Hybrid inflation with a non-minimally coupled scalar field

Transcription

1 Hybrid inflation with a non-minimally coupled scalar field Seoktae Koh CQUeST, Sogang University, Korea COSMO/CosPA Univ., JAPAN 27 Sept. - 1 Oct., 2010 work with M. Minamitsuji

2 PART I: Non-minimally coupledχfield Hybrid inflation with a nonminimally coupled χ field S = d 4 x [ 1 g 2κ 2(1 κ2 ξχ 2 )R 1 2 µ φ µ φ 1 ] 2 µ χ µ χ V(φ,χ) Potential of Hybrid inflation Effective Potential V(φ,χ) = λ 4 (χ2 v 2 ) m2 φ g2 φ 2 χ 2 V eff (φ,χ) = V(φ,χ)+ 1 2 ξχ2 R = V(φ,χ)+6ξχ 2 H 2 effective mass ofχfield during inflation m 2 eff (φ) = d2 V eff dχ 2 = λv 2 (1 κ 2 ξv 2 )+(g 2 +2ξκ 2 m 2 )φ 2 χ=0

3 Non-minimally coupled χ field critical value when the tachyonic stage begins Comments m 2 eff = 0 = φ2 c = λv2 (1 κ 2 ξv 2 ) g 2 + 2κ 2 ξm 2 Forξ < g2 2κ 2 m 2, theχfield is always tachyonic, irrespective of the value ofφ, and inflation never happens. Forxi > 1 κ 2 v 2, inflation is never terminated byχfield. non-minimal vs. minimal hybrid Inflation Forξ > 0, inflation is terminated later than standard hybrid inflatio (ξ = 0). (φ c < φ c,ξ=0 ) Forξ < 0, inflation is terminated earlier than standard hybrid inflatio (ξ = 0). (φ c > φ c,ξ=0 )

4 φ field dynamics in non-minimal hybrid inflation EoM for φ during inflation (χ = 0) φ+3h φ+m 2 φ 0 background solution for φ with a slow-roll approximtions φ = φ c e rh(t tc), r = 3 ( ) 1 1 4m2 2 9H 2 Conditions for Hybrid inflation in non-minimal hybrid inflation m χ 2 H 2 1, (m 2 χ = λv2 (1 κ 2 ξv 2 )), λv 4 m 2 φ 2 c = m 2 κ 2 ξχ 2 1 g 2 v 2 2(1 2κ 2 ξv 2 )

5 χ field dynamics in non-minimal hybrid inflation Equation of motion ofχfield (Ignoring nonlinear interaction terms) χ,nn + 3χ,n +(β 12ξ)(e 2rn 1)χ = 0, n H(t t c ) Beforeφ = φ c, χ field is massive χ(n) = e 3 2 n [c 1 J ν (z)+c 2 Y ν (z)], ν = 1 9 β 12ξe nr r 4 +β 12ξ, z = r Afterφ = φ c with rn 1 ( ) ( )] χ(n) e [d n 1 Ai 4δ +δ2/3 n +d 2 Bi 4δ +δ2/3 n, δ = 2(β+12 ξ )r Since Ai(x) is a decreasing function, we choosed 1 = 0

6 χ-field perturbations For a large scale perturbations (k ah), since δχ χ, before phase transition (φ > φ c ) δχ e 1 2 (3 r)(n N ) e i(θ θ ) δχ After phase transition (φ e < φ < φ c ) ( ) 1 δχ e 1 2 (3 r)n e 3 2 N φi 2δχ c e i(θ θc) e 3nBi( 9 +δ 3/2 n) 2 4δ 4/3 φ 9 c Bi( ) 4δ 4/3 In the subhorizon scales (k ah), WKB approximation provides assuming the adiabatic initial vaccum state at η δχ 1 [ ( 4 k 2 + β 12ξ ( ) φ 2 )] 1/4 a η 2 e i ωdη φ c At the horizon crossing time,δχ becomes δχ = H (2k 3 ) 1/2 (β+12 ξ ) 1/4 ( ) 1/2 φ φ c

7 χ-field perturbations During superhorizon evolution, δχ χ = ζ rφ i 0 (β 12ξ) 1/4 χ i ( φc φ ) 1/2 ( ) k 3(1 r)/2 e i(ψ ψ i ) H whereζ 0 is the amplitude of the adiabatic curvature perturbation H ζ 0 = (2k 3 ) 1/2 rφ i e rn

8 Non-adiabatic curvature perturbations Evolution of the curvature perturbation in Einstein frame ζ = H k 2 ( δφ H a 2Φ+H φ δχ ) χ Υ, d dˆt, d dt Υ = 1 2 ( ) ( ) 2 Ω 2 φ 2 χ 2 Ω 2 φ (lnω) 2 + χ 2,χ χ Ω 2 φ 2 χ 2 +Ω 2 φ, 2 conformal transformation:ĝ µν = Ω 2 (χ)g µν,ω 2 = 1 κ 2 ξχ 2 t : cosmic time in Jordan frame, ˆt : cosmic time in Einstein frame ζ is invariant under the conformal transformation curvature perturbation for the superhorzion scales rφ i ζ = ζ 0 [1+ (β 12ξ) 1 4χ i ( φc φ ) 1 2 ( ) 3 k 2 (1 r) nf e i(ψ ψ i ) dn Ωχv ] H 0 χ

