Oscillation and Future Detection of Failed Supernova Neutrinos
from Black Hole Forming Collapse

Here, we review the neutrino oscillation by the MSW effect in brief kotake06. Neutrino oscillation is caused by the discrepancy between the mass eigenstate and flavor eigenstate of neutrino. In general, the flavor eigenstate is related with the mass eigenstate as below,

where $s_{ij}=\sin {\theta }_{ij}$, $c_{ij}=\cos {\theta }_{ij}$ and ${\theta }_{ij}$ is the mixing angle for $i,j=1,2,3$ ($i<j$). While ${\sin }^2{\theta }_{12}\approx 0.32$ and ${\sin }^2{\theta }_{23}\approx 0.5$ are well measured from recent experiments, only the upper limit is given for ${\sin }^2{\theta }_{13}\le 2.0\times {10}^{-2}$ gongar08. Neutrino propagation is described by the Dirac equation for the mass eigenstate and we can take an extremely relativistic limit here owing to smaller masses of neutrinos comparing with their energy. Moreover, electron type neutrinos get the effective mass from the interaction with electrons when they propagate inside matter. Thus, the time evolution equation of the neutrino wave functions can be written as,

where $G_F$, $n_e\left(t\right)$ and $E$ are the Fermi constant, the electron number density and the neutrino energy, respectively. For the anti-neutrino sector, the sign of $n_e\left(t\right)$ changes. $t$ is an Affine parameter along the neutrino worldline. $\Delta m_{ji}^2$ are the mass squared differences and they are experimentally determined as $\Delta m_{21}^2=8\times {10}^{-5}$ eV and $|\Delta m_{31}^2|=3\times {10}^{-3}$ eV. Whether the sign of $\Delta m_{31}^2$ is plus (normal mass hierarchy) or minus (inverted mass hierarchy) is unclear under the current status gongar08. Solving Eq. (2) along the neutrino propagation, we can follow the neutrino flavor eigenstate $|\psi ⟩$ and evaluate the survival probability of ${\nu }_f$ ($f=e,\mu ,\tau$ and their anti-particles) as $p_{{\nu }_f}=|⟨{\psi }_{{\nu }_f}|\psi ⟩{|}^2$.

A calculation of the survival probability can be divided to two parts, namely a probability that a neutrino generated as flavor $f$ reaches to the earth with $i$-th mass eigenstate, $P^{\star }({\nu }_f\to {\nu }_i)$, and a probability that a neutrino entering the earth with $i$-th mass eigenstate is detected as ${\nu }_f$, $P^{\oplus }({\nu }_i\to {\nu }_f)$. Using them, the survival probabilities are expressed as,

where ${\nu }_x$ represents the sum of ${\nu }_{\mu }$ and ${\nu }_{\tau }$, and $\overline{{\nu }_x}$ represents the sum of $\overline{{\nu }_{\mu }}$ and $\overline{{\nu }_{\tau }}$. In other words, $p_{{\nu }_x}$ is a probability that the neutrino generated as ${\nu }_{\mu }$ is detected as ${\nu }_{\mu }$ or ${\nu }_{\tau }$ on the earth. It is noted that we can assume $\mu$-$\tau$ symmetry owing to ${\sin }^2{\theta }_{13}\approx 0$ and ${\sin }^2{\theta }_{23}\approx 0.5$.

Neutrinos emitted from the core of black hole progenitor propagate through the stellar envelope, which has the density gradient. In other words, the electron number density, $n_e\left(t\right)$, varies along the neutrino worldline. It is noted that the conversions occur mainly in the resonance layers. The matter density at the resonance layer is given as

where $Y_e$ is the electron fraction. In the stellar envelope, there are two resonances, namely H resonance and L resonance, where H resonance lies at higher density ($\sim 2\times {10}^3$ g/cm${}^3$) than L resonance does at lower density ($\sim 20$ g/cm${}^3$). H resonance corresponds to the case of $\Delta m^2=\Delta m_{31}^2$ and $\theta ={\theta }_{13}$ in Eq. (4), while L resonance does to $\Delta m^2=\Delta m_{21}^2$ and $\theta ={\theta }_{12}$. Thus, in case of the normal mass hierarchy, both resonances reside for the neutrino sector and the anti-neutrino sector has no resonance. On the other hand, in case of the inverted mass hierarchy, H resonance shifts to the anti-neutrino sector. If the density profile is gradual at the resonance layer, the resonance is “adiabatic” and the conversion occurs completely. When the resonance layer has a steep density profile, the resonance is “non-adiabatic” and the conversion does not occur. The flip probability of the mass eigenstates, $P_F$, is analytically estimated bilen87 as

where $r$ and $r_{res.}$ are the radial coordinate and the position of the resonance layer, respectively.

