Supernova neutrino signals based on long-term axisymmetric simulations
MMNRAS , 1–19 (2020) Preprint 24 February 2021 Compiled using MNRAS L A TEX style file v3.0
Supernova neutrino signals based on long-term axisymmetricsimulations
Hiroki Nagakura (cid:63) , Adam Burrows , David Vartanyan Department of Astrophysical Sciences, Princeton University, 4 Ivy Lane, Princeton, NJ 08544, USA Astronomy Department and Theoretical Astrophysics Center, University of California, Berkeley, CA 94720, USA
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
We study theoretical neutrino signals from core-collapse supernova (CCSN) computed using axisymmetric CCSNsimulations that cover the post-bounce phase up to ∼ ∼ × erg. By combining the recent constraints on the equation-of-state, we further estimate the gravitationalmass of PNS in the remnant of SN 1987A, which is ∼ . M (cid:12) . Key words: neutrinos - supernovae: general.
Core-collapse supernovae (CCSNe) are catastrophic explo-sions of massive stars ( (cid:38) M (cid:12) ) and cosmic factories of neu-tron stars (NSs) and black holes (BHs). The physical stateof a NS has been a mystery since the discovery of pulsars(Hewish et al. 1968). Observations suggest that there is a di-versity in the microscopic and macroscopic physics character-izing NSs, depending on the mass, spin, isospin-asymmetry,and temperature (see, e.g., ¨Ozel & Freire 2016). The forma-tion processes during CCSN explosions seem to account forthis diversity. The next nearby CCSN is expected to pro-vide detailed information on the dynamical features in theexplosion mechanism and NS (or BH) formation process vianeutrinos and gravitational waves. This has motivated multi-decade efforts to develop realistic theoretical models. It re-quires, however, large-scale numerical simulations of CCSN,incorporating multi-scale and multi-physics processes. It isalso essential that these simulations need to cover long-termpost-bounce evolution for a wide range of progenitors to de-velop a comprehensive understanding of CCSN dynamics andthe formation process of compact remnants, which is still a (cid:63) E-mail: [email protected] grand challenge in computational astrophysics. This paper isdevoted to this effort.Theoretically, there is an indication that fluid-dynamics,nucleosynthesis, and neutrinos/gravitational waves emis-sions, which are key ingredients in CCSN physics, stronglydepend upon dimension, except for the very light progen-itors (perhaps (cid:46) M (cid:12) ). This insight is due to the re-markable progress in multi-dimensional (multi-D) modelingof CCSN during the last decades. Three-dimensional (3D)models (see, e.g., Lentz et al. 2015; Roberts et al. 2016;O’Connor & Couch 2018; Kuroda et al. 2018; Vartanyanet al. 2019a; Nagakura et al. 2019a; M¨uller et al. 2019; Glaset al. 2019; Walk et al. 2019; Burrows et al. 2020; Nagakuraet al. 2020; Iwakami et al. 2020; Pan et al. 2020; Bollig et al.2020) are nowadays available boasting different implemen-tations of the input physics. In the meantime, great efforthas been made in axisymmetric (2D) models with full Boltz-mann (multi-energy, multi-angle and multi-species) neutrinotransport (Nagakura et al. 2018, 2019b; Harada et al. 2020),covering various types of progenitors (Nakamura et al. 2015;Summa et al. 2016; Burrows & Vartanyan 2021), and long-term ( > © a r X i v : . [ a s t r o - ph . H E ] F e b H. Nagakura et al. as turbulence and convection), they are much more realis-tic than those in spherical symmetry. For instance, we havewitnessed that they show similar explodability and neutrinoemission characteristics to those in 3D (see, e.g., Vartanyanet al. 2019a; Nagakura et al. 2019a) . This motivates us to uselong-term 2D models to study some CCSN physics of which3D models are not yet available.Recently we conducted a comprehensive study of neutrinosignals based on our 3D CCSN models (Nagakura et al. 2021),in which theoretical neutrino signals from CCSNe in somerepresentative terrestrial detectors were provided. However,our previous study was limited to the early post-bounce phase( (cid:46) ∼ But see also (Hanke et al. 2012; Burrows et al. 2012; Couch 2013;Nagakura et al. 2013; Takiwaki et al. 2014) for the argument re-garding the difference of explodability between 2D and 3D models. progenitor continuum, and these are scrutinized in this paper.We also provide some useful fitting formulae which can be di-rectly applied in real observations to estimate the total neu-trino energy (TONE) emitted at CCSN sources from purelyobserved quantities. We demonstrate how our approach canbe used in real observations by analyzing the neutrino datafrom SN 1987A. By combining other recent observational con-straints regarding NS equation-of-state (EOS), we place aconstraint on the mass of a NS in SN1987A. It should also bementioned that our neutrino data are publicly available (aswell as our 3D models) . These data will be useful for moredetailed detector simulations and to develop new methodsand pipelines with which to analyze neutrino signals.This paper is organized as follows. We first describe someessential information on our methods and models in Sec. 2.Sec. 3 corresponds to the body of this paper, in which allresults are encapsulated. Finally, we conclude this paper witha summary and discussion in Sec. 4. We briefly summarize our 2D CCSN models. The simula-tions were carried out by using our neutrino-radiation hy-drodynamic code F ornax , which was designed to capturerealistic multi-D features of the dynamics by incorporatingup-to-date input physics and numerical techniques. The neu-trino transport is calculated using a multi-energy and multi-species two-moment (M1) scheme with essential neutrino-matter interactions (Burrows et al. 2006) including some ad-vanced rates with recoils/weak-magnetism (Horowitz 2002)and axial-vector many body corrections to neutrino-nucleonscattering (Horowitz et al. 2017). Fluid-velocity dependenceand general relativistic corrections are included to least orderin the neutrino transport; for hydrodynamics, a general rela-tivistic correction is added in a monopole-term of the gravita-tional potential following the method in Marek et al. (2006).We refer readers to Skinner et al. (2019) for more details onthe characteristics and capabilities of the code.During the last few years, we have investigated many as-pects of CCSN dynamics by performing CCSN simulations inboth 2D (Skinner et al. 2016; Radice et al. 2017; Vartanyanet al. 2018) and 3D (Vartanyan et al. 2019a; Burrows et al.2019, 2020; Vartanyan et al. 2019b; Nagakura et al. 2019a,2020; Vartanyan & Burrows 2020). In our new 2D simula-tions, the same input physics is employed, but we simulate to ∼ Ni mass , NS mass, and explosive nu-cleosynthesis. This paper focuses on a detailed analysis of theneutrino signals based on these 2D models. from the link, . Sawada & Suwa (2020), suggesting that the simulations to atleast (cid:38) Ni mass.MNRAS , 1–19 (2020)
CSN neutrinos by long-term 2D simulations A ve r a g e s ho ck r a d i u s [ k m ] Time [s]12 M ⊙
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The time trajectory of angle-averaged shock radii for ourCCSN models. The solid and dotted lines represent models whicheither succeed or fail, respectively.
We started our CCSN simulations from the onset of gravi-tational collapse. We employed matter profiles at the presu-pernova phase computed in Sukhbold et al. (2018) as initialconditions. We note that the progenitor models are differentfrom those used in our previous 3D CCSN simulations (Bur-rows et al. 2020), except for the 25 M (cid:12) model. In this study,we analyze 15 models over a mass range of 12 to 26 . M (cid:12) .It should be mentioned that the models with low-mass pro-genitors are not considered here, since the multi-D effects areless prominent for them (see also Nagakura et al. 2021). Wewitnessed successful explosions in most of our CCSN models,except for 12 and 15 M (cid:12) progenitors, seen in the time evolu-tion of the angle-averaged shock radii displayed in Fig. 1. Forall exploding models, shock revival occurs at < . . The primary cause of the failure ofshock revival for the non-exploding models seems to be theirless prominent Si/O interfaces and their shallower initial den-sity profiles (see Fig. 1 in Burrows & Vartanyan 2021). Thistrend is consistent with what we found in our previous study(Vartanyan et al. 2018).These self-consistent simulations enable us to identify theingredients that characterize the neutrino signals and theirdimensional dependence. In our neutrino analysis based on3D models (Nagakura et al. 2021), we concluded that PNSconvection is one of the major elements leading to differ-ences with the 1D models in their neutrino signals. Thereis a caveat, however; the conclusion may be valid only inearly post-bounce phase ( (cid:46) We note, however, that the non-exploding progenitors behavedifferently (see, e.g., Burrows et al. 2019). P N S B a r y on - M ass [ M ⊙ ] Time [s]
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The time evolution of the baryon-mass of the PNS. Sim-ilar to Fig. 1, the solid and dashed lines distinguish the explosionand non-explosion models. signals. It should be stressed that the dynamical feature ofmass accretion onto PNS in multi-D is qualitatively differ-ent from that in 1D (see also Wongwathanarat et al. 2010).This is also associated with how and when the shock waveis revived, indicating that the neutrino emissions bear thestamp of the matter dynamics in post-shock-revival phases.As we will discuss in Sec. 3.1, the temporal variation in neu-trino signals is the one that contains such a hydrodynamicalinformation.Importantly, the non-explosion outcomes in the 12 and15 M (cid:12) models should not necessarily be considered thedefinitive answer concerning their explodability. This is sim-ply because there still remain uncertainties in CCSN simula-tions. One is the stellar evolution model, its rotational, mag-netic field, and multi-D stellar profiles. Another is the needfor improved input physics regarding general relativity, multi-angle neutrino transport, neutrino oscillation, and neutrino-matter interactions. Numerical methods and grid and groupresolution also affect the final outcome. Therefore, we need tokeep in mind such uncertainties as we proceed with the follow-ing analysis. Nevertheless, our simulations provide up-to-dateCCSN models, and non-exploding models still capture someessential characteristics. Based on our 2D CCSN models, we estimate event countsin some representative terrestrial neutrino detectors. Themethod is essentially the same as that used in our neutrinoanalysis of 3D CCSN models. We refer readers to Nagakuraet al. (2021) for the details of our method , and we brieflydescribe the essential elements of our method here.We employ the detector software, SNOwGLoBES , to es-timate the neutrino counts. In SNOwGLobes, cross section See also Seadrow et al. (2018), although the analysis pipeline isdifferent from that used in this paper. The software is available at https://webhome.phy.duke.edu/~schol/snowglobes/ . MNRAS000
The time evolution of the baryon-mass of the PNS. Sim-ilar to Fig. 1, the solid and dashed lines distinguish the explosionand non-explosion models. signals. It should be stressed that the dynamical feature ofmass accretion onto PNS in multi-D is qualitatively differ-ent from that in 1D (see also Wongwathanarat et al. 2010).This is also associated with how and when the shock waveis revived, indicating that the neutrino emissions bear thestamp of the matter dynamics in post-shock-revival phases.As we will discuss in Sec. 3.1, the temporal variation in neu-trino signals is the one that contains such a hydrodynamicalinformation.Importantly, the non-explosion outcomes in the 12 and15 M (cid:12) models should not necessarily be considered thedefinitive answer concerning their explodability. This is sim-ply because there still remain uncertainties in CCSN simula-tions. One is the stellar evolution model, its rotational, mag-netic field, and multi-D stellar profiles. Another is the needfor improved input physics regarding general relativity, multi-angle neutrino transport, neutrino oscillation, and neutrino-matter interactions. Numerical methods and grid and groupresolution also affect the final outcome. Therefore, we need tokeep in mind such uncertainties as we proceed with the follow-ing analysis. Nevertheless, our simulations provide up-to-dateCCSN models, and non-exploding models still capture someessential characteristics. Based on our 2D CCSN models, we estimate event countsin some representative terrestrial neutrino detectors. Themethod is essentially the same as that used in our neutrinoanalysis of 3D CCSN models. We refer readers to Nagakuraet al. (2021) for the details of our method , and we brieflydescribe the essential elements of our method here.We employ the detector software, SNOwGLoBES , to es-timate the neutrino counts. In SNOwGLobes, cross section See also Seadrow et al. (2018), although the analysis pipeline isdifferent from that used in this paper. The software is available at https://webhome.phy.duke.edu/~schol/snowglobes/ . MNRAS000 , 1–19 (2020)
H. Nagakura et al. and detector responses in various types of detectors and re-action channels are provided. Assuming a distance to theCCSN source for a neutrino oscillation model (see Sec. 2.3),our analysis starts by constructing mock data of flavor- andenergy-dependent neutrino fluxes (fluences) at the Earth byusing the neutrino data from our 2D CCSN simulations. Wefocus only on the angle-averaged neutrino signals, which arevery similar as those in our 3D models (see, e.g., Vartanyanet al. 2019a; Nagakura et al. 2019a), indicating that they areappropriate stand-ins for 3D. On the other hand, the angularvariation in the neutrino signals of our 2D models is not asaccurate as in 3D, since the 2D simulations are artificiallyaxisymmetric. Hence, we postpone a detailed analysis of theangular dependence of neutrino signals until long-term 3Dsimulations are available. We also refer readers to Nagakuraet al. (2021), in which the detailed analysis of the angulardependence in the early post-bounce phase ( (cid:46) ornax , is equipped with multi-energy (spectral) and multi-species neutrino transport, theenergy spectrum of each flavor of neutrinos can be obtainedwithout any additional prescriptions . Note that our CCSNsimulations do not distinguish mu- and tau- neutrinos (andtheir anti-particles), which are, hence, collectively treated as“heavy” leptonic neutrinos ( ν x ) in the signal analysis . Itshould be noted, however, that they are not identical in re-ality due to slightly different neutrino-matter interactions.Indeed, the deviation slightly increases with energy. The col-lective treatment of heavy leptonic neutrinos is, however, areasonable approximation for (cid:46)
50 MeV neutrinos. We notethat the detection of CCSN neutrinos will be dominated byneutrinos in the energy range of (cid:46)
20 MeV, indicating thatour bundling approach captures all qualitative trends in theneutrino signals. As a final remark, we note that we focusonly on the major interaction channel each detector, whichis enough to see the overall trends in the neutrino signal. Itshould be mentioned that other channels would be importantfor the analysis of high energy neutrinos; we refer readers toNagakura & Hotokezaka (2020) for the analyses in the energyrange of >
50 MeV and with for other reaction channels.In this study, we consider four (five) representative ter-restrial detectors: Super-Kamiokande (SK) (Abe et al. 2016)or Hyper-Kamiokande (HK) (Hyper-Kamiokande Proto-Collaboration et al. 2018), the deep underground neutrinoexperiment (DUNE) (Acciarri et al. 2016; Ankowski et al.2016; Abi et al. 2020), the Jiangmen Underground Neutrino In this paper, we assume that the distance is 10 kpc, unlessotherwise stated. The energy spectrum of neutrinos from CCSN has been usuallyfit by a Gamma distribution with an average energy and a pinchingparameter (see, e.g., Keil et al. 2003), which is very useful for thespectrum analysis of neutrino signals with statistical methods (see,e.g., Barger et al. 2002; Minakata 2002; Minakata et al. 2008; GalloRosso et al. 2017; Laha & Beacom 2014; Lu et al. 2016; Gallo Rossoet al. 2018; Nikrant et al. 2018). However, such an analytic fit dis-cards fine structures in the energy spectrum; hence, raw neutrinodata of energy spectra extracted from CCSN simulations would bepreferable. Note that we do distinguish the heavy leptonic neutrinos fromtheir anti-partners at the Earth. This is because they undergo dif-ferent flavor conversions (see also Sec. 2.3).
