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Neutrino Cosmology Neutrino cosmology Yvonne Y. Y. Wong The University of New South Wales, Sydney, Australia Invisibles’19 School, LSC Canfranc, June 3 – 7, 2019 Or, how neutrinos fit into this grand scheme? The grand lecture plan... Lecture 1: Neutrinos in homogeneous cosmology 1. The homogeneous and isotropic universe 2. The hot universe and the relic neutrino background 3. Measuring the relic neutrino background via Neff Lecture 2: Neutrinos in inhomogeneous cosmology Lecture 1: Neutrinos in homogeneous cosmology 1. The homogeneous and isotropic universe 2. The hot universe and the relic neutrino background 3. Measuring the relic neutrino background via Neff Useful references... ● Lecture notes – A. D. Dolgov, Neutrinos in cosmology, Phys. Rept. 370 (2002) 333 [hep-ph/0202122] – J. Lesgourgues & S. Pastor, Massive neutrinos and cosmology, Phys. Rep. 429 (2006) 307 [astro-ph/0603494]. – ● Textbooks – J. Lesgourgues, G. Mangano, G. Miele & S. Pastor, Neutrino cosmology 1. The homogeneous and isotropic universe... The concordance flat ΛCDM model... ● The simplest model consistent with present observations. Cosmological (Nearly) constant Massless Neutrinos (3 families) 5% 69% 26% Composition today 13.4 billion years ago (at photon decoupling) Plus flat spatial geometry+initial conditions from single-field inflation Friedmann-Lemaître-Robertson-Walker universe... ● Cosmological principle: our universe is spatially homogeneous and isotropic on sufficiently large length scales (i.e., we are not special). – Homogeneous → same everywhere – Isotropic → same in all directions Size of visible universe – Sufficiently large scales → > O(100 Mpc) ~ O(10 Gpc) Isotropic but Homogeneous but Homogeneous not homogeneous not isotropic and isotropic Friedmann-Lemaître-Robertson-Walker universe... ● Homogeneity and isotropy imply maximally symmetric 3-spaces (3 translational and 3 rotational symmetries). – A spacetime geometry that satisfies these requirements: 2 2 2 2 dr 2 2 2 2 ds =−dt +a (t) +r (d θ +sin θ d ϕ ) FLRW metric [ 1−K r 2 ] Spatial geometry a(t) = Scale factor K = -1 (hyperbolic), 0 (flat), +1 (spherical) a(t 2) = Factor by which a physical length scale increases between time t1 and t2. a(t 1) ● An observer at rest with the FLRW spatial coordinates is a comoving observer. 2 2 2 2 dr 2 2 2 2 ds =−dt +a (t) +r (d θ +sin θ d ϕ ) [ 1−K r 2 ] Comoving observers scale factor a → The physical distance between two comoving observers increases with time, but the coordinate distance between them remains unchanged. Cosmological redshift... ● All test particles (massive or massless) moving on geodesics of an FLRW universe suffer cosmological redshift: Momentum of a point −1 Physical momentum of a particle measured by |⃗p|∝a particle decreases with comoving observers expanding space. scale factor a ● Alternatively, in terms of wavelength: λ∝1/|⃗p| Wavelength measured t0= today by comoving observer z = Redshift λ a(t ) parameter 0 = 0 ≡1+z λe a(te) Wavelength of particle (usually photon) emitted by comoving emitter ● In an FLRW universe, there is a one-to-one correspondence between t, a, and z: redshift z time t scale factor a → We use them interchangeably as a measure of time. μ ν Matter/energy content: conservation law... ∇ μ T (α)=0 ● Local conservation of energy-momentum in an FLRW universe implies: Energy Pressure density d ρ a˙ α +3 (ρ +P )=0 Continuity equation d t a α α – There is one such equation for each substance α. ● We need in addition to specify a relation between ρ(t) and P(t) (a property of the substance). P (t ) α w = Equation of state parameter wα (t)≡ ρα (t ) ● Assuming a constant wα: How energy density −3(1+wα) ρα (t)∝a evolves with the scale factor. −3(1+wα) Matter/energy content: what is out there? ρα (t)∝a ● Nonrelativistic matter – Atoms (or constituents thereof); “baryons” in cosmology-speak. wm≃0 – Dark matter (does not emit light but −3 ⇒ρ ∝a Volume expansion feels gravity); GR people call it “dust”. m ● Ultra-relativistic radiation (at least for a significant part of their evolution history) wr=1/3 – Photons (mainly the CMB) −4 Volume expansion – ⇒ρ ∝a Relic neutrinos (analogous to CMB) r + momentum redshift – Gravitational waves?? ● Other funny things wΛ=−1 – More space, Cosmological constant/vacuum energy more energy – ?? ⇒ρΛ ∝constant log(ρ) Radiation domination Matter domination Λ domination −4 ρr ∝a ρ ∝a−3 Matter-radiation m equality Matter-Λ equality ρΛ=const log(a) redshift z time t