Multi-field inflationary trajectories with a fast-turn feature

Transcription

1 Multi-field inflationary trajectories with a fast-turn feature

2 H const Zero spa)al curvature is an a1rac)ve fixed point of the evolu)on equa)ons Comoving scales con)nuously leave the causally connected patch All par)cle densi)es quickly washed out Introduc)on H a n,n>1 Zero spa)al curvature is an repulsive fixed point of the evolu)on equa)ons Comoving scales con)nuously enter the causally connected patch Reasons to contemplate infla)on; there is an extra bonus: genera)on of cosmological perturba)ons.

3 H const Zero spa)al curvature is an a1rac)ve fixed point of the evolu)on equa)ons Comoving scales con)nuously leave the causally connected patch All par)cle densi)es quickly washed out Introduc)on H a n,n>1 Zero spa)al curvature is an repulsive fixed point of the evolu)on equa)ons Comoving scales con)nuously enter the causally connected patch Got it! Will check Reasons to contemplate infla)on; there is an extra bonus: genera)on of cosmological perturba)ons.

4 H const Zero spa)al curvature is an a1rac)ve fixed point of the evolu)on equa)ons Comoving scales con)nuously leave the causally connected patch All par)cle densi)es quickly washed out Introduc)on H a n,n>1 Zero spa)al curvature is an repulsive fixed point of the evolu)on equa)ons Comoving scales con)nuously enter the causally connected patch Got it! valida)on interpreta)on Will check Reasons to contemplate infla)on; there is an extra bonus: genera)on of cosmological perturba)ons.

5 V (σ) 1. Quasi- de- Si1er solu)on A scalar field slowly rolls down a poten)al Ḣ H 2 1 P R = V 24π 2 M 4 P σ 1 2M 2 P η 1 M 2 P Vσ V σσ V V 2

6 VΣ 3 Σ M 2 P 2 Vσ V P Φ P Φ k 1 final P R = 3 1 V 24π 2 MP 4 = Nlogaa 0. k e k Τ V 3 12π 2 M 2 P V 2 σ log( kτ) Assume P R = A (k/k 0 ) n s 1 n s 1= d log P R d log k = 6 +2η

7 V Future? V 1/ GeV 1/4 V P R = 24π 2 MP 4 n s =1 6 +2η φ infl Is this interpreta)on SAFE? φ r P T P R = 16 α s f NL g NL

8 Mul)- field infla)on On super- Hubble scales, along a turn in the infla)onary trajectory, there is a coupling between the adiaba)c and entropy modes and the la0er can source the former. Curiously, some)mes ignored; see eg. account in Avgous)dis, Cremonini, Davis, Ribeiro, KT & Watson 11 Hubble radius crossing Gordon et al, 2000 instantaneous entropy (isocurvature) perturba)on instantaneous adiaba)c (curvature) perturba)on

9 Mul)- field infla)on Hubble radius crossing e.g. Groot Nibbelink & Van Tent 2000, 2001, DiMarco & Finelli, 2005, Tolley & Wyman 2009, Achucarro et al. 2010, Cremonini, Lalak, KT 2010, 2011 On super- Hubble scales, when the infla)onary trajectory turns away from geodesic lines in the field space, there is a coupling between the adiaba)c and entropy modes and the la0er can source the former. instantaneous entropy (isocurvature) perturba)on instantaneous adiaba)c (curvature) perturba)on

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11 Violent events in the past?

12 Violent events in the past? θ Hubble radius crossing The largest mode we can see

13 Violent events in the past? P R /P 0 =1+2 θ sin(2k/k 0 ) Shiu & Xu, 2011 P R /P 0 =1+e H2 t 2 θ 2 Gao, Langlois & Mizuno, 2012 P R /P 0 =1 EOM s unaffected for k ph θ

14 v σ + Moderately fast turns EOM s neglec)ng terms suppressed by the slow- roll parameter = Ḣ/H2 k 2 + µ2 σ ρ 2 2 v s + η 2 k 2 + µ2 s ρ 2 2 η 2 v σ + 2ρ η v s 2ρ η 2 v s =0 v s 2ρ η v σ 2ρ η 2 v σ =0 (v σ = aq σ,v s = a δs, ρ = θ/h)

15 v σ + Moderately fast turns EOM s neglec)ng terms suppressed by the slow- roll parameter = Ḣ/H2 k 2 + µ2 σ ρ 2 2 v s + vσ v s η 2 k 2 + µ2 s ρ 2 2 η 2 v σ + 2ρ η v s 2ρ η 2 v s =0 v s 2ρ η v σ 2ρ η 2 v σ =0 (v σ = aq σ,v s = a δs, ρ = θ/h) have the following solu)ons for = e ıkη cos θ sin θ k ρ µ 2 σ/k, µ 2 s/k,e ıkη sin θ cos θ

