On the occurrence of fast neutrino flavor conversions in multidimensional supernova models

TL;DR

This study investigates fast neutrino flavor conversions in multidimensional supernova models, revealing that asymmetric neutrino flux patterns can lead to rapid flavor transformation with significant implications for supernova physics.

Contribution

First analysis of crossing phenomena and fast neutrino flavor conversions in multi-D supernova models using Boltzmann transport solutions.

Findings

Existence of unstable oscillation modes with rapid growth rates.

Asymmetric flux patterns facilitate neutrino flavor crossing.

Potential impact on supernova explosion mechanisms and nucleosynthesis.

Abstract

The dense neutrino medium in a core-collapse supernova or a neutron-star merger event can experience fast flavor conversions on time/distance scales that are much smaller than those of vacuum oscillations. It is believed that fast neutrino flavor transformation occurs in the region where the angular distributions of $\nu_e$ and $\bar\nu_e$ cross each other. We present the first study of this crossing phenomenon and the fast neutrino flavor conversions in multidimensional (multi-D) supernova models. We examine the neutrino distributions obtained by solving the Boltzmann transport equation for several fixed profiles which are representative snapshots taken from separate 2D and 3D supernova simulations with an $11.2 M_\odot$ progenitor model. Our research shows that the spherically asymmetric patterns of the $\nu_e$ and $\bar\nu_e$ fluxes in multi-D models can assist the appearance of the crossing between the $\nu_e$ and $\bar\nu_e$ angular distributions. In the models that we have studied, there exist unstable neutrino oscillation modes in and beyond the neutrino decoupling region which have amplitude growth rates as large as an $e$-fold per nanosecond. This finding can have important consequences for the explosion mechanism, nucleosynthesis, and neutrino signals of core-collapse supernovae.

Equations

$${\bar{\nu }}_e+p\rightleftharpoons n+e^{+}\text{and}{\nu }_e+n\rightleftharpoons p+e^{-}$$

$$\varrho =\frac{f_{{\nu }_e}+f_{{\nu }_x}}{2}+\frac{f_{{\nu }_e}-f_{{\nu }_x}}{2}\left[\begin{array}{cc}s& S\\ S^{\ast }& -s\end{array}\right],$$

$$i({\partial }_t+\mathbf{v}\cdot \nabla ){\varrho }_{\mathbf{p}}=\left[\frac{{\mathsf{M}}^2}{2E_{\nu }}+\frac{\lambda }{2}{\sigma }_3+{\mathsf{H}}_{\nu \nu ,\mathbf{p}},{\varrho }_{\mathbf{p}}\right]+\mathcal{C}[{\varrho }_{\mathbf{p}}],$$

$${\mathsf{H}}_{\nu \nu ,\mathbf{p}}=\sqrt{2}{G}_{\mathrm{F}}\int \frac{{\mathrm{d}}^3{p}^{\prime }}{(2\pi {)}^3}(1-\mathbf{v}\cdot {\mathbf{v}}^{\prime })({\varrho }_{{\mathbf{p}}^{\prime }}-{\bar{\varrho }}_{{\mathbf{p}}^{\prime }}).$$

$$G_{\mathbf{v}}=\sqrt{2}{G}_{\mathrm{F}}{\int }_0^{\infty }\frac{E_{\nu }^2\mathrm{d}{E}_{\nu }}{(2\pi {)}^3}[f_{{\nu }_e}(\mathbf{p})-f_{{\bar{\nu }}_e}(\mathbf{p})],$$

$$i({\partial }_t+\mathbf{v}\cdot \nabla )S_{\mathbf{v}}=({ϵ}_0+\mathbf{v}\cdot \mathbf{ϵ})S_{\mathbf{v}}-\int \mathrm{d}{\Gamma }_{{\mathbf{v}}^{\prime }}(1-\mathbf{v}\cdot {\mathbf{v}}^{\prime })G_{{\mathbf{v}}^{\prime }}{S}_{{\mathbf{v}}^{\prime }},$$

$$\tau (s)={\int }_s^{\infty }\chi (s^{\prime })\mathrm{d}{s}^{\prime },$$

Full text

I Introduction

At the exhaustion of its nuclear fuel, the stellar core of a massive star collapses under its own gravity into a neutron star or black hole, and the rest of the star either explodes as a supernova or falls back to become a part of the black hole. In a successful explosion, the hot proto-neutron star (PNS) formed at the center of the (core-collapse) supernova quickly cools down by emitting $\sim 10^{58}$ neutrinos in all flavors in just tens of seconds. Although the exact mechanism of the supernova explosion remains elusive, the neutrino reactions