Observatory (JUNO) (An et al. 2016), and IceCube (Abbasiet al. 2011). SK is currently in operation, and others are com-ing online and will be available in several years. A reactionchannel with inverse beta decay on proton channel (IBD-p):¯ ν e + p → e + + n, (1)is a major reaction channel for neutrinos from CCSN on SK(HK), JUNO, and IceCube. DUNE is, on the other hand,sensitive to ν e through a charged-current reaction channelwith Argon (CCAre): ν e + Ar → e − + K ∗ , (2)which is the major channel for the detector. We consider neu-trino event counts regarding the above two reaction channelsin this paper.We assume that SK and HK have identical detector con-figurations, except for the fiducial volume, which is set as32 . . . Although this shouldbe taken into account when retrieving the energy spectrumof neutrinos, the spectrum reconstructions are not the mainfocus of this paper. As an example of the spectrum recon-struction, we refer readers to Nagakura (2021), in which theenergy spectra of all flavors of neutrino are retrieved by usingdata in multiple reaction channels and detectors, taking intoaccount detector response and Poisson noise. We employ the simplest, but widely used, neutrino oscillationmodel. Neutrinos are assumed to execute flavor conversionsadiabatically by Mikheyev-Smirnov-Wolfenstein (MSW) ef-fects with matter. The flavor conversion depends on themass-hierarchy; hence, we study the two cases: normal-and inverted-mass hierarchy. By following Dighe & Smirnov(2000), the neutrino flux at the Earth, F i , where the subscript i represents the neutrino flavor, can be computed using thosewithout flavor conversion F i as F e = pF e + (1 − p ) F x , (3)¯ F e = ¯ p ¯ F e + (1 − ¯ p ) ¯ F x , (4) F x = 12 (1 − p ) F e + 12 (1 + p ) F x , (5)¯ F x = 12 (1 − ¯ p ) ¯ F e + 12 (1 + ¯ p ) ¯ F x , (6)where p denotes the survival probability which depends onthe neutrino oscillation model and the neutrino mass hierar-chy. The upper bar denotes the anti-neutrino quantities. Inthe case of the normal-mass hierarchy, p and ¯ p can be writtenas p = sin θ , (7)¯ p = cos θ cos θ . (8) It should be noted, however, that we take into account thePoisson noise to discuss the detectability of temporal variations inneutrino signals. See Sec. 3.2 for more details.MNRAS , 1–19 (2020)
CSN neutrinos by long-term 2D simulations L [ e r g s - ]
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26 M ⊙ ⊙ L ν e ε ν e < > [ M e V ] L ν e < ε ν e > L [ e r g s - ] L ν e < ε ν e >L ν - e ε ν - e < > [ M e V ] L ν e < ε ν e >L ν - e < ε ν - e > L [ e r g s - ] Time [s] L ν e < ε ν e >L ν - e < ε ν - e >L ν x ε ν x < > [ M e V ] Time [s] L ν e < ε ν e >L ν - e < ε ν - e >L ν x < ε ν x > Figure 3.
Time evolution of angle-averaged neutrino luminosity (left) and average energy (right). They are evaluated in the laboratoryframe and measured at 250 km in CCSN simulations. From top to bottom, they are ν e , ¯ ν e , and ν x , respectively. The color represents theprogenitor models. The solid and dashed lines distinguish the explosion and non-explosion models, respectively.MNRAS000
Time evolution of angle-averaged neutrino luminosity (left) and average energy (right). They are evaluated in the laboratoryframe and measured at 250 km in CCSN simulations. From top to bottom, they are ν e , ¯ ν e , and ν x , respectively. The color represents theprogenitor models. The solid and dashed lines distinguish the explosion and non-explosion models, respectively.MNRAS000 , 1–19 (2020) H. Nagakura et al.
In the case of the inverted hierarchy, they are p = sin θ cos θ , (9)¯ p = sin θ . (10)Following Capozzi et al. (2017), we adopt the neutrino mix-ing parameters, θ and θ as sin θ = 2 . × − andsin θ = 2 . × − , which are the same as those used inNagakura et al. (2021). First, we analyze neutrino emissions at at the supernovae.Fig. 3 displays the time evolution of the neutrino luminosi-ties and average energies. There emerge rich flavor-, time-, and progenitor dependent features. The difference in themass accretion rates onto the PNS is primarily responsiblefor generating this diversity. As evidence, both mass accre-tion rate and neutrino luminosity (regardless of flavor) duringthe early post-bounce phase ( (cid:46) . M (cid:12) modelare the highest among all the models shown. We also notethat the higher mass accretion rate also builds a more mas-sive PNS (see Fig. 2), which boosts the neutrino luminosity inthe late phase. It should be mentioned that those qualitativetrends are consistent with what has been reported in previ-ous studies based on 1D models (see, e.g., Nakazato et al.2013; Warren et al. 2020; Segerlund et al. 2021). However, wealso find that there emerge strong temporal variations in thelate phase, which have not been reported in previous studies.The variation amplitude depends on progenitor and neutrinoflavor. The temporal characteristics seems to be one of themissing elements in previous studies. Below we perform anin-depth analysis of its physical origin.Here, we first consider the effects of PNS convection onneutrino signals, since this multi-D fluid instability is a ma-jor missing element in 1D models and actually affects theneutrino emissions at (cid:46) M (cid:12) . We confirm that PNSconvection commonly occurs for all progenitors in the region10 (cid:46) r (cid:46)
25 km at (cid:46) (cid:38) Y e ) and entropy per baryon( S ) around the surface of PNS ∼
10 km decrease with time.It should be mentioned that PNS convection facilitates itsdeleptonization and cooling (see also Roberts et al. 2012),indicating that the quasi-steady evolution is different fromthat in 1D models. As shown in the top panels of Fig. 5, thelocation of sharp negative Y e gradient gradually sinks into the inner region. Eventually, the inner edge of PNS convectionreaches the coordinate center (mass center of PNS), whichcan be seen in Fig. 5 for the 12 and 16 M (cid:12) models at the 3and 4 s snapshots . Once it reaches the center, the negativelepton number gradient starts to disappear, that correspondsto the time when PNS convection nearly ceases. Hence, thevigor of PNS convection becomes weaker with time in thelate post-bounce phase.Based on these results, let us consider the impact of PNSconvection on the neutrino emissions. First, we point out thatthe temporal variations in neutrino emissions are seen evenafter PNS convection has subsided (at (cid:38) ν x emissions is remarkablyweak when compared to that of the other species regardless ofCCSN models (see left panels in Fig. 3), which is inconsistentwith the above argument. For these reasons, we conclude thatthe temporal variation in neutrino emission is not primarilydriven by PNS convection .It is mass accretion onto the PNS that is the cause of thetemporal variations in neutrino signals. This conclusion isbuttressed by the fact that we observe long-lasting mass ac-cretion onto PNS in all models. The mass of the PNS mono-tonically increases up to the end of all of our simulations,regardless of progenitor (see Fig. 2) . One may think thatthis is due simply to weak explosions accompanied by a largeamount of fallback accretion. However, our models includecases with strong explosions . This trend is qualitativelydifferent from that observed in 1D models. In 1D, strong ex-plosions unbind most of post-shock matter above the PNS.As a result, the late time accretion rate is subtle . On theother hand, shock revival in multi-D occurs rather asymmet-rically, indicating that the vigor of shock expansion dependson the geometry. It is, hence, possible to have a weak shockexpansion in some directions even when the overall explosionis strong. This actually happens in our CCSN models; forinstance, the 26 M (cid:12) model has a strong dipole explosion,and it is also accompanied by large amounts of early fall-back accretion around the equator. We note that this trendis common in multi-D CCSN models, and the mass inflowcan last for more than a few seconds (see, e.g., Young et al.2006; Fryer 2009; Wongwathanarat et al. 2010; Chan et al. The arrival time of the inner edge of PNS convection to thecenter depends on the model; the decay of PNS convection tendsto take longer for heavier proto-neutron stars. We note that PNS convection potentially affects the weak tem-poral variation in the ν x s. See below for more details. It should be noted that very light progenitors such as the 9 M (cid:12) model, which is not included in our study, may be exceptions, forwhich mass accretion almost ceases after the shock revival. Thisis mainly due to the steep density gradient outside its progenitorcore. For instance, the explosion energy of 26 M (cid:12) is ∼ . × erg.See Burrows & Vartanyan (2021) for more details. We note that strong fallback accretion may occur even in 1Ddue to the reverse shock generated by the deceleration of the shockwave in the hydrogen envelope (see, e.g., Chevalier 1989; Wong-wathanarat et al. 2015). However, this would occur at a very latephase ( ∼ hours), which is not the phase we consider in this paper.MNRAS , 1–19 (2020) CSN neutrinos by long-term 2D simulations Figure 4.
