Lecture III

Supernova A is Born

1500 km

Shock wave ~1051ergs 3X107 km B. E. ~2-3 1053 ergs 10 km

proto-: t~1-2 s 100 km Epochs in the evolution of a neutron star.

Prakash et al., Phys. Rept. (1990) Traversing the Phase Diagram T (MeV) 170 QGP

50 Novel High Density Phases: 10 Hyperons, Kaons Quark Matter .. Nuclear Matter Nuclei Neutron

930 ~1200 ? ? µΒ (MeV) Traversing the Phase Diagram T (MeV) 170 QGP

50 Novel High Density Phases: 10 Hyperons, Kaons Quark Matter .. Nuclear Collapse Matter Nuclei Neutron Stars

930 ~1200 ? ? µΒ (MeV) Traversing the Phase Diagram T (MeV) 170 QGP

PNS Evolution 50 Novel High Density Phases: 10 Hyperons, Kaons Quark Matter .. Nuclear Collapse Matter Nuclei Neutron Stars

930 ~1200 ? ? µΒ (MeV) Neutrinos Future: 3 ×1053 ergs = 1058 × 20 MeV Neutrinos Can detect ~10,000 neutrinos from galactic supernova Past: SN 1987a: ~ 20 neutrinos ..in support of supernova theory

28 np~ 6.7 × 10 =number of per ton Mtons= detector mass in tons -44 2 σref ~ 2 × 10 cm = weak cross-section D ~ 10 kpc = distance to supernova Future: 3 ×1053 ergs = 1058 × 20 MeV Neutrinos Can detect ~10,000 neutrinos from galactic supernova Past: SN 1987a: ~ 20 neutrinos ..in support of supernova theory

28 np~ 6.7 × 10 =number of protons per ton Mtons= detector mass in tons -44 2 σref ~ 2 × 10 cm = weak cross-section D ~ 10 kpc = distance to supernova Dimensional Analysis with G, M, and R

R 2GM Compactness: s = 0.3 0.4 R Rc2 ￿ −

GM m 1 R Binding : n = s m = 140 200 MeV R 2 R n −

R3 R R R 3/2 τFree Fall = = 80 ms − GM c R ￿ 1000 km ￿ ￿ s ￿ ￿

R R R τHydro = τFree Fall = c − c R s ￿ ￿ s Time-Scales

2 2 2 2 4GF me E 44 E 2 σweak = 2 =1.7 10− 2 cm π me × me

2 1 5 me λν (T )= =3 10 2 cm nσweak(T ) × T

2 2 2 R 4 R T τ = = 10− s Diffusion cλ 10 km m2 ν ￿ ￿ e Protoneutron Star Evolution

Initial state of the PNS. Pons et al. (1999) Snap shots of the evolution diffusion deleptonizes and cools the PNS.

Typical time-scales:

Pons et al. (1999) Snap shots of the evolution Neutrino diffusion deleptonizes and cools the PNS.

Typical time-scales:

⎛ t ⎞ T(t) ≈ T(t = 0) ⎜1 − ⎟ ⎝ τC ⎠ R2 τC ≈ CV c λν

3 µ Y (t) ≈ ν ν 6π 2 ⎛ t ⎞ ≈ Y(t = 0) exp⎜ − ⎟ ⎝ τD ⎠ 3 ∂Y R2 τ = L D 2 Y c π ∂ ν λνe

Pons et al. (1999)

€ Snap shots of the evolution Neutrino diffusion deleptonizes and cools the PNS.

