ρ 2 ) • Then below some H<<M we can integrate out ρ to get 2( ∂φ ) 2 + ( ∂φ ) 4 L eff = 1 + ... + V ( φ ) − 1 2 ξ R φ 2 M 4 where M → M - ξ R/2 • Other examples might be DBI with more general ξ R φ 2

/~20 Motivation observational data 1 st order derivatives in φ (cf. Horndeski/G- inflation at 2 nd order ) coupling ξ? large non-Gaussianity (e.g., equilateral limit): 8 • Obtain constraints on inflation models via • k-inflation with non-minimal coupling • Most general single field theory with up to • Can we get constraints on conformal • Due to variable sound speed, k-inflation can generate f NL ∝ 1 c 2 s

/~20 Comment on E and J frames 9 Jordan frame Einstein frame (non-minimal) (minimal) action action S ˆ S mode function mode function ˆ u ˆ u k k b R = R b power spectrum power spectrum P R P b R non-Gaussianity non-Gaussianity � 0 | ˆ u k 3 | 0 ⇥ � 0 | u k 1 u k 2 u k 3 | 0 ⇥ u k 1 ˆ u k 2 ˆ T. Kubota, N. Misumi, W. Naylor and N. Okuda, JCAP 02 (2012) 034, arXiv: 1112.5233 [gr-qc]

/~20 Standard k-inflation inflationary attractor solutions 10 (Armendariz-Picon, Damour, Mukhanov, 1999 ) � 1 ⇤ X = − 1 ⇥ d 4 x √− g S = 2 κ 2 R + P ( φ , X ) 2 g µ ν ∂ µ φ∂ ν φ P ( φ , X ) = K ( φ ) X + X 2 + · · · T µ ν = ∂ P ∂ X ⇥ µ φ ⇥ ν φ � Pg µ ν E ( φ , X ) = 2 XP ,X − P • Consider FLRW universe described by line element ds 2 = − dt 2 + a 2 ( t ) δ ij dx i dx j • Friedmann equation and the continuity equation are 3 H 2 = E master equation √ ˙ E = − 3 E ( E + P ) • K(φ) changing from <0 to >0 essentially leads to

/~20 Attractors in k-inflation In k-inflation, the universe can expand exponentially using only master equation 11 (Armendariz-Picon, et al., PLB 458 1999) Solid lines are stable attractors √ ˙ E = − 3 E ( E + P ) P P=E E P=‐E

/~20 k-inflation phase diagram We obtain similar attractor/slow-roll solutions in the non-minimal case (see later if time permits) 12 (Armendariz-Picon, et al., PLB 458 1999)

/~20 non-minimal coupling ξ<0 obtained by substituting ξ→ -|ξ| Non-minimal k-inflation Conformal transformation via field redefinition 13 � 1 2 κ 2 R − 1 ⇤ ⇥ d 4 x √− g 2 ξφ 2 R + P ( φ , X ) S = g µ ν = Ω 2 g µ ν Ω 2 ≡ | 1 − ξκ 2 φ 2 | ˆ X = − 1 g µ ν ∂ µ φ∂ ν φ = Ω 2 ˆ 2 Ω 2 ˆ X � 1 ⇤ ⇥ ˆ 2 κ 2 ˆ R + ˆ ⌅ d 4 x S = − ˆ P ( φ , X ) g P ( φ , X ) = ˆ ˆ K ( φ ) ˆ X + ˆ L ( φ ) ˆ X 2 K ( φ ) = K ( φ ) − ξ ( K ( φ ) − 6 ξ ) κ 2 φ 2 ˆ ˆ L ( φ ) = L ( φ ) = 1 (1 − ξκ 2 φ 2 ) 2

/~20 14 E.g., setting K( φ )=-1 • Actually even for K( φ )<0 effect of ξ >0 coupling leads ˆ to a sign change in K( φ ) ` H f L K 10 8 6 4 2 f - 4 - 2 2 4 • ξ＝1/100,1/10 and 1/3

/~20 15 K( φ )=-1 • Speed of sound c s stays small until dφ/dt (∝X) drops ` ` 2 X c 1.0 b b P , b P , b X X c 2 b 0.8 s = = b b X + 2 b X b E , b P , b P , b X b X X 0.6 0.4 X = 1 0.2 ˆ g µ ν ∂ µ φ∂ ν φ 2 ˆ f 0.5 1.0 1.5 f NL ∝ 1 c 2 • Note that level of non-Gaussianity s

/~20 16 ξ=1/3) Example attractor solution • We see an attractor for E=p and E=-p (here for E 0.05 0.25 R 0.05 0.10 0.15 0.20 - 0.05 - 0.10 • More plots in progress ... - 0.15 - 0.20 - 0.25

