The first (Pop III) stars formed only out of H and He and were likely more massive than present-day stars. Massive Pop III stars in the range 140-260 M$_\odot$ are predicted to end their lives as pair-instability supernovae (PISNe), enriching the environment with a unique abundance pattern, with high ratios of odd to even elements. Recently, the most promising candidate for a pure descendant of a zero-metallicity massive PISN (260 M$_{\odot}$) was discovered by the LAMOST survey, the star J1010+2358. However, the key elements to verify the high PISN contribution, C and Al, were missing from the analysis. To rectify this, we obtained and analyzed a high-resolution VLT/UVES spectrum, correcting for 3D and/or non-LTE effects. Our measurements of both C and Al give much higher values (~1 dex) than expected from a 260 M$_{\odot}$ PISN. Furthermore, we find significant discrepancies with the previous analysis, and therefore a much less pronounced odd-even pattern. Thus, we show that J1010+2358 cannot be a pure descendant of a 260 M$_{\odot}$ PISN. Instead, we find that the best fit model consists of a 13 M$_{\odot}$ Pop II core-collapse supernova combined with a Pop III supernova. Alternative, less favoured solutions $(χ^2/χ^2_{\rm best}\approx2.3)$ include a 50% contribution from a 260 M$_{\odot}$ PISN, or a 40% contribution from a Pop III type Ia supernova. Ultimately, J1010+2358 is certainly a unique star giving insights into the earliest chemical enrichment, however, this star is not a pure PISN descendant.
ULTIMATE-Subaru (Ultra-wide Laser Tomographic Imager and MOS with AO for Transcendent Exploration on Subaru) and WFIRST (Wide Field Infra-Red Survey Telescope) are the next generation near-infrared instruments that have a large field-of-view. They allow us to conduct deep and wide transient surveys in near-infrared. Such a near-infrared transient survey enables us to find very distant supernovae that are redshifted to the near-infrared wavelengths. We have performed the mock transient surveys with ULTIMATE-Subaru and WFIRST to investigate their ability to discover Population III pair-instabili…
We present 2D simulations of pair-instability supernovae considering rapid rotation during their explosion phases. Recent studies of the Pop III star formation suggested that these stars could be born with a mass scale about 100 Msun and with a strong rotation. Based on stellar evolution models, these massive Pop III stars might have died as highly energetic pair-instability supernovae. We perform 2D calculations to investigate the impact of rotation on pair-instability supernovae. Our results suggest that rotation leads to an aspherical explosion due to an anisotropic collapse. If the first s…
A very massive star with a carbon-oxygen core in the range of $64$ M$_{\odot}<M_{\mathrm{CO}}<133$ M$_{\odot}$ is expected to undergo a very different kind of explosion known as a pair instability supernova. Pair instability supernovae are candidates for superluminous supernovae due to the prodigious amounts of radioactive elements they create. While the basic mechanism for the explosion is understood, how a star reaches a state is not, thus observations of a nearby pair-instability supernova would allow us to test current models of stellar evolution at the extreme of stellar masses. Much will…
Interactions between massive stars in binaries are thought to be responsible for much of the observed diversity of supernovae. As surveys probe rarer populations of events, we should expect to see supernovae arising from increasingly uncommon progenitor channels. Here we examine a scenario in which massive stars merge after they have both formed a hydrogen-exhausted core. We suggest this could produce stars which explode as pair-instability supernovae (PISNe) with significantly more hydrogen, at a given metallicity, than in single-star models with the same pre-explosion oxygen-rich core mass. …
In certain mass ranges, massive stars can undergo a violent pulsation triggered by the electron/positron pair instability that ejects matter, but does not totally disrupt the star. After one or more of these pulsations, such stars are expected to undergo core-collapse to trigger a supernova explosion. The mass range susceptible to this pulsational phenomena may be as low as 50-70 Msun if the progenitor is of very low metallicity and rotating sufficiently rapidly to undergo nearly homogeneous evolution. The mass, dynamics, and composition of the matter ejected in the pulsation are important asp…
It was recently proposed that exotic particles can trigger a new stellar instability which is analogous to the e-e+ pair instability if they are produced and reach equilibrium in the stellar plasma. In this study, we construct axion instability supernova (AISN) models caused by the new instability to predict their observational signatures. We focus on heavy axion-like particles (ALPs) with masses of ~400 keV--2 MeV and coupling with photons of g_{ag}~10^{-5} GeV^{-1}. It is found that the 56Ni mass and the explosion energy are significantly increased by ALPs for a fixed stellar mass. As a resu…