Using flip probabilities at H resonance ($P_H$) and L resonance ($P_L$), $P^{\star }({\nu }_f\to {\nu }_i)$ in Eqs. (3) are calculated. Here we define matrixes $P^{\star }$ and $\overline{P^{\star }}$ whose elements, $P_{if}^{\star }$ and $\overline{P_{if}^{\star }}$, are $P^{\star }({\nu }_f\to {\nu }_i)$ and $P^{\star }(\overline{{\nu }_f}\to \overline{{\nu }_i})$, respectively, and they are calculated as dighe00

in case of the normal mass hierarchy. On the other hand, in case of the inverted mass hierarchy, H resonance resides for the anti-neutrino sector and the survival probabilities are given as,

In the absence of the earth effects, $P^{\oplus }({\nu }_i\to {\nu }_f)$ is displaced to $|U_{fi}{|}^2$, where $U_{fi}$ are elements of matrix $U$ given in Eq. (1b), and Eq. (3) are rewritten as,

in case of the normal mass hierarchy and

in case of the inverted mass hierarchy. In Fig. 1, we show the results for the survival probabilities as a function of ${\sin }^2{\theta }_{13}$ for the reference model with $E=20$ MeV. In this figure, the analytic formula, Eqs. (8) and (9), are compared with the numerical solutions of Eq. (2) by Runge-Kutta method. We can recognize that the analytic estimations well coincide with numerical results. This trend holds true for other progenitor models and neutrino energies, $E$, and we can safely use the analytic formula hereafter. In addition, L resonance is perfectly adiabatic (i.e., $P_L\approx 0$) for all of the progenitor models adopted here and for an energy range of our interest ($E=1$-250 MeV). Thus, in case of the normal mass hierarchy $p_{\overline{{\nu }_e}}\approx 0.68$ and $p_{\overline{{\nu }_x}}\approx 0.84$ are robust while $p_{{\nu }_e}\approx 0.32$ and $p_{{\nu }_x}\approx 0.66$ are in the inverted mass hierarchy.

In Paper II, the progenitor dependence of neutrino emission is investigated. Since the neutrino survival probabilities depend on the density profile of stellar envelope, we utilize the same stellar models of evolutionary calculations with Paper II. $40M_{\odot }$ model by Ref. woosley95 (the reference model), $40M_{\odot }$ model by Ref. hashi95 and $50M_{\odot }$ model by Refs. umeda05; tomi07 are named WW95, H95 and TUN07, respectively, in Paper II. We use the same notation in our paper. Incidentally, these models do not take into account the mass-loss whereas massive stars are suggested to lose their mass during the quasi-static evolutions. If strong a stellar wind sheds a large amount of the envelope, the density profile at the resonance layer will be affected. The mass-loss rate is thought to depend on the stellar metalicity, $Z$. Since TUN07 is a model with $Z=0$, the effects of mass-loss are absent. Incidentally the same model with TUN07 but $Z=0.02$ (solar metalicity) has also been computed umeda07. In this paper, we adopt this model as TUN07Z in order to discuss its sensitivity. In addition, WW95 and H95 are solar metalicity models but without the mass loss.

In Fig. 2, we show $p_{{\nu }_e}$ and $p_{{\nu }_x}$ for the normal mass hierarchy and $p_{\overline{{\nu }_e}}$ and $p_{\overline{{\nu }_x}}$ for the inverted mass hierarchy as functions of ${\sin }^2{\theta }_{13}$ evaluated from the analytic formula. The neutrino energy is fixed to $E=20$ MeV for all models. We can see that the results of model WW95 differs from others. This is because the H resonance lies at the boundary of oxygen layer and carbon layer for model WW95 and its density gradient is steeper than those of other models.

As already seen in Eq. (4), the resonance layer depends on the neutrino energy, $E$. In Fig. 3, the energy dependences of $p_{{\nu }_e}$ for each progenitor model are shown for various values of ${\sin }^2{\theta }_{13}$ in case of the normal mass hierarchy. $p_{{\nu }_e}$ is insensitive to the progenitor model for ${\sin }^2{\theta }_{13}\ge {10}^{-3}$ or ${\sin }^2{\theta }_{13}\le {10}^{-6}$ whereas it is sensitive for ${10}^{-6}\le {\sin }^2{\theta }_{13}\le {10}^{-3}$. In case of intermediate value of ${\sin }^2{\theta }_{13}$, there are energies which give a peak of $p_{{\nu }_e}$ for some models. Roughly speaking, H resonance of neutrinos with this energy lies at the boundary of layers. For instance, model TUN07 has the boundary of convective region and non-convective region near He shell burning layer. On the other hand, the envelope of model TUN07Z is already stripped and there are no such a boundary near the resonance layer. Thus, its result does not have a peak of $p_{{\nu }_e}$ even for the case of intermediate value of ${\sin }^2{\theta }_{13}$. These trends hold true for $p_{{\nu }_x}$ in case of the normal mass hierarchy, and for $p_{\overline{{\nu }_e}}$ and $p_{\overline{{\nu }_x}}$ in case of the inverted mass hierarchy.

