Axion Inflation: Naturally thermal Ricardo Zambujal Ferreira Institut de Ci´ encies del Cosmos, Universitat de Barcelona In collaboration with Alessio Notari (JCAP 1709 (2017) no.09, 007, 1711.07483) Ricardo Zambujal Ferreira Axion Inflation: Naturally thermal
Axions in inflation • Appealing way of realizing inflation; mass is protected by the (discrete) shift symmetry. E.g.: Natural Inflation [Freese, Frieman and Olinto ’90] L φ = ( ∂ µ φ ) 2 + Λ 4 (1 + cos( φ/f )) • Axions ( φ ) are expected to couple to gauge fields through an axial coupling F µν ≡ ǫ µναβ φ f F µν ˜ F µν , ˜ √− g F αβ where f is the axion decay constant. Moreover, the universe has to reheat. This coupling is an efficient and safe way to do it. • When φ develops a VEV, parity is broken and the eom for the massless gauge field ( A ± ) during inflation becomes [Tkachev 86’, Anber&Sorbo 06’] ˙ � k 2 ± 2 kξ � φ A ′′ ± ( τ, k ) + A ± ( τ, k ) = 0 , ξ = τ 2 fH Ricardo Zambujal Ferreira Axion Inflation: Naturally thermal
• Instability band: (8 ξ ) − 1 H < k/a < 2 ξH . If ξ ≃ constant: [Anber and Sorbo 06’] 1 e − ikτ , A k ( τ ) ≃ √ subhorizon 2 k 1 2 √ πkξ e πξ , A k ( τ ) ≃ superhorizon • Phenomenology: • Large loop corrections to ζ induced through the coupling 2 ξζF ˜ F P ζ O (10 − 4 ) P 2 obs e 4 πξ 2-point function : = 1-loop f equi O (10 − 7 ) P obs e 6 πξ non-Gaussianity: NL | 1-loop = • Large tensor modes, flatenning of the potential by backreaction, preheating, ... • Non-Gaussianity constraints ξ � 2 . 5 ( ξ < 2 . 2 ) on CMB scales which imposes a lower bound on f and excludes some of the interesting phenomenology. [Anber&Sorbo 09’, Sorbo 11’, Barnaby&Peloso 11’, Linde et al. 13’, Bartolo et al. 14’, Mukohyama et al. 14’, RZF &Sloth 14’, Adshead et al. 15’, Planck 15’, RZF et. al 15’, ...] Ricardo Zambujal Ferreira Axion Inflation: Naturally thermal
Particle production and thermalization [ RZF &Notari 1706.00373] • But what happens when ξ ≫ 1 ? • Instability band covers subhorizon modes where particle interpretation is meaningfull. Instability ⇒ particle production of modes • Gauge field effective particle number ( N γ ) per mode k : � A ′ 2 k + k 2 A 2 1 2 + N γ ( k ) = ρ γ ( k ) N γ ( k ) ≃ 0 , k/a ≫ H � k = ⇒ N γ ( k ) ≃ e 2 πξ k 2 k 8 πξ , k/a ≪ H pol What happens when there are many particles around...? Ricardo Zambujal Ferreira Axion Inflation: Naturally thermal
If gauge field is Abelian (e.g. photons) interactions are For example, the scatering rate of γγ → γγ 4 d 3 p i � � 1 � | M n | 2 × � S γγ → γγ = (2 π ) 3 (2 E i ) E 1 i =2 × B γγ → γγ ( k, p 2 , p 3 , p 4 )(2 π ) 4 δ (4) ( k µ + p µ 2 − p µ 3 − p µ 4 ) where B γγ → γγ ( p 1 , p 2 , p 3 , p 4 ) contains the phase space factors given by B γγ → γγ ( p 1 , p 2 , p 3 , p 4 ) = N γ ( p 1 ) N γ ( p 2 ) [1 + N γ ( p 3 )] [1 + N γ ( p 4 )] − ( p 1 ↔ p 3 , p 2 ↔ p 4 ) . which is ∝ N 3 γ . Ricardo Zambujal Ferreira Axion Inflation: Naturally thermal
• All the scatterings are enhanced by powers of N γ . Therefore, when N γ reaches a given threshold t scatterings, decays ≪ H − 1 ⇒ thermalization • To estimate the conditions for thermalization we derive, from the eom, Boltzmann-like eqs. for N γ + ( k ) , N γ − ( k ) and N φ ( k ) : − 4 kξ Re [ g ( k, τ )] N ′ � � γ + ( k, τ ) = N γ + ( k, τ ) + 1 / 2 | g ( k, τ ) | 2 + k 2 τ N ′ N ′ γ − ( k, τ ) ≃ φ ( k, τ ) ≃ 0 where g ( k, τ ) = A ′ ( k, τ ) /A ( k, τ ) . Then, add the scatterings and decays + S ++ + S + φ + D + φ + S + − N ′ γ + ( k ) = − 4 kξ Re [ g A ( k,τ )] � � N γ + ( k ) + 1 / 2 , τ | g A ( k,τ ) | 2 + k 2 γ − ( k ) = − S + − , φ ( k ) = − S + φ − D + φ N ′ N ′ , Ricardo Zambujal Ferreira Axion Inflation: Naturally thermal
• Numerically we verify that and verify that the distribution of particles approaches a Bose-Einstein distribution when � f � ξ � 0 . 44 log + 3 . 4 = ⇒ ξ � 5 . 8 H Observational Constraint ( P obs ) ζ In the backreacting and non-perturbative regime ⇒ unclear. • But if gauge fields belong to the SM thermalization is much more efficient • Many ( γψ , gg ), fixed and unsuppressed interactions (more predictive). More realistic, inflaton has to couple to SM. • For example, γψ scatterings or gluon self-interactions thermalization requires � 2 � πα EM � 2 � HN 2 H γγ → e − e + ≫ N γγ → e − e + H ⇒ ξ � 2 . 9 2 k ∗ � 9 πα S � 2 � 2 � H HN 3 gg → gg ≫ N gg → gg H ⇒ ξ � 2 . 9 32 k ∗ Under control ! Ricardo Zambujal Ferreira Axion Inflation: Naturally thermal
What happens next? • After thermalization gauge field develops a thermal mass m T = ¯ g T + m 2 � � k 2 ± 2 kξ A ′′ ± + ω 2 ω 2 T T ( k ) A ± = 0 , T ( k ) = . H 2 τ 2 τ • If m T > ξH the instability disappears and thermal bath redshifts. However, if T � H the thermal mass disappears and the instability restarts • Therefore, the system should reach an equilibrium (or oscillate around it) which balances the two terms: T eq ≃ ξH ω 2 T ( k ) � 0 ⇒ ¯ g The equilibrium temperature is linear in ξ and thus all predictions are changed! Ricardo Zambujal Ferreira Axion Inflation: Naturally thermal
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