The final evolution of stars in the mass range 70 - 140 solar masses is explored. Depending upon their mass loss history and rotation rates, these stars will end their lives as pulsational pair-instability supernovae producing a great variety of observational transients with total durations ranging from weeks to millennia and luminosities from 10$^{41}$ to over 10$^{44}$ erg s$^{-1}$. No non-rotating model radiates more than $5 \times 10^{50}$ erg of light or has a kinetic energy exceeding $5 \times 10^{51}$ erg, but greater energies are possible, in principle, in magnetar-powered explosions w…
An increasing number of so-called superluminous supernovae (SLSNe) are discovered. It is believed that at least some of them with slowly fading light curves originate in stellar explosions induced by the pair instability mechanism. Recent stellar evolution models naturally predict pair instability supernovae (PISNe) from very massive stars at wide range of metallicities (up to Z=0.006, Yusof et al. 2013). In the scope of this study we analyse whether PISN models can match the observational properties of SLSNe with various light curve shapes. Specifically, we explore the influence of different …
University of Florida（佛罗里达大学）；Center for Astrophysics | Harvard & Smithsonian（哈佛-史密森尼天体物理中心）；California Institute of Technology（加州理工学院）；The University of Tokyo（东京大学）；The University of Texas at Austin（德克萨斯大学奥斯汀分校）；National Astronomical Observatory of Japan, National Institutes of Natural Sciences（日本国立天文台，国立自然科学研究院）
We discuss the relevance of the magnetorotational instability (MRI) in core-collapse supernovae (CCSNe). Our recent numerical studies show that in CCSNe, the MRI is terminated by parasitic instabilities of the Kelvin-Helmholtz type. To determine whether the MRI can amplify initially weak magnetic fields to dynamically relevant strengths in CCSNe, we performed three-dimensional simulations of a region close to the surface of a differentially rotating proto-neutron star in non-ideal magnetohydrodynamics with two different numerical codes. We find that under the conditions prevailing in proto-neu…
Recent stellar evolution models show consistently that very massive metal-free stars evolve into red supergiants shortly before they explode. We argue that the envelopes of these stars, which will form pair-instability supernovae, become pulsationally unstable and that this will lead to extreme mass-loss rates despite the tiny metal content of the envelopes. We investigate the pulsational properties of such models and derive pulsationally induced mass-loss rates, which take the damping effects of the mass loss on the pulsations selfconsistently into account. We find that the pulsations may ind…
In recent years, the viability of the pair-instability supernova (PISN) scenario for explaining superluminous supernovae has all but disappeared except for a few slowly-evolving examples. However, PISN are not predicted to be superluminous throughout the bulk of their mass range. In fact, it is more likely that the first PISN we see (if we have not seen one already) will not be superluminous. Here, we present hydrodynamic simulations of PISNe for four stellar models with unique envelope properties spanning the PISN mass range. In addition, we compute synthetic light curves for comparison with …
The origin of high-redshift quasars and their supermassive black hole engines is unclear. One promising solution is the collapse of a primordial supermassive star. Observational confirmation of this scenario may be challenging, but a general relativistic instability supernova provides one avenue for such. Previous studies have found that a general relativistic instability supernova has a potentially decades-long plateau phase visible to JWST at high redshift. In this work, we examine stars with mass just below the general relativistic instability supernova mass range. These stars pulsate, ejec…
Context. Pulsational pair-instabilitye supernovae (PPISNe) and pair instability supernovae (PISNe) are the result of a thermonuclear runaway in the presence of a background electron-positron pair plasma. As such, their evolution and resultant black hole (BH) masses could possibly be affected by screening corrections due to the electron pair plasma. Aims. Sensitivity of PISNe and PPISNe to relativistic weak screening has been explored. Methods. In this paper a weak screening model that includes effects from relativistic pair production has been developed and applied at temperatures approaching …
Observed supermassive black holes in the early universe have several proposed formation channels, in part because most of these channels are difficult to probe. One of the more promising channels, the direct collapse of a supermassive star, has several possible probes including the explosion of a helium-core supermassive star triggered by a general relativistic instability. We develop a straightforward method for evaluating the general relativistic radial instability without simplifying assumptions and apply it to population III supermassive stars taken from a post Newtonian stellar evolution …
