Univesità di Bologna Inflation and Reheating in Spontaneously Geneated Gavity (A. Ceioni, F. Finelli, A. Tonconi, G. Ventui) Phys.Rev.D81:123505,2010
Motivations
Inflation (FTV Phys.Lett.B681:383-386,2009) S = d 4 x g gµν 2 µσ ν σ + γ 2 σ2 R V (σ) δ 0 σ/σ(t i ) 0 H(t i )/H δ n : n : n+1 1 δn+1 δn 1 dδ n /dn n d n /dn N =ln a(t) a(t i ) γ σ V dv eff dσ σ V dv dσ 4 1, σ n V d n V eff dσ n 1 V (σ) σ n δ 1 = γ (n 4) 1+γ (n + 2), 1 = γ (n 2) (n 4) 2+2γ (n + 2)
Inflationay specta f i,k + α i f i,k + k2 a 2 H 2 f i,k =0, = d dn,i= s, t f s (x) =R(x) = Hδσ(x) σ α s =3 1 +2δ 1 +2δ 2 2 δ 1δ 2 1+δ 1 f t(x) =h (x),h + (x) α t =3 1 +2δ 1 P R (k) k3 2π 2 R k 2 P R (k ) k k ns 1 P h (k) 2k3 π 2 h+,k 2 + h,k 2 P h (k ) k k nt
Inflationay specta f i,k + α i f i,k + k2 a 2 H 2 f i,k =0, = d dn,i= s, t f s (x) =R(x) = Hδσ(x) σ α s =3 1 +2δ 1 +2δ 2 2 δ 1δ 2 1+δ 1 f t(x) =h (x),h + (x) α t =3 1 +2δ 1 P R (k) k3 k 2π 2 R k 2 P R (k ) k ns 1 Compaed to data = P h(k) P R (k) = 8 n t 1 n t 2 P h (k) 2k3 π 2 h+,k 2 + h,k 2 P h (k ) k k nt
Slow Roll Appoximation δ i 1, i 1 γ (δ 1 6γ) 1 = δ 1 (3 + 24γ δ 1 + δ 2 )+3γ 1+2δ 1 δ2 1 σ dv eff 6γ V dσ δ1 2(1+δ 1 ) 1 = δ 1 γ 2+4δ 1 +2δ 2 γ 1 γ 1 1= 2(δ 1 + δ 2 + 1 ) = 2γσ2 V eff,σσ 3V eff,σ V 2 eff,σ 3 V σ V V 2 2 + γρ δ = 16 (δ 1 + 1 ) = 8γσ2 1+ 2γρ δ V 2 eff,σ ρ δ δ 2 V 2 δ 1 H 2 P R (k ) 4π 2 (1 + 6γ) δ1 2 σ2 P R (k )=(2.445 ± 96) 10 9
Slow Roll Appoximation δ i 1, i 1 γ (δ 1 6γ) 1 = δ 1 (3 + 24γ δ 1 + δ 2 )+3γ 1+2δ 1 δ2 1 σ dv eff 6γ V dσ δ1 2(1+δ 1 ) 1 = δ 1 γ 2+4δ 1 +2δ 2 γ 1 γ 1 γ 1 1= 2(δ 1 + δ 2 + 1 ) = 2γσ2 V eff,σσ 3V eff,σ V 2 eff,σ 3 V σ V V 2 2 + γρ δ = 16 (δ 1 + 1 ) = 8γσ2 1+ 2γρ δ V 2 eff,σ ρ δ δ 2 V 2 δ 1 H 2 P R (k ) 4π 2 (1 + 6γ) δ1 2 σ2 P R (k )=(2.445 ± 96) 10 9