Color map of the angle-averaged lateral speed of fluids displayed as functions of radius and time. We selected four representativemodels: 12 (top-left), 16 (top-right), 20 (bottom-left) and 26 M (cid:12) (bottom-right). We also display angle-averaged isodensity radii with10 , , , and 10 g / cm (from large to small radii) as white line in each panel. . Since asymmet-ric shock expansion and fluid-instabilities alter matter motionin the post-shock region, the accretion inflow onto the PNSis highly disorganized, causing the temporal variation of theneutrino emissions.Motivated by the above considerations, we take a lookat the time evolution of accretion rate onto PNS, which isdisplayed in Fig. 6 for some selected models. As expected,this plot clearly displays both the long-lasting accretion ontoPNS and the strong temporal variation for the explosionmodels (solid lines). The amplitudes of temporal variation It should be noted, however, that the detailed accretion struc-ture would depend on dimension. In 3D, the shock morphology isgenerally more spherical than in 2D, indicating that the asymme-try of the accretion flows may be reduced. are roughly tens of percent of the short-time-average (quasi-steady) component (see below for the definition of the quasi-steady component). On the other hand, the temporal varia-tion is rather mild in the 12 M (cid:12) non-exploding model (seebelow for more details). At a first glance, the amplitude ofthe temporal variations has a positive correlation with theneutrino luminosities. For instance, the mass accretion rateof the 26 M (cid:12) model has large temporal variations at ∼ A qs ) with re- MNRAS000
Color map of the angle-averaged lateral speed of fluids displayed as functions of radius and time. We selected four representativemodels: 12 (top-left), 16 (top-right), 20 (bottom-left) and 26 M (cid:12) (bottom-right). We also display angle-averaged isodensity radii with10 , , , and 10 g / cm (from large to small radii) as white line in each panel. . Since asymmet-ric shock expansion and fluid-instabilities alter matter motionin the post-shock region, the accretion inflow onto the PNSis highly disorganized, causing the temporal variation of theneutrino emissions.Motivated by the above considerations, we take a lookat the time evolution of accretion rate onto PNS, which isdisplayed in Fig. 6 for some selected models. As expected,this plot clearly displays both the long-lasting accretion ontoPNS and the strong temporal variation for the explosionmodels (solid lines). The amplitudes of temporal variation It should be noted, however, that the detailed accretion struc-ture would depend on dimension. In 3D, the shock morphology isgenerally more spherical than in 2D, indicating that the asymme-try of the accretion flows may be reduced. are roughly tens of percent of the short-time-average (quasi-steady) component (see below for the definition of the quasi-steady component). On the other hand, the temporal varia-tion is rather mild in the 12 M (cid:12) non-exploding model (seebelow for more details). At a first glance, the amplitude ofthe temporal variations has a positive correlation with theneutrino luminosities. For instance, the mass accretion rateof the 26 M (cid:12) model has large temporal variations at ∼ A qs ) with re- MNRAS000 , 1–19 (2020)
H. Nagakura et al. Y e ⊙ (No-exp)12M ⊙ (No-exp) 16M ⊙ (Exp)12M ⊙ (No-exp) 16M ⊙ (Exp) 20M ⊙ (Exp)12M ⊙ (No-exp) 16M ⊙ (Exp) 20M ⊙ (Exp) 26M ⊙ (Exp) S Radius [km] ⊙ (No-exp) 16M ⊙ (Exp) 20M ⊙ (Exp) 26M ⊙ (Exp) ⊙ (No-exp) 16M ⊙ (Exp) 20M ⊙ (Exp) 26M ⊙ (Exp) ⊙ (No-exp) 16M ⊙ (Exp) 20M ⊙ (Exp) 26M ⊙ (Exp) ⊙ (No-exp) 16M ⊙ (Exp) 20M ⊙ (Exp) 26M ⊙ (Exp) Figure 5.
Angle-averaged electron fraction ( Y e ) and entropy (S) profiles are shown in top and bottom panels, respectively. From left toright, they are 12, 16, 20, and 26 M (cid:12) models, respectively. Color represents the different time snapshots. M ass acc r e t i on r a t e [ M ⊙ / s ] Time [s]
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The time evolution of mass accretion rate measured at100 km for selected models, 12, 16, 20, and 26 M (cid:12) models, whichare distinguished by color. The line type denotes the explosion(solid) and non-exploding (dashed) model. spect to an arbitrary time-dependent quantity A as, A qs ( t ) = 1∆ t (cid:90) t +0 . tt − . t dτ A ( τ ) , (11)and we set ∆ t to 300 ms in this study . By using A qs , we We checked the dependence of the choice of ∆ t , and we con-firmed that our results are insensitive to this choice unless ∆ t is (cid:46)
100 ms. -0.4-0.2 0 0.2 0.4 0.6 0.8 1 1 1.5 2 2.5 3 3.5 4 N o r m a li z e d t e m po r a l va r i a t i on s Time [s] ν e ν - e ν x M.
26 M ⊙ Figure 7.
Time variable component of the neutrino luminosity andmass accretion rate as a function of time. The vertical axis is nor-malized such that the peak amplitude is unity. See Eq. 12 and textfor the defenition of the variable. define the time variable component ( A tv ): A tv ( t ) = A ( t ) − A qs ( t ) . (12)Fig. 7 shows A tv as a function of time for neutrino luminosi-ties and mass accretion rates in the 26 M (cid:12) model. Note thatthe vertical axis is normalized so that the peak amplitude of A tv of each quantity in the interval 1 − ν e (¯ ν e ) luminosity and mass accretionrate. We can also see that the temporal variations of ν x are MNRAS , 1–19 (2020)
CSN neutrinos by long-term 2D simulations -1-0.5 0 0.5 1 1 2 3 4 X Time [s] ν e ν - e ν x
26 M ⊙ -1-0.5 0 0.5 1 1 2 3 4 X Time [s]
12 M ⊙
16 M ⊙
20 M ⊙
26 M ⊙
26 M ⊙ ν e Figure 8.
Correlation function of the temporal variation of neutrino luminosity and mass accretion rate. The left panel shows the result forthe 26 M (cid:12) model. The color distinguishes the neutrino species. The right panel displays the same as the left one, but shows the progenitordependence. In the panel, we display only the result of the ν e and ˙ M correlation. See the text for more details. correlated with the mass accretion rate, albeit more weaklythan ν e and ¯ ν e .To assess the correlation more quantitatively, we furthercompute a normalized correlation function of the temporalvariation between each species of neutrino and the mass ac-cretion rate (see Kuroda et al. 2017, for the similar definitionof the correlation function), which can be written as:X( t, ∆T) = Y ν ˙ M ( t, ∆T)Y ν ( t, ∆T) × Y ˙ M ( t ) , (13)where Y ν ˙ M ( t, ∆T) = (cid:90) dτ H ( t − τ ) A tv ν ( τ + ∆T) A tv˙ M ( τ ) , Y ν ( t, ∆T) = (cid:115)(cid:90) dτ H ( t − τ ) ( A tv ν ( τ + ∆T)) , Y ˙ M ( t ) = (cid:115)(cid:90) dτ H ( t − τ ) (cid:16) A tv˙ M ( τ ) (cid:17) , (14)(15)In the expression, H denotes the Hann window function. Thesize of the time window is set to 300 ms. ∆T represents thetime delay of the response of the neutrino luminosity fromthe temporal variation of the mass accretion rate. Since ∆Tis not known a priori, it is varied in the range of 0 −
10 ms inthis study . We note that our employed correlation functionmay be improved by more sophisticated prescriptions to ex-tract the quasi-steady component (see, e.g., Chen et al. 2018). We note that ∆T also depends on where we measure neutrinosignals (250 km in this study) and mass accretion rate (here at 100km). Taking into account neutrino propagation, the actual delaytime of the response would be smaller than ∆T.
However, our method suffices for the purposes of this paper;indeed it captures the qualitative trend of the correlation, aswe now show.Fig. 8 portrays the time evolution of the correlation func-tion of the temporal variations between neutrino luminosityand mass accretion rate. In the plot, ∆T is chosen so that theabsolute value of the correlation function is the maximum.Roughly speaking, ∆T is ∼ M (cid:12) model in the left panel and show the flavor-dependent feature of the correlation function. We confirm astrong correlation of the temporal variation between ν e (¯ ν e )luminosity and mass accretion rate. On the other hand, ν x has the least correlation among them. The weak correlationmay indicate that PNS convection affects the temporal vari-ations, albeit subdominantly. In the right panel of Fig. 8, weshow the progenitor dependence for selected models, focus-ing on ν e . The positive correlation is clearly shown in otherCCSN models, except for the 12 M (cid:12) model. Hence, we con-clude that the mass accretion onto PNS is the most influentialplayer in the temporal variations of the neutrino signals.There are a few caveats in the conclusion. Although wereveal that inhomogeneous mass accretion flows play a domi-nant role in determining the temporal characteristics of neu-trino signals, the correlation becomes weaker with time; in-deed, it is less than 0.5 for all models at 4 s (see Fig. 8).Meanwhile, temporal variations still exist at that time (seeFig. 7). The weak correlation with the mass accretion rate in-dicates that there is another driving force creating temporalvariation in neutrino signals. We suggest this is due to a fluidinstability right above the PNS . Fig. 9 displays the entropydistribution in the central region for the 26 M (cid:12) model at 4 s. It should be mentioned that the fluid instability is differentMNRAS000
However, our method suffices for the purposes of this paper;indeed it captures the qualitative trend of the correlation, aswe now show.Fig. 8 portrays the time evolution of the correlation func-tion of the temporal variations between neutrino luminosityand mass accretion rate. In the plot, ∆T is chosen so that theabsolute value of the correlation function is the maximum.Roughly speaking, ∆T is ∼ M (cid:12) model in the left panel and show the flavor-dependent feature of the correlation function. We confirm astrong correlation of the temporal variation between ν e (¯ ν e )luminosity and mass accretion rate. On the other hand, ν x has the least correlation among them. The weak correlationmay indicate that PNS convection affects the temporal vari-ations, albeit subdominantly. In the right panel of Fig. 8, weshow the progenitor dependence for selected models, focus-ing on ν e . The positive correlation is clearly shown in otherCCSN models, except for the 12 M (cid:12) model. Hence, we con-clude that the mass accretion onto PNS is the most influentialplayer in the temporal variations of the neutrino signals.There are a few caveats in the conclusion. Although wereveal that inhomogeneous mass accretion flows play a domi-nant role in determining the temporal characteristics of neu-trino signals, the correlation becomes weaker with time; in-deed, it is less than 0.5 for all models at 4 s (see Fig. 8).Meanwhile, temporal variations still exist at that time (seeFig. 7). The weak correlation with the mass accretion rate in-dicates that there is another driving force creating temporalvariation in neutrino signals. We suggest this is due to a fluidinstability right above the PNS . Fig. 9 displays the entropydistribution in the central region for the 26 M (cid:12) model at 4 s. It should be mentioned that the fluid instability is differentMNRAS000 , 1–19 (2020) H. Nagakura et al.
Figure 9.
The entropy per baryon in color with fluid velocities in vectors for 26 M (cid:12) model at 4 s. As shown in the figure, the accretion shock wave emerges inthe northern hemisphere. The shock wave fluctuates with atime scale of a few ms, and the overall morphology is changeson a ∼
100 ms timescale. On the other hand, the shock dy-namics (instability) strongly depends upon dimension, indi-cating that it is unclear if our finding is generic in 3D. We,hence, postpone the detailed study of this dynamics to fu-ture work. Nevertheless, it may be that shock instability byfallback accretion may emerge much earlier than previouslythought (see Chevalier 1989; Houck & Chevalier 1992).As another caveat, we note that the small temporal vari-ation found in non-explosion models may in part be due tonumerical artifacts introduced by using 2D. This is becausespiral Standing Accretion Shock Instability (SASI) is, in gen-eral, observed in 3D non-exploding models, which induces astrong quasi-periodic temporal variations in the neutrino sig-nals (see also Nagakura et al. 2021). We note that 2D models from PNS convection that we have discussed, which happens moredeeply inside the PNS. suppress the non-axisymmetric mode, indicating that the spi-ral SASI may be suppressed artificially. On the contrary, wespeculate that the temporal variation found in our 2D ex-plosion models may be overestimated compared to that in3D. This is attributed to the fact that the explosion geome-try may be too asymmetric in 2D, which overestimates theasymmetry in mass accretion rates and neutrino signals. Ad-dressing these issues also requires sophisticated 3D long-termsimulations, which are postponed to future work. Having inmind these caveats, we move on to the analysis of neutrinosignals at the Earth.
