Warm inflation after Planck

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1 Warm inflation after Planck Cold inflation/warm inflation Dissipative coefficient: Low T regime Primordial spectrum: λφ 4 Work done in collab. With: A. Berera, R. Ramos, J. Rosa, R. Cerezo, G. Vicente Mar Bastero Gil University of Granada

2 [ Planck collab.: astro ph/ ]

3 Slow Roll Inflation Scalar field rolling down its (flat) potential P= ϕ 2 / 2 V(ϕ) V(ϕ) negative pressure Flat potential The curvature and the slope smaller than the (Hubble) expansion rate H Kinetic energy << potential energy H 2 ~ V/3m P 2 Slow roll parameters 2 =m V ' ' P curvature V 1 = m 2 P 2 V ' 2 V 1 slope Inflation Reheating Radiation Matter Inflaton interacts with other particles

4 Cold inflation Interactions negligible during Inflation Warm inflation Reheating Radiation Inflaton decay into light d. of f. Dissipation Radiation A (small) fraction of the vacuum energy is converted into radiation during inflation = R, R ϕ+(3h+υ) ϕ+v ϕ =0 ρ R +4 Hρ R =Υ ϕ 2 Source term Decay into light dof= extra friction ρ R 1/4 <H ρ R 1/4 >H Cold inflation Warm inflation Moss PLB154, 85; Yokoyama & Maeda PLB207 '88; Berera PRL75 '95

5 Extra friction term: Q<<1, T<<H Q=Υ/(3H) Standard Cold Inflation ( fluctuations: δ ϕ H ) 2 π Q< 1, T > H Weak Dissipative Regime Standard slow roll Q>1, T > H 3 H V, 4 H r 2 Strong Dissipative Regime Thermal fluctuations: 1/2 T 1/2 δ ϕ k F 2 π k F H (1+Q) Slow roll : 3 H(1+Q) ϕ V ϕ (ϕ, T), 4 Hρ r Υ ϕ 2 η ϕ <(1+Q), ϵ ϕ <(1+Q), β Υ <(1+Q), δ T <1 β Υ =m P 2 (Υ ϕ V ϕ )/(Υ V) Thermal corrections: T =T V T / V

6 Dissipative coefficient heavy m χ = gφ > H, T light fermions L = 1 2 m 2 2 g h Inflaton moves down the potential, it excites χ (massive field), which decays into light dof (thermal bath) φ y χ d 4 x 1 x x 1 t 1 V =0 self energy Slowly varying: t 1 = t 1 t t... V =0 = d 4 x x t Berera & Ramos PRD 01, 03, 05; Hall & Moss PRD 05 Dissipative coefficient

7 Dissipative channel: generic in inflationary models ex: Chaotic sneutrino inflation W= M R 2 N N +h H L N +h H Q U +... R R N u R t u 3 3 sneutrino Higgs (s)top (light: M R < H) (heavy: M Hu >> H) (massless) Murayama et al. PRL 93 Ellis, Raidal, Yanagida PLB 04 Adiabatic approximation: Macroscopic ϕ/ϕ<γ χ h 2 m χ /(8 π) H<Γ χ Microscopic (low T regime: T >H, mχ>t) Γ χ ϕ/ϕ > Γ χ H >( Γ χ m χ )( m χ T )( T H )>1 Thermalization: nσ i v >H Moss & Graham 2008

8 Dissipative coefficient: Υ= 4 T ( g2 2 2 ) φ 2 d 4 p (2 π) ρ 2 4 χ n B (1+n B ) Spectral function: 4 p p, p 0 = p p p φ y χ Decay rate: = h 2 N Y 6 4 m 2 p F T p, p 0 Low T ( mχ/t > 1) Υ g 2 T 32 2 π g 2 N χ h 2 N Y ( m χ T ) 1/2 e m χ/t h 2 N Y N χ ( T 2 ) m χ MBG, Berera, Ramos, Rosa 2012 Pole: p 0 p Low momentum: p, p 0 m