Slow Roll Appoximation δ i 1, i 1 γ (δ 1 6γ) 1 = δ 1 (3 + 24γ δ 1 + δ 2 )+3γ 1+2δ 1 δ2 1 σ dv eff 6γ V dσ δ1 2(1+δ 1 ) 1 = δ 1 γ 2+4δ 1 +2δ 2 γ 1 γ 1 1= 2(δ 1 + δ 2 + 1 ) = 2γσ2 V eff,σσ 3V eff,σ V 2 eff,σ 3 V σ V V 2 2 + γρ δ = 16 (δ 1 + 1 ) = 8γσ2 1+ 2γρ δ V 2 eff,σ ρ δ δ 2 V 2 δ 1 H 2 P R (k ) 4π 2 (1 + 6γ) δ1 2 σ2 P R (k )=(2.445 ± 96) 10 9
Slow Roll Appoximation δ i 1, i 1 γ (δ 1 6γ) 1 = δ 1 (3 + 24γ δ 1 + δ 2 )+3γ 1+2δ 1 δ2 1 σ dv eff 6γ V dσ δ1 2(1+δ 1 ) 1 = δ 1 γ 2+4δ 1 +2δ 2 γ 1 γ 1 γ 1 1= 2(δ 1 + δ 2 + 1 ) = 2γσ2 V eff,σσ 3V eff,σ V 2 eff,σ 3 V σ V V 2 2 + γρ δ = 16 (δ 1 + 1 ) = 8γσ2 1+ 2γρ δ V 2 eff,σ ρ δ δ 2 V 2 δ 1 H 2 P R (k ) 4π 2 (1 + 6γ) δ1 2 σ2 P R (k )=(2.445 ± 96) 10 9
Slow Roll Appoximation δ i 1, i 1 γ (δ 1 6γ) 1 = δ 1 (3 + 24γ δ 1 + δ 2 )+3γ 1+2δ 1 δ2 1 σ dv eff 6γ V dσ δ1 2(1+δ 1 ) 1 = δ 1 γ 2+4δ 1 +2δ 2 γ 1 γ 1 1= 2(δ 1 + δ 2 + 1 ) = 2γσ2 V eff,σσ 3V eff,σ V 2 eff,σ 3 V σ V V 2 2 + γρ δ = 16 (δ 1 + 1 ) = 8γσ2 1+ 2γρ δ V 2 eff,σ ρ δ δ 2 V 2 δ 1 H 2 P R (k ) 4π 2 (1 + 6γ) δ1 2 σ2 P R (k )=(2.445 ± 96) 10 9
SB Potentials Geneal Featues γ 0 γ 1 2m V σ σ 0 σ V 0 1 = δ 2 1, σ 0 2N 1 δ2 1 γ 2δ 2 m +1 N 8 δ2 1 γ 8 m N γ 1 V σ 4 V LG (σ) = µ 4 V CW (σ) = µ 8 σ4 log σ4 σ 4 0 σ 2 σ 2 0 2 1 + µ 8 σ4 0 1 2 N, 12 N 2 1 3 2N, 4 N
SB Potentials Numeical Analysis γ 0 V (σ) = µ 4 ( σ 2 σ 2 0) 2 V (σ) = µ 8 σ4 ( log σ4 σ 4 0 ) 1 + µ 8 σ4 0 68% C.L. 95% C.L. L.F. N = 50 V (σ) = Λ 8 ( σ 2 σ 2 0) 4 V (σ) =Λ [ )] 1 + cos (π σσ0 S.F. N = 50 L.F. N = 70 S.F. N = 70 16 = 1 3
SB Potentials Numeical Analysis γ 0 V (σ) = µ 4 ( σ 2 σ 2 0) 2 V (σ) = µ 8 σ4 ( log σ4 σ 4 0 ) 1 + µ 8 σ4 0 γ = 10 5 SF γ = 10 5 SF 68% C.L. 95% C.L. L.F. N = 50 V (σ) = Λ 8 ( σ 2 σ 2 0) 4 γ = 10 5 SF V (σ) =Λ [ )] 1 + cos (π σσ0 γ = 10 5 SF S.F. N = 50 L.F. N = 70 S.F. N = 70 16 = 1 3