Fig. 10 displays the angle-averaged neutrino event count as afunction of time for selected models: 12, 16, 20, and 26 M (cid:12) .In the early phase ( (cid:46) MNRAS , 1–19 (2020)
CSN neutrinos by long-term 2D simulations E ve n t r a t e [ s - ] NOOSCNORMALInV
12 M ⊙ at 10kpc (SK-IBD-p) 16 M ⊙ at 10kpc (SK-IBD-p) 20 M ⊙ at 10kpc (SK-IBD-p) 26 M ⊙ at 10kpc (SK-IBD-p)12 M ⊙ at 10kpc (DUNE-CCAre) 16 M ⊙ at 10kpc (DUNE-CCAre) 20 M ⊙ at 10kpc (DUNE-CCAre) 26 M ⊙ at 10kpc (DUNE-CCAre)12 M ⊙ at 10kpc (JUNO-IBD-p) 16 M ⊙ at 10kpc (JUNO-IBD-p) 20 M ⊙ at 10kpc (JUNO-IBD-p) 26 M ⊙ at 10kpc (JUNO-IBD-p)12 M ⊙ at 10kpc (SK-IBD-p) 16 M ⊙ at 10kpc (SK-IBD-p) 20 M ⊙ at 10kpc (SK-IBD-p) 26 M ⊙ at 10kpc (SK-IBD-p)12 M ⊙ at 10kpc (DUNE-CCAre) 16 M ⊙ at 10kpc (DUNE-CCAre) 20 M ⊙ at 10kpc (DUNE-CCAre) 26 M ⊙ at 10kpc (DUNE-CCAre)12 M ⊙ at 10kpc (JUNO-IBD-p) 16 M ⊙ at 10kpc (JUNO-IBD-p) 20 M ⊙ at 10kpc (JUNO-IBD-p) 26 M ⊙ at 10kpc (JUNO-IBD-p)12 M ⊙ at 10kpc (SK-IBD-p) 16 M ⊙ at 10kpc (SK-IBD-p) 20 M ⊙ at 10kpc (SK-IBD-p) 26 M ⊙ at 10kpc (SK-IBD-p)12 M ⊙ at 10kpc (DUNE-CCAre) 16 M ⊙ at 10kpc (DUNE-CCAre) 20 M ⊙ at 10kpc (DUNE-CCAre) 26 M ⊙ at 10kpc (DUNE-CCAre)12 M ⊙ at 10kpc (JUNO-IBD-p) 16 M ⊙ at 10kpc (JUNO-IBD-p) 20 M ⊙ at 10kpc (JUNO-IBD-p) 26 M ⊙ at 10kpc (JUNO-IBD-p)12 M ⊙ at 10kpc (SK-IBD-p) 16 M ⊙ at 10kpc (SK-IBD-p) 20 M ⊙ at 10kpc (SK-IBD-p) 26 M ⊙ at 10kpc (SK-IBD-p)12 M ⊙ at 10kpc (DUNE-CCAre) 16 M ⊙ at 10kpc (DUNE-CCAre) 20 M ⊙ at 10kpc (DUNE-CCAre) 26 M ⊙ at 10kpc (DUNE-CCAre)12 M ⊙ at 10kpc (JUNO-IBD-p) 16 M ⊙ at 10kpc (JUNO-IBD-p) 20 M ⊙ at 10kpc (JUNO-IBD-p) 26 M ⊙ at 10kpc (JUNO-IBD-p) E ve n t r a t e [ s - ]
12 M ⊙ at 10kpc (SK-IBD-p) 16 M ⊙ at 10kpc (SK-IBD-p) 20 M ⊙ at 10kpc (SK-IBD-p) 26 M ⊙ at 10kpc (SK-IBD-p)12 M ⊙ at 10kpc (DUNE-CCAre) 16 M ⊙ at 10kpc (DUNE-CCAre) 20 M ⊙ at 10kpc (DUNE-CCAre) 26 M ⊙ at 10kpc (DUNE-CCAre)12 M ⊙ at 10kpc (JUNO-IBD-p) 16 M ⊙ at 10kpc (JUNO-IBD-p) 20 M ⊙ at 10kpc (JUNO-IBD-p) 26 M ⊙ at 10kpc (JUNO-IBD-p)12 M ⊙ at 10kpc (SK-IBD-p) 16 M ⊙ at 10kpc (SK-IBD-p) 20 M ⊙ at 10kpc (SK-IBD-p) 26 M ⊙ at 10kpc (SK-IBD-p)12 M ⊙ at 10kpc (DUNE-CCAre) 16 M ⊙ at 10kpc (DUNE-CCAre) 20 M ⊙ at 10kpc (DUNE-CCAre) 26 M ⊙ at 10kpc (DUNE-CCAre)12 M ⊙ at 10kpc (JUNO-IBD-p) 16 M ⊙ at 10kpc (JUNO-IBD-p) 20 M ⊙ at 10kpc (JUNO-IBD-p) 26 M ⊙ at 10kpc (JUNO-IBD-p)12 M ⊙ at 10kpc (SK-IBD-p) 16 M ⊙ at 10kpc (SK-IBD-p) 20 M ⊙ at 10kpc (SK-IBD-p) 26 M ⊙ at 10kpc (SK-IBD-p)12 M ⊙ at 10kpc (DUNE-CCAre) 16 M ⊙ at 10kpc (DUNE-CCAre) 20 M ⊙ at 10kpc (DUNE-CCAre) 26 M ⊙ at 10kpc (DUNE-CCAre)12 M ⊙ at 10kpc (JUNO-IBD-p) 16 M ⊙ at 10kpc (JUNO-IBD-p) 20 M ⊙ at 10kpc (JUNO-IBD-p) 26 M ⊙ at 10kpc (JUNO-IBD-p)12 M ⊙ at 10kpc (SK-IBD-p) 16 M ⊙ at 10kpc (SK-IBD-p) 20 M ⊙ at 10kpc (SK-IBD-p) 26 M ⊙ at 10kpc (SK-IBD-p)12 M ⊙ at 10kpc (DUNE-CCAre) 16 M ⊙ at 10kpc (DUNE-CCAre) 20 M ⊙ at 10kpc (DUNE-CCAre) 26 M ⊙ at 10kpc (DUNE-CCAre)12 M ⊙ at 10kpc (JUNO-IBD-p) 16 M ⊙ at 10kpc (JUNO-IBD-p) 20 M ⊙ at 10kpc (JUNO-IBD-p) 26 M ⊙ at 10kpc (JUNO-IBD-p) E ve n t r a t e [ s - ] Time [s]
12 M ⊙ at 10kpc (SK-IBD-p) 16 M ⊙ at 10kpc (SK-IBD-p) 20 M ⊙ at 10kpc (SK-IBD-p) 26 M ⊙ at 10kpc (SK-IBD-p)12 M ⊙ at 10kpc (DUNE-CCAre) 16 M ⊙ at 10kpc (DUNE-CCAre) 20 M ⊙ at 10kpc (DUNE-CCAre) 26 M ⊙ at 10kpc (DUNE-CCAre)12 M ⊙ at 10kpc (JUNO-IBD-p) 16 M ⊙ at 10kpc (JUNO-IBD-p) 20 M ⊙ at 10kpc (JUNO-IBD-p) 26 M ⊙ at 10kpc (JUNO-IBD-p)
12 M ⊙ at 10kpc (SK-IBD-p) 16 M ⊙ at 10kpc (SK-IBD-p) 20 M ⊙ at 10kpc (SK-IBD-p) 26 M ⊙ at 10kpc (SK-IBD-p)12 M ⊙ at 10kpc (DUNE-CCAre) 16 M ⊙ at 10kpc (DUNE-CCAre) 20 M ⊙ at 10kpc (DUNE-CCAre) 26 M ⊙ at 10kpc (DUNE-CCAre)12 M ⊙ at 10kpc (JUNO-IBD-p) 16 M ⊙ at 10kpc (JUNO-IBD-p) 20 M ⊙ at 10kpc (JUNO-IBD-p) 26 M ⊙ at 10kpc (JUNO-IBD-p)
12 M ⊙ at 10kpc (SK-IBD-p) 16 M ⊙ at 10kpc (SK-IBD-p) 20 M ⊙ at 10kpc (SK-IBD-p) 26 M ⊙ at 10kpc (SK-IBD-p)12 M ⊙ at 10kpc (DUNE-CCAre) 16 M ⊙ at 10kpc (DUNE-CCAre) 20 M ⊙ at 10kpc (DUNE-CCAre) 26 M ⊙ at 10kpc (DUNE-CCAre)12 M ⊙ at 10kpc (JUNO-IBD-p) 16 M ⊙ at 10kpc (JUNO-IBD-p) 20 M ⊙ at 10kpc (JUNO-IBD-p) 26 M ⊙ at 10kpc (JUNO-IBD-p)
12 M ⊙ at 10kpc (SK-IBD-p) 16 M ⊙ at 10kpc (SK-IBD-p) 20 M ⊙ at 10kpc (SK-IBD-p) 26 M ⊙ at 10kpc (SK-IBD-p)12 M ⊙ at 10kpc (DUNE-CCAre) 16 M ⊙ at 10kpc (DUNE-CCAre) 20 M ⊙ at 10kpc (DUNE-CCAre) 26 M ⊙ at 10kpc (DUNE-CCAre)12 M ⊙ at 10kpc (JUNO-IBD-p) 16 M ⊙ at 10kpc (JUNO-IBD-p) 20 M ⊙ at 10kpc (JUNO-IBD-p) 26 M ⊙ at 10kpc (JUNO-IBD-p) Figure 10.