/~20 Power spectrum & spectral index new variables plots currently in progress (including e-foldings)... 17 a 3 ˆ � � ˆ S (2) = � � ( ⇥ R ) 2 � ˆ ˙ d ˆ td 3 x R 2 − ˆ ˆ a ˆ c 2 ˆ s ˙ X b b b ˙ ˙ P , b b b H c s ✏ X b b b , , s = ✏ = − H 2 = ⌘ = b b ✏ b c s b • Slow roll parameters: H 2 b b H H s ⇥ 2 v � ¨ z v � c 2 ¨ z v = 0 a 2 b z 2 = 2 b ✏ v = z b ⇣ ⌘ R , k 2 − ¨ z s b c 2 c 2 b v b v b ¨ k + k = 0 s z (Note that ĉ s is a function of time ⇒ d ĉ s /dt <<1 ) • Standard quantization leads to: d ln P b R b b E 2 H 2 1 1 b b b R k b = − 2 b ✏ − b ⌘ − b P k = = ✏ , n s − 1 = s b E + b b d ln b 36 ⇡ 2 8 ⇡ 2 c s b b P k

18 /~20 Appendix: Conformal properties d ˆ a = Ω a d ˆ τ = d τ t = Ω dt ˆ � 2 ˆ � ( Ω 2 ) � R = Ω 2 � � Ω 2 ≡ | 1 − ξκ 2 φ 2 | ˆ R + 3 X Ω 2 � ˙ � ( Ω 2 ) � Ω 2 Ω 2 + 3 K = K � 2 P = P Ω 4 + 3 � 2 ˆ ˆ 2 Ω 2 Ω 2 4 ˙ Ω 2 H = 1 H + 1 � � b ˆ R = R Ω 2 Ω 2 � ˙ ˙ ¨ Ω 2 Ω 2 Ω 2 H = 1 H − 1 Ω 2 − 3 + 1 ˆ � � � ˙ ˙ 2 H Ω 2 Ω 2 Ω 2 4 2

/~20 (to be confirmed) Preheating in non-minimal K-inflation interesting? • permitting) contribution coming from ξ (later slide, time Shape of non-Gaussianity is important and has • minimal coupling broadens allowed parameter space Final Comments investigation, even for DBI non-minimal models (next slide) Classical stability (backreaction) needs further • characteristic signal, different from other models? In k-inflation with non-minimal coupling, is there a • 19 • Non-canonical models lead to non-Gaussianity & non-

/~20 20 • In Jordan frame easy to see , where Classical stability? • However another instability can be found from EOM: • G G e ff = 1 − φ 2 / φ 2 φ c2 = ( κ 2 ξ ) -1 =M pl2 /(8 πξ ) c R µ ν − 1 1 P X 2 φ + P φ + ( P XX r µ X + P X φ r µ φ ) r µ φ = ξφ R 2 g µ ν R = T µ ν M 2 pl g µ ν 2 ( φ 2 ) � r µ r ν ( φ 2 ) + ( R µ ν � 1 h 2 g µ ν R ) φ 2 T µ ν = P X r µ φ r ν φ + g µ ν P + ξ Taking trace of T µ ν and subbing back into EOM ⇒ ˜ φ 2 P X + 6 ξ � 6 ξφ P X φ r µ φ r µ φ � 6 ξφ P XX r µ X r µ φ + 4 P � 6 ξφ P φ h⇣ ⌘ i c R = pl ( φ 2 � ˜ M 2 P X P X P X φ 2 c ) M 2 where ( P x =1 , standard result) ˜ pl φ 2 c = ξ (1 − 6 ξ /P X ) ℋ constraints ⇒ only for anistropic spacetimes; cf. φ > ˜ φ c Futamase et al., Phys.Rev. D 39 (1989) 405-411

/~20 equilateral limit k-inflation with non-minimal coupling 21 squeezed limit • Shape of non-Gaussianity from ξ-contribution F(1,x 2 ,x 3 )x 22 x 32 x 2 =k 2 /k 1 x 3 =k 3 /k 1 • In all limits non-minimal part has nonzero value F ( k 1 , k 2 , k 3 ) ∝ 3 k 1 k 2 k 3 + 1 X k i k j 2 k 3 k 2 k 1 k 2 k 3 i>j • Result from N. Misumi’s Master’s thesis (cf. Qiu & Yang)

/~20 Appendix: Other shapes 22 The shape of non-Gaussianity depends on model of inflation; however, current data not sensitive to shape; only overall amplitude f NL F(1,x 2 ,x 3 )x 22 x 32 x 3 =k 3 /k 1 , x 2 =k 2 /k 1 (Babich et al. 2004)

/~20 23 Appendix: ADM Decomposition are given by • In what follows drop the hat and for Einstein frame ds 2 = − N 2 dt 2 + h ij ( dx i + N i dt )( dx j + N j dt ) N i ∆ t S = 1 t + ∆ t � √ hN [ R (3) + 2 P + N − 2 ( E ij E ij − E 2 )] dx i d 4 x 2 N ∆ t E ij = 1 2(˙ h ij � ⇥ i N j � ⇥ j N i ) t • Hamiltonian and momentum constraints equations R (3) + 2 P − 2 N − 2 P ,X ( ˙ φ − N i ∂ i φ ) 2 − N − 2 ( E ij E ij − E 2 ) = 0 i ) � ⇥ i ( N − 1 E ) = P ,X N − 1 ∂ i φ ( ˙ ⇥ j ( N − 1 E j φ � N i ∂ i φ ) • Solve constraints equations to first order only for N and N i (Chen et al. JCAP 01); in unitary gauge h ij = a 2 e 2 R δ ij δφ = 0 ,