SB Potentials Numeical Analysis γ 0 V (σ) = µ 4 ( σ 2 σ 2 0) 2 V (σ) = µ 8 σ4 ( log σ4 σ 4 0 ) 1 + µ 8 σ4 0 γ = 10 7 LF γ = 10 5 SF γ = 10 7 LF γ = 10 5 SF 68% C.L. 95% C.L. L.F. N = 50 γ = 10 5 LF V (σ) = Λ 8 ( σ 2 σ 2 0) 4 γ = 10 5 SF γ = 10 5 LF V (σ) =Λ [ )] 1 + cos (π σσ0 γ = 10 5 SF S.F. N = 50 L.F. N = 70 S.F. N = 70 16 = 1 3
SB Potentials Geneal Featues γ 0 γ 1 2m V σ σ 0 σ V 0 1 = δ 2 1, σ 0 2N 1 δ2 1 γ 2δ 2 m +1 N 8 δ2 1 γ 8 m N γ 1 V σ 4 V LG (σ) = µ 4 V CW (σ) = µ 8 σ4 log σ4 σ 4 0 σ 2 σ 2 0 2 1 + µ 8 σ4 0 1 2 N, 12 N 2 1 3 2N, 4 N
SB Potentials Numeical Analysis 2m V σ σ 0 σ γ 1 V 0 1 N σ 0 V (σ) = µ 4 ( σ 2 σ 2 0) 2 V (σ) = µ 8 σ4 ( log σ4 σ 4 0 ) 1 + µ 8 σ4 0 68% C.L. 95% C.L. L.F. N = 50 V (σ) = Λ 8 ( σ 2 σ 2 0) 4 V (σ) =Λ [ )] 1 + cos (π σσ0 S.F. N = 50 L.F. N = 70 S.F. N = 70 16 = 1 3
SB Potentials Numeical Analysis 2m V σ σ 0 σ γ 1 V 0 1 N σ 0 V (σ) = µ 4 ( σ 2 σ 2 0) 2 V (σ) = µ 8 σ4 ( log σ4 σ 4 0 ) 1 + µ 8 σ4 0 m =1 m =1 68% C.L. 95% C.L. L.F. N = 50 V (σ) = Λ 8 ( σ 2 σ 2 0) 4 m =1 V (σ) =Λ [ )] 1 + cos (π σσ0 S.F. N = 50 L.F. N = 70 S.F. N = 70 16 = 1 3
SB Potentials Numeical Analysis 2m V σ σ 0 σ γ 1 V 0 1 N σ 0 V (σ) = µ 4 ( σ 2 σ 2 0) 2 V (σ) = µ 8 σ4 ( log σ4 σ 4 0 ) 1 + µ 8 σ4 0 m =1 m =1 68% C.L. 95% C.L. L.F. N = 50 m =2 V (σ) = Λ 8 ( σ 2 σ 2 0) 4 m =1 V (σ) =Λ [ )] 1 + cos (π σσ0 S.F. N = 50 L.F. N = 70 S.F. N = 70 16 = 1 3
SB Potentials Geneal Featues γ 0 γ 1 2m V σ σ 0 σ V 0 1 = δ 2 1, σ 0 2N 1 δ2 1 γ 2δ 2 m +1 N 8 δ2 1 γ 8 m N γ 1 V σ 4 V LG (σ) = µ 4 V CW (σ) = µ 8 σ4 log σ4 σ 4 0 σ 2 σ 2 0 2 1 + µ 8 σ4 0 1 2 N, 12 N 2 1 3 2N, 4 N
SB Potentials Numeical Analysis γ 1 V σ 4 V (σ) = µ 4 ( σ 2 σ 2 0) 2 V (σ) = µ 8 σ4 ( log σ4 σ 4 0 ) 1 + µ 8 σ4 0 68% C.L. 95% C.L. L.F. N = 50 V (σ) = Λ 8 ( σ 2 σ 2 0) 4 V (σ) =Λ [ )] 1 + cos (π σσ0 S.F. N = 50 L.F. N = 70 S.F. N = 70 16 = 1 3