Event rates detected in the major reaction channels for each detector as a function of time. The results for SK, DUNE, andJUNO are displayed from top to bottom. From left to right, we show different CCSN models: 12, 16, 20, and 26 M (cid:12) . The color distinguishesthe neutrino oscillation models: red (no oscillations), blue (normal mass hierarchy with adiabatic MSW), green (inverted mass hierarchywith adiabatic MSW). of detector. The dependence on neutrino oscillation model in2D models is exactly the same as in 3D. At the late phase,some new features appear in the neutrino signals. First, thequasi-steady component of neutrino event count rate becomesless sensitive to neutrino oscillation model, which is consis-tent with that reported in previous 1D studies (see, e.g., Suwaet al. 2019) . This is attributed to the fact that neutrinoemission at the source evolves into a common luminosityand spectrum across the three flavors. This can be under-stood as follows. The neutrino emissions in the late phasehave quasi-thermal (Fermi-Dirac) spectra and are character-ized roughly by a temperature and chemical potential at theneutrinosphere. The neutrinosphere is less sensitive to flavordue to the sharp density gradient in PNS envelope , indicat-ing that the difference of neutrino temperature among flavors Although the trend is common, the event counts in 1D mod-els are different from those in multi-D models (Nagakura et al.2021). In the early phase, PNS convection is mainly responsiblefor the difference. In the late phase, the difference is remarkable,in particular for successful explosion models, in which long-lastingasymmetric mass accretion in multi-D models boosts the eventcounts (see also Sec. 3.1). The PNS envelope contracts with time due to energy loss byneutrino emissions. The sharp density gradient at the outer PNSboundary in the late phase can be seen in Fig. 4. As shown in this is small. Furthermore, the lepton loss from the PNS by neu-trino emissions reduces Y e inside of the PNS, which makesthe chemical potential of ν e (and ¯ ν e ) start to approach zero.Since the chemical potential of heavy leptonic neutrinos iszero (unless (stable) muons appear in matter (Bollig et al.2017; Fischer et al. 2020)), the difference of the chemical po-tential among three flavors of neutrinos reduces with time.For these reasons, all neutrinos evolve towards the identicalspectrum.There emerges a rich diversity in the count rates as a func-tion of both neutrino oscillation and CCSN models. As shownin Fig. 10, the event counts at (cid:38) M (cid:12) models, unlessthe neutrino mass hierarchy is inverted order. This is con-sistent with our discussion in Sec. 3.1 that ¯ ν e emissions atCCSN source strongly vary with time for these models (seeSec. 3.1) . Similarly, the event count in DUNE has a strongtime variable component, unless the neutrino mass hierarchy figure, the isodensity radii for different densities converge to thesame radius, indicating that the density gradient is very steep. For the inverted mass hierarchy, on the other hand, SK, HK,and JUNO are the sensitive to ν x emissions at the supernova.Therefore, the event count time variability is the least among dif-ferent oscillation models. MNRAS000
Event rates detected in the major reaction channels for each detector as a function of time. The results for SK, DUNE, andJUNO are displayed from top to bottom. From left to right, we show different CCSN models: 12, 16, 20, and 26 M (cid:12) . The color distinguishesthe neutrino oscillation models: red (no oscillations), blue (normal mass hierarchy with adiabatic MSW), green (inverted mass hierarchywith adiabatic MSW). of detector. The dependence on neutrino oscillation model in2D models is exactly the same as in 3D. At the late phase,some new features appear in the neutrino signals. First, thequasi-steady component of neutrino event count rate becomesless sensitive to neutrino oscillation model, which is consis-tent with that reported in previous 1D studies (see, e.g., Suwaet al. 2019) . This is attributed to the fact that neutrinoemission at the source evolves into a common luminosityand spectrum across the three flavors. This can be under-stood as follows. The neutrino emissions in the late phasehave quasi-thermal (Fermi-Dirac) spectra and are character-ized roughly by a temperature and chemical potential at theneutrinosphere. The neutrinosphere is less sensitive to flavordue to the sharp density gradient in PNS envelope , indicat-ing that the difference of neutrino temperature among flavors Although the trend is common, the event counts in 1D mod-els are different from those in multi-D models (Nagakura et al.2021). In the early phase, PNS convection is mainly responsiblefor the difference. In the late phase, the difference is remarkable,in particular for successful explosion models, in which long-lastingasymmetric mass accretion in multi-D models boosts the eventcounts (see also Sec. 3.1). The PNS envelope contracts with time due to energy loss byneutrino emissions. The sharp density gradient at the outer PNSboundary in the late phase can be seen in Fig. 4. As shown in this is small. Furthermore, the lepton loss from the PNS by neu-trino emissions reduces Y e inside of the PNS, which makesthe chemical potential of ν e (and ¯ ν e ) start to approach zero.Since the chemical potential of heavy leptonic neutrinos iszero (unless (stable) muons appear in matter (Bollig et al.2017; Fischer et al. 2020)), the difference of the chemical po-tential among three flavors of neutrinos reduces with time.For these reasons, all neutrinos evolve towards the identicalspectrum.There emerges a rich diversity in the count rates as a func-tion of both neutrino oscillation and CCSN models. As shownin Fig. 10, the event counts at (cid:38) M (cid:12) models, unlessthe neutrino mass hierarchy is inverted order. This is con-sistent with our discussion in Sec. 3.1 that ¯ ν e emissions atCCSN source strongly vary with time for these models (seeSec. 3.1) . Similarly, the event count in DUNE has a strongtime variable component, unless the neutrino mass hierarchy figure, the isodensity radii for different densities converge to thesame radius, indicating that the density gradient is very steep. For the inverted mass hierarchy, on the other hand, SK, HK,and JUNO are the sensitive to ν x emissions at the supernova.Therefore, the event count time variability is the least among dif-ferent oscillation models. MNRAS000 , 1–19 (2020) H. Nagakura et al. E ve n t r a t e [ s - ] Frequency [Hz]12 M ⊙
16 M ⊙
20 M ⊙
26 M ⊙ SK (NORMAL) at 10 kpc 0 10 20 30 40 50 60 3 10 30 100 300 E ve n t r a t e [ s - ] Frequency [Hz]SK (NORMAL) at 10 kpc DUNE (InV) at 10 kpc
Figure 11.
The Fourier transform of the event rate in the late post-bounce phase. The time window is chosen from 1 s after the core bounceto the end of simulations. In the left (right) panel, we show the case with SK (DUNE) in normal (inverted) mass hierarchy for CCSNe at10 kpc. The color distinguishes CCSN models. The dashed line indicates the 12 M (cid:12) non-exploding model, while solid lines are used forexploding models. is normal order. This originates from the strong time varia-tions in the ν e emissions at the supernova source. The timevariability is less remarkable for the 12 and 16 M (cid:12) modelsregardless of neutrino oscillations. As mentioned already inSec. 3.1, the non-exploding models tend to have weak timevariability (ignoring a possible 3D spiral SASI), which is re-sponsible for the weak variations we see here in the 12 M (cid:12) model. For the 16 M (cid:12) model the temporal variation in themass accretion rate is strong (see Fig. 6). However, it has thesmallest mean mass accretion rate among our models (seeFigs. 2 and 6). This implies that the accretion component ofthe neutrino luminosity is also small, indicating the tempo-ral variation is smeared out by the core diffusion componentof the neutrino luminosity. We note, however, that the posi-tive correlation of the temporal correlation between neutrinosignals and mass accretion rate remains strong even in thosemodels with weak time variability in the neutrino signals (seeFig. 8).To see the temporal structure more clearly, Fig. 11 portraysthe Fourier transform of the event rate after 1 s in the caseof SK with the normal mass hierarchy (left) and DUNE withthe inverted mass hierarchy (right). Although there emergeno strongly characteristic time frequencies, we find that thelow frequencies ( (cid:46)
20 Hz) dominate the temporal structure.The dominance by low-frequency variations supports the con-clusion that temporal variations in neutrino signals are notprimarily driven by matter dynamics in the vicinity of PNS,but rather by external factors such as accretion flows ontoPNS.Below we assess the detectability of these temporal varia-tions. We note that in reality the time variations of the eventcounts may be smeared out by various sources of noise; hence,we take them into account in this discussion. The goal of this estimation is to see the minimum time bin (∆T bin ) for whichthe time variations dominate the noise, thus providing thehighest resolution possible for the time and frequency for theFourier analysis. Based on this estimation, we discuss the de-tectability of the temporal variations.We start by extracting the quasi-steady component of theevent count, which can be done by using Eq. 12, i.e., n qs ( t ) = 1∆ t (cid:90) t +0 . tt − . t dτ n ( τ ) , (16)where n and n qs denote the (raw) event count and its quasi-steady component, respectively. The selection of ∆ t can berather arbitrary, but we would suggest that a few hundredsmilliseconds is appropriate for extracting the quasi-steadycomponent (see Sec. 3.1).We first consider the case of SK (HK), DUNE, and JUNO,in which Poisson noise is the dominant source of detectornoise. The Poisson noise in event counts with a time windowof ∆T bin can be estimated as N noise ( t ) ∼ ( n qs ( t ) ∆T bin ) . . (17)The temporal component of the neutrino signal can be esti-mated as N tv ( t ) ∼ | n ( t ) − n qs ( t ) | ∆T bin ≡ α ( t ) n qs ( t ) ∆T bin , (18)where α denotes the degree of temporal variation. Thus, thesignal-to-noise ratio (SNR) can be given asSNR( t ) = N tv ( t ) N noise ( t ) ∼ α ( t ) ( n qs ( t ) ∆T bin ) . . (19)Let us estimate the required ∆T bin by inserting a typicalvalue for each parameter. n qs is higher than (cid:38) MNRAS , 1–19 (2020)
CSN neutrinos by long-term 2D simulations assumed to be 10 kpc). α depends on neutrino oscillation andprogenitor models, but it is ∼ . M (cid:12) model. If we set the threshold SNR to 5, ∆T bin can be estimated as∆T bin ∼
300 [ms] (cid:18)
SNR5 (cid:19) (cid:16) α . (cid:17) − (cid:18) n qs × (cid:19) − . (20)It should be mentioned that ∼ ∼ α ismuch higher than 0 .
2. On the other hand, HK will be regis-tering ∼ ∼ N noiseIC ( t ) ∼ (cid:0) (1 . × + n qs ( t ))∆T bin (cid:1) . . (21) n qs is roughly 100 times higher than that in SK (it is ∼ × at this phase), indicating that the background noisedominates when the CCSNe is at a distance of 10 kpc. Hence,the SNR can be given asSNR IC ∼ − α n qs √ ∆T bin . (22)Thus, the required time width to resolve the temporal varia-tion can be estimated as∆T bin(IC) ∼
20 [ms] (cid:18)
SNR5 (cid:19) (cid:16) α . (cid:17) − (cid:18) n qs × (cid:19) − . (23)This estimate suggests that IceCube is capable of resolv-ing temporal variations of ∼
10 Hz even when the CCSNis at 10 kpc; hence, IceCube would provide the most detailedmeasurement among detectors of the temporal variations inneutrino signal. It should be mentioned, however, that ∆Tfor IceCube increases more rapidly with decreasing n qs thanthat of other detectors (compare the n qs dependence betweenEqs. 20 and 23) . This indicates that other detectors, in par-ticular HK, would eventually become more sensitive to tem-poral variations at the very late phases ( (cid:38)
10 s). It shouldalso be mentioned that the threshold time (or frequency)bin strongly depends upon the distance to the CCSN; forinstance, if the source is at 5 kpc (the background noise stilldominates in this situation), the threshold time frequency ismore than 10 times that at 10 kpc, i.e., ∼
