Fast Turn Feature in Mul.- Field Infla.on. Krzysztof Turzyński Faculty of Physics, Warsaw University

Transcription

1 Fast Turn Feature in Mul.- Field Infla.on Krzysztof Turzyński Faculty of Physics, Warsaw University

2 Introduc.on String theory Just 2σ, please! interpreta.on Infla%on Bwahaha! valida.on Par%cle physics

3 VΣ 3 Σ M 2 P a la Mukhanov 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) Parametrize P R = A (k/k 0 ) n s 1 n s 1= d log P R d log k = 6 +2η

4 φ Observa.onal results infla.on ini.al condi.ons standard cosmo evolu.on CMB observa.ons for the perturba.ons P R = V n s =1 6 +2η 24π 2 MP 4 r P T = 16 P R V infla.on parameters of the infla.onary poten.al V 1/ GeV 1/4 Aspects of mul.- field infla.on that don t/do change this picture Planck 15.5mo φ infl

5 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 la:er 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

6 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 la:er can source the former. instantaneous entropy (isocurvature) perturba.on instantaneous adiaba.c (curvature) perturba.on

7 Violent events in the past?

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

9 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

10 Moderately fast turns v σ + 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)

11 Moderately fast turns v σ + 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

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

13 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

14 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

15 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 2011) 1. Remember about sta.s.cal independence of v σ,v s 2. Only 1st equa.on contains ρ 3. Worry about terms ρ 2 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

16 Perform a field redefini.on vσ v s = Very fast turns cos θ sin θ sin θ cos θ u1 Groot Nibbelink & van Tent 2001, see also Lalak, Langlois, Pokorski, KT 2007 u 2 > removes ρ 2 terms > removes kine.c mixing u 1 + θ sin 2θ u 1 θ cos 2θ u =0 u 2 θ cos 2θ u 1 θ sin 2θ u =0

17 Perform a field redefini.on vσ v s = Very fast turns cos θ sin θ sin θ cos θ u1 Groot Nibbelink & van Tent 2001, see also Lalak, Langlois, Pokorski, KT 2007 u 2 > removes ρ 2 terms > removes kine.c mixing u 1 + θ sin 2θ u 1 θ cos 2θ u =0 u 2 θ cos 2θ u 1 θ sin 2θ u =0 Finite jump in u 1,u 2 for vσ v s cos θ sin θ, sin θ cos θ No effect on P R, either

18 Very fast turns: par.cle produc.on χ M m M m θ φ Non- adiaba.c change of direc.on for: H t 1 Par%cle produc%on! Side remark: Par.cle produc.on with non- adiaba.c change of the metric in the field space explored in Avgous.dis, Cremonini, Davis, Ribeiro, KT, Watson 2012

19 Very fast turns: par.cle produc.on with M. Konieczka & R.H. Ribeiro, in prep. a n = M m χ M m θ θ = θδ(t t 0 ) I = 1 2 Number density J = 1 2 of produced heavy par.cles (M m) a = αa + βa Bogolyubov transforma.on α = ıωα ωω 1 β Iα Jβ β = ıωβ ωω 1 α Iβ Jα cos θ sin θ C = sin θ cos θ φ CM 2 ωc T Ċ C T ω 1 + diagonal ω 1 C T Ċ ω ωc T Ċ ω 1 ω 1 C T Ċ ω Nilles, Peloso & Sorbo, 2001 d 3 k (2π) 3 N(k) 1 24π 2 M 3 sin 2 θ

20 Very fast turns: par.cle produc.on working hypothesis or wishful thinking χ * Just aaer the turn without par.cle produc.on: damped oscilla.ons in the direc.on propor.onal to the slow- roll with par.cle produc.on: * kine.c energy of the oscilla.ons radiated away as par.cles φ hence smoothing of the trajectory and even less features in the power spectrum

21 It is conceivable that a simple pa:ern of predic%ons consistent with observa%ons result from intricate workings of the fundamental theory: fast turns decouple Ø Several examples show that the effects of sub- horizon fast turns decouple as k 2 (at least). Ø Par.cle produc.on makes turning infla.onary trajectories smoother, which suppresses observable features.

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

24 Quasi- single field infla.on

25 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. roulele infla.on, Bond et al applica.on with a weak coupling Cremonini, Lalak & KT, 2010

26 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. roulele infla.on, Bond et al. 2006

27 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. roulele infla.on, Bond et al. 2006

28 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 n solve semi- analy.cally (alas, no.me) n 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. roulele infla.on, Bond et al MP b ξ 2

29 Quasi- single field infla.on 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. roulele infla.on, Bond et al M 2 P b 2 ξ 2

30 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

31 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,

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

33 It is conceivable that a simple pa:ern of predic%ons consistent with observa%ons result from intricate workings of the fundamental theory: fast turns decouple Ø Several examples show that the effects of sub- horizon fast turns decouple as k 2 (at least). Ø Par.cle produc.on makes turning infla.onary trajectories smoother, which suppresses observable features. Summary or resemble minimal models Ø Constant- rate fast- turn infla.on