[TABLE]

are known to be important in the heating and cooling of the matter deep inside the supernova and can play an important role in exploding the star Bethe and Wilson (1985); Janka (2012); Burrows (2013). Supernova ejecta are also one of the few places in the universe where the heavy elements can be abundantly produced. An important factor that regulates the nucleosynthesis in the supernova ejecta is its electron fraction $Y_{e}$ (defined as the ratio of the net number density of the electrons to that of the baryons or $n_{e}/n_{\mathrm{b}}$) Qian and Woosley (1996) which in turn is determined by the neutrino processes in Eq. (1). Because these processes involve only $\nu_{e}$ and $\bar{\nu}_{e}$, and because the neutrinos in different flavors are emitted in different intensities and energies in a supernova, the transformation or oscillations between different neutrino flavors ($\nu_{e}\rightleftharpoons\nu_{\mu/\tau}$ and $\bar{\nu}_{e}\rightleftharpoons\bar{\nu}_{\mu/\tau}$) near or even inside the surface of the PNS can have a significant impact on the nucleosynthesis in the supernova ejecta and the supernova dynamics. Understanding the flavor transformation of the neutrinos inside the supernova is also crucial to the prediction of the neutrino signals from future galactic supernovae and the neutrino diffuse background Gava et al. (2009); Horiuchi et al. (2009); Beacom (2010); Mirizzi et al. (2016); Horiuchi et al. (2018).

Because of the coherent forward scattering by the dense neutrino medium surrounding the PNS, supernova neutrinos can experience collective flavor transformation Pastor and Raffelt (2002); Duan et al. (2006a, b, 2010); Chakraborty et al. (2016a) which occurs much deeper than does the flavor conversion induced by matter alone through the Mikheyev-Smirnov-Wolfenstein (MSW) mechanism Wolfenstein (1978, 1979); Mikheyev and Smirnov (1985). However, based on the stationary neutrino bulb model Duan et al. (2006a) in which all neutrinos are decoupled from matter at a single sharp neutrino sphere, it was found that collective flavor transformation is suppressed during the accretion phase of the supernova due to the large density of the ambient matter Esteban-Pretel et al. (2008); Sarikas et al. (2012); Chakraborty et al. (2011); Zaizen et al. (2018) and/or even the neutrinos themselves Duan and Friedland (2011). Although, in the more realistic models that evolve with time and does not have the spherical symmetry, this suppression can be lifted for the collective modes that oscillate rapidly in space and time Duan and Shalgar (2015); Chakraborty et al. (2016b); Abbar et al. (2015); Abbar and Duan (2015); Dasgupta and Mirizzi (2015), the physical conditions can change significantly before these modes have a sufficient growth in amplitudes to engender significant flavor conversions.

In a real supernova, the neutrinos of different flavors (and energies) decouple from matter at different depths of the PNS and have different angular distributions outside the PNS. As a result, fast neutrino flavor conversions can be driven by the neutrino density $n_{\nu}$ which occur on a characteristic length scale $l_{\text{fast}}\sim(\hbar c)^{-2}G_{\mathrm{F}}^{-1}n_{\nu}^{-1}$ with $G_{\mathrm{F}}$ being the Fermi coupling constant of the weak interaction Sawyer (2005, 2016); Chakraborty et al. (2016c); Izaguirre et al. (2017); Wu and Tamborra (2017); Capozzi et al. (2017); Dasgupta et al. (2017); Abbar and Duan (2018); Abbar and Volpe (2019). Such flavor transformation is rightly called fast because $l_{\text{fast}}$ is much shorter than the typical wavelength of vacuum oscillations ($\sim\mathcal{O}(1)$ km for a 10 MeV neutrino with the atmospheric neutrino mass splitting) and even the mean free path of the neutrino inside the PNS Capozzi et al. (2019). It is believed that fast neutrino flavor conversions occur only where the angular distributions of $\nu_{e}$ and $\bar{\nu}_{e}$ cross each other Dasgupta et al. (2017); Izaguirre et al. (2017). However, an earlier study of the one-dimensional (1D) supernova models of the Garching group with the Boltzmann neutrino transport did not reveal any scenario with such crossed neutrino angular distributions or fast neutrino flavor conversions Tamborra et al. (2017). It has been speculated Chakraborty et al. (2016c); Dasgupta et al. (2017) that the multi-D models exhibiting the lepton-emission self-sustained asymmetry (LESA) Tamborra et al. (2014) may have regions that foster fast flavor conversions. But the neutrino transports in most of the existing multi-D supernova simulations are implemented with various approximations and do not provide detailed angular distributions of the neutrinos.