9 Dissipative (low momentum) coefficient (Adiabatic, close to equilibrium approx.) T/H > 1 T 3 / 2 Low T, mχ/t > 10 T Intermediate/High T, mχ/t < 1 Low T: m χ > T g 2 T N h 4 2 T m 2 g 0 m T 1 Very high T, mχ/t << 1 Large thermal corrections, a few e folds... 4 (Berera, Gleiser, Ramos '98; Yokoyama & Linde '98) General: c= 1,0,1,3 T c Moss & Xiong hep ph/ ; MBG, Berera & Ramos, ;

10 Fluctuations & primordial spectrum Thermal fluctuations become the source for the adiabatic perturbations (inflaton) System Environment (light d. of f., integrated out ) Dissipation Υ transfer of energy fluctuation force ξ diss δ ϕ k +(3 H+Υ)δ ϕ k + ϕ δ Υ+( k 2 a +V 2 ϕ ϕ)δ ϕ k =ξ diss q k +ξ k Fluctuation Dissipation rel.: ξ diss (k)ξ diss (k ') =2 Υ T a 3 (2 π) 3 δ (3) (k k ')δ(t t ') Quantum noise: ξ q (k) ξ q (k ') =H 2 (2 n(k/ T)+1)a 3 (2 π) 3 δ (3) (k k ')δ(t t ') Berera & Fang PRL74 (1995); Taylor& Berera PRD62 (2000); Oliveira & Joras, PRD64 (2001); Hall, Moss &Berera PRD69 (2004) Ramos & da Silva,

11 Primordial spectrum P R ( Ḣ ϕ ) 2 ( H 2 π ) 2[ T H 2 π 3 Q 3+4 π Q +coth( H 2 T )] Primordial spectrum (WDR) T/H>1, Q<1: P R ( Ḣ ϕ ) 2 ( H 2 π ) 2 ( 2 T H ) (Thermal spectrum) Tensor to scalar ratio r 4 ϵ ϕ π(1+q) 5/ 2 ( H T ) 16 ϵ ϕ

12 Weak dissipation: λφ 4 potential Ne=50 CI Ne=60 g*=15/4 g*= T/H: 1, 10, 100 Low T regime: λ g 4 N χ 10 6

13 Υ T c Primordial spectrum: Q >> 1 c =3 c =1 c =0 c = 1 P R = h h T P R h r h T P R r P R r P R h = p = 2 h r Q h P R ~P R c =0 1 Q 5 c 1 2 (Graham & Moss 2009) (MBG, Berera, Ramos 2011) P R (c=0) ( Ḣ ϕ ) 2 ( H 2 π ) 2[ T H 2 π 3 Q 3+4 π Q +coth( H 2 T )]

14 Dissipation & Viscosity The radiation bath is expected to depart from a perfect fluid due to particle production T μ ν =ρu μ u ν +(P+π b )(u μ u ν +g μ ν )+π μ ν (u σ =fluid flow velocity) Adiabatic pressure p T 4, Bulk vis. π b 3 ζ b μ u μ Shear vis. π μ ν 2 ζ s σ μ ν Light fields: ζ b (T), ζ s (T) T 3 Jeon & Yaffe PRD53 ('96) Shear tends to damp the growth of the radiation fluctuations (before freeze out), and therefore that of the field F[ ζ s 1] 1 P R P R (c=0)f[q] F s [ ζ s ] ζ s = 4 9 ζ s H F[ ζ s >1] 0 ρ r +p r ζ s 1 No growing mode

15 Strong dissipation: λφ 4 potential Ne=60 g*=15/4 g*= ζ s 0 n s >1 Blue tilted spectrum! (dq/dne > 0) ζ s 1 avoids the growing mode, and makes the spectrum red tilted

16 Extra friction Υ(T,φ) (Q=Υ(T,φ)/(3H)): Summary Dissipative effects due to decaying fields can be relevant during inflation, and modify the inflationary predictions slow roll: < (1 + Q), < (1 + Q), < (1 + Q) spectrum: T/H>1, Q<1: P R ( Ḣ ϕ ) 2 ( H 2 π ) 2 ( 2 T H ) Chaotic models: smaller values of the inflationary scale are needed to fit the primordial spectrum smaller tensor to scalar ratio λφ 4 compatible with data!