Typical time-scales:

⎛ t ⎞ T(t) ≈ T(t = 0) ⎜1 − ⎟ ⎝ τC ⎠ R2 τC ≈ CV c λν

3 µ Y (t) ≈ ν ν 6π 2 ⎛ t ⎞ ≈ Y(t = 0) exp⎜ − ⎟ ⎝ τD ⎠ 3 ∂Y R2 τ = L D 2 Y c π ∂ ν λνe

Pons et al. (1999)

€ Response of Interacting System

ν ν emission ω,q ω =q ω scattering

∆R q 2π 1 q = > λ ∆R

2π 2π ω = > τ = τ τcollision collision Collision Time 2π 2π ω = < τ τcollision Response of Interacting System

ν ν emission ω,q T ω =q ω scattering

∆R q 2π 1 q = > λ ∆R

2π 2π ω = > τ = τ τcollision collision Collision Time 2π 2π ω = < τ τcollision Response of Interacting System

ν ν emission ω,q T ω =q ω scattering

∆R q 2π 1 q = > λ ∆R 2π 1 q = < λ ∆R ∆R 2π 2π ω = > τ = τ τcollision collision Collision Time 2π 2π ω = < τ τcollision Response of Interacting System

ν ν emission ω,q T ω =q ω scattering

∆R q 2π 1 q = > λ ∆R 2π 1 q = < λ ∆R ∆R 2π 2π ω = > τ = τ τcollision collision Collision Time 2π 2π ω = < τ τcollision Linear Response Theory

Consider a weak perturbation of the system by an external field

The external field can couple to any property of the system. For illustration lets assume it couples to density, then

This will produce a density fluctuation

Eigenstate of Htot Eigenstate of Hsystem

If Hext is weak it is possible to show that Density-density correlation function

ΠR is the Fourier transform of DR and it is called the polarization function or the generalized susceptibility. Neutrino Rates

ν ν emission ω,q ω =q ω scattering

q d 2σ E ≈ G2 Im L (k,k + q) Πµν (q) V dcosθ dE ʹ F E ʹ [ µν ]

Lµν (k,k + q) = Tr [ lµ (k) lν (k + q) ] r d 4 p µν ( ,q ) Tr [ j µ (p) jν (p q) ] Π ω = ∫ 4 + (2π) Dynamic structure factor: spectrum of density, spin and current fluctuations. € Neutrino Scattering

µ µ Neutrinos couple to j (x) = ψ(x) γ (cV − cAγ5) ψ(x) density and spin NR + µ0 + i µi → cV ψ ψ δ − cA ψ σ ψ δ

€

Burrows,Reddy, Thomspon (2006) Sum Rules: Thermodynamic (compressibility) sum Rule:

dω (1− eβhω ) ⎛ ∂ρ⎞ lim ∫ Sρ (q,ω) = ρ ⎜ ⎟ q →0 2π hω ∂p ⎝ ⎠T This relates response to the equation of state - provides a good consistency check. € F-Sum Rule:

dω (1− eβhω ) 1 S (q, ) [[H, (q)], (q)] ∫ ω ρ ω = 2 ρ ρ 2π 2h 2h

€ S(q,ω) of a classical dense liquid

The density-density correlation for N particles is

Ensemble average Positions at t

Positions at t=0

Need to specify equations of motion ie rj(t). Classical limit:

One simple method is Molecular Dynamics. Response in Strongly Coupled Systems

Charged ions in a screened plasma.

Z2 e2 Γ= =8 akT Correlations are ωplasmon =0.3MeV q =3MeV important.

(Fortunately) nuclear interactions are relatively weaker.

Carlson & Reddy (2004) RPA - vertex corrections

Self-consistent approximation to a mean-field ground state Mean Free Paths in the RPA

r r p + q Γµ Γν = r p € € € r r G (p + q ) Γµ µν Γν € r G (p) € νµ € € € + r r r r G (p +€q ) µσ p + q Γµ Γν r r V p Horowitz & Wehrberger, (1991) G (p) στ Burrows & Sawyer, (1999) € σµ Reddy, Pons, Prakash, Lattimer, (1999) € € € € € € First-Order Transitions Glendenning (1992) (nuclear -> quark) Alford, Rajagopal, Reddy & Wilczek (2001) (nuclear ->CFL)

exotic a A Mixed phase µ e = −µQ − − + b nuclear + € B

µ B

Mixed Phase: Heterogeneous phase with Nuclear Exotic € structures of size 5-10 fm. Neutrino Mean Free Path in a Mixed Phase

ν’

ν

Scattering from quark droplets in a quark-hadron transition. 2 dσ GF 2 2 = ND Sq QW Eν(1 + cos(θ)) dcos (θ) 16π Reddy, Bertsch & Prakash, (2000)

€ Neutrino Propagation in Superconducting phases

r 4 Πµν(q, qo) = ∫ d p Tr [ G(p) ΓµG(p + q)Γν] = p+q

Γµ Γν € p + p+q Γµ Γν

p Gap modifies excitation spectrum.