log(ρ) Radiation domination Matter domination Λ domination Big bang nucleosynthesis Last scattering surface (CMB) −4 ρr ∝a Structure −3 Matter-radiation ρm ∝a formation equality Matter-Λ equality Today ρΛ=const -9 -6 -3 0 log(a) redshift z a =1 By convention time t 0 Friedmann equation... Newton's constant ● Derived from the Einstein equation: R = Ricci scalar and tensor 1 (nonlinear functions of the Rμ ν− g μ ν R=8 π G T μ ν 2nd derivative of the 2 Stress-energy tensor spacetime metric) ● The Friedmann equation is an evolution equation for the scale factor a(t): 2 2 a˙ 8 πG Friedmann H (t )≡ = ∑ ρα−K equation ( a ) 3 α H(t) = Hubble parameter ● Friedmann+continuity equations → specify the whole system. Friedmann equation... ● You may also have seen the Friedmann equation in this form: 2 2 −3 −4 −2 H (t )=H (t0 )[ Ωm a +Ωr a +ΩΛ +ΩK a ] 2 ¯ρα (t0 ) 3 H (t ) K Ωα= , ρcrit (t )≡ , Ω K ≡− 2 2 ρcrit (t 0 ) 8 πG a H (t0 ) Critical density ● A flat universe means: From measuring the CMB temperature a and energy spectrum. Ωm+Ωr +ΩΛ≃Ωm +ΩΛ=1 −5 Radiation energy density is negligibly small today: Ωr∼10 Friedmann equation... ● You may also have seen the Friedmann equation in this form: 2 2 −3 −4 −2 H (t )=H (t0 )[ Ωm a +Ωr a +ΩΛ +ΩK a ] 2 ¯ρα (t0 ) 3 H (t ) K Ωα= , ρcrit (t )≡ , Ω K ≡− 2 2 ρcrit (t 0 ) 8 πG a H (t0 ) Critical density ● Current observations: I'll tell you how Ωm∼0.3, ΩΛ∼0.7, |Ω K|<0.01 e.g., Ade et al. you measure −1 −1 [Planck collaboration] these in a bit... arXiv:1502.01589 H 0≡H (t 0)∼70 km s Mpc Open, flat, K = 0 Λ-dominated Open, curved, K = -1 Our universe! Matter-dominated Scale factor a(t) Open, flat, K = 0 Matter-dominated Closed, curved, K = +1 Matter-dominated Time t 2. The hot universe and the relic neutrino background... The hot universe... ● The early universe was a very hot and dense place. → Frequent particle interactions. ● If the interaction rate (per particle) is so large that Interaction rate Γ >> Expansion rate, H → the interaction process is in a state of equilibrium. Equilibrium... −5 −2 G F∼10 GeV 19 mpl∼10 GeV ● Consider the weak interaction. The interaction rate per particle is: Fermi constant Number density n ~ T 3 2 5 3 Γ=n⟨σ v⟩∼G T 2 3 T F Γ ∼m G T ∼ H pl F (1 MeV ) T = 2 2 temperature Cross section σ ~ GF T Relative velocity v ~ 1 ● The Hubble expansion goes like → When the temperature exceeds O(1) MeV, even 2 weak interaction processes 8πG T are in a state of equilibrium! H = ∑ ρα∼ 3 α m √ pl Natural units k =1 Planck mass B Weak EM log(ρ) Radiation domination Matter domination Λ domination Big bang nucleosynthesis Last scattering surface (CMB) −4 ρr ∝a −3 Structure ρ ∝a Matter-radiation formation m equality Matter-Λ equality Today ρΛ=const -9 -6 -3 0 log(a) redshift z a =1 By convention time t 0 g* for the standard model of particle physics: Weak interaction goes out of equilibrium Nucleosynthesis starts Electroweak What's left? g* (T ) phase transition Mainly ● Photons ● Neutrinos Energy density Quark-hadron normalised to the phase transition Small numbers* of photon energy density ● Electrons in one polarisation state ● Nucleons ● Nuclei * Small means −9 <10 nγ Photon temperature Time Particle content & interactions at 0.1 < T < 100 MeV... ● QED plasma ● 3 families of neutrinos + antineutrinos + - e ,e , γ ν e ,ν¯e , ν μ , ν¯μ , ν τ , ν¯τ EM interactions: Weak interactions @ T > O(1) MeV: + - + - e e ↔ e e ν i ν j ↔ ν i ν j e e↔ e e Coupled ν i ν¯j ↔ ν i ν¯j γ e↔ γ e ν i ν i ↔ ν i ν i + - γ γ ↔e e ν i ν¯i ↔ ν j ν¯j etc . ν e ↔ ν e etc . 2 5 + - Γ∼GF T ν ν¯ ↔ e e 2 Weak interactions @ T > O(1) MeV: H ∼T /mpl Particle content & interactions at 0.1 < T < 100 MeV... ● QED plasma ● 3 families of neutrinos + antineutrinos + - e ,e , γ ν e ,ν¯e , ν μ , ν¯μ , ν τ , ν¯τ EM interactions: Weak interactions @ T > O(1) MeV: + - + - e e ↔ e e ν i ν j ↔ ν i ν j e e↔ e e Decoupled ν i ν¯j ↔ ν i ν¯j γ e↔ γ e ν i ν i ↔ ν i ν i + - γ γ ↔e e ν i ν¯i ↔ ν j ν¯j etc . ν e ↔ ν e etc . 2 5 ν ν¯ ↔ e+ e- Γ∼GF T Not efficient 2 Weak interactions @ T > O(1) MeV: H ∼T /m pl T< O(1) MeV Thermal history of neutrinos... Events Time Photon temperature, T γ Neutrino temperature Phase space density Thermal history of neutrinos... Events Neutrinos in thermal contact with QED plasma 2 5 Weak interaction: ~G F T T 2 H Expansion rate: H ~ mplanck Time Photon temperature, T γ 1 MeV Neutrino temperature T =T 1 Phase space f E≃ density Relativistic Fermi-Dirac exp[ p /T ] 1 Thermal history of neutrinos..
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