In this paper, we evaluate the earth effects, $P^{\oplus }({\nu }_i\to {\nu }_f)$, solving Eq. (2) along the neutrino worldline numerically by Runge-Kutta method using the realistic density profile of the earth dziewo81. It is noted that the density variation along the neutrino worldline depends the nadir angle of the black hole progenitor. In Fig. 4, we show some examples for the results of the earth effects. We find that the spectral shape is deformed to wave-like and its shape varies with the nadir angle. This is because the typical length of neutrino oscillation becomes comparable to the size of the earth and the probabilities $P^{\oplus }({\nu }_i\to {\nu }_f)$ become sensitive to the neutrino energy. The deformation of the spectrum occurs for most of the cases of the parameter sets. However, when the value of ${\sin }^2{\theta }_{13}$ is larger ($\gtrsim {10}^{-3}$) and H resonance is perfectly adiabatic, the deformation disappears for the neutrino sector in case of the normal mass hierarchy and for the anti-neutrino sector in case of the inverted mass hierarchy. Incidentally, these features of deformation are very similar with the case of the ordinary supernovae luna01; ktaka02 and ONeMg supernovae luna07.

In this study, results in Paper I and II are adopted as the number spectra of the failed supernova neutrinos before the oscillation, $d{N}_{{\nu }_f}/dE$. The numerical code of general relativistic $\nu$-radiation hydrodynamics adopted in these references solves the Boltzmann equation for neutrinos together with Lagrangian hydrodynamics under spherical symmetry. Four species of neutrinos, namely ${\nu }_e$, ${\overline{\nu }}_e$, ${\nu }_{\mu }$ and ${\overline{\nu }}_{\mu }$, are considered assuming that ${\nu }_{\tau }$ and ${\overline{\nu }}_{\tau }$ are the same as ${\nu }_{\mu }$ and ${\overline{\nu }}_{\mu }$, respectively. It is noted that the luminosities and spectra of ${\nu }_{\mu }$ and ${\overline{\nu }}_{\mu }$ at the emission (before the oscillation) are almost identical because they have the same reactions and the difference of coupling constants is minor. 14 grid points are used for energy distribution of neutrinos. We interpolate the energy spectra linearly, conserving the total neutrino number. We calculate the probabilities of flavor conversion and event numbers at the detector. Taking into account the oscillation, spectra of neutrino flux at the detector, $d{F}_{{\nu }_f}/dE$, can be expressed as:

where $R$ is a distance from a black hole progenitor to the earth. In this study, we set $R=10$ kpc, which is typical length of our Galaxy.

In Papers I and II, the core collapse of massive stars with 40 and $50M_{\odot }$ woosley95; hashi95; umeda05; tomi07 have been computed using two sets of supernova EOS’s. They have concluded that the EOS at high density and temperature can be constructed from the duration of failed supernova neutrinos. In these references, they have adopted the EOS by Lattimer & Swesty (1991) (LS-EOS) lati91 and EOS by Shen et al. (1998) (Shen-EOS) shen98a; shen98b. LS-EOS is obtained by the extension of the compressible liquid model with three choices of the incompressibility. The case of 180 MeV, which is a representative of soft EOS, has been chosen in Paper I and II. Shen-EOS is based on the relativistic mean field theory and a rather stiff EOS with the incompressibility of 281 MeV. Progenitor models adopted in Paper I and II are already stated in Sec. II.3 and their calculated models are listed in Table 1. We use the same names for the models given in Paper I and II hereafter. Since Shen-EOS is stiffer than LS-EOS, the maximum mass of the neutron star is larger and it takes longer time to form a black hole. Therefore the total energy of emitted neutrinos is larger for Shen-EOS than LS-EOS. In Fig. 5, time-integrated spectra without the neutrino oscillation are shown for models W40S and W40L. The plots for ${\overline{\nu }}_x$ are not shown because they are almost the same with those for ${\nu }_x$. However, this feature is not the case after the oscillation is taken into account.

SuperKamiokande is a water Cherenkov detector, which is a descendant of Kamiokande, located at the site of Kamioka mine in Japan. The neutrino reactions in the detector which we take into account are as follows:

We adopt the cross sections for the reaction (11a) from Ref. strumia03, for the reactions (11f) and (11g) from Ref. haxton87 and for others from Ref. totsuka92. It is noted that the reaction (11a) makes dominant contribution to the total event number. Incidentally, the reaction (11f) has the largest cross section for neutrinos with the energy $\gtrsim 80$ MeV although the ambiguities exist the theoretical cross sections of the neutrino-nucleus reactions. We assume that the fiducial volume is 22.5 kton and the trigger efficiency is 100% at 4.5 MeV and 0% at 2.9 MeV, which are the values at the end of SuperKamiokande I hosaka06. The energy resolution was 14.2% for $E_e=10$ MeV at this phase hosaka06 and roughly proportional to $\sqrt{E_e}$ luna01, where $E_e$ is the observed kinetic energy of scattered electrons and positrons. In this study, we choose an energy bin width of 1 MeV.

Here we focus on model W40S only since the qualitative features are similar among other models. In Fig. 6, we show the time-integrