# On the Time Distribution of Supernova Antineutrino Flux

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## Abstract

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## 1. Introduction

What do we know for sure about the various phases of gravitational collapse?How to learn more using neutrino observations?

## 2. Open Issues after Supernova 1987A

- that the supernova precursor did not fit fully expectations, and perhaps was non-standard, with unexpected implications;
- that the neutron star had not been observed, even casting a shadow over the significance of the observed events;
- that the observed neutrino events were more directional than expected—directed from the supernova forward-raising similar concerns as in the previous point;
- that also, their average energies were lower than those calculated, perhaps indicating an instrumental problem;
- that the comparability of the energy spectra of Kamiokande-II, IMB and Baksan was not entirely clear;

- There is a growing consensus towards the idea that the precursor could have been a two-star system that had recently merged, although it is not clear whether this impacts expectations about core collapse and neutrinos significantly.
- The direction of the neutrino events, studied e.g., in Ref. [58], seems less problematic than occasionally claimed.
- It has been widely recognized that the theoretical uncertainties in the mean energies are much larger than those estimated in the past, and therefore, it is not currently claimed that there is any serious incompatibility with the theory.
- A complete and systematic study of the energy spectra has verified the compatibility of the energy spectra and confirmed the stability and substantial validity [44] of a standard interpretation, as the one initially summarized by Bahcall. There is a well-defined region of the parameter space that allows the interpretation of the events as due to gravity collapse, as being due to a non-atypical gravitational collapse; the average energy of the antineutrinos is ${\overline{E}}_{\nu}=12\mathrm{M}\mathrm{e}\phantom{\rule{-0.21251pt}{0ex}}\mathrm{V}$ and the total radiated energy is of the order of $5\times {10}^{52}\mathrm{erg}$, with errors of 10% and 30%, respectively.
- The only problem that remains unsolved is the meaning of the 5 low-energy events seen by the Mont Blanc/LSD detector [51], which precede those seen by the other three detectors, and which do not seem easy to attribute to the supernova.

## 3. Parameterized Spectrum of Electronic Antineutrinos

#### 3.1. Generalities

#### 3.2. Model with Two Emission Phases

we will assume that at any given time, the flux can be described by a sum of the accretion and cooling components,

each of which is quantified by a temperature and an intensity of the emission (in the way discussed in the next section) each of which is a function of time.

#### 3.2.1. Emission from Thermal Cooling

#### 3.2.2. Emission from Processes around the Accretion Zone

#### 3.3. Expectations

- The luminosity, the number of irradiated neutrinos and signal rates in the detectors are much higher during the accretion phase than during the cooling phase. This result is consistent with what has been discussed in the previous literature [42,43,79], and it is due to the volumetric character of the accretion emission highlighted above.
- If the cooling luminosity is about the same for all six neutrino types of neutrinos and antineutrinos, then in about 10 s $3\times {10}^{53}$ $\mathrm{erg}$ will be extracted from the core of the star.
- The number of electron antineutrino events from the accretion phase, which we expect to last a fraction of a second, will be a bit smaller but comparable with that of the cooling phase.

#### 3.4. Remark on Neutrino Flavor Transformation

## 4. A Model for the Time Evolution

#### 4.1. Description of Luminosity

- the position of the maximum of the curve ${t}_{0}$;
- the two timescales that drive the decrease in luminosity, ${\mathsf{\tau}}_{a}$ and ${\mathsf{\tau}}_{c}$, for accretion and cooling emission, respectively.

#### 4.2. Description of the Flux

## 5. Tests and Applications

#### 5.1. Illustration of the Expected Flux

#### 5.2. Comparison with SN1987A

#### 5.3. Predictions

## 6. Variants and Possible Improvements

#### 6.1. Variants Concerning the Cooling Component

#### 6.2. Variants Concerning the Accretion Component

#### 6.3. Variants Concerning the Other Neutrino Flavors

## 7. Discussion and Outlook

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Detector Response

#### Appendix A.1. Kinematics

- The threshold of the reaction, which is obtained when ${E}_{\mathrm{e}}^{CM}={m}_{\mathrm{e}}$, is$${E}_{\mathrm{thr}}={\mathsf{\delta}}_{+}\phantom{\rule{2.em}{0ex}}\mathrm{where}\phantom{\rule{1.em}{0ex}}{\mathsf{\delta}}_{\pm}=\frac{{({m}_{\mathrm{n}}\pm {m}_{\mathrm{e}})}^{2}-{m}_{\mathrm{p}}^{2}}{2{m}_{\mathrm{p}}}.$$
- For ${E}_{\nu}>{E}_{\mathrm{thr}}$ we have$${E}_{\mathrm{e}}^{CM}=\frac{{E}_{\nu}-\mathsf{\delta}}{\sqrt{1+2{E}_{\nu}/{m}_{\mathrm{p}}}}\phantom{\rule{2.em}{0ex}}\mathrm{where}\phantom{\rule{1.em}{0ex}}\mathsf{\delta}=\frac{{m}_{\mathrm{n}}^{2}-{m}_{\mathrm{e}}^{2}-{m}_{\mathrm{p}}^{2}}{2{m}_{\mathrm{p}}}.$$
- The corresponding momentum ${p}_{\mathrm{e}}^{CM}$ can be written as$${p}_{\mathrm{e}}^{CM}=\sqrt{\frac{({E}_{\nu}-{\mathsf{\delta}}_{+})({E}_{\nu}-{\mathsf{\delta}}_{-})}{1+2{E}_{\nu}/{m}_{\mathrm{p}}}},$$