SB Potentials Numeical Analysis γ 1 V σ 4 V (σ) = µ 4 ( σ 2 σ 2 0) 2 V (σ) = µ 8 σ4 ( log σ4 σ 4 0 ) 1 + µ 8 σ4 0 γ = 10 3 SF γ = 10 3 SF 68% C.L. 95% C.L. L.F. N = 50 V (σ) = Λ 8 ( σ 2 σ 2 0) 4 γ = 10 3 SF V (σ) =Λ [ )] 1 + cos (π σσ0 γ = 10 3 SF S.F. N = 50 L.F. N = 70 S.F. N = 70 16 = 1 3
SB Potentials Numeical Analysis VIABLE γ 1 V σ 4 γ = 10 1 LF V (σ) = µ 4 ( σ 2 σ 2 0) 2 γ = 10 3 SF VIABLE γ = 10 1 LF V (σ) = µ 8 σ4 ( log σ4 σ 4 0 ) 1 + µ 8 σ4 0 γ = 10 3 SF 68% C.L. 95% C.L. L.F. N = 50 γ = 10 3 LF V (σ) = Λ 8 ( σ 2 σ 2 0) 4 γ = 10 3 SF γ = 10 3 LF V (σ) =Λ [ )] 1 + cos (π σσ0 γ = 10 3 SF S.F. N = 50 L.F. N = 70 S.F. N = 70 16 = 1 3
(P)Reheating ρ 2 R = 3H (ρ R + P R)+Γ σ ρ σ = 3H (ρ σ + P σ ) Γ σ 2 ÿ +[1+ q (t)sin2t] y =0
Multiple Scale Analysis (MSA) H(t) Γ σ + 1 dv (σ) dσ 4 V (σ) σ 2 σ2 σ + σ σ 3 2γ [2 V (σ)+() σ2 ]=0 V (σ) σ = σ 0 + δσ δσ + ω0δσ 2 + n 4m2 /σ 0 δσ 2(1 + 6γ) δσ2 2 2 σ 0 δσ + 3(1 + 6γ) σ 0 2γ O δσ 2, O δ σ 2 ω 2 0 δσ2 + δσ 2 =0 V (σ) m2 2 δσ2 + O δσ 3 dv (σ) m 2 δσ + n dσ 2 δσ2 + O δσ 3 ω 0 = m 2 LG 10 13 GeV
Multiple Scale Analysis (MSA) H(t) Γ σ + 1 dv (σ) dσ 4 V (σ) σ 2 σ2 σ + σ σ 3 2γ [2 V (σ)+() σ2 ]=0 V (σ) σ = σ 0 + δσ δσ + ω0δσ 2 + n 4m2 /σ 0 δσ 2(1 + 6γ) δσ2 2 2 fast pat σ 0 δσ + 3(1 + 6γ) σ 0 2γ O δσ 2, O δ σ 2 ω 2 0 δσ2 + δσ 2 =0 V (σ) m2 2 δσ2 + O δσ 3 dv (σ) m 2 δσ + n dσ 2 δσ2 + O δσ 3 ω 0 = m 2 LG 10 13 GeV
Multiple Scale Analysis (MSA) H(t) Γ σ + 1 dv (σ) dσ 4 V (σ) σ 2 σ2 σ + σ σ 3 2γ [2 V (σ)+() σ2 ]=0 V (σ) σ = σ 0 + δσ δσ + ω0δσ 2 + n 4m2 /σ 0 δσ 2(1 + 6γ) δσ2 2 2 fast pat σ 0 δσ + σ 0 3(1 + 6γ) 2γ slow pat O δσ 2, O δ σ 2 ω 2 0 δσ2 + δσ 2 =0 V (σ) m2 2 δσ2 + O δσ 3 dv (σ) m 2 δσ + n dσ 2 δσ2 + O δσ 3 ω 0 = m 2 LG 10 13 GeV