100 Hz temporalvariations may be resolved. For such a nearby CCSN, Ice-Cube would be capable of resolving ∼
20 Hz temporal vari-ation even if α = 0 .
1, which corresponds to CCSN modelswith weak temporal variations such as the 16 M (cid:12) model.We now turn our attention to properties of the cumula-tive number of events. Fig. 12 shows the energy spectra with This is attributed to the fact that the background noise doesnot depend on n qs . respect to each major reaction channel at each detector forselected post-bounce times: 1, 2, and 4 s. We note that thesmearing effects by detector response are taken into accountin these plots (although Poisson noise is neglected). We findthat the peak energy at the spectrum shifts to higher energywith time regardless of progenitor and neutrino oscillationmodel, as expected. It should be mentioned that the shape ofthe energy spectrum is similar for different CCSN models .Fig. 13 shows the time evolution of the cumulative num-ber of events at each detector. In the previous studies, it hasbeen pointed out that the event count has a positive corre-lation with the PNS mass (see, e.g., Suwa et al. 2020). Wefind, however, that the correlation is not definitive and de-pends upon neutrino oscillation model and detector. For in-stance, the cumulative number of events at (cid:38) M (cid:12) models) tends tobe higher than that of exploding models in the case withoutflavor conversions. We also note that the cumulative eventcounts at (cid:38) M (cid:12) model is the highest amongexploding models, although the PNS mass is not the largest(see Fig. 2). These trends can be understood as follows. Thesethree models commonly have high mass accretion rates at thelate phases. This indicates that the accretion component forthe ν e and ¯ ν e emissions is also higher. In the meantime, theaverage neutrino energy of ν e s and ¯ ν e s at the late phases isalso higher than during the early phase. As a result, the de-tection efficiency of the neutrinos (in the case without flavorconversions) becomes higher. We also note that the CCArecross section in DUNE is more sensitive to high-energy neu-trinos than is the IBD-p in SK, indicating that this average-energy difference makes more of a difference in DUNE thanSK.Interestingly, the positive correlation between PNS massand event rate in each detector tends to be recovered inthe cases with flavor conversions (see, e.g., the case withthe normal-mass hierarchy at DUNE in Fig. 13). This is at-tributed to the fact that all detectors have sensitivities tonot only ν e s and ¯ ν e s, but also to ν x s at the CCSN source byvirtue of neutrino mixings. We note that the ν x count has astrong correlation with the mass of the PNS (see Fig. 3).Finally, we discuss the correlation between the cumulativenumber of events at each detector and the total neutrino en-ergy (TONE) emitted by the supernova. This analysis is anextension of our previous discussion in Sec. 3.4 of Nagakuraet al. (2021). In our previous study, we found an interestingcorrelation between them, and provided fitting formulae forthe relation. We note, however, that the provided formulaemight be valid only in the early phase ( ∼ ∼ There may be, however, rich diversity in the high-energy com-ponents ( (cid:38)
50 MeV) among progenitors. See Nagakura & Ho-tokezaka (2020) for more details. MNRAS000
50 MeV) among progenitors. See Nagakura & Ho-tokezaka (2020) for more details. MNRAS000 , 1–19 (2020) H. Nagakura et al. C u m u l a t i ve E ve n t s [ / M e V ] T = 1sT = 2sT = 4sNormalInverted
12 M ⊙ at 10 kpc (SK-IBD-p) 16 M ⊙ at 10 kpc (SK-IBD-p) 20 M ⊙ at 10 kpc (SK-IBD-p) 26 M ⊙ at 10 kpc (SK-IBD-p)12 M ⊙ at 10 kpc (DUNE-CCAre) 16 M ⊙ at 10 kpc (DUNE-CCAre) 20 M ⊙ at 10 kpc (DUNE-CCAre) 26 M ⊙ at 10 kpc (DUNE-CCAre)12 M ⊙ at 10 kpc (JUNO-IBD-p) 16 M ⊙ at 10 kpc (JUNO-IBD-p) 20 M ⊙ at 10 kpc (JUNO-IBD-p) 26 M ⊙ at 10 kpc (JUNO-IBD-p)12 M ⊙ at 10 kpc (SK-IBD-p) 16 M ⊙ at 10 kpc (SK-IBD-p) 20 M ⊙ at 10 kpc (SK-IBD-p) 26 M ⊙ at 10 kpc (SK-IBD-p)12 M ⊙ at 10 kpc (DUNE-CCAre) 16 M ⊙ at 10 kpc (DUNE-CCAre) 20 M ⊙ at 10 kpc (DUNE-CCAre) 26 M ⊙ at 10 kpc (DUNE-CCAre)12 M ⊙ at 10 kpc (JUNO-IBD-p) 16 M ⊙ at 10 kpc (JUNO-IBD-p) 20 M ⊙ at 10 kpc (JUNO-IBD-p) 26 M ⊙ at 10 kpc (JUNO-IBD-p)12 M ⊙ at 10 kpc (SK-IBD-p) 16 M ⊙ at 10 kpc (SK-IBD-p) 20 M ⊙ at 10 kpc (SK-IBD-p) 26 M ⊙ at 10 kpc (SK-IBD-p)12 M ⊙ at 10 kpc (DUNE-CCAre) 16 M ⊙ at 10 kpc (DUNE-CCAre) 20 M ⊙ at 10 kpc (DUNE-CCAre) 26 M ⊙ at 10 kpc (DUNE-CCAre)12 M ⊙ at 10 kpc (JUNO-IBD-p) 16 M ⊙ at 10 kpc (JUNO-IBD-p) 20 M ⊙ at 10 kpc (JUNO-IBD-p) 26 M ⊙ at 10 kpc (JUNO-IBD-p)12 M ⊙ at 10 kpc (SK-IBD-p) 16 M ⊙ at 10 kpc (SK-IBD-p) 20 M ⊙ at 10 kpc (SK-IBD-p) 26 M ⊙ at 10 kpc (SK-IBD-p)12 M ⊙ at 10 kpc (DUNE-CCAre) 16 M ⊙ at 10 kpc (DUNE-CCAre) 20 M ⊙ at 10 kpc (DUNE-CCAre) 26 M ⊙ at 10 kpc (DUNE-CCAre)12 M ⊙ at 10 kpc (JUNO-IBD-p) 16 M ⊙ at 10 kpc (JUNO-IBD-p) 20 M ⊙ at 10 kpc (JUNO-IBD-p) 26 M ⊙ at 10 kpc (JUNO-IBD-p) C u m u l a t i ve E ve n t s [ / M e V ]
12 M ⊙ at 10 kpc (SK-IBD-p) 16 M ⊙ at 10 kpc (SK-IBD-p) 20 M ⊙ at 10 kpc (SK-IBD-p) 26 M ⊙ at 10 kpc (SK-IBD-p)12 M ⊙ at 10 kpc (DUNE-CCAre) 16 M ⊙ at 10 kpc (DUNE-CCAre) 20 M ⊙ at 10 kpc (DUNE-CCAre) 26 M ⊙ at 10 kpc (DUNE-CCAre)12 M ⊙ at 10 kpc (JUNO-IBD-p) 16 M ⊙ at 10 kpc (JUNO-IBD-p) 20 M ⊙ at 10 kpc (JUNO-IBD-p) 26 M ⊙ at 10 kpc (JUNO-IBD-p)12 M ⊙ at 10 kpc (SK-IBD-p) 16 M ⊙ at 10 kpc (SK-IBD-p) 20 M ⊙ at 10 kpc (SK-IBD-p) 26 M ⊙ at 10 kpc (SK-IBD-p)12 M ⊙ at 10 kpc (DUNE-CCAre) 16 M ⊙ at 10 kpc (DUNE-CCAre) 20 M ⊙ at 10 kpc (DUNE-CCAre) 26 M ⊙ at 10 kpc (DUNE-CCAre)12 M ⊙ at 10 kpc (JUNO-IBD-p) 16 M ⊙ at 10 kpc (JUNO-IBD-p) 20 M ⊙ at 10 kpc (JUNO-IBD-p) 26 M ⊙ at 10 kpc (JUNO-IBD-p)12 M ⊙ at 10 kpc (SK-IBD-p) 16 M ⊙ at 10 kpc (SK-IBD-p) 20 M ⊙ at 10 kpc (SK-IBD-p) 26 M ⊙ at 10 kpc (SK-IBD-p)12 M ⊙ at 10 kpc (DUNE-CCAre) 16 M ⊙ at 10 kpc (DUNE-CCAre) 20 M ⊙ at 10 kpc (DUNE-CCAre) 26 M ⊙ at 10 kpc (DUNE-CCAre)12 M ⊙ at 10 kpc (JUNO-IBD-p) 16 M ⊙ at 10 kpc (JUNO-IBD-p) 20 M ⊙ at 10 kpc (JUNO-IBD-p) 26 M ⊙ at 10 kpc (JUNO-IBD-p)12 M ⊙ at 10 kpc (SK-IBD-p) 16 M ⊙ at 10 kpc (SK-IBD-p) 20 M ⊙ at 10 kpc (SK-IBD-p) 26 M ⊙ at 10 kpc (SK-IBD-p)12 M ⊙ at 10 kpc (DUNE-CCAre) 16 M ⊙ at 10 kpc (DUNE-CCAre) 20 M ⊙ at 10 kpc (DUNE-CCAre) 26 M ⊙ at 10 kpc (DUNE-CCAre)12 M ⊙ at 10 kpc (JUNO-IBD-p) 16 M ⊙ at 10 kpc (JUNO-IBD-p) 20 M ⊙ at 10 kpc (JUNO-IBD-p) 26 M ⊙ at 10 kpc (JUNO-IBD-p) C u m u l a t i ve E ve n t s [ / M e V ] Energy [MeV]
12 M ⊙ at 10 kpc (SK-IBD-p) 16 M ⊙ at 10 kpc (SK-IBD-p) 20 M ⊙ at 10 kpc (SK-IBD-p) 26 M ⊙ at 10 kpc (SK-IBD-p)12 M ⊙ at 10 kpc (DUNE-CCAre) 16 M ⊙ at 10 kpc (DUNE-CCAre) 20 M ⊙ at 10 kpc (DUNE-CCAre) 26 M ⊙ at 10 kpc (DUNE-CCAre)12 M ⊙ at 10 kpc (JUNO-IBD-p) 16 M ⊙ at 10 kpc (JUNO-IBD-p) 20 M ⊙ at 10 kpc (JUNO-IBD-p) 26 M ⊙ at 10 kpc (JUNO-IBD-p)
12 M ⊙ at 10 kpc (SK-IBD-p) 16 M ⊙ at 10 kpc (SK-IBD-p) 20 M ⊙ at 10 kpc (SK-IBD-p) 26 M ⊙ at 10 kpc (SK-IBD-p)12 M ⊙ at 10 kpc (DUNE-CCAre) 16 M ⊙ at 10 kpc (DUNE-CCAre) 20 M ⊙ at 10 kpc (DUNE-CCAre) 26 M ⊙ at 10 kpc (DUNE-CCAre)12 M ⊙ at 10 kpc (JUNO-IBD-p) 16 M ⊙ at 10 kpc (JUNO-IBD-p) 20 M ⊙ at 10 kpc (JUNO-IBD-p) 26 M ⊙ at 10 kpc (JUNO-IBD-p)
12 M ⊙ at 10 kpc (SK-IBD-p) 16 M ⊙ at 10 kpc (SK-IBD-p) 20 M ⊙ at 10 kpc (SK-IBD-p) 26 M ⊙ at 10 kpc (SK-IBD-p)12 M ⊙ at 10 kpc (DUNE-CCAre) 16 M ⊙ at 10 kpc (DUNE-CCAre) 20 M ⊙ at 10 kpc (DUNE-CCAre) 26 M ⊙ at 10 kpc (DUNE-CCAre)12 M ⊙ at 10 kpc (JUNO-IBD-p) 16 M ⊙ at 10 kpc (JUNO-IBD-p) 20 M ⊙ at 10 kpc (JUNO-IBD-p) 26 M ⊙ at 10 kpc (JUNO-IBD-p)
12 M ⊙ at 10 kpc (SK-IBD-p) 16 M ⊙ at 10 kpc (SK-IBD-p) 20 M ⊙ at 10 kpc (SK-IBD-p) 26 M ⊙ at 10 kpc (SK-IBD-p)12 M ⊙ at 10 kpc (DUNE-CCAre) 16 M ⊙ at 10 kpc (DUNE-CCAre) 20 M ⊙ at 10 kpc (DUNE-CCAre) 26 M ⊙ at 10 kpc (DUNE-CCAre)12 M ⊙ at 10 kpc (JUNO-IBD-p) 16 M ⊙ at 10 kpc (JUNO-IBD-p) 20 M ⊙ at 10 kpc (JUNO-IBD-p) 26 M ⊙ at 10 kpc (JUNO-IBD-p) Figure 12.