16 Moderately fast turns χ t =0.0002H 1 θ = π/4 analysis of the model by Gao, Langlois & Mizuno, 2012 φ

17 Moderately fast turns k 2 P R P k No enhancement with respect to single- field result at large k θ = π/4 t =0.0002H 1 analysis of the model by Gao, Langlois & Mizuno, P 0 = H2 8π 2 M 2 p k k 0

18 P R P 0,P S P 0 Moderately fast turns P R cos 2 θ sin 2 θ primordial Hubble radius crossing P R total θ = π/4 t =0.0002H 1 analysis of the model by Gao, Langlois & Mizuno, P R induced k 0 Η

19 v σ + k 2 + µ2 σ ρ 2 2 Very fast turns EOM s neglec)ng terms suppressed by the slow- roll parameter = Ḣ/H2 v s + η 2 k 2 + µ2 s ρ 2 2 η 2 v σ + Assume ρ/η = θδ(η η 0 ) (Shiu & Xu 2001) No effect on P R, either 2ρ η v s 2ρ η 2 v s =0 v s 2ρ η v σ 2ρ η 2 v σ =0 ρ = θ/h

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21 Quasi- single field infla)on Χ Φ geodesic lines infla)onary trajectory weak coupling Chen & Wang, 2009 θ/h

22 Quasi- single field infla)on

23 Quasi- single field infla)on Χ Χ Φ geodesic lines infla)onary trajectory Φ b(φ) φ weak coupling Chen & Wang, 2009 θ/h possibility of a strong coupling 1 M 2 P b 2 e.g. roule1e infla)on, Bond et al applica)on with a weak coupling Cremonini, Lalak & KT, 2010

24 Quasi- single field infla)on Χ Toy model Cremonini, Lalak & KT, 2011 b(φ) = φ/m, M MP V (φ, χ) = V0 1+α φ φ0 MP 2 χ +β MP small, slow- roll = 2β 2 Φ b(φ) φ 2 2 large, m H or ηss = m /3H 1 possibility of a strong coupling 1 MP2 b 2 e.g. roule1e infla)on, Bond et al. 2006

25 Quasi- single field infla)on Χ Toy model Cremonini, Lalak & KT, 2011 Homogeneous EOMs φ + 3H φ + Vφ b e2b χ 2 = 0, Φ χ + 3H χ + 2b φ χ + e 2b Vχ = 0 b(φ) φ possibility of a strong coupling 1 MP2 b 2 e.g. roule1e infla)on, Bond et al. 2006

26 Quasi- single field infla)on Χ Toy model parameters ξ, m /H Cremonini, Lalak & KT, 2011 Homogeneous EOMs φ + 3H φ + Vφ b e2b χ 2 = 0, Φ χ + 3H χ + 2b φ χ + e 2b Vχ = 0 EOMs for perturba)ons d k 2 τ2 dτ ξ τ 2ξ τ 0 d + dτ solve semi- analy)cally (alas, no )me) solve numerically Lalak, Langlois, Pokorski & KT, τ2ξ2 τ4ξ2 2 1 m τ 2 H2 b(φ) φ ucur uiso possibility of a strong coupling 1 MP2 b 2 1 =0 e.g. roule1e infla)on, Bond et al MP b ξ 2

27 Quasi- single field infla)on Later development For k τ < ξ and m ξh curvature perturba)ons described by an effec)ve theory with a single degree of freedom and a modified dispersion rela)on. ω 2 = m2 4ξ 2 H 2 k2 ph + 1 4ξ 2 H 2 k4 ph Baumann & Green, 2011; first term: Tolley & Wyman, 2009, Achucarro et al, 2010 see also: Ashoorioon et al Χ parameters ξ, m /H Φ b(φ) φ possibility of a strong coupling 1 M 2 P b 2 e.g. roule1e infla)on, Bond et al M 2 P b 2 ξ 2

28 k τ = ξ ξ = 300, m 2 = 5000H A C B single- field infla)on, c s <1 Tolley & Wyman, k τ = m /H P R P sf k τ =1/c s Cremonini, Lalak & KT, curvature perturba)ons single- field infla)on, c s =1 N 1

29 ξ = 300, m A C k τ = ξ P R P sf k τ = ξ 10 2 curvature perturba)ons 10 2 Cremonini, Lalak & KT, single- field infla)on, c s = N π σ model of Baumann & Green,

30 Summary Several examples show that the effects of sub- horizon fast turns decouple as k 2 (at least). Some new things possible with non- canonical kine)c terms Cremonini, Lalak, KT 2010, 2011 Avgous)dis, Cremonini, Davis, Ribeiro, KT, Watson 2012 Work in progress

31 Just one backup slide

32 10 4 m 2 gel/h 2 = Temporary instability at the Hubble radius crossing Enhancement of the curvature perturba)ons by many orders of magnitude P R, P S , Cremonini, Lalak, KT Ξ N