Recently, the full information of the neutrino angular distributions became available to multi-D supernova models by directly solving the Boltzmann equation in three momentum dimensions Sumiyoshi and Yamada (2012). This approach has been used to study the neutrino transport in supernova cores Sumiyoshi et al. (2015) and its impact on the explosion dynamics Nagakura et al. (2018).

In this paper, we present the first survey of the neutrino angular distributions in multi-D supernova models to ascertain the physical conditions under which fast neutrino flavor conversions may occur.

II Neutrino transport and fast flavor conversions

At each space-time point $(t,\mathbf{r})$, the flavor content of the neutrino medium in the momentum mode $\mathbf{p}$ can be specified by its flavor density matrix $\varrho_{\mathbf{p}}(t,\mathbf{r})$ Sigl and Raffelt (1993) which, for two flavors ($e$ and $x$) and in the weak-interaction basis, is Banerjee et al. (2011)

[TABLE]

where $f_{\nu_{e}/\nu_{x}}$ are the initial occupation numbers in the corresponding flavors, and the complex and real scalar fields $S$ and $s$ describe the flavor coherence and the flavor conversion of the neutrino, respectively. A similar expression exists for the flavor density matrix $\bar{\varrho}$ of the antineutrino. The flavor evolution of the neutrino medium is governed by the Liouville-von Neumann equation Sigl and Raffelt (1993); Strack and Burrows (2005); Cardall (2008); Volpe et al. (2013); Vlasenko et al. (2014)

[TABLE]

where we have adopted the natural units $c=\hbar=1$. In the above equation, $E_{\nu}=|\mathbf{p}|$, $\mathbf{v}=\mathbf{p}/E_{\nu}$, and $\mathsf{M}^{2}$ are the energy, velocity, and mass-square matrix of the neutrino, respectively, $\lambda=\sqrt{2}G_{\mathrm{F}}n_{e}$ is the matter potential Wolfenstein (1978); Mikheyev and Smirnov (1985), $\sigma_{3}$ is the third Pauli matrix,

[TABLE]

is the neutrino potential stemming from the neutrino-neutrino forward scattering Fuller et al. (1987); Nötzold and Raffelt (1988); Pantaleone (1992), and $\mathcal{C}[\varrho_{\mathbf{p}}]$ denotes the collision, emission and absorption of the neutrinos.

To study fast neutrino flavor conversions, we ignore the vacuum oscillation Hamiltonian $\mathsf{M}^{2}/2E_{\nu}$ so that the flavor transformation of the neutrinos becomes energy independent. We also ignore the collision term $\mathcal{C}[\varrho_{\mathbf{p}}]$ and assume that the physical conditions are homogeneous and stationary for the distance and time scales of interest except for the small but rapidly varying flavor mixing amplitude $S_{\mathbf{v}}(t,\mathbf{r})$. This approximation is valid before significant flavor conversion has occurred so that $|S_{\mathbf{v}}|\ll 1$ and $s\approx 1$. In this scenario, it is useful to define the angular distribution of the electron lepton number (ELN) of the neutrino flux as Izaguirre et al. (2017)