Numerical studies of primordial star formation suggest that the first stars in the universe may have been very massive. Stellar models indicate that non-rotating Population III stars with initial masses of 140-260 Msun die as highly energetic pair-instability supernovae. We present new two-dimensional simulations of primordial pair-instability supernovae done with the CASTRO code. Our simulations begin at earlier times than previous multidimensional models, at the onset of core collapse, to capture any dynamical instabilities that may be seeded by collapse and explosive burning. Such instabili…
Massive stars that end their lives with helium cores in the range of 35 to 65 Msun are known to produce repeated thermonuclear outbursts due to a recurring pair-instability. In some of these events, solar masses of material are ejected in repeated outbursts of several times 10$^{50}$ erg each. Collisions between these shells can sometimes produce very luminous transients that are visible from the edge of the observable universe. Previous 1D studies of these events produce thin, high-density shells as one ejection plows into another. Here, in the first multidimensional simulations of these coll…
Both recent observations and stellar evolution models suggest that pair-instability supernovae (PISNe) could occur in the local Universe, at metallicities below Z_Sun/3. Previous PISN models were mostly produced at very low metallicities in the context of the early Universe. We present new PISNe models at a metallicity of Z=0.001, which are relevant for the local Universe. We take the self-consistent stellar evolutionary models of pair-instability progenitors with initial masses of 150 and 250 solar masses at metallicity of Z=0.001 by Langer et al. (2007) and follow the evolution of these mode…
Massive stars having a CO core of $\sim$40-60 M$_\odot$ experience pulsational pair-instability (PPI) after carbon-burning. This instability induces strong pulsations of the whole star and a part of outer envelope is ejected. We investigate the evolution and mass ejection of metal-poor very massive stars which experience PPI. We use stellar models with initial masses of 140, 200, and 250 M$_\odot$ and the metallicity Z=0.004. Their masses decrease to 54.09, 58.65, and 61.03 M$_\odot$ before the neon-burning owing to mass-loss and He mass fraction at the surface becomes about 20%. During the PP…
Since the emergence of the new class of extremely bright transients, super-luminous supernovae (SLSNe), three main mechanisms to power their light curves (LCs) have been discussed. They are the spin-down of a magnetar, interaction with circumstellar material, and the decay of large amounts of radioactive nickel in pair-instability supernovae (PISNe). Given the high degree of diversity seen within the class, it is possible that all three mechanisms are at work. PISN models can be self- consistently simulated from the main sequence phase of very massive stars (VMS) through to their explosion. Th…
Understanding the link between massive ($\gtrsim 30$ M$_{\odot{}}$) stellar black holes (BHs) and their progenitor stars is a crucial step to interpret observations of gravitational-wave events. In this paper, we discuss the final fate of very massive stars (VMSs), with zero-age main sequence (ZAMS) mass $>150$ M$_{\odot}$, accounting for pulsational pair-instability supernovae (PPISNe) and for pair-instability supernovae (PISNe). We describe an updated version of our population synthesis code SEVN, in which we added stellar evolution tracks for VMSs with ZAMS mass up to $350 $M$_{\odot{}}$ an…
Present time-domain astronomy efforts will unveil a variety of rare transients. We focus here on pulsational pair-instability evolution, which can result in signatures observable with electromagnetic and gravitational waves. We simulate grids of bare helium stars to characterize the resulting black hole (BH) masses and ejecta composition, velocity, and thermal state. The stars do not react "elastically" to the thermonuclear explosion: there is not a one-to-one correspondence between pair-instability driven ignition and mass ejections, causing ambiguity in what is an observable pulse. In agreem…
The Effect of the Fast-Flavor Instability on Core-Collapse Supernova Models: II. Quasi-Equipartition and the Impact of Various Angular Reconstruction Methods
We infer the expected detection number of pair instability supernovae (PISNe) during the operation of the Euclid space telescope based on binary population models. Our models reproduce the global maximum of the rate at the primary BH mass of $\sim 9-10$ $M_\odot$, and the overall gradient of the primary BH mass distribution in the binary BH merger rate consistent with recent observations. We consider different PISN conditions depending on the $^{12}$C$(α, γ)^{16}$O reaction rate. The fiducial and $3σ$ models adopt the standard and $3σ$-smaller reaction rate, respectively. Our fiducial model pr…