Energy spectrum of the cumulative number of events for the major reaction channel of each detector. We show the result forSK, DUNE, and JUNO from top to bottom. From left to right, we display the 12, 16, 20, and 26 M (cid:12) models. The color represents thetime. The line type distinguishes the neutrino oscillation models. (12 and 15 M (cid:12) ) from the behavior of the explosion modelsin the case without flavor conversion (see the left column ofFig. 14). The systematic deviation of non-exploding modelscan be understood as follows: Non-exploding models manifesta high accretion component for ν e and ¯ ν e emissions in thelater phases, while the average energy is remarkably higherthan during the early phases (see top-right and middle-rightpanels of Fig. 3), which increases the detection efficiency forall detectors. As a result, the event counts tend to be higherwith respect to the same TONE. On the other hand, the de-viation is smaller in the cases with flavor conversions. For in-stance, an almost progenitor-independent correlation emergesat DUNE for the normal-mass hierarchy. This is attributed tothe fact that the event counts reflect ν x at the CCSN sourcein the neutrino oscillation model. We note that ν x consti-tutes the dominant contribution to TONE . In the caseswith other detectors (SK, HK, JUNO and IceCube), theyalso have the similar trend as seen in DUNE. It should bementioned that the progenitor dependence of the correlationis much smaller in the inverted-mass hierarchy than in thenormal one for these detectors, since ¯ ν e at the Earth mostlyreflects the properties of ν x at the supernova. We note that the neutrino luminosity of the individual species ofheavy leptonic neutrinos is smaller than that of ν e or ¯ ν e neutrinos.However, we have four such species. Below we provide approximate formulae for the correla-tions for the neutrino oscillation models. We first point outthat the quadratic fit used in Nagakura et al. (2021) can notcapture the simulation results adequately; hence, we fit themwith a higher-order quartic polynominal. It should be notedthat, although the fit can be improved by using cubic func-tions, we find that the functions break the monotonic relationbefore TONE reaches 6 × erg. This is actually unphysi-cal. Hence, we employ quartic functions in the fit. We confirmthat monotonicity is guaranteed up to a TONE of 10 erg,which is a firm upper limit to the total emission of CCSNneutrinos (see also Reed & Horowitz 2020).The fitting formulae are given in the case of normal mass MNRAS , 1–19 (2020)
CSN neutrinos by long-term 2D simulations hierarchy as:[SK − IBDp − NORMAL] N Cum = (cid:0) E + 5 E − . E + 0 . E (cid:1)(cid:18) V . (cid:19) (cid:18) d
10 kpc (cid:19) − , (24)[DUNE − CCAre − NORMAL] N Cum = (cid:0) E + 4 . E − . E + 0 . E (cid:1)(cid:18) V
40 ktons (cid:19) (cid:18) d
10 kpc (cid:19) − , (25)[JUNO − IBDp − NORMAL] N Cum = (cid:0) E + 5 . E − . E + 0 . E (cid:1)(cid:18) V
20 ktons (cid:19) (cid:18) d
10 kpc (cid:19) − , (26)[IceCube − IBDp − NORMAL] N Cum = (cid:0) E + 600 E − E + 0 . E (cid:1)(cid:18) V . (cid:19) (cid:18) d
10 kpc (cid:19) − , (27)and in the case with the inverted mass hierarchy as[SK − IBDp − InV] N Cum = (cid:0) E + 4 E − . E + 0 . E (cid:1)(cid:18) V . (cid:19) (cid:18) d
10 kpc (cid:19) − , (28)[DUNE − CCAre − InV] N Cum = (cid:0) E + 4 . E − . E + 0 . E (cid:1)(cid:18) V
40 ktons (cid:19) (cid:18) d
10 kpc (cid:19) − , (29)[JUNO − IBDp − InV] N Cum = (cid:0) E + 3 E − . E + 0 . E (cid:1)(cid:18) V
20 ktons (cid:19) (cid:18) d
10 kpc (cid:19) − , (30)[IceCube − IBDp − InV] N Cum = (cid:0) E + 430 E − E + 0 . E (cid:1)(cid:18) V . (cid:19) (cid:18) d
10 kpc (cid:19) − , (31)where N Cum , E , and V denote the cumulative number ofevents, TONE in the units of 10 ergs, and the detector vol-ume, respectively. We note that Eqs. 24 and 28 with V = 220ktons represent the HK case.There is a caveat regarding the fitting formulae. Althoughthey are capable of reproducing the results of explosion mod-els, there is a systematic deviation for non-exploding modelsfor all the detectors for the normal mass hierarchy, and forDUNE with the inverted mass hierarchy (see Fig. 14). Thisis attributed to the fact that the accretion component of ν e sor ¯ ν e s (at the supernova) at late times contributes then sub-stantially to the event counts (as discussed already). As aresult, the event counts tend to be higher than other caseswith respect to the same TONE (see also Fig. 13 and rele-vant discussions). On the other hand, the systematic error isroughly ∼ ∼
50 kpc) is (cid:46) ∼
700 kpc) with ∼
10% errors. This indicates that the statis-tical noise does not compromise the accuracy of the estima-tion when compared with the estimate from the retrieved en-ergy spectrum of all the flavors of neutrino (Nagakura 2021).As an interesting demonstration, we apply our fitting for-mulae to estimate a TONE of SN 1987A from event counts inKamiokande-II (Hirata et al. 1987). We assume that all eventswere detected through the IBD-p reaction channel, and thatthe detector configuration is the same as that in SK exceptfor the fiducial volume, which is ∼ ∼ × erg.By using the obtained TONE, we further estimate the massof the neutron star in SN 1987A. Here, we assume that theTONE is the same as the binding energy of the NS. Wealso assume that the dimensionless tidal polarizability at M = 1 . M (cid:12) (Λ . ) is ∼ . By employing the result of(Reed & Horowitz 2020), the gravitational mass of the neu-tron star can be estimated as ∼ . M (cid:12) . The result seemsconsistent with that of other observed neutron stars (see, e.g.,¨Ozel & Freire 2016), albeit smaller than the canonical values( ∼ . M (cid:12) ). Our long-term 2D CCSN models for a wide range of progen-itor mass reveal some new features in CCSN dynamics andconcerning neutrino signals at the late phases. The matterdynamics in the vicinity of the PNS is highly dynamical evenduring the later phases, due not only to PNS convection,but also asymmetrical fallback mass accretions and fluid in-stabilities. It should be stressed that not excising the innerregion of the CCSN core in a simulation is crucial for cap-turing all possible feedback effects on the neutrino signals.Using a self-consistent treatment throughput, we found thatthe temporal variations in the neutrino emissions mainly cor-relate with those in mass-accretion rate (see Fig. 8). We alsofound that the correlation is generic for all explosion models,although the actual impact on the neutrino signals dependson models. We stress that the dynamical features in neutrino This corresponds to the case using the SFHo EOS which weemploy in our CCSN simulations (see, e.g., Steiner et al. 2013;Han & Steiner 2019). It is also within the observational constraints(Λ . = 190 +390 − ) placed by Abbott (2018).MNRAS000
10% errors. This indicates that the statis-tical noise does not compromise the accuracy of the estima-tion when compared with the estimate from the retrieved en-ergy spectrum of all the flavors of neutrino (Nagakura 2021).As an interesting demonstration, we apply our fitting for-mulae to estimate a TONE of SN 1987A from event counts inKamiokande-II (Hirata et al. 1987). We assume that all eventswere detected through the IBD-p reaction channel, and thatthe detector configuration is the same as that in SK exceptfor the fiducial volume, which is ∼ ∼ × erg.By using the obtained TONE, we further estimate the massof the neutron star in SN 1987A. Here, we assume that theTONE is the same as the binding energy of the NS. Wealso assume that the dimensionless tidal polarizability at M = 1 . M (cid:12) (Λ . ) is ∼ . By employing the result of(Reed & Horowitz 2020), the gravitational mass of the neu-tron star can be estimated as ∼ . M (cid:12) . The result seemsconsistent with that of other observed neutron stars (see, e.g.,¨Ozel & Freire 2016), albeit smaller than the canonical values( ∼ . M (cid:12) ). Our long-term 2D CCSN models for a wide range of progen-itor mass reveal some new features in CCSN dynamics andconcerning neutrino signals at the late phases. The matterdynamics in the vicinity of the PNS is highly dynamical evenduring the later phases, due not only to PNS convection,but also asymmetrical fallback mass accretions and fluid in-stabilities. It should be stressed that not excising the innerregion of the CCSN core in a simulation is crucial for cap-turing all possible feedback effects on the neutrino signals.Using a self-consistent treatment throughput, we found thatthe temporal variations in the neutrino emissions mainly cor-relate with those in mass-accretion rate (see Fig. 8). We alsofound that the correlation is generic for all explosion models,although the actual impact on the neutrino signals dependson models. We stress that the dynamical features in neutrino This corresponds to the case using the SFHo EOS which weemploy in our CCSN simulations (see, e.g., Steiner et al. 2013;Han & Steiner 2019). It is also within the observational constraints(Λ . = 190 +390 − ) placed by Abbott (2018).MNRAS000 , 1–19 (2020) H. Nagakura et al. C u m u l a t i ve E ve n t s [ ]
12 M ⊙
13 M ⊙
14 M ⊙
15 M ⊙
16 M ⊙
17 M ⊙
18 M ⊙
19 M ⊙ SK-IBD-p (NOOSC) SK-IBD-p (NORMAL) SK-IBD-p (InV)DUNE-CCAre (NOOSC) DUNE-CCAre (NORMAL) DUNE-CCAre (InV)JUNO-IBD-p (NOOSC) JUNO-IBD-p (NORMAL) JUNO-IBD-p (InV)IceCube-IBD-p (NOOSC) IceCube-IBD-p (NORMAL) IceCube-IBD-p (InV)10 kpc
20 M ⊙
21 M ⊙
22 M ⊙
23 M ⊙
25 M ⊙
26 M ⊙ ⊙ SK-IBD-p (NOOSC) SK-IBD-p (NORMAL) SK-IBD-p (InV)DUNE-CCAre (NOOSC) DUNE-CCAre (NORMAL) DUNE-CCAre (InV)JUNO-IBD-p (NOOSC) JUNO-IBD-p (NORMAL) JUNO-IBD-p (InV)IceCube-IBD-p (NOOSC) IceCube-IBD-p (NORMAL) IceCube-IBD-p (InV)10 kpcSK-IBD-p (NOOSC) SK-IBD-p (NORMAL) SK-IBD-p (InV)DUNE-CCAre (NOOSC) DUNE-CCAre (NORMAL) DUNE-CCAre (InV)JUNO-IBD-p (NOOSC) JUNO-IBD-p (NORMAL) JUNO-IBD-p (InV)IceCube-IBD-p (NOOSC) IceCube-IBD-p (NORMAL) IceCube-IBD-p (InV)10 kpc02468 C u m u l a t i ve E ve n t s [ ] SK-IBD-p (NOOSC) SK-IBD-p (NORMAL) SK-IBD-p (InV)DUNE-CCAre (NOOSC) DUNE-CCAre (NORMAL) DUNE-CCAre (InV)JUNO-IBD-p (NOOSC) JUNO-IBD-p (NORMAL) JUNO-IBD-p (InV)IceCube-IBD-p (NOOSC) IceCube-IBD-p (NORMAL) IceCube-IBD-p (InV)10 kpcSK-IBD-p (NOOSC) SK-IBD-p (NORMAL) SK-IBD-p (InV)DUNE-CCAre (NOOSC) DUNE-CCAre (NORMAL) DUNE-CCAre (InV)JUNO-IBD-p (NOOSC) JUNO-IBD-p (NORMAL) JUNO-IBD-p (InV)IceCube-IBD-p (NOOSC) IceCube-IBD-p (NORMAL) IceCube-IBD-p (InV)10 kpcSK-IBD-p (NOOSC) SK-IBD-p (NORMAL) SK-IBD-p (InV)DUNE-CCAre (NOOSC) DUNE-CCAre (NORMAL) DUNE-CCAre (InV)JUNO-IBD-p (NOOSC) JUNO-IBD-p (NORMAL) JUNO-IBD-p (InV)IceCube-IBD-p (NOOSC) IceCube-IBD-p (NORMAL) IceCube-IBD-p (InV)10 kpc02468101214 C u m u l a t i ve E ve n t s [ ] SK-IBD-p (NOOSC) SK-IBD-p (NORMAL) SK-IBD-p (InV)DUNE-CCAre (NOOSC) DUNE-CCAre (NORMAL) DUNE-CCAre (InV)JUNO-IBD-p (NOOSC) JUNO-IBD-p (NORMAL) JUNO-IBD-p (InV)IceCube-IBD-p (NOOSC) IceCube-IBD-p (NORMAL) IceCube-IBD-p (InV)10 kpcSK-IBD-p (NOOSC) SK-IBD-p (NORMAL) SK-IBD-p (InV)DUNE-CCAre (NOOSC) DUNE-CCAre (NORMAL) DUNE-CCAre (InV)JUNO-IBD-p (NOOSC) JUNO-IBD-p (NORMAL) JUNO-IBD-p (InV)IceCube-IBD-p (NOOSC) IceCube-IBD-p (NORMAL) IceCube-IBD-p (InV)10 kpcSK-IBD-p (NOOSC) SK-IBD-p (NORMAL) SK-IBD-p (InV)DUNE-CCAre (NOOSC) DUNE-CCAre (NORMAL) DUNE-CCAre (InV)JUNO-IBD-p (NOOSC) JUNO-IBD-p (NORMAL) JUNO-IBD-p (InV)IceCube-IBD-p (NOOSC) IceCube-IBD-p (NORMAL) IceCube-IBD-p (InV)10 kpc0369121518 0 0.5 1 1.5 2 2.5 3 3.5 4 C u m u l a t i ve E ve n t s [ ] Time [s]SK-IBD-p (NOOSC) SK-IBD-p (NORMAL) SK-IBD-p (InV)DUNE-CCAre (NOOSC) DUNE-CCAre (NORMAL) DUNE-CCAre (InV)JUNO-IBD-p (NOOSC) JUNO-IBD-p (NORMAL) JUNO-IBD-p (InV)IceCube-IBD-p (NOOSC) IceCube-IBD-p (NORMAL) IceCube-IBD-p (InV)10 kpc 0 0.5 1 1.5 2 2.5 3 3.5 4Time [s]SK-IBD-p (NOOSC) SK-IBD-p (NORMAL) SK-IBD-p (InV)DUNE-CCAre (NOOSC) DUNE-CCAre (NORMAL) DUNE-CCAre (InV)JUNO-IBD-p (NOOSC) JUNO-IBD-p (NORMAL) JUNO-IBD-p (InV)IceCube-IBD-p (NOOSC) IceCube-IBD-p (NORMAL) IceCube-IBD-p (InV)10 kpc 0 0.5 1 1.5 2 2.5 3 3.5 4Time [s]SK-IBD-p (NOOSC) SK-IBD-p (NORMAL) SK-IBD-p (InV)DUNE-CCAre (NOOSC) DUNE-CCAre (NORMAL) DUNE-CCAre (InV)JUNO-IBD-p (NOOSC) JUNO-IBD-p (NORMAL) JUNO-IBD-p (InV)IceCube-IBD-p (NOOSC) IceCube-IBD-p (NORMAL) IceCube-IBD-p (InV)10 kpc
Figure 13.
