Massive Neutrinos in Cosmology Part II: Cosmology Carlo Giunti - PowerPoint PPT Presentation

Massive Neutrinos in Cosmology Part II: Cosmology Carlo Giunti INFN, Torino, Italy giunti@to.infn.it Neutrino Unbound: http://www.nu.to.infn.it Torino Graduate School in Physics and Astrophysics Torino, May 2018 C. Giunti and C.W. Kim Fundamentals of Neutrino Physics and Astrophysics Oxford University Press 15 March 2007 – 728 pages C. Giunti − Massive Neutrinos in Cosmology – II − Torino PhD Course − May 2018 − 1/52

Thermodynamics of the Early Universe ◮ Thermal equilibrium: g χ � p ) d 3 p n χ = f χ ( � (2 π ) 3 g χ � p ) d 3 p ρ χ = E χ ( � p ) f χ ( � (2 π ) 3 p | 2 | � g χ � p ) d 3 p p χ = p ) f χ ( � (2 π ) 3 3 E χ ( � 1 ◮ Statistical distribution: f χ ( � p ) = p ) − µ χ ) / T χ ± 1 e ( E χ ( � ◮ Chemical potential: ◮ a + b ⇆ c + d = ⇒ µ a + µ b = µ c + µ d ◮ µ γ = 0 and χ + ¯ χ → γγ = ⇒ µ χ = − µ ¯ χ ◮ Conserved charge = ⇒ µ χ � = 0 if n χ � = n ¯ χ C. Giunti − Massive Neutrinos in Cosmology – II − Torino PhD Course − May 2018 − 2/52

1 ◮ Relativistic limit: T χ ≫ m χ and T χ ≫ µ χ = ⇒ f χ ( � p ) ≃ p | / T χ ± 1 e | �  ζ (3) π 2 g χ T 3 ( χ = boson)   χ n χ ≃ 3 ζ (3) π 2 g χ T 3 ( χ = fermion) ,   χ 4 π 2  30 g χ T 4 ( χ = boson)   χ ρ χ ≃ π 2 7 30 g χ T 4 ( χ = fermion) ,   χ 8 p χ ≃ 1 3 ρ χ , ◮ Average energy: π 4  30 ζ (3) T χ ≃ 2 . 701 T χ ( χ = boson)    � E χ � ≃ �| � p χ |� ≃ 7 π 4 180 ζ (3) T χ ≃ 3 . 151 T χ ( χ = fermion)    C. Giunti − Massive Neutrinos in Cosmology – II − Torino PhD Course − May 2018 − 3/52

Neutrino Decoupling CNB BBN CMB C. Giunti − Massive Neutrinos in Cosmology – II − Torino PhD Course − May 2018 − 4/52

◮ Neutrinos are in equilibrium in the early Universe through weak interactions: ( − ) ( − ) ( − ) ( − ) ν ⇆ e + e − ν ¯ ν e ⇆ ν e ν N ⇆ ν N ν e n ⇆ pe − ν e p ⇆ ne + n ⇆ pe − ¯ ¯ ν e ◮ Interaction rate: Γ ν = n ν � σ v � ∼ G 2 F T 5 n ν ∼ T 3 σ ∼ G 2 F T 2 v ≃ 1 ◮ The rate of expansion is given by the Friedmann equation: ˙ 8 π R ( t ) ρ − k H 2 = H ( t ) ≡ R 2 3 M P R ( t ) ρ rad with ρ rad = π 2 8 π ◮ In the radiation-dominated era: H 2 ≃ 30 g ∗ T 4 3 M 2 P 2 π 3 / 2 √ g ∗ T 2 g χ + 7 � � √ H ≃ g ∗ = g χ 8 3 5 M P χ =relativistic χ =relativistic bosons fermions = 2 + 7 8 4 + +7 + g ( e ± ) ◮ Before ν decoupling: g ∗ = g ( γ ) + g ( ν ) 8 6 = 10 . 75 ∗ ∗ ∗ C. Giunti − Massive Neutrinos in Cosmology – II − Torino PhD Course − May 2018 − 5/52