Super-luminous supernovae that radiate more than 10^44 ergs per second at their peak luminosity have recently been discovered in faint galaxies at redshifts of 0.1-4. Some evolve slowly, resembling models of 'pair-instability' supernovae. Such models involve stars with original masses 140-260 times that of the Sun that now have carbon-oxygen cores of 65-30 solar masses. In these stars, the photons that prevent gravitational collapse are converted to electron-positron pairs, causing rapid contraction and thermonuclear explosions. Many solar masses of 56Ni are synthesized; this isotope decays to…
Very massive 140-260 Msun stars can die as highly-energetic pair-instability supernovae (PI SNe) with energies of up to 100 times those of core-collapse SNe that can completely destroy the star, leaving no compact remnant behind. These explosions can synthesize $0.1-30$ Msun of radioactive Ni56, which can cause them to rebrighten at later times when photons due to Ni56 decay diffuse out of the ejecta. However, heat from the decay of such large masses of Ni56 could also drive important dynamical effects deep in the ejecta that are capable of mixing elements and affecting the observational signa…
A model of a supernova explosion inside a dense extended hydrogenless envelope is proposed to explain the properties of the light curve for one of the superluminous supernovae PTF12dam. It is argued in the literature that the flux of this supernova rises too fast to be explained by the explosion model due to the instability associated with the electron-positron pair production (pair-instability supernova, PISNe), but it is well described by the models with energy input by a magnetar. We show that the PTF12dam-type supernovae can be explained without a magnetar in a model with a radiative shock…
New particles coupled to the Standard Model can equilibrate in stellar cores if they are sufficiently heavy and strongly coupled. In this work, we investigate the astrophysical consequences of such a scenario for massive stars by incorporating new contributions to the equation of state into a state of the art stellar structure code. We focus on axions in the "cosmological triangle", a region of parameter space with $300{\rm\,keV} \lesssim m_a \lesssim 2$ MeV, $g_{aγγ}\sim 10^{-5}$ GeV$^{-1}$ that is not presently excluded by other considerations. We find that for axion masses $m_a \sim m_e $, …
So called superluminous supernovae have been recently discovered in the local Universe. It appears possible that some of them originate from stellar explosions induced by the pair instability mechanism. Recent stellar evolution models also predict pair instability supernovae (PISNe) from very massive stars at fairly high metallicities (i.e. Z~0.004). We provide supernova (SN) models and synthetic light curves for two progenitor models, a 150 Msun red-supergiant and a 250 Msun yellow-supergiant at a metallicity of Z=0.001, for which the evolution from the main sequence to collapse, and the init…
We explore a variety of models in which SN~1961V, one of the most enigmatic supernovae (SNe) ever observed, was a pulsational pair-instability supernova (PPISN). Successful models reproduce the bolometric light curve of the principal outburst and, in some cases, the emission one year before and several years afterward. All models have helium-rich ejecta, bulk hydrogenic velocities near 2000 km s$^{-1}$, and total kinetic energies from 4 to 8 $\times 10^{50}$ erg. Each eventually leaves behind a black hole remnant. Three subclasses of PPISN models are explored, each with two different choices o…
Energetics of nuclear reaction is fundamentally important to understand the mechanism of pair instability supernovae (PISNe). Based on the hydrodynamic equations and thermodynamic relations, we derive exact expressions for energy conservation suitable to be solved in simulation. We also show that some formulae commonly used in the literature are obtained as approximations of the exact expressions. We simulate the evolution of very massive stars of ~100-320 Msun with zero- and 1/10 Zsun, and calculate further explosions as PISNe, applying each of the exact and approximate formulae. The calculat…
Pair-instability and pulsational pair-instability supernovae (PPISN) have not been unambiguously observed so far. They are, however, promising candidates for the progenitors of the heaviest binary black hole (BBH) mergers detected. If these BBHs are the product of binary evolution, then PPISNe could occur in very close binaries. Motivated by this, we discuss the implications of a PPISN happening with a close binary companion, and what impact these events have on the formation of merging BBHs through binary evolution. For this, we have computed a set of models of metal-poor ($Z_\odot/10$) singl…