17 Summary For a T dependent dissipative coefficient, the field and radiation perturbation EOM form a coupled system The radiation backreacts on the field perturbations, and the field fluctuations are amplified before freeze out. The larger the dissipation, the larger the growth 5 c 1 / 2 P R ~P R c =0 1 Q This can be counterbalanced by the dissipative effects within the radiation fluid (shear viscous pressure) P R ~P R c =0 when s 1 (freeze out value, model independent)

18 Field EOM: Shear viscosity: fluctuations & primordial spectrum G I G k 3 H I k G I k 2 a V G I 2 k 2 H T 1 /2 k Radiation: G I G r 4 H I r k 2 a G I 2 r 2 G I 2 G I G I r 3 H 1 k 2 a 2 H G I G 2 s r I r /3 G I Shear parameter s = 4 9 s H r p r G I = c 4 G I r 1 c G I r Coupled system

19 Primordial spectrum: no shear y k = k 3 / 2 G I 2 H T 1 /2 Q = / 3 H =1 0 0 c =3 r Q k 3 /2 G I w k = r 2 H T 1 /2 c =0 Growing mode! k 3 /2 G I u k = 2 H T 1 /2 freeze out Fluctuations coupled before freeze out: z c Q c z F 2 =9 Q/4 z 1 = a H/k c=0: y k 2 0 = Q 1 3 Q Q 1 /2 c>0: y k 2 c Q 5 c 1 / 2 Graham & Moss JCAP 07(2009); MBG, Berera & Ramos JCAP 07 (2011)

20 Q > 1 : too large amplitude spectral index: smaller inflationary scale r<<1 n s 1= d l np R d N e = n s 1 c =0 5 c 1 2 d l n Q /d N e n s 1? d Q /d N e 0 Blue tilted?

21 Primordial spectrum: shear effects Shear tends to damp the growth of the radiation fluctuations (before freeze out), and therefore that of the field G I r 3 H 1 k 2 a 2 H G I G 2 s r I r /3 G I s 1 y k 2 y k 2 c =0 Independent of the shear T dependence Susy models: 0. 1 g 4 N N 2 T 3 / 2 s 1 2 7N T 3 /g 4 s 1 g 4 6 9H / T s = s H / 3 r

22 Primordial spectrum: P R Ḣ 2 P R is constant after horizon crossing (freeze out) Cold inflation Warm inflation (Υ(φ,H)) Freeze out k F =k/a=h k F =k/a(t F )>H (extra friction) Field spectrum P d 3 k P k F H F k k F 2 3 Thermal spectrum n B k T F k 2 2 Primordial spectrum P P R C 1 Q 5 / 2 T R 1 H 2 H P R Q =0 2 m P 2 Tensor to scalar ratio r = P T P R =1 6 r = 4 1 Q 5 /2 H T 1 6 Non gaussianity f N L ~O, (Single, canonical field) f N L 1 5 l n 1 Q /1 4 Moss &Xiong JCAP 04(2007)

23 Gauge invariant Fluctuations at linear order Field G I = /H Energy denstiy r r G I = r r /H Momentum r r G I = r r p r /H Dissipation G I = /H Curvature Perturbation Primordial spectrum: P R = k 3 R = H T p T T Comoving curvature Perturbation 2 2 R k 2 (<...>: average over noise realizations )

24 (a) Monomial Potentials: V =V 0 /m P n (CI: large field models φ >m p, r~o(0.1), n S <1 ) P R 1/2 =5x10 5, T/H>1 Q 1 /3 1 Q 2 4 C /3 1 4 n / 3 m P / T /H Q /C 1 /3 n 2 /3 m P / C φ Weak DR increases Strong DR WMAP Warm φ 4 models compatible with data Inflation ends in the SDR, before Rad~V Smaller values of φ required T /H 1 C ~