9 Classical Backreaction I Nonlinear terms in χ field equation becomes important at χ = χ (1) χ 2 (1) = v 2 1 κ 2 ξv 2 Nonlinear terms in φ field equation becomes important at χ = χ (2) χ (2) = m2 g 2 Since χ 2 (2) χ2 (1), the nonlinear effects in φ field direction appear earlier than those in χ field direction. χ 2 (2) χ 2 (1) = (1 κ2 ξv 2 )m 2 g 2 v 2 1

10 Classical Backreaction II ξ 1: ζ ζ 0 [1 2βm 2 (g 2 + 2m 2 κ 2 ξ) r 1/2 λv 2 g 2 (β 12ξ) 3/4 ei( ψ +ψ i ) ( φ χ )( ) 3 k 2 r] H (Can re-produce the minimal case (ξ = 0) results due to classical back-reaction (Abolhasani & Firouzjahi, )) ξ 1: [ ζ ζ 0 1 πr11/6 e i(ψ 3ψ i +2ψ e ) 3 2 5/6 κ 2 ξ 2 (β 12ξ) 13/12 φ3/r i χ 3 i ( φc φ ) 1/2 ( k H ) 3(1 r)/2 ] ( β(g 2 + 2κ 2 ξm 2 ) 3/2r ) λv 2 (β 12ξ) It is necessary to consider the quantum backreaction effect to check the effct of the entropy perturbations on the adiabatic perturbations during tachyonic stage. (In Progress!)

11 PART II: Hybrid Inflation with a Non-minimally Coupled Inflaton Hybrid inflation with a nonminimally coupled inflaton S = d 4 x [ 1 g 2κ 2(1 κ2 ξφ 2 )R 1 2 µ φ µ φ 1 ] 2 µ χ µ χ V(φ,χ) Potential of Hybrid inflation V(φ,χ) = λ 4 (χ2 v 2 ) m2 φ µφ g2 φ 2 χ 2 global minima: (φ,χ) = (0,±v) During inflation, χ field stays at the false vacuum (χ = 0) effective mass ofχ: m 2 eff = λv2 +g 2 φ 2 ; tachyonic instability occurs at φ 2 = φ 2 c = λv2 /g 2 and inflation ends

12 Classification of the potential in Einstein frame I Class I m 4 µλv 4 < 1 m i) 2 < ξ < µ : one κ 2 λv 4 κ 2 m 2 local minimum at φ = φ e m ii) 2 > ξ > 0: κ 2 λv 4 monotonically increasing iii) ξ > µ : monotonically κ 2 m 2 decreasing φ e : local minimum φ 2 e = m2 +κ 2 λv 4 ξ µ+κ 2 m 2 ξ ) Atφ = φ c ( λ1/2 v g, tachyonic instability occurs and inflation ends. φ e φ c ξ = 0.01 ξ = ξ = ξ = V Ξ 0.01 φ/m pl Ξ Ξ Φ

13 ε Class I: m 4 µλv 4 < 1 (λ = 1,v 2 = 10 2 m 2 pl,m2 = 10 6 m 2 pl,µ = 10 6 ) ξ = ξ = ξ = ξ = ξ = 0.01 ξ = ξ = ξ = η φ/m pl { > 0 blue titled spectrum n s 1 = 2η 6ǫ 2η < 0 red tilted spectrum φ/m pl Coupling parameter Region Spectrum m 2 κ 2 λv < ξ < µ 4 κ 2 m 2 Vacuum-dominated Red Local minimum Blue Large field Red 0 < ξ < m2 Vacuum-dominated Blue κ 2 λv 4 Large field Red ξ > µ κ 2 m 2 Vacuum-dominated Red

14 Classification of the potential in Einstein frame II Class II i) ii) µ κ 2 m 2 m 4 µλv 4 > 1 < ξ < m2 κ 2 λv 4 : one local maximum µ > ξ > 0: κ 2 m 2 monotonically increasing iii) ξ > m2 κ 2 λv 4 : monotonically decreasing φ/m pl φ e : local maximum φ 2 e = m2 +κ 2 λv 4 ξ µ+κ 2 m 2 ξ ξ = -1x10-4 ξ = -5x10-4 ξ = -1x10-3 ξ = -5x10-3 V Ξ N (= ln a) 0.05 Ξ Ξ Φ

15 ε Class II: Spectrum (λ = 1,v 2 = 10 2 m 2 pl,m2 = 10 2 m 2 pl,µ = 10 4 ) slow-roll parameter ξ = -1x10-4 ξ = -5x10-4 ξ = -1x10-3 ξ = -5x10-3 η ξ = -1x10-4 ξ = -5x10-4 ξ = -1x10-3 ξ = -5x φ/m pl φ/m pl Coupling parameter Region Spectrum µ κ 2 m 2 < ξ < m2 Vacuum-dominated Blue κ 2 λv 4 Local maximum Red 0 < ξ < µ κ 2 m 2 Vacuum-dominated Blue Large field Red ξ > m2 κ 2 λv 4 Vacuum-dominated Red

16 Summary We consider hybrid inflation when an inflaton or a waterfall field couples to gravity nonminimally. For a waterfall field coupled to gravity ( ξ 1): 1 We investigate the effect of the entropy perturbations due to the waterfall field on the adiabatic perturbations. 2 We are investigating the effect of the classical and quantum backreaction effect on the curvature perturbations which are induced by the entropy perturbations. For an inflaton coupled to gravity: 1 In Einstein frame the potential has a local minimum or is monotonically increasing as φ increases for m4 < 1 µλv 4 depending on ξ. For m4 > 1, the potential has a local µλv 4 maximum or is monotonically increasing which can generate inflation. 2 The scalar spectral index shows blue-tilted or red-tilted spectrum depending on the coupling parameter ξ.