� − 1 / 3 ∼ 1 MeV T ν -dec ∼ M P G 2 ◮ Neutrino decoupling: Γ ν ∼ H � = ⇒ F ◮ A more precise calculation takes into account that the dominant processes for T � 100 MeV are ( − ) ( − ) ν ⇆ e + e − ν ¯ ν e ⇆ ν e ( − ) ( − ) ν e,µ,τ ν e,µ,τ e − , e + e + , e − ν e , ¯ ν e ν e , ¯ ν e W Z W e − , e + ν e , ¯ ν e e + , e − ν e , ¯ ν e e ± e ± ◮ Since the rates of these processes depend on neutrino energy E ≃ p , the decoupling temperature is not instantaneous and depends on p : � p � p � − 1 / 3 � − 1 / 3 T ν e -dec ( p ) ≃ 2 . 7 T ν µ,τ -dec ( p ) ≃ 4 . 5 T T ◮ Taking into account that � E � ≃ 3 T , one obtains: T ν e -dec ≃ 1 . 9 MeV T ν µ,τ -dec ≃ 3 . 1 MeV C. Giunti − Massive Neutrinos in Cosmology – II − Torino PhD Course − May 2018 − 6/52

1 ⇒ f ν -dec ◮ Hot relics: relativistic at decoupling = ( � p ) ≃ p | / T ν -dec + 1 ν e | � d r 2 � d θ 2 + sin 2 θ d φ 2 �� ◮ FRW metric: d τ 2 = d t 2 − R 2 ( t ) 1 − k r 2 + r 2 � � − 1 � R ◮ Momentum scaling with expansion: | � p | = | � p | ν -dec R ν -dec � − 1 � � | � p | ( R / R ν -dec ) � 1 f ν ( � p ) ≃ exp + 1 = T ν -dec p | / T ν + 1 e | � � − 1 � R Effective temperature scales with expansion: T ν = T ν -dec R ν -dec C. Giunti − Massive Neutrinos in Cosmology – II − Torino PhD Course − May 2018 − 7/52

Electron-Positron Annihilation ◮ After neutrino decoupling at T ≃ 1 MeV e ± and γ are the only relativistic particles in thermal equilibrium. ◮ At m e / 3 ≃ 0 . 2 MeV electrons and positrons became nonrelativistic: out-of-equilibrium e − e + → γγ heat the photon distribution. ◮ During this phase the photon temperature does not scate as R − 1 . = 2 π 2 s = ρ + p 45 g s T 3 ◮ Entropy density: γ T g χ + 7 � � g s = g χ 8 χ =interacting χ =interacting relativistic relativistic bosons fermions T γ ∝ g − 1 / 3 ◮ Entropy conservation: s ∝ R − 3 R − 1 = ⇒ s C. Giunti − Massive Neutrinos in Cosmology – II − Torino PhD Course − May 2018 − 8/52

� R after � − 1 T after ◮ Before and after e − e + annihilation: ν = T before R before ν � g after � − 1 / 3 � R after � g after � − 1 / 3 T after � − 1 T after γ s s ν = = T before g before R before g before T before γ ν s s 1.5 ◮ T before = T before T ! = 1.401 T " γ ν 1.4 = 2 + 7 + g ( e ± ) = g ( γ ) ◮ g before 1.3 8 4 s s s T ! / T " e + e - # !! 1.2 = g ( γ ) ◮ g after = 2 s s 1.1 � 4 � 1 / 3 T ! = T " 1 ◮ T after T after ≃ 0 . 7138 T after = ν γ γ 11 10 1 0.1 T ! (MeV) � 4 � 1 / 3 γ = 1 . 945 ± 0 . 001 K = (1 . 676 ± 0 . 001) × 10 − 4 eV T 0 T 0 ν = ◮ 11 C. Giunti − Massive Neutrinos in Cosmology – II − Torino PhD Course − May 2018 − 9/52

Effective Number of Relativistic Degrees Of Freedom � 4 � � � 4 / 3 1 + 7 ◮ Radiation density: ρ rad = ρ γ N eff 8 11 N ν Why N ν ◮ Three standard neutrinos: eff = 3 . 046 eff > 3? [Mangano et al, NPB 729 (2005) 221] → N ν eff = 3 . 045 [de Salas, Pastor, JCAP 1607 (2016) 051] ◮ Neutrino decoupling was not instantaneous at T ν -dec . ◮ Higher-energy neutrinos decoupled later and were not completely decoupled during e − e + annihilation. ( − ) ( − ) ◮ This effect is different for ν e and ν µ,τ because of the additional ( − ) charged-current interactions of ν e : ( − ) ( − ) ν e,µ,τ ν e,µ,τ e − , e + e + , e − ν e , ¯ ν e ν e , ¯ ν e W Z W e − , e + ν e , ¯ ν e e + , e − ν e , ¯ ν e e ± e ± C. Giunti − Massive Neutrinos in Cosmology – II − Torino PhD Course − May 2018 − 10/52