#### Appendix A.2. Description of Some Supernova Neutrino Telescopes

## References

- Murdin, P.; Murdin, L. Supernovae; Cambridge University Press: Cambridge, UK, 1978. [Google Scholar]
- Clark, D.H.; Stephenson, F.R. The Historical Supernovae; Pergamon: Oxford, UK, 1977. [Google Scholar]
- Woosley, S.E.; Weaver, T.A. The Evolution and explosion of massive stars. 2. Explosive hydrodynamics and nucleosynthesis. Astrophys. J. Suppl.
**1995**, 101, 181–235. [Google Scholar] [CrossRef] [Green Version] - Thielemann, F.K.; Nomoto, K.; Hashimoto, M.A. Core-Collapse Supernovae and Their Ejecta. Astrophys. J.
**1996**, 460, 408. [Google Scholar] [CrossRef] - Kobayashi, C.; Karakas, A.I.; Umeda, H. The Evolution of Isotope Ratios in the Milky Way Galaxy. Mon. Not. R. Astron. Soc.
**2011**, 414, 3231. [Google Scholar] [CrossRef] [Green Version] - Nomoto, K.; Kobayashi, C.; Tominaga, N. Nucleosynthesis in Stars and the Chemical Enrichment of Galaxies. Ann. Rev. Astron. Astrophys.
**2013**, 51, 457–509. [Google Scholar] [CrossRef] - Branch, D.; Wheeler, J.C. Supernova Explosions; Springer: Berlin/Heidelberg, Germany, 2017. [Google Scholar]
- Kobayashi, C.; Karakas, A.I.; Lugaro, M. The Origin of Elements from Carbon to Uranium. Astrophys. J.
**2020**, 900, 179. [Google Scholar] [CrossRef] - Bethe, H.A. Supernovae. Phys. Today
**1990**, 43, 24–27. [Google Scholar] [CrossRef] - Woosley, S.; Janka, T. The physics of core-collapse supernovae. Nat. Phys.
**2005**, 1, 147. [Google Scholar] [CrossRef] - Janka, H.T.; Langanke, K.; Marek, A.; Martinez-Pinedo, G.; Mueller, B. Theory of Core-Collapse Supernovae. Phys. Rep.
**2007**, 442, 38–74. [Google Scholar] [CrossRef] [Green Version] - Raffelt, G.G. Neutrinos and the stars. Proc. Int. Sch. Phys. Fermi
**2012**, 182, 61–143. [Google Scholar] [CrossRef] - Janka, H.T. Explosion Mechanisms of Core-Collapse Supernovae. Ann. Rev. Nucl. Part. Sci.
**2012**, 62, 407–451. [Google Scholar] [CrossRef] [Green Version] - Burrows, A. Colloquium: Perspectives on core-collapse supernova theory. Rev. Mod. Phys.
**2013**, 85, 245. [Google Scholar] [CrossRef] [Green Version] - Foglizzo, T.; Kazeroni, R.; Guilet, J.; Masset, F.; González, M.; Krueger, B.K.; Novak, J.; Oertel, M.; Margueron, J.; Faure, J.; et al. The explosion mechanism of core-collapse supernovae: Progress in supernova theory and experiments. Publ. Astron. Soc. Aust.
**2015**, 32, e009. [Google Scholar] [CrossRef] [Green Version] - Mirizzi, A.; Tamborra, I.; Janka, H.T.; Saviano, N.; Scholberg, K.; Bollig, R.; Hudepohl, L.; Chakraborty, S. Supernova Neutrinos: Production, Oscillations and Detection. La Rivista del Nuovo Cimento
**2016**, 39, 1–112. [Google Scholar] [CrossRef] - Janka, H.T. Neutrino-driven Explosions. In Handbook of Supernovae; Alsabti, A., Murdin, P., Eds.; Springer: Berlin/Heidelberg, Germany, 2017. [Google Scholar] [CrossRef] [Green Version]
- Janka, H.T. Neutrino Emission from Supernovae. In Handbook of Supernovae; Alsabti, A., Murdin, P., Eds.; Springer: Berlin/Heidelberg, Germany, 2017. [Google Scholar] [CrossRef] [Green Version]
- Horiuchi, S.; Kneller, J.P. What can be learned from a future supernova neutrino detection? J. Phys. G
