Linear Analysis of Fast Pairwise Neutrino Oscillations
InCore-Collapse Supernovae Based on Realistic Data
MILAD DELFAN AZARI

Department of Physics, Waseda University
December 21, 2018
Yukawa Institute for Theoretical Physics, Kyoto University

第31回 理論懇シンポジウム 宇宙物理の標準理論：未来へ向けての再考Introduction
Fate of a star depends on its

MassMstar ≳ 8 Msun
Core-Collapse

SN 1987A

Supernovae

Cassiopeia A

Core-Collapse

𝜈 − trapping

Core bounce𝜈𝜈Fe𝜈𝜈𝜈𝜈SN explosion and NS cooling

Shock in envelope & PNS cooling𝜈

Shock propagation in coreNS𝜈𝜈𝜈Why neutrinos are important ?
✓ almost all of the binding energy of NS liberated in the gravitational collapse
is emitted in the form of neutrinos and the kinetic energy of matter in thesupernova explosion is just 1% of this energy.

✓ a fraction of the electron-type neutrinos and anti-neutrinos are
re-absorbed by the matter between the shock front and the so-called gained
radius and deposit their energy to push the stagnated shock again.

Core-Collapse

𝜈 − trapping𝜈𝜈Fe𝜈𝜈𝜈𝜈

SN explosion and NS coolingNSCore bounce

Shock in envelope & PNS cooling𝜈Shock propagation in corer >~ 103 km

r ~ 102 km

Matter effect𝜈 − 𝜈 self interaction
Duan et al., PRD 74, 105014, 2006

Wolfenstein PRD 17, 2369, 1978
𝜈𝜈𝜈What fast pairwise means?
✓ Collective pair conversion 𝜈𝑒 𝜈𝑒ҧgeneric for SN neutrino transport

𝜈𝑥 𝜈𝑥ҧ by forward scattering maybe✓ Depending on the angular intensity, this rate is “Fast”

Motivation✓ Problems remaining;• No successful explosion model for CCSNeand• Collective neutrino oscillation has been ignored✓ Our goal;
Study the collective neutrino oscillations quantitatively.Basic Equations and Formulae

Equation of Motion:
𝜕𝑡 + 𝑣. 𝛻𝑟 𝜌 = 𝑖 𝜌, 𝐻

𝑓𝜐𝑒 +𝑓𝜐𝑥
𝑓𝜐𝑒 −𝑓𝜐𝑥𝜌=+22𝑠𝑆∗

𝑆,−𝑠𝑀21𝜇𝐻=+ 𝑣 Λ𝜇 𝜎3 + 2𝐺𝐹 න 𝑑Γ ′ 𝑣 𝜇 𝑣′𝜇 𝜌′2𝐸
2Kinetic Hamiltonian

Matter Hamiltonian

Neutrino-Neutrino Hamiltonian

Vacuum Oscillation

MSW Oscillation

Collective Oscillation

𝑀2 : Mass-squared matrix𝑣 𝜇 ∶ (1,v)Λ𝜇 ∶ 2𝐺𝐹 (𝑛𝑒 − 𝑛𝑒 + )𝑢𝜇𝑑Γ ′ =dv’/4𝜋

Basic Equations and Formulae
Linearized Equation of Motion:𝑖 𝜕𝑡 + 𝑣. 𝛻𝑟 𝑆𝑣 = 𝑣 𝜇 Λ𝜇 + Φ𝜇 𝑆𝑣 −dv’ 𝜇‫ ׬‬4𝜋 𝑣 𝑣′𝜇 𝐺𝑣 ′ 𝑆𝑣 ′𝐺𝑣 =∞ 𝑑𝐸𝐸 22𝐺𝐹 ‫׬‬02𝜋2𝑣 𝜇 𝑘𝜇 𝑄𝑣 = 𝑎𝜇dv’ 𝜇where ;‫ ׬‬4𝜋 𝑣 𝑣′𝜇 𝐺𝑣 ′ 𝑄𝑣 ′𝑘𝜇 = 𝐾𝜇 − Λ𝜇 − Φ𝜇 with 𝑘𝜇 = (𝜔,k)𝑎𝜇 ≡ −𝑓𝜈𝑒 𝐸, v − 𝑓𝜈ഥ𝑒 (𝐸, v)

dv
𝜇Φ ≡𝐺𝑣 𝑣 𝜇4𝜋Assuming the solutions in the form of ;𝑆𝑣 𝑡, 𝑟 = 𝑄𝑣 Ω, 𝐾 𝑒 −𝑖(Ω𝑡−𝐾.𝑟)

Π 𝜇𝜈 (𝜔,k) 𝑎𝜈 = 0Polarization tensor Π 𝜇𝜈 = 𝜂 𝜇𝜈 +𝑑𝑉𝑣 𝜇𝑣 𝜈‫ ׬‬4𝜋 𝐺𝑉 𝜔−𝑉.𝑘D(𝜔,k) ≡ det Π = 0

Background Numerical Model✓ Results of the realistic 2D simulations on theK-Supercomputer

Nagakura et al., ApJ 854, 136 (2018)✓ For non-rotating progenitor model of Mstar=11.2 𝑀⊙
Woosley et al., Reviews of Modern Physics 74, 1015 (2002)✓With the Boltzmann equation for neutrinotransport being solved and special relativistic effectwith a two energy grid technique. Nagakura et al., ApJS 214,16 (2014)✓ Newtonian hydrodynamical equations & the Poisson
equation for self-gravity were solved simultaneously.

Method

Entropy distributions at the central partof the core

Outer product of 𝜈𝑒 and 𝜈𝑒ҧNeutrino fluxes with compared to
the radial direction at r= 44.8 km

Method

Angular distributions of 𝜈𝑒 , 𝜈𝑒ҧ & 𝜈𝑥 in meridian section at different azimuthal axis.,r= 44.8 km and t= 15.0 ms post bounce

Results
Dispersion relation

Rendering of the neutrino distribution

det Π = 0
𝜔 = 10𝜔 = 10 + 𝑖 10

Real part of det Π = 0
Imaginary part of det Π = 0
𝜔 = 10 + 𝑖 20

det Π = 0
𝜔 = 10𝜔 = 10 + 𝑖 10

Real part of det Π = 0
Imaginary part of det Π = 0
𝜔 = 10 + 𝑖 20

NO crossing in Re k , Means NO instability is expected𝜔 = 100

𝜔 = 100 + 𝑖 50

𝜔 = 100 + 𝑖 100

𝜔 = −100

𝜔 = −100 + 𝑖 50

𝜔 = −100 + 𝑖 50

No oscillation so far ….

No oscillation so far ….
How & under what condition canwe obtain the oscillation?

Modified data
Dispersion relation

Rendering of the neutrino distribution

det Π = 0

Real part of det Π = 0
Imaginary part of det Π = 0

𝜔 = −7

𝜔 = −7 + 𝑖 1.5

𝜔 = −7 + 𝑖 2det Π = 0

Real part of det Π = 0
Imaginary part of det Π = 0

𝜔 = −7

𝜔 = −7 + 𝑖 1.5

𝜔 = −7 + 𝑖 2

There is crossing in Re k , Means; instability is expectedSummary and future works✓Collective neutrino oscillations in core-collapse supernovae has
been studied quantitatively.
✓We confirm that there is no sign of instability which leads to theconversion in case of the realistic data analysis.
✓The possible oscillation conditions in different radii and time steps
has been studied.
✓This method might be applicable to other astrophysical objects
such as NS-mergers disks, etc. We plan to investigate theconversion possibilities in these objects in near future.