Multiple Scale Analysis (MSA) δσ(t) A (t)e i ω 0 t + c.c. A (t) e ±i ω 0 t δσ = σ 0 2 f 1+ ω 0 t cos (ω 0 t + θ 0 ) t+t/2 f, θ 0 2γ 3() A(t) = 1 T t T/2 A(t )dt, T =2π/ω 0 H(t) 2/(3t) Ḣ + 32 H2 ρ σ =3γσ 2 0H 2 ρ R, P σ = 2γσ 2 0 ρ R P R w P σ ρ σ = 3γ 1+9γ
Reheating ρ σ + 1 dv (σ) dσ 4 V (σ) σ R = 3H (ρ R + P R )+Γ σ 2 2 σ2 σ + σ σ σ = σ 0 + δσ 3 2γ [2 V (σ)+() σ2 )+2ρ R ]+ Γ ρ R V (σ) δσ 2, Γ H δσ/σ 0 σ =0 δσ + ω 2 0δσ + n 4m2 /σ 0 δσ 2(1 + 6γ) δσ2 2 2 σ 0 + Γ δσ δσ + σ 0 3(1 + 6γ) 2γ ω 2 0 δσ2 + δσ 2 + 2 ρ R =0 ρ δσ(t) A(t) cos (ω 0 t + θ 0 ) 2 A + Γ A +3 R Γ A ω 2 0 +4 ρ R +() A 2 ω 2 0 3γσ 2 0 ρ R +() A 2 ω 2 0 3γσ 2 0 A =0 ρ R =0 H = ρ R +() A 2 ω 2 0 3γσ 2 0 ρ σ = A 2 ω 2 0 (1 + 9γ)
Reheating ρ σ + 1 dv (σ) dσ 4 V (σ) σ R = 3H (ρ R + P R )+Γ σ 2 2 σ2 σ + σ σ σ = σ 0 + δσ 3 2γ [2 V (σ)+() σ2 )+2ρ R ]+ Γ ρ R V (σ) δσ 2, Γ H δσ/σ 0 σ =0 δσ + ω 2 0δσ + n 4m2 /σ 0 fast pat δσ 2(1 + 6γ) δσ2 2 2 σ 0 + Γ δσ δσ + σ 0 3(1 + 6γ) 2γ ω 2 0 δσ2 + δσ 2 + 2 ρ R =0 ρ δσ(t) A(t) cos (ω 0 t + θ 0 ) 2 A + Γ A +3 R Γ A ω 2 0 +4 ρ R +() A 2 ω 2 0 3γσ 2 0 ρ R +() A 2 ω 2 0 3γσ 2 0 A =0 ρ R =0 H = ρ R +() A 2 ω 2 0 3γσ 2 0 ρ σ = A 2 ω 2 0 (1 + 9γ)
Reheating ρ σ + 1 dv (σ) dσ 4 V (σ) σ R = 3H (ρ R + P R )+Γ σ 2 2 σ2 σ + σ σ σ = σ 0 + δσ 3 2γ [2 V (σ)+() σ2 )+2ρ R ]+ Γ ρ R V (σ) δσ 2, Γ H δσ/σ 0 σ =0 δσ + ω 2 0δσ + n 4m2 /σ 0 fast pat δσ 2(1 + 6γ) δσ2 2 2 σ 0 + Γ δσ + δσ 3(1 + 6γ) σ 0 2γ slow pat ω 2 0 δσ2 + δσ 2 + 2 ρ R =0 ρ δσ(t) A(t) cos (ω 0 t + θ 0 ) 2 A + Γ A +3 R Γ A ω 2 0 +4 ρ R +() A 2 ω 2 0 3γσ 2 0 ρ R +() A 2 ω 2 0 3γσ 2 0 A =0 ρ R =0 H = ρ R +() A 2 ω 2 0 3γσ 2 0 ρ σ = A 2 ω 2 0 (1 + 9γ)