**2018**, 45, 043002. [Google Scholar] [CrossRef] [Green Version] - Müller, B. Neutrino Emission as Diagnostics of Core-Collapse Supernovae. Ann. Rev. Nucl. Part. Sci.
**2019**, 69, 253–278. [Google Scholar] [CrossRef] [Green Version] - Mezzacappa, A.; Endeve, E.; Messer, O.E.B.; Bruenn, S.W. Physical, numerical, and computational challenges of modeling neutrino transport in core-collapse supernovae. Living Rev. Relativ.
**2020**, 6, 4. [Google Scholar] - Burrows, A.; Vartanyan, D. Core-Collapse Supernova Explosion Theory. Nature
**2021**, 589, 29–39. [Google Scholar] [CrossRef] [PubMed] - Ott, C. The Gravitational Wave Signature of Core-Collapse Supernovae. Class. Quantum Gravity
**2009**, 26, 063001. [Google Scholar] [CrossRef] - Fryer, C.L.; New, K.C.B. Gravitational waves from gravitational collapse. Living Rev. Relativ.
**2011**, 14, 1. [Google Scholar] [CrossRef] [Green Version] - Ott, C.D.; Abdikamalov, E.; Mösta, P.; Haas, R.; Drasco, S.; O’Connor, E.P.; Reisswig, C.; Meakin, C.A.; Schnetter, E. General-Relativistic Simulations of Three-Dimensional Core-Collapse Supernovae. Astrophys. J.
**2013**, 768, 115. [Google Scholar] [CrossRef] [Green Version] - Kuroda, T.; Takiwaki, T.; Kotake, K. Gravitational Wave Signatures from Low-mode Spiral Instabilities in Rapidly Rotating Supernova Cores. Phys. Rev. D
**2014**, 89, 044011. [Google Scholar] [CrossRef] [Green Version] - Radice, D.; Morozova, V.; Burrows, A.; Vartanyan, D.; Nagakura, H. Characterizing the Gravitational Wave Signal from Core-Collapse Supernovae. Astrophys. J. Lett.
**2019**, 876, L9. [Google Scholar] [CrossRef] [Green Version] - Arimoto, M.; Asada, H.; Cherry, M.L.; Fujii, M.S.; Fukazawa, Y.; Harada, A.; Hayama, K.; Hosokawa, T.; Ioka, K.; Itoh, Y.; et al. Gravitational Wave Physics and Astronomy in the Nascent Era. arXiv
**2021**, arXiv:2104.02445. [Google Scholar] - Janka, H.T.; Melson, T.; Summa, A. Physics of Core-Collapse Supernovae in Three Dimensions: a Sneak Preview. Ann. Rev. Nucl. Part. Sci.
**2016**, 66, 341–375. [Google Scholar] [CrossRef] [Green Version] - Müller, B. Hydrodynamics of core-collapse supernovae and their progenitors. Astrophysics
**2020**, 6, 3. [Google Scholar] [CrossRef] - Hirata, K.; Kajita, T.; Koshiba, M.; Nakahata, M.; Oyama, Y.; Sato, N.; Suzuki, A.; Takita, M.; Totsuka, Y.; Kifune, T.; et al. Observation of a Neutrino Burst from the Supernova SN 1987a. Phys. Rev. Lett.
**1987**, 58, 1490–1493. [Google Scholar] [CrossRef] [PubMed] - Hirata, K.; Kajita, T.; Koshiba, M.; Nakahata, M.; Oyama, Y.; Sato, N.; Suzuki, A.; Takita, M.; Totsuka, Y.; Kifune, T.; et al. Observation in the Kamiokande-II Detector of the Neutrino Burst from Supernova SN 1987a. Phys. Rev. D
**1988**, 38, 448–458. [Google Scholar] [CrossRef] - Bionta, R.; Blewitt, G.; Bratton, C.B.; Casper, D.; Ciocio, A.; Claus, R.; Cortez, B.; Crouch, M.; Dye, S.T.; Errede, S.; et al. Observation of a Neutrino Burst in Coincidence with Supernova SN 1987a in the Large Magellanic Cloud. Phys. Rev. Lett.
**1987**, 58, 1494. [Google Scholar] [CrossRef] [Green Version] - Bratton, C.B.; Casper, D.; Ciocio, A.; Claus, R.; Crouch, M.; Dye, S.T.; Errede, S.; Gajewski, W.; Goldhaber, M.; Haines, T.J.; et al. Angular Distribution of Events From Sn1987a. Phys. Rev. D
**1988**, 37, 3361. [Google Scholar] [CrossRef] - Alekseev, E.; Alekseeva, L.; Krivosheina, I.; Volchenko, V. Detection of the Neutrino Signal From SN1987A in the LMC Using the Inr Baksan Underground Scintillation Telescope. Phys. Lett. B
**1988**, 205, 209–214. [Google Scholar] [CrossRef] - Colgate, S.A.; Johnson, M.H. Hydrodynamic Origin of Cosmic Rays. Phys. Rev. Lett.
**1960**, 5, 235–238. [Google Scholar] [CrossRef] - Colgate, S.A.; White, R.H. The Hydrodynamic Behavior of Supernovae Explosions. Astrophys. J.