The time evolution of cumulative number of events in the major reaction channel of each detector.Color and line type distinguishes CCSN models and explosion/non-explosion, respectively, which are the sameas those used in Fig. 1. From top to bottom, we show the results of SK, DUNE, JUNO, and IceCube. Fromleft to right, a different neutrino oscillation model is assumed. signals during the late post-bounce phase are the missing inthe previous toy or spherically-symmetric models.In this study, we employed SNOwGLoBES, taking into ac-count neutrino oscillations with an adiabatic approach. Weprovided some basic results for the neutrino signals, such astime evolution of the event rate (see Fig. 10) and energyspectra for the cumulative number of events (see Fig. 12) ateach detector. We also assessed the detectability of temporalvariations of the event rate at each detector by employing anoise model (see Eqs. 20 and 23). Not unexpectedly, we findthat IceCube would be the best detector with which to study temporal variations. We have updated our fitting formulaefor the correlation between cumulative number of events ateach detector and the total neutrino energy (TONE) emit-ted at a CCSN source. Such formulae will prove very usefulfor low-statistic detections, i.e., distant CCSNe. Indeed, wepresent an interesting demonstration by using the real datafor SN 1987A at Kamiokande-II, and we find the TONE is ∼ × erg and the corresponding (gravitational) NS masscould be near ∼ . M (cid:12) . We note that CCSNe at the An-dromeda galaxy will also be targets once HK is available.There remain interesting issues to be addressed. It has been MNRAS , 1–19 (2020)
CSN neutrinos by long-term 2D simulations C u m u l a t i ve E ve n t s [ ]
12 M ⊙
13 M ⊙
14 M ⊙
15 M ⊙
16 M ⊙
17 M ⊙
18 M ⊙
19 M ⊙ SK-IBD-p (NOOSC) SK-IBD-p (NORMAL) SK-IBD-p (InV)DUNE-CCAre (NOOSC) DUNE-CCAre (NORMAL) DUNE-CCAre (InV)JUNO-IBD-p (NOOSC) JUNO-IBD-p (NORMAL) JUNO-IBD-p (InV)IceCube-IBD-p (NOOSC) IceCube-IBD-p (NORMAL) IceCube-IBD-p (InV)10 kpc
20 M ⊙
21 M ⊙
22 M ⊙
23 M ⊙
25 M ⊙
26 M ⊙ ⊙ SK-IBD-p (NOOSC) SK-IBD-p (NORMAL) SK-IBD-p (InV)DUNE-CCAre (NOOSC) DUNE-CCAre (NORMAL) DUNE-CCAre (InV)JUNO-IBD-p (NOOSC) JUNO-IBD-p (NORMAL) JUNO-IBD-p (InV)IceCube-IBD-p (NOOSC) IceCube-IBD-p (NORMAL) IceCube-IBD-p (InV)10 kpcSK-IBD-p (NOOSC) SK-IBD-p (NORMAL) SK-IBD-p (InV)DUNE-CCAre (NOOSC) DUNE-CCAre (NORMAL) DUNE-CCAre (InV)JUNO-IBD-p (NOOSC) JUNO-IBD-p (NORMAL) JUNO-IBD-p (InV)IceCube-IBD-p (NOOSC) IceCube-IBD-p (NORMAL) IceCube-IBD-p (InV)10 kpc02468 C u m u l a t i ve E ve n t s [ ] SK-IBD-p (NOOSC) SK-IBD-p (NORMAL) SK-IBD-p (InV)DUNE-CCAre (NOOSC) DUNE-CCAre (NORMAL) DUNE-CCAre (InV)JUNO-IBD-p (NOOSC) JUNO-IBD-p (NORMAL) JUNO-IBD-p (InV)IceCube-IBD-p (NOOSC) IceCube-IBD-p (NORMAL) IceCube-IBD-p (InV)10 kpcSK-IBD-p (NOOSC) SK-IBD-p (NORMAL) SK-IBD-p (InV)DUNE-CCAre (NOOSC) DUNE-CCAre (NORMAL) DUNE-CCAre (InV)JUNO-IBD-p (NOOSC) JUNO-IBD-p (NORMAL) JUNO-IBD-p (InV)IceCube-IBD-p (NOOSC) IceCube-IBD-p (NORMAL) IceCube-IBD-p (InV)10 kpcSK-IBD-p (NOOSC) SK-IBD-p (NORMAL) SK-IBD-p (InV)DUNE-CCAre (NOOSC) DUNE-CCAre (NORMAL) DUNE-CCAre (InV)JUNO-IBD-p (NOOSC) JUNO-IBD-p (NORMAL) JUNO-IBD-p (InV)IceCube-IBD-p (NOOSC) IceCube-IBD-p (NORMAL) IceCube-IBD-p (InV)10 kpc02468101214 C u m u l a t i ve E ve n t s [ ] SK-IBD-p (NOOSC) SK-IBD-p (NORMAL) SK-IBD-p (InV)DUNE-CCAre (NOOSC) DUNE-CCAre (NORMAL) DUNE-CCAre (InV)JUNO-IBD-p (NOOSC) JUNO-IBD-p (NORMAL) JUNO-IBD-p (InV)IceCube-IBD-p (NOOSC) IceCube-IBD-p (NORMAL) IceCube-IBD-p (InV)10 kpcSK-IBD-p (NOOSC) SK-IBD-p (NORMAL) SK-IBD-p (InV)DUNE-CCAre (NOOSC) DUNE-CCAre (NORMAL) DUNE-CCAre (InV)JUNO-IBD-p (NOOSC) JUNO-IBD-p (NORMAL) JUNO-IBD-p (InV)IceCube-IBD-p (NOOSC) IceCube-IBD-p (NORMAL) IceCube-IBD-p (InV)10 kpcSK-IBD-p (NOOSC) SK-IBD-p (NORMAL) SK-IBD-p (InV)DUNE-CCAre (NOOSC) DUNE-CCAre (NORMAL) DUNE-CCAre (InV)JUNO-IBD-p (NOOSC) JUNO-IBD-p (NORMAL) JUNO-IBD-p (InV)IceCube-IBD-p (NOOSC) IceCube-IBD-p (NORMAL) IceCube-IBD-p (InV)10 kpc0369121518 0 10 20 30 40 C u m u l a t i ve E ve n t s [ ] TONE [10 erg] SK-IBD-p (NOOSC) SK-IBD-p (NORMAL) SK-IBD-p (InV)DUNE-CCAre (NOOSC) DUNE-CCAre (NORMAL) DUNE-CCAre (InV)JUNO-IBD-p (NOOSC) JUNO-IBD-p (NORMAL) JUNO-IBD-p (InV)IceCube-IBD-p (NOOSC) IceCube-IBD-p (NORMAL) IceCube-IBD-p (InV)10 kpc 0 10 20 30 40
TONE [10 erg] SK-IBD-p (NOOSC) SK-IBD-p (NORMAL) SK-IBD-p (InV)DUNE-CCAre (NOOSC) DUNE-CCAre (NORMAL) DUNE-CCAre (InV)JUNO-IBD-p (NOOSC) JUNO-IBD-p (NORMAL) JUNO-IBD-p (InV)IceCube-IBD-p (NOOSC) IceCube-IBD-p (NORMAL) IceCube-IBD-p (InV)10 kpc 0 10 20 30 40
TONE [10 erg] SK-IBD-p (NOOSC) SK-IBD-p (NORMAL) SK-IBD-p (InV)DUNE-CCAre (NOOSC) DUNE-CCAre (NORMAL) DUNE-CCAre (InV)JUNO-IBD-p (NOOSC) JUNO-IBD-p (NORMAL) JUNO-IBD-p (InV)IceCube-IBD-p (NOOSC) IceCube-IBD-p (NORMAL) IceCube-IBD-p (InV)10 kpc
Figure 14.
Cumulative number of events for the major reaction channel of each detector as a function of thetotal neutrino energy (TONE) emitted at CCSN sources (10 kpc). The position of each panel, color, line typeare the same as those in Fig. 13. reported that stellar rotation affects the neutrino signal (see,e.g., Summa et al. 2018) and we have yet to ascertain thedegree to which our fitting formulae might be altered to ac-commodate it (Eqs. 24-31). It should be mentioned, however,that the effect should be minor, unless the rotation is remark-ably faster than expected from stellar evolution. Another con-cern is with possible collective neutrino oscillations; indeed,there have been many reports that fast pairwise conversioncould occur in both the preshock and post shock regions (Ab-bar et al. 2019; Nagakura et al. 2019c; Morinaga et al. 2020;Delfan Azari et al. 2020; Glas et al. 2020; Abbar et al. 2020;Capozzi et al. 2020). Although this could give a significant im- pact on neutrino signals in the early post bounce phase ( (cid:46)
ACKNOWLEDGEMENTS
The authors acknowledge Kate Scholberg for help using theSNOwGLoBES software. We are also grateful for ongoing
MNRAS000
MNRAS000 , 1–19 (2020) H. Nagakura et al. contributions to the effort of CCSN simulation projects byDavid Radice, Josh Dolence, Aaron Skinner, Matthew Cole-man, and Chris White. We acknowledge support from theU.S. Department of Energy Office of Science and the Officeof Advanced Scientific Computing Research via the Scien-tific Discovery through Advanced Computing (SciDAC4) pro-gram and Grant DE-SC0018297 (subaward 00009650). In ad-dition, we gratefully acknowledge support from the U.S. NSFunder Grants AST-1714267 and PHY-1804048 (the lattervia the Max-Planck/Princeton Center (MPPC) for PlasmaPhysics). An award of computer time was provided by theINCITE program. That research used resources of the Ar-gonne Leadership Computing Facility, which is a DOE Of-fice of Science User Facility supported under Contract DE-AC02-06CH11357. In addition, this overall research projectis part of the Blue Waters sustained-petascale computingproject, which is supported by the National Science Founda-tion (awards OCI-0725070 and ACI-1238993) and the stateof Illinois. Blue Waters is a joint effort of the University ofIllinois at Urbana-Champaign and its National Center for Su-percomputing Applications. This general project is also partof the “Three-Dimensional Simulations of Core-Collapse Su-pernovae” PRAC allocation support by the National ScienceFoundation (under award
DATA AVAILABILITY
The data underlying this article will be shared on reasonablerequest to the corresponding author.
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