[TABLE]

where we have assumed $f_{\nu_{x}}(\mathbf{p})=f_{\bar{\nu}_{x}}(\mathbf{p})$. Keeping only the terms in Eq. (3) of magnitude $\mathcal{O}(|S_{\mathbf{v}}|)$ or larger, one obtains Banerjee et al. (2011); Väänänen and Volpe (2013); Izaguirre et al. (2017)

[TABLE]

where $\mathrm{d}\Gamma_{\mathbf{v}^{\prime}}$ is the differential solid angle in the direction of $\mathbf{v}^{\prime}$, $\epsilon_{0}=\lambda+\int\!\mathrm{d}\Gamma_{\mathbf{v}^{\prime}}G_{\mathbf{v}^{\prime}}$, and $\bm{\epsilon}=\int\!\mathrm{d}\Gamma_{\mathbf{v}^{\prime}}G_{\mathbf{v}^{\prime}}\mathbf{v}^{\prime}$.

Collective flavor transformation is induced by the normal modes in the neutrino medium which, in the linear regime (where $|S_{\mathbf{v}}|\ll 1$), are of the form $S_{\mathbf{v}}(t,\mathbf{r})=Q_{\mathbf{v}}\,e^{-i\Omega t+i\mathbf{K}\cdot\mathbf{r}},$ where $Q_{\mathbf{v}}$, $\Omega$ and $\mathbf{K}$ are constant with the latter two being also independent of $\mathbf{v}$. The flavor mixing amplitude $S_{\mathbf{v}}$ remains small unless for a real wave vector $\mathbf{K}$ the corresponding frequency has a positive imaginary component, i.e. $\Omega_{\mathrm{i}}=\mathrm{Im}(\Omega)>0$. In this case the wave amplitude can grow exponentially on the time scale of $l_{\text{fast}}/c$. Such unstable normal modes with fast growing amplitudes can exist when the ELN distribution $G_{\mathbf{v}}$ crosses 0 at some angle(s) Dasgupta et al. (2017); Izaguirre et al. (2017).

III ELN crossings in multi-D supernova models

We study the angular distributions of the neutrinos which are obtained by solving the Boltzmann transport equation (without any flavor transformation) for several fixed supernova profiles Sumiyoshi et al. (2015). These profiles were taken from the representative snapshots at $t_{\text{pb}}=100$, 150 and 200 ms post the core bounce, respectively, of a 2D and a 3D supernova simulations by using the Lattimer & Swesty equation of state Lattimer and Swesty (1991) with an approximate neutrino transport and with an $11.2M_{\odot}$ progenitor model Takiwaki et al. (2012, 2014). The spatial resolutions of the Boltzmann calculations are $(256,64,1)$ and $(256,64,32)$ for the 2D and 3D models, respectively, for radius up to $2613$ km from the original simulations, where $(N_{r},N_{\Theta},N_{\Phi})$ are the numbers of spatial zones in the spherical coordinates $(r,\Theta,\Phi)$. The momentum resolution $(N_{E_{\nu}},N_{\theta_{\nu}},N_{\phi_{\nu}})$ of the neutrino flux in each spatial zone is $(14,6,12)$, where $\theta_{\nu}$ and $\phi_{\nu}$ are the zenith and azimuthal angles of the neutrino velocity $\mathbf{v}$ with respect to the radial direction, respectively.

Out of the three snapshots of the 2D model we find regions with ELN crossings within and above the decoupling region in the one at $t_{\text{pb}}=200$ ms. At this time, the deformed shock has reached over 500 km and is poised to explode the star with the help of a bipolar growth of the hydrodynamic instabilities. Depending on their flavors and energies, neutrinos decouple from matter at radius $\sim 50-70$ km which can be viewed as the “surface” or neutrino sphere of the PNS. In the first two panels of Fig. 1 we show the number densities $n_{\nu}=\int\!\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}f_{\nu}(\mathbf{p})$ and average flux densities $\mathbf{j}_{\nu}=\int\!\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}f_{\nu}(\mathbf{p})\mathbf{v}$ in this snapshot for $\nu=\nu_{e}$ and $\bar{\nu}_{e}$, respectively. Both $n_{\nu}$ and $\mathbf{j}_{\nu}$ are mostly spherically symmetric in this snapshot with $\mathbf{j}_{\nu}$ generally pointing in the radial direction with some cases of non-radial fluxes. Naively, one may think that ELN crossings may occur below the neutrino sphere where $\mathbf{j}_{\nu_{e}}$ and $\mathbf{j}_{\bar{\nu}_{e}}$ can point in very different directions. This is not the case, however, because $f_{\nu}(\mathbf{p})$ is highly isotropic in this region, and we find no ELN crossing here.