**1966**, 143, 626. [Google Scholar] [CrossRef] - Wilson, J.R. A Numerical Study of Gravitational Stellar Collapse. Astrophys. J.
**1971**, 163, 209. [Google Scholar] [CrossRef] - Nadyozhin, D. The neutrino radiation for the hot neutron star formation and the envelope outburst problem. Astrophys. Space Sci.
**1978**, 53, 131–153. [Google Scholar] [CrossRef] - Bethe, H.A.; Wilson, J.R. Revival of a stalled supernova shock by neutrino heating. Astrophys. J.
**1985**, 295, 14–23. [Google Scholar] [CrossRef] - Bahcall, J.N. Neutrino Astrophysics; Cambridge University Press: Cambridge, UK, 1989. [Google Scholar]
- Loredo, T.J.; Lamb, D.Q. Bayesian analysis of neutrinos observed from supernova SN-1987A. Phys. Rev. D
**2002**, 65, 063002. [Google Scholar] [CrossRef] [Green Version] - Pagliaroli, G.; Vissani, F.; Costantini, M.L.; Ianni, A. Improved analysis of SN1987A antineutrino events. Astropart. Phys.
**2009**, 31, 163–176. [Google Scholar] [CrossRef] [Green Version] - Vissani, F. Comparative analysis of SN1987A antineutrino fluence. J. Phys. G
**2015**, 42, 013001. [Google Scholar] [CrossRef] - Scholberg, K. Supernova Neutrino Detection. Ann. Rev. Nucl. Part. Sci.
**2012**, 62, 81–103. [Google Scholar] [CrossRef] [Green Version] - Rozwadowska, K.; Vissani, F.; Cappellaro, E. On the rate of core collapse supernovae in the milky way. New Astron.
**2021**, 83, 101498. [Google Scholar] [CrossRef] - Fukuda, S.; Fukuda, Y.; Hayakawa, T.; Ichihara, E.; Ishitsuka, M.; Itow, Y.; Kajita, T.; Kameda, J.; Kaneyuki, K.; Kasuga, S.; et al. The Super-Kamiokande detector. Nucl. Instrum. Methods A
**2003**, 501, 418–462. [Google Scholar] [CrossRef] - Beacom, J.F.; Vagins, M.R. GADZOOKS! Anti-neutrino spectroscopy with large water Cherenkov detectors. Phys. Rev. Lett.
**2004**, 93, 171101. [Google Scholar] [CrossRef] [Green Version] - Simpson, C.; Abe, K.; Bronner, C.; Hayato, Y.; Ikeda, M.; Ito, H.; Iyogi, K.; Kameda, J.; Kataoka, Y.; Kato, Y.; et al. Sensitivity of Super-Kamiokande with Gadolinium to Low Energy Anti-neutrinos from Pre-supernova Emission. Astrophys. J.
**2019**, 885, 133. [Google Scholar] [CrossRef] [Green Version] - Abbasi, R.; Abdou, Y.; Abu-Zayyad, T.; Ackermann, M.; Adams, J.; Aguilar, J.A.; Ahlers, M.; Allen, M.M.; Altmann, D.; Andeen, K.; et al. IceCube Sensitivity for Low-Energy Neutrinos from Nearby Supernovae. Astron. Astrophys.
**2011**, 535, A109, Erratum in**2014**, 563, C1. [Google Scholar] [CrossRef] [Green Version] - Dadykin, V.L.; Zatsepin, G.T.; Korchagin, V.B.; Korchagin, P.V.; Mal’Gin, A.S.; Ryazhskaya, O.G.; Ryasny, V.G.; Talochkin, V.P.; Khal’Chukov, F.F.; Yakushev, V.F.; et al. Detection of a Rare Event on 23 February 1987 by the Neutrino Radiation Detector under Mont Blanc. JETP Lett.
**1987**, 45, 593–595. [Google Scholar] - Ryazhskaya, O.G. Problems of Neutrino Radiation from SN 1987A: 30 Years Later. Phys. At. Nucl.
**2018**, 81, 113–119. [Google Scholar] [CrossRef] - Galeotti, P.; Pizzella, G. New analysis for the correlation between gravitational wave and neutrino detectors during SN1987A. Eur. Phys. J. C
**2016**, 76, 426. [Google Scholar] [CrossRef] [Green Version] - Vissani, F.; Costantini, M.L.; Fulgione, W.; Ianni, A.; Pagliaroli, G. What is the issue with SN1987A neutrinos? Ital. Phys. Soc. Proc.
**2011**, 103, 611–619. [Google Scholar] - Orlando, S. Linking Core-Collapse Supernova Explosions to Supernova Remnants through 3D MHD Modeling: The Case of SN 1987A. Talk at the Workshop “Core-Collapse Supernovae in the Multi-Messenger Era”, L’Aquila, Italy, 2–3 July 2018. [Google Scholar]
- Ono, M.; Nagataki, S.; Ferrand, G.; Takahashi, K.; Umeda, H.; Yoshida, T.; Orlando, S.; Miceli, M. Matter Mixing in Aspherical Core-collapse Supernovae: Three-dimensional Simulations with Single Star and Binary Merger Progenitor Models for SN 1987A. Astrophys. J.