We calculate the evolution of massive stars, which undergo pulsational pair-instability (PPI) when the O-rich core is formed. The evolution from the main-sequence through the onset of PPI is calculated for stars with the initial masses of $80 - 140$ $M_{\odot}$ and metallicities of $Z = 10^{-3} - 1.0$ $Z_\odot$. Because of mass loss, $Z \leq 0.5$ $Z_\odot$ is necessary for stars to form He cores massive enough (i.e., mass $>40 ~M_\odot$) to undergo PPI. The hydrodynamical phase of evolution from PPI through the beginning of Fe core collapse is calculated for the He cores with masses of $40 - 6…
We present the expected observational properties of a general relativistic instability supernova (GRSN) from the 55,500 Msun primordial (Population III) star. Supermassive stars exceeding 1e4 Msun may exist in the early Universe. They are generally considered to collapse through the general relativistic instability to be seed black holes to form supermassive (~ 1e9 Msun) black holes observed as high-redshift quasars. Some of them, however, may explode as GRSNe if the explosive helium burning unbinds the supermassive stars following the collapse triggered by the general relativistic instability…
The aim of this paper is to examine the effects of the horizontal turbulence in differentially rotating stars on the GSF instability and apply our results to pre-supernova models. For this purpose we derive the expression for the GSF instability with account of the thermal transport and smoothing of the mu-gradient by the horizontal turbulence. We apply the new expressions in numerical models of a 20 solar mass star. We show that if N^2_{Omega} < 0 the Rayleigh-Taylor instability cannot be killed by the stabilizing thermal and mu-gradients, so that the GSF instability is always there and we de…
Gravitational-wave detections are now starting to probe the mass distribution of stellar-mass black holes (BHs). Robust predictions from stellar models are needed to interpret these. Theory predicts the existence of a gap in the BH mass distribution because of pair-instability supernova. The maximum BH mass below the gap is the result of pulsational mass loss. We evolve massive helium stars through their late hydrodynamical phases of evolution using the open-source MESA stellar evolution code. We find that the location of the lower edge of the mass gap at 45$M_\odot$ is remarkably robust again…
If very massive stars (M >~ 100 Msun) can form and avoid too strong mass loss during their evolution, they are predicted to explode as pair-instability supernovae (PISNe). One critical test for candidate events is whether their nucleosynthesis yields and internal ejecta structure, being revealed through nebular-phase spectra at t >~ 1 yr, match those of model predictions. Here we compute theoretical spectra based on model PISN ejecta at 1-3 years post-explosion to allow quantitative comparison with observations. The high column densities of PISNe lead to complete gamma-ray trapping for t >~ 2 …
Models of pair-instability supernovae (PISNe) predict a gap in black hole (BH) masses between $\sim 45M_\odot-120M_\odot$, which is referred to as the upper BH mass-gap. With the advent of gravitational-wave astrophysics it has become possible to test this prediction, and there is an important associated effort to understand what theoretical uncertainties modify the boundaries of this gap. In this work we study the impact of rotation on the hydrodynamics of PISNe, which leave no compact remnant, as well as the evolution of pulsational-PISNe (PPISNe), which undergo thermonuclear eruptions befor…
We study the effects of rotation on the dynamics, energetics and Ni-56 production of Pair Instability Supernova explosions by performing rotating two-dimensional ("2.5-D") hydrodynamics simulations. We calculate the evolution of eight low metallicity (Z = 10^-3, 10^-4 Zsun) massive (135-245 Msun) PISN progenitors with initial surface rotational velocities 50% that of the critical Keplerian value using the stellar evolution code MESA. We allow for both the inclusion and the omission of the effects of magnetic fields in the angular momentum transport and in chemical mixing, resulting in slowly-r…
SN 2007bi is an extremely luminous Type Ic supernova. This supernova is thought to be evolved from a very massive star, and two possibilities have been proposed for the explosion mechanism. One possibility is a pair-instability supernova with an M_{CO} ~ 100 M_sun CO core progenitor. Another possibility is a core-collapse supernova with M_{CO} ~ 40 M_sun. We investigate the evolution of very massive stars with main-sequence mass M_{MS} = 100 - 500 M_sun and Z_0 = 0.004, which is in the metallicity range of the host galaxy of SN 2007bi, to constrain the progenitor of SN 2007bi. The supernova ty…
We study the transition to runaway expansion of an initially stalled core-collapse supernova shock. The neutrino luminosity, mass accretion rate, and neutrinospheric radius are all treated as free parameters. In spherical symmetry, this transition is mediated by a global non-adiabatic instability that develops on the advection time and reaches non-linear amplitude. Here we perform high-resolution, time-dependent hydrodynamic simulations of stalled supernova shocks with realistic microphysics to analyze this transition. We find that radial instability is a sufficient condition for runaway expan…