In our study, ELN crossings usually begin to appear in the region where the neutrinos begin to decouple from matter as shown in the right panels of Fig. 1. Furthermore, we find that, at the radii where ELN crossings do occur, they usually appear in the angular zones with the $\bar{\nu}_{e}$-to-$\nu_{e}$ ratio $\alpha=n_{\bar{\nu}_{e}}/n_{\nu_{e}}$ close to 1. The correlation between $\alpha$ and ELN crossings is not really a surprise. In the neutrino decoupling region, $f_{\nu}(\mathbf{p})$ becomes more and more peaked in the forward direction. Because the PNS is rich in neutrons, $\bar{\nu}_{e}$’s decouple from matter at smaller radii than $\nu_{e}$’s do and thus obtain a more forwardly peaked distribution. However, this difference in $f_{\nu_{e}}(\mathbf{p})$ and $f_{\bar{\nu}_{e}}(\mathbf{p})$ is usually not large enough to result in an ELN crossing unless $\bar{\nu}_{e}$ has a flux density very close to that of $\nu_{e}$. This is likely the reason why no ELN crossing was found in a previous study of 1D supernova models Tamborra et al. (2017). In the 2D model presented in Fig. 1, however, $\alpha$ can vary across angular zones at the same radius which leads to ELN crossings in some regions.

To check the sensitivity of our results on the angular resolution of the neutrino distributions, we also solve the neutrino transport for the $t_{\text{pb}}=200$ ms snapshot of the 2D model with $N_{\theta_{\nu}}=36$. We find that, although the angle averaged properties of the neutrinos such as $n_{\nu}$, $\mathbf{j}_{\nu}$ and $\alpha$ are almost identical for the two calculations, the region with ELN crossings is much wider in the high-resolution calculation than in the one with a lower resolution (right panels of Fig. 1). To find out the reason why some spatial zones show ELN crossings in the high-resolution calculation only, we compare the ELN distributions $G_{\mathbf{v}}$ of these zones in the two calculations. Because $f(\mathbf{p})$ are approximately axially symmetric about the radial direction in our models, we integrated them over $\phi_{\nu}$ and calculated $f_{\nu}(\theta_{\nu})=\iint\frac{E_{\nu}^{2}\mathrm{d}E_{\nu}\mathrm{d}\phi_{\nu}}{(2\pi)^{3}}f_{\nu}(\mathbf{p})$ and $G(\theta_{\nu})=\int_{0}^{2\pi}\mathrm{d}\phi_{\nu}G_{\mathbf{v}}$. We plot $f_{\nu_{e}}(\theta_{\nu})$, $f_{\bar{\nu}_{e}}(\theta_{\nu})$ and $G(\theta_{\nu})$ for the spatial zone centered at $r=65.6$ km and $\cos\Theta=0.96$ in the first two panels of Fig. 2. From this figure one sees that the coarse angular resolution is sufficient to capture the overall shape of $f_{\nu}(\theta_{\nu})$, and it is more accurate in the backward directions than in the forward directions. However, because the ELN distribution $G(\theta_{\nu})$ is sensitive to the small difference between $f_{\nu_{e}}(\theta_{\nu})$ and $f_{\bar{\nu}_{e}}(\theta_{\nu})$, and because it is likely to cross 0 near the radial direction, a high angular resolution in the forward directions is needed to accurately describe the crossing in the neutrino distributions. This issue becomes more severe at large radii where $f_{\nu_{e}}(\theta_{\nu})$ and $f_{\bar{\nu}_{e}}(\theta_{\nu})$ are both highly peaked in the forward direction, and they may cross each other at an angle beyond the last $\theta_{\nu}$ bin.