1 ◮ Equilibrium distribution: f eq ( � p ) ≃ e p / T + 1 ◮ Nonthermal distortions: f ν α ( � p , t ) = f eq ( � p ) (1 + δ ν α ( � p , t )) ◮ Boltzmann equation: � ∂ � ∂ t − Hp ∂ � � f ν α ( � p , t ) = C f ν α ; f ν β , f e ± ∂ p 1.05 No osc ν e θ 13 =0 1.04 T γ s 2 13 =0.047 ν e R = T − 1 f ν / f eq for y = 10 1.03 ν ν µ ν τ 1.02 x = m e R = m e / T ν ν x 1.01 y = pR = p / T ν 1 0.1 1 10 x = m e R [Mangano et al, NPB 729 (2005) 221, arXiv:hep-ph/0506164] C. Giunti − Massive Neutrinos in Cosmology – II − Torino PhD Course − May 2018 − 11/52

◮ Neutrino oscillations mix the flavor distributions. √ √ 2 G F ( n ℓ − − n ℓ + ) − 8 2 G F p ◮ Matter potential: V ( ℓ ) CC = ( ρ ℓ − + ρ ℓ + ) 3 m 2 W ∆ m 2 31 / 2 p ν e ⇆ ν µ,τ ( ϑ 13 ) ν µ ⇆ ν τ ( ϑ 23 ) ∆ m 2 21 / 2 p ν e ⇆ ν µ,τ ( ϑ 12 ) V ( ρ ) CC V ( n ) CC ν µ,τ dec ν e dec [Lesgourgues, Mangano, Miele, Pastor, Neutrino Cosmology, Cambridge University Press, 2013] C. Giunti − Massive Neutrinos in Cosmology – II − Torino PhD Course − May 2018 − 12/52

1.05 1.06 No osc No osc ν e θ 13 =0 1.05 T γ θ 13 =0 1.04 s 2 13 =0.047 s 2 13 =0.047 ν e 1.04 Frozen f ν (y) / f eq (y) f ν / f eq for y = 10 1.03 1.03 1.02 1.02 ν x ν x 1.01 1.01 1 1 0 2 4 6 8 10 12 0.1 1 10 x = m e R y = p R [Mangano et al, NPB 729 (2005) 221, arXiv:hep-ph/0506164] R = T − 1 x = m e R = m e / T ν y = pR = p / T ν ν N eff = 3 + δρ ν e + δρ ν µ + δρ ν τ = 3 . 046 ρ ν ρ ν ρ ν  f ν 1 ≃ 0 . 7 f ν e + 0 . 3 f ν µ,τ  � | U α k | 2 f ν α f ν k = ⇒ f ν 2 ≃ 0 . 3 f ν e + 0 . 7 f ν µ,τ f ν 3 ≃ f ν µ,τ α = e ,µ,τ  C. Giunti − Massive Neutrinos in Cosmology – II − Torino PhD Course − May 2018 − 13/52

Energy Density  ρ c ≃ ρ R ∝ R − 4   ∝ R − 3  ρ M   R − 4 ∝ R Rad: ρ c ρ Λ 1  R − 4 ∝ R 4  ∝   ρ c   ρ c ≃ ρ M ∝ R − 3   ∝ R − 4  ρ R m 3 ≃ 0 . 05 eV  R − 3 ∝ R − 1  Mat: ρ c ρ Λ 1 m 2 ≃ 0 . 008 eV  R − 3 ∝ R 3  ∝    ρ c m 1 ≪ m 2  ρ c ≃ ρ Λ = const. ρ R  R / R 0  ∝ R − 4  Λ: ρ c ρ c = 3 M P ρ M 8 π H 2 ∝ R − 3 Ω i = ρ i /ρ c    ρ c C. Giunti − Massive Neutrinos in Cosmology – II − Torino PhD Course − May 2018 − 14/52