Reheating Tempeatue ρ R = 4Hρ R + Γ 1+9γ ρ σ dρ σ = 3Hρ σ Γ dt ρ σ ρ R + 1+9γ H = ρ σ 3γσ0 2 ρ R ρ σ ρ R Γ 1 ρ R ρ σ t Γ H 0 ρ R (t) Γ2 γσ0 2 20 (1 + 6γ) 2 3e e Γ t 2() +5 Γ t 2() 1
Reheating Tempeatue ρ R = 4Hρ R + Γ 1+9γ ρ σ dρ σ = 3Hρ σ Γ dt ρ Deviations σ fom GR ρ R + 1+9γ H = ρ σ 3γσ0 2 Recove GR Equations Γ = Γ ρ σ 1+9γ ρ σ ρ R ρ σ ρ R Γ 1 ρ R ρ σ t Γ H 0 ρ R (t) Γ2 γσ0 2 20 (1 + 6γ) 2 3e e Γ t 2() +5 Γ t 2() 1
Reheating Tempeatue ρ R = 4Hρ R + Γ 1+9γ ρ σ dρ σ = 3Hρ σ Γ dt ρ Deviations σ fom GR ρ R + 1+9γ H = ρ σ 3γσ0 2 Recove GR Equations Γ = Γ ρ σ 1+9γ ρ σ ρ R ρ σ ρ R Γ 1 ρ R ρ σ t Γ H 0 ρ R (t) γσ0 20 (1 + 6γ) 2 3e ) 3 Γ2 2 M P (1 + 6γ) 2 e Γ t 2() +5 Γ t 2() 1
Reheating Tempeatue ρ R = 4Hρ R + Γ 1+9γ ρ σ dρ σ = 3Hρ σ Γ dt ρ Deviations σ fom GR ρ R + 1+9γ H = ρ σ 3γσ0 2 Recove GR Equations Γ = Γ ρ σ 1+9γ ρ σ ρ R ρ σ ρ R Γ 1 ρ R ρ σ t Γ H 0 3 ρ R (t) γσ0 20 (1 + 6γ) 2 3e ) 3 Γ2 2 Γ M MP P (1 + 6γ) 2 e T (IG) eh Γ t 2() +5 Γ t 2() 1
Reheating Tempeatue ρ R = 4Hρ R + Γ 1+9γ ρ σ dρ σ = 3Hρ σ Γ dt ρ Deviations σ fom GR ρ R + 1+9γ H = ρ σ 3γσ0 2 Recove GR Equations Γ = Γ ρ σ 1+9γ ρ σ 1.0 ρ R ρ σ 0.8 ρ R Γ 1 ρ R ρ σ t Γ H 0 TIGTEG 0.6 Analytic esult Numeical esult Γ t 2() +5 ρ R (t) γσ0 20 (1 + 6γ) 2 3e ) 3 Γ2 2 M P Γ t 2 e 2() (1 + 6γ) 1 T (IG) eh 3 Γ MP 6 5 4 3 2 1 0 1 Log 10 Γ
Pe-heating q/4 ÿ(t) +[A(t) +2q(t) sin2t] y(t) =0 Stability bands A/4 Instability bands: lage fo lage q (BROAD/STOCHASTIC RESONANCE) 1 q q2 8 O(q3 ) < A < 1+q q2 8 + O(q3 )
MSA & Pe-heating ÿ +[1+ q (t)sin2t] y =0 y(t) A (t) e it +c.c. q(t) = p t n 4 drea dt 4 dima dt q(t)rea =0 + q(t)ima =0 n>1= Re A t Re A (t 0 )exp Re A (t) =ReA (t 0 ) exp Decaying pat n<1= Re A t p Re A (t 0 )exp 4(1 n) t1 n p/4 t n =1= Re A =ReA (t 0 ) t 0 p 4(n 1) t1 n 0 t t 0 Exponential behavio Powe law behavio Constant behavio p 4 t n d t