We calculate the exponential growth rates $\Omega_{\mathrm{i}}$ of the unstable fast oscillation modes as functions of the real wave number $K$ for the (interpolated) ELN distributions plotted in Fig. 2, and the results are shown in the right panel in the same figure. We assume the axial symmetry about the radial direction in calculating $\Omega_{\mathrm{i}}$. We also assume that the wave vector of the collective flavor oscillation wave is along the radial direction. In this particular example, $S_{\mathbf{v}}$ can grow by an $e$-fold within $\sim 1$ nanosecond which is indeed much faster than the typical changing rates of the physical conditions inside the supernova.

We also find ELN crossings in the 3D model which seem to be more common than in the 2D model. This observation can be understood by noting that the spatial asymmetries in the neutrino emission might evolve faster in the case of the 3D simulations Vartanyan et al. (2019). They appear in all the three snapshots that we have studied. In the $t_{\text{pb}}=200$ ms snapshot, ELN crossings begin to occur at $r=46$ km in the neutrino decoupling region. We show $\alpha$ in this snapshot at $r=65.6$ km in the left panel of Fig. 3 and mark with crosses the spatial zones where ELN crossings occur. As in the 2D model, ELN crossings are more likely to appear in the regions with $\alpha$ close to 1. The asymmetry pattern in the emissions of $\nu_{e}$ and $\bar{\nu}_{e}$ is associated with a similar pattern in $Y_{e}$ which is present even well below the neutrino sphere (middle and right panels of Fig. 3). The region with lower $Y_{e}$ and more neutron-rich matter emits more $\bar{\nu}_{e}$’s and absorbs more $\nu_{e}$’s through the reactions in Eq. (1). This pattern of $Y_{e}$ and $\alpha$ is likely caused by the active matter convection inside the PNS as in the LESA phenomenon Tamborra et al. (2014).

Our study highlights the need of more multi-D supernova simulations with the neutrino Boltzmann transport like those performed in Ref. Nagakura et al. (2018). Our results show that the neutrino distributions with even a very coarse angular resolution can be used to identify the ELN crossings in the neutrino decoupling region, although a better resolution in the radial direction is needed to accurately describe the crossings especially at large radii. Although we have found fast growing neutrino oscillations in the multi-D models that we have studied, it remains to be seen if the conditions for the ELN crossing will be met in the more self-consistent multi-D supernova simulations with accurate neutrino transport, and if the amplitudes of the fast neutrino oscillation modes can continue to grow into the nonlinear regime and cause significant flavor conversions deep inside the supernova. If fast neutrino flavor conversions are indeed found to occur near the surface of the PNS, they will have profound implications for the dynamics, nucleosynthesis, and neutrino signals of the supernova.

Acknowledgments

We thank J. Martin and C. Yi for valuable discussions. This work is supported by the US DOE EPSCoR grant DE-SC0008142 and the NP grant DE-SC0017803 at UNM for S. A. and H. D., “Gravitation et physique fondamentale” (GPHYS) of the Observatoire de Paris for S. A. and M. C. V., and JSPS and MEXT KAKENHI Grant Numbers (JP15K05093, JP17H01130, JP17K14306, 18H01212) and (JP26104006, 17H05206, JP17H06357, JP17H06364, JP17H06365) for K. S. and T. T., resepctively. K. S. and T. T. acknowledge the support on the computing resources at NAOJ, KEK, JLDG, YITP, RCNP, UT as well as Post-K and K-computer of the RIKEN AICS.

Appendix A The optical depths

In this appendix, we provide the information of the optical depths in the $t_{\text{pb}}=200$ ms snapshot of the 2D supernova model shown in Fig. 1. The optical depth of a neutrino is defined as

[TABLE]

where $\chi(s)$ is the neutrino opacity, and the integration is performed along the radial coordinate for the neutrinos in the forward angle bin. In the first three panels of Fig. 4, we show the optical depths of the neutrinos of different flavors and all with energy $E_{\nu}=20.9$ MeV. The very right panel of the figure shows the matter density in this snapshot and the location of the neutrinospheres (where $\tau=2/3$) for these neutrinos. The average energies of the neutrinos are in the range of $10-25$ MeV in this snapshot.