No excitations below the gap ? Carter & Reddy (2000) Neutrino Propagation in Superconducting phases

r 4 Πµν(q, qo) = ∫ d p Tr [ G(p) ΓµG(p + q)Γν] = p+q

Γµ Γν € p + p+q Γµ Γν

p Gap modifies excitation spectrum.

No excitations below the gap ? Carter & Reddy (2000) Collective(Goldstone) modes p+q = q + γ Γµ Γµ µ λ(q) p

Generalized Ward Identity : NG ˜ −1 ˜ −1 NG ∑qµΓµ = τ3 G (p) − G (p + q)τ3 µ if lt λ(q) is finite q→0 r Γ(q = (q ,qo) is singular when

r Bogoliubov, Nuovo Cimento, 7, 6 (1958) vF Anderson, Phys. Rev. 112, 1900 (1958) qo = |q | 3 Nambu, Phys. Rev. 117, 648 (1960)

€ Collective(Goldstone) modes p+q = q + γ Γµ Γµ µ λ(q) p

Generalized Ward Identity : Low energy response NG ˜ −1 ˜ −1 NG ∑qµΓµ = τ3 G (p) − G (p + q)τ3 dominated by the µ Goldstone (a.k.a. if lt λ(q) is finite q→0 Bogoliubov-Anderson ) r Γ(q = (q ,qo) is singular when mode

r Bogoliubov, Nuovo Cimento, 7, 6 (1958) vF Anderson, Phys. Rev. 112, 1900 (1958) qo = |q | 3 Nambu, Phys. Rev. 117, 648 (1960)

€ Spectrum of Density Fluctuations in Superfluids

Goldstone Mode Pair Breaking

2Δ Color-Flavor Locked Phase

BCS pairing of all 9 quarks: Δ ≈ 100 MeV ! E ≈ gµ gluon

Equark ≈ 2Δ €

E € GB: SUC(3)⊗SUL (3)⊗SUR (3)

Energy Δ ≈ mlight ms µ

€ E (1) = 0 GB: UB Excitation Spectrum SU(3)color ⊗ SU(3)L ⊗ SU(3)R ⊗ U(1)B ⇓ € SU(3)color+L+R ⊗ Z2

€ Alford, Rajagopal & Wilczek, (1999) Goldstone Modes in CFL

dispersion relations : 1 E = v p (v ≈ ) H 3 E = m2 + v2 p2 Schafer Phys. Rev .D 65, 074006 (2002) π π Manuel & Tytgat, Phys. Lett. B 479, 190 (2000) 2 ms 2 2 2 Son & Stephanov, Phys. Rev. D 61, 074012 (2000) EK m = ± + mk + v p 2µ

€ Coupling to Neutrinos

φo ν φ- e- Z W- ν ν

e- φ ν φ Z W- ν

ν φ φ

dispersion relations : 1 E = v p (v ≈ ) H 3 2 2 2 Eπ = mπ + v p 2 ms 2 2 2 EK m = ± + mk + v p 2µ Reddy, Sadszikowski & Tachibana (2003)

€ Neutrino Emit Cherenkov

+ φo φ ν νe

GFfH G f ν’ F π e-

Dominant Process CFL matter is opaque.

Reddy, Sadszikowski & Tachibana (2003) CFL matter is opaque.

Reddy, Sadszikowski & Tachibana (2003) Neutron Star Tomography - The Dream !