**2020**, 888, 111. [Google Scholar] [CrossRef] - Page, D.; Beznogov, M.V.; Garibay, I.; Lattimer, J.M.; Prakash, M.; Janka, H.T. NS 1987A in SN 1987A. Astrophys. J.
**2020**, 898, 125. [Google Scholar] [CrossRef] - Costantini, M.L.; Ianni, A.; Vissani, F. SN1987A and the properties of neutrino burst. Phys. Rev. D
**2004**, 70, 043006. [Google Scholar] [CrossRef] [Green Version] - Pantaleone, J.T. Neutrino oscillations at high densities. Phys. Lett. B
**1992**, 287, 128–132. [Google Scholar] [CrossRef] - Samuel, S. Neutrino oscillations in dense neutrino gases. Phys. Rev. D
**1993**, 48, 1462–1477. [Google Scholar] [CrossRef] - Sigl, G.; Raffelt, G. General kinetic description of relativistic mixed neutrinos. Nucl. Phys. B
**1993**, 406, 423–451. [Google Scholar] [CrossRef] - Dighe, A.S.; Smirnov, A.Y. Identifying the neutrino mass spectrum from the neutrino burst from a supernova. Phys. Rev. D
**2000**, 62, 033007. [Google Scholar] [CrossRef] [Green Version] - Duan, H.; Kneller, J.P. Neutrino flavour transformation in supernovae. J. Phys. G
**2009**, 36, 113201. [Google Scholar] [CrossRef] [Green Version] - Duan, H.; Fuller, G.M.; Qian, Y.Z. Collective Neutrino Oscillations. Ann. Rev. Nucl. Part. Sci.
**2010**, 60, 569–594. [Google Scholar] [CrossRef] [Green Version] - Volpe, C. Neutrino Quantum Kinetic Equations. Int. J. Mod. Phys. E
**2015**, 24, 1541009. [Google Scholar] [CrossRef] [Green Version] - Glas, R.; Janka, H.T.; Capozzi, F.; Sen, M.; Dasgupta, B.; Mirizzi, A.; Sigl, G. Fast Neutrino Flavor Instability in the Neutron-star Convection Layer of Three-dimensional Supernova Models. Phys. Rev. D
**2020**, 101, 063001. [Google Scholar] [CrossRef] [Green Version] - Tamborra, I.; Shalgar, S. New Developments in Flavor Evolution of a Dense Neutrino Gas Annual. Rev. Nucl. Part. Sci.
**2020**, 71, 165–188. [Google Scholar] [CrossRef] - Wolfenstein, L. Neutrino Oscillations in Matter. Phys. Rev. D
**1978**, 17, 2369–2374. [Google Scholar] [CrossRef] - Mikheyev, S.P.; Smirnov, A.Y. Resonance Amplification of Oscillations in Matter and Spectroscopy of Solar Neutrinos. Sov. J. Nucl. Phys.
**1985**, 42, 913–917. [Google Scholar] - Bahcall, J.N.; Krastev, P.I.; Smirnov, A.Y. Is large mixing angle MSW the solution of the solar neutrino problems? Phys. Rev. D
**1999**, 60, 093001. [Google Scholar] [CrossRef] [Green Version] - Minakata, H.; Nunokawa, H. Inverted hierarchy of neutrino masses disfavored by supernova 1987A. Phys. Lett. B
**2001**, 504, 301–308. [Google Scholar] [CrossRef] [Green Version] - Barger, V.; Marfatia, D.; Wood, B.P. Supernova 1987A did not test the neutrino mass hierarchy. Phys. Lett. B
**2002**, 532, 19–28. [Google Scholar] [CrossRef] [Green Version] - Minakata, H.; Nunokawa, H.; Tomas, R.; Valle, J.W.F. Probing supernova physics with neutrino oscillations. Phys. Lett. B
**2002**, 542, 239–244. [Google Scholar] [CrossRef] [Green Version] - Minakata, H.; Nunokawa, H.; Tomas, R.; Valle, J.W.F. Parameter Degeneracy in Flavor-Dependent Reconstruction of Supernova Neutrino Fluxes. JCAP
**2008**, 12, 006. [Google Scholar] [CrossRef] - Nagakura, H.; Hotokezaka, K. Non-thermal neutrinos created by shock acceleration in successful and failed core-collapse supernova. Mon. Not. R. Astron. Soc.