The formation of supermassive Population III stars with masses $\gtrsim$ 10,000 Msun in primeval galaxies in strong UV backgrounds at $z \sim$ 15 may be the most viable pathway to the formation of supermassive black holes by $z \sim$ 7. Most of these stars are expected to live for short times and then directly collapse to black holes, with little or no mass loss over their lives. But we have now discovered that non-rotating primordial stars with masses close to 55,000 Msun can instead die as highly energetic thermonuclear supernovae powered by explosive helium burning, releasing up to 10$ ^{55…
This chapter concentrates on the deaths of very massive stars, the events leading up to their deaths, and how mass loss affects the resulting death. The previous three chapters emphasized the theory of wind mass loss, eruptions, and core collapse physics, but here we emphasize mainly the observational properties of the resulting death throes. Mass loss through winds, eruptions, and interacting binaries largely determines the wide variety of different types of supernovae that are observed, as well as the circumstellar environments into which the supernova blast waves expand. Connecting these ob…
We show that the luminous supernovae (SNe) associated with ultra-long gamma-ray bursts (GRBs) can be related to the slow cooling from the explosions of hydrogen-free progenitors extended by pulsational pair-instability. In the accompanying paper (Marchant & Moriya 2020), we have shown that some rapidly-rotating hydrogen-free GRB progenitors that experience pulsational pair-instability can keep an extended structure caused by pulsational pair-instability until the core collapse. Such progenitors have large radii exceeding 10 Rsun and they sometimes reach beyond 1000 Rsun at the time of the core…
In standard stellar evolution, stars with masses ranging from approximately $150$ to $240 M_\odot$ are expected to evolve to a pair instability supernova with no black hole (BH) remnant. This evolutionary behavior leads to a predicted gap in the black hole mass function from approximately $50$ to $140 M_\odot$. Yet the LIGO and Virgo Collaborations[1] recently discovered black holes of masses $66 M_\odot$ and $85 M_\odot$ in the gravitational wave event GW190521. We propose a new method to populate the BH mass gap. If an energy source is added throughout the star in addition to nuclear fusion,…
The first full-scale three-dimensional (3D) core-collapse supernova (SN) simulations with sophisticated neutrino transport show pronounced effects of the standing accretion shock instability (SASI) for two high-mass progenitors (20 and 27 M_sun). In a low-mass progenitor (11.2 M_sun), large-scale convection is the dominant nonradial hydrodynamic instability in the postshock accretion layer. The SASI-associated modulation of the neutrino signal (80 Hz in our two examples) will be clearly detectable in IceCube or the future Hyper-Kamiokande detector, depending on progenitor properties, distance,…
Superluminous supernovae have been proposed to arise from Population III progenitors that explode as pair-instability supernovae. Pop III stars are the first generation of stars in the Universe, and are thought to form as late as $z \sim 6$. Future near-infrared imaging facilities such as ULTIMATE-Subaru can potentially detect and identify these PISNe with a dedicated survey. Gravitational lensing by intervening structure in the Universe can aid in the detection of these rare objects by magnifying the high-$z$ source population into detectability. We perform a mock survey with ULTIMATE-Subaru,…
Near-solar metallicity (and low-redshift) Pair-Instability Supernova (PISN) candidates challenge stellar evolution models. Indeed, at such a metallicity, even an initially very massive star generally loses so much mass by stellar winds that it will avoid the electron-positron pair-creation instability. We use recent results showing that a magnetic field at the surface of a massive star can significantly reduce its effective mass-loss rate to compute magnetic models of very massive stars (VMSs) at solar metallicity and explore the possibility that such stars end as PISNe. We implement the quenc…
Neutrino-neutrino refraction causes self-induced flavor conversion in dense neutrino fluxes. For the first time, we include the azimuth angle of neutrino propagation as an explicit variable and find a new generic multi-azimuth-angle (MAA) instability which, for simple spectra, occurs in the normal neutrino mass hierarchy. Matter suppression of this instability in supernovae requires larger densities than the traditional bimodal case. The new instability shows explicitly that solutions of the equations for collective flavor oscillations need not inherit the symmetries of initial or boundary con…