MSA & Pe-heating ÿ +[1+ q (t)sin2t] y =0 fast pat y(t) A (t) e it +c.c. q(t) = p t n 4 drea dt 4 dima dt q(t)rea =0 + q(t)ima =0 n>1= Re A t Re A (t 0 )exp Re A (t) =ReA (t 0 ) exp Decaying pat n<1= Re A t p Re A (t 0 )exp 4(1 n) t1 n p/4 t n =1= Re A =ReA (t 0 ) t 0 p 4(n 1) t1 n 0 t t 0 Exponential behavio Powe law behavio Constant behavio p 4 t n d t
MSA & Pe-heating ÿ +[1+ q (t)sin2t] y =0 fast pat slow pat y(t) A (t) e it +c.c. q(t) = p t n 4 drea dt 4 dima dt q(t)rea =0 + q(t)ima =0 n>1= Re A t Re A (t 0 )exp Re A (t) =ReA (t 0 ) exp Decaying pat n<1= Re A t p Re A (t 0 )exp 4(1 n) t1 n p/4 t n =1= Re A =ReA (t 0 ) t 0 p 4(n 1) t1 n 0 t t 0 Exponential behavio Powe law behavio Constant behavio p 4 t n d t
Scala Petubations δσ k = a 3 (1 + δ 1 ) 2 δσ k d 2 δσ k d(ω 0 t) 2 +[A σ +2q σ 1 sin (2ω 0 t)+2q σ 2 sin (ω 0 t)] δσ k =0 A σ = k2 a 2 ω 2 0 +1, q σ 1 = 2 ω 0 t, q σ 2 = 27γ 2() ω 0 t R k = Ḣ σ δσ k t 1 t 1 δσ k = t 1 a3/2 t 1 t t 2/3 3/2 = const
Scala Petubations δσ k = a 3 (1 + δ 1 ) 2 δσ k d 2 δσ k d(ω 0 t) 2 +[A σ +2q σ 1 sin (2ω 0 t)+2q σ 2 sin (ω 0 t)] δσ k =0 A σ = k2 a 2 ω 2 0 +1, q σ 1 = 2 ω 0 t, q σ 2 = 27γ 2() ω 0 t p =4, n =1 = δσ k t R k = Ḣ σ δσ k t 1 t 1 δσ k = t 1 a3/2 t 1 t t 2/3 3/2 = const
Tenso Petubations h k a 3/2 σ h k d 2 hk d(ω 0 t/2) 2 +[A h +2q h sin (ω 0 t)] h k =0 A h (t) = 4k2 a 2 ω 2 0, q h = f 2ω 0 t A h (t) 1 = k a ω 0 2
Scala Field S χ = dx 4 g gµν 2 µχ ν χ m2 χ 2 χ2 ξ 2 Rχ2 + g2 2 σ2 χ 2 ξ > 0 χ k = a 3/2 χ d 2 χ k k dt 2 + m eff,χ (t) 2 χ k =0 m eff,χ (t) 2 = ω 2 0 k 2 a 2 ω 2 0 + 3 (4ξ 1) f 1+ ω 0 t sin ω 0 t +2g 2 M P 2 γω 2 0 f 1+ ω 0 t sin ω 0t + m2 χ ω 2 0 + g2 2 M P 2 γω 2 0 + O 1 t 2 m 2 χ + g 2 M P 2 2γ ω2 0 4 = m 2 4 (1 + 6γ) m χ g 0 ξ
Conclusions γ γ 1 γ 1 V σ 4