**2021**, 502, 89–107. [Google Scholar] [CrossRef] - Mirizzi, A.; Raffelt, G.G. New analysis of the sn 1987a neutrinos with a flexible spectral shape. Phys. Rev. D
**2005**, 72, 063001. [Google Scholar] [CrossRef] [Green Version] - Lunardini, C. The diffuse supernova neutrino flux, supernova rate and sn1987a. Astropart. Phys.
**2006**, 26, 190–201. [Google Scholar] [CrossRef] [Green Version] - Buras, R.; Rampp, M.; Janka, H.T.; Kifonidis, K. Two-dimensional hydrodynamic core-collapse supernova simulations with spectral neutrino transport. 1. Numerical method and results for a 15 solar mass star. Astron. Astrophys.
**2006**, 447, 1049–1092. [Google Scholar] [CrossRef] [Green Version] - Vissani, F.; Pagliaroli, G. How much can we learn from SN1987A events? Or: An analysis with a two-Component model for the antineutrino signal. In Proceedings of the 4th International Workshop on Neutrino Oscillations in Venice: Ten Years after the Neutrino Oscillations, Venice, Italy, 15–18 April 2008. [Google Scholar]
- Smarr, L.; Wilson, J.R.; Barton, R.T.; Bowers, R.L. Rayleigh-taylor overturn in supernova core collapse. Astrophys. J.
**1981**, 246, 515–525. [Google Scholar] [CrossRef] - Fantini, G.; Gallo Rosso, A.; Vissani, F.; Zema, V. Introduction to the Formalism of Neutrino Oscillations. Adv. Ser. Dir. High Energy Phys.
**2018**, 28, 37–119. [Google Scholar] [CrossRef] - Capozzi, F.; Di Valentino, E.; Lisi, E.; Marrone, A.; Melchiorri, A.; Palazzo, A. The unfinished fabric of the three neutrino paradigm. arXiv
**2021**, arXiv:2107.00532. [Google Scholar] - Pagliaroli, G.; Vissani, F.; Coccia, E.; Fulgione, W. Neutrinos from Supernovae as a Trigger for Gravitational Wave Search. Phys. Rev. Lett.
**2009**, 103, 031102. [Google Scholar] [CrossRef] [Green Version] - Halzen, F.; Raffelt, G.G. Reconstructing the supernova bounce time with neutrinos in IceCube. Phys. Rev. D
**2009**, 80, 087301. [Google Scholar] [CrossRef] [Green Version] - Iida, T. Search for Supernova Relic Neutrino at Super-Kamiokande. Ph.D. Thesis, The University of Tokyo, Tokyo, Japan, 2010. [Google Scholar]
- Kowarik, T.; Griesel, T.; Piegsa, A. Supernova Search with the AMANDA / IceCube Detectors. In Proceedings of the 31st ICRC, Lodz, Poland, 7–15 July 2009. [Google Scholar]
- An, F.; An, G.; An, Q.; Antonelli, V.; Baussan, E.; Beacom, J.; Bezrukov, L.; Blyth, S.; Brugnera, R.; Avanzini, M.B.; et al. Neutrino Physics with JUNO. J. Phys. G
**2016**, 43, 030401. [Google Scholar] [CrossRef] - Abi, B.; Acciarri, R.; Acero, M.A.; Adamov, G.; Adams, D.; Adinolfi, M.; Ahmad, Z.; Ahmed, J.; Alion, T.; Monsalve, S.A.; et al. Supernova neutrino burst detection with the Deep Underground Neutrino Experiment. Eur. Phys. J. C
**2021**, 81, 423. [Google Scholar] [CrossRef] - Aiello, S.; Albert, A.; Garre, S.A.; Aly, Z.; Ambrosone, A.; Ameli, F.; Andre, M.; Androulakis, G.; Anghinolfi, M.; Anguita, M.; et al. The KM3NeT potential for the next core-collapse supernova observation with neutrinos. Eur. Phys. J. C
**2021**, 81, 445. [Google Scholar] [CrossRef] - Monzani, M.E. Supernova neutrino detection in Borexino. Il Nuovo Cimento C
**2006**, 29, 269–280. [Google Scholar] [CrossRef] - Lang, R.F.; McCabe, C.; Reichard, S.; Selvi, M.; Tamborra, I. Supernova neutrino physics with xenon dark matter detectors: A timely perspective. Phys. Rev. D
**2016**, 94, 103009. [Google Scholar] [CrossRef] [Green Version] - Gallo Rosso, A. Supernova neutrino fluxes in HALO-1kT, Super-Kamiokande, and JUNO. JCAP
**2021**, 06, 046. [Google Scholar] [CrossRef] - Kistler, M.D.; Yuksel, H.; Ando, S.; Beacom, J.F.; Suzuki, Y. Core-Collapse Astrophysics with a Five-Megaton Neutrino Detector. Phys. Rev. D
**2011**, 83, 123008. [Google Scholar] [CrossRef] [Green Version] - Burrows, A.; Klein, D.; Gandhi, R. The Future of supernova neutrino detection. Phys. Rev. D
**1992**, 45, 3361–3385. [Google Scholar] [CrossRef] [PubMed] - Skadhauge, S.; Zukanovich Funchal, R. Determining neutrino and supernova parameters with a galactic supernova. JCAP
**2007**, 04, 014. [Google Scholar] [CrossRef] [Green Version] - Keehn, J.G.; Lunardini, C. Neutrinos from failed supernovae at future water and liquid argon detectors. Phys. Rev. D
**2012**, 85, 043011. [Google Scholar] [CrossRef] [Green Version] - Lujan-Peschard, C.; Pagliaroli, G.; Vissani, F. Spectrum of Supernova Neutrinos in Ultra-pure Scintillators. JCAP
**2014**, 07, 051. [Google Scholar] [CrossRef] [Green Version] - Gallo Rosso, A.; Vissani, F.; Volpe, M.C. What can we learn on supernova neutrino spectra with water Cherenkov detectors? JCAP
**2018**, 4, 040. [Google Scholar] [CrossRef] [Green Version] - Nagakura, H. Retrieval of energy spectra for all flavours of neutrinos from core-collapse supernova with multiple detectors. Mon. Not. R. Astron. Soc.
**2020**, 500, 319–332. [Google Scholar] [CrossRef] - Al Kharusi, S.; BenZvi, S.Y.; Bobowski, J.S.; Bonivento, W.; Brdar, V.; Brunner, T.; Caden, E.; Clark, M.; Coleiro, A.; Colomer-Molla, M.; et al. SNEWS 2.0: A next-generation supernova early warning system for multi-messenger astronomy. New J. Phys.
**2021**, 23, 031201. [Google Scholar] [CrossRef] - Abbott, L.F.; De Rujula, A.; Walker, T.P. Constraints on the Neutrino Mass From the Supernova Data: A Systematic Analysis. Nucl. Phys. B
**1988**, 299, 734–756. [Google Scholar] [CrossRef] - Nardi, E.; Zuluaga, J.I. Exploring the sub-eV neutrino mass range with supernova neutrinos. Phys. Rev. D
**2004**, 69, 103002. [Google Scholar] [CrossRef] [Green Version] - Pagliaroli, G.; Rossi-Torres, F.; Vissani, F. Neutrino mass bound in the standard scenario for supernova electronic antineutrino emission. Astropart. Phys.
**2010**, 33, 287–291. [Google Scholar] [CrossRef] [Green Version] - Leonor, I.; Cadonati, L.; Coccia, E.; D’Antonio, S.; Di Credico, A.; Fafone, V.; Frey, R.; Fulgione, W.; Katsavounidis, E.; Ott, C.D.; et al. Searching for prompt signatures of nearby core-collapse supernovae by a joint analysis of neutrino and gravitational-wave data. Class. Quantum Gravity
**2010**, 27, 084019. [Google Scholar] [CrossRef] [Green Version] - Abbott, B.P.; Abbott, R.; Abbott, T.D.; Abernathy, M.R.; Acernese, F.; Ackley, K.; Adams, C.; Adams, T.; Addesso, P.; Adhikari, R.X.; et al. A First Targeted Search for Gravitational-Wave Bursts from Core-Collapse Supernovae in Data of First-Generation Laser Interferometer Detectors. Phys. Rev. D
**2016**, 94, 102001. [Google Scholar] [CrossRef] [Green Version] - Abdikamalov, E.; Pagliaroli, G.; Radice, D. Gravitational Waves from Core-Collapse Supernovae. arXiv
**2020**, arXiv:2010.04356. [Google Scholar] - Abe, K.; Adrich, P.; Aihara, H.; Akutsu, R.; Alekseev, I.; Ali, A.; Ameli, F.; Anghel, I.; Anthony, L.H.V.; Antonova, M.; et al. Supernova Model Discrimination with Hyper-Kamiokande. Astrophys. J.
**2021**, 916, 15. [Google Scholar] [CrossRef] - Strumia, A.; Vissani, F. Precise quasielastic neutrino/nucleon cross-section. Phys. Lett. B
**2003**, 564, 42–54. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**(

**a**) Sketch of one typical shape of the antineutrino luminosity, as indicated by a simulation by the Garching Group, as reported in Ref. [16]. After a fast growth phase (red), there is a very intense emission lasting a fraction of a second (highlighted in orange) followed by a less intense, slowly decreasing and long-lasting emission (from yellow to blue). (

**b**) Conceptual diagram for the emission. Around the nascent neutron star there are many emitting centres, due to the reaction between positrons and neutrons. From Ref. [79].

**Figure 2.**Expected neutrino flux (1), differential in time and neutrino energy, from Equations (9) and (11), given the parameters (33) and the model emission (28) and (29). The reference distance $D=10\mathrm{k}\mathrm{pc}$ is assumed. (

**a**) First second, flux in linear scale. (

**b**) First 100 s, time in logarithmic scale.

**Figure 4.**Top row: (

**a**) Time position of the events from SN1987A, as observed in Kamiokande-II (black vertical lines) compared with the differential counting rate predicted in the model (blue curve, signal + background) for the value of the parameters indicated in the text [42,43,44]. (

**b**) Same but with respect to the energy. Bottom row: Cumulative distribution functions (CDF) for the same flux, regarded as a model of SN1987A emission. We show in black the data, in blue the theoretical expectation. (

**c**) Cumulative time distributions (${t}_{\mathrm{off}}=50$ ms, p-value = 56%). (

**d**) Cumulative energy distribution (p-value 51%).

**Figure 5.**Expected differential counting rate of electron antineutrino events in Super–Kamiokande differential in neutrino energy, as per definitions (5) and (6), given the parameters (33) and the model emission (28) and (29). The reference distance $D=10\mathrm{k}\mathrm{pc}$ is assumed. (

**a**) The first second of emission; each contour marks steps of 25 s

^{‒1}MeV

^{‒1}. (

**b**) Global distribution. Note the similarity with the flux, as shown in Figure 2.

**Figure 6.**Counting rate in IceCube, as expected from model emission (28) and (29) given the parameters (33). The gray line marks the 30 $\mathrm{k}$$\mathrm{Hz}$ threshold given by background fluctuations in $\mathsf{\delta}t=1.6384\mathrm{m}\mathrm{s}$ time bins (see Appendix A.2). Note the similarity with the luminosity curve shown in Figure 3, driven by the similarities in the definitions (4) and (6).

**Table 1.**Reference values for luminosity ($\mathcal{L}$), number of neutrinos per second (${\dot{N}}_{\nu}$), average energy (${\overline{E}}_{\nu}$), rates in detectors (${\mathcal{R}}_{\mathrm{SK}}$, ${\mathcal{R}}_{\mathrm{Ice}3}$), for the two phases of cooling and of accretion defined in Section 3.2. All the quantities are referred to as the electron antineutrinos (${\nu}_{\mathrm{e}}$) species. For the given quantities, we specify benchmark values and power law indices, as defined in Equations (15) and (17) respectively.

${\mathcal{L}}^{\u2605}$ | ${\dot{\mathit{N}}}_{\mathit{\nu}}^{\u2605}$ | ${\overline{\mathit{E}}}_{\mathit{\nu}}^{\u2605}$ | ${\mathcal{R}}_{\mathbf{SK}}^{\u2605}$ | ${\mathcal{R}}_{\mathbf{Ice}3}^{\u2605}$ | |
---|---|---|---|---|---|

benchmark values [equation (15)] | |||||

[$\mathrm{erg}$/$\mathrm{s}$] | [${\overline{\nu}}_{\mathrm{e}}$/$\mathrm{s}$] | [$\mathrm{M}\mathrm{e}\phantom{\rule{-0.21251pt}{0ex}}\mathrm{V}$] | [$\mathrm{Hz}$] | [$\mathrm{Hz}$] | |

cooling | $5.2\times {10}^{51}$ | $2.3\times {10}^{56}$ | $14.2$ | $6.7\times {10}^{2}$ | $7.9\times {10}^{4}$ |

accretion | $5.0\times {10}^{52}$ | $2.4\times {10}^{57}$ | $13.0$ | $5.2\times {10}^{3}$ | $4.7\times {10}^{5}$ |

power law indices [equation (17)] | |||||

cooling | $4.0$ | $3.0$ | $1.0$ | $5.1$ | $6.0$ |

accretion | $5.5$ | $4.6$ | $0.9$ | $6.7$ | $7.5$ |

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Vissani, F.; Gallo Rosso, A.
On the Time Distribution of Supernova Antineutrino Flux. *Symmetry* **2021**, *13*, 1851.
https://doi.org/10.3390/sym13101851

**AMA Style**

Vissani F, Gallo Rosso A.
On the Time Distribution of Supernova Antineutrino Flux. *Symmetry*. 2021; 13(10):1851.
https://doi.org/10.3390/sym13101851

**Chicago/Turabian Style**

Vissani, Francesco, and Andrea Gallo Rosso.
2021. "On the Time Distribution of Supernova Antineutrino Flux" *Symmetry* 13, no. 10: 1851.
https://doi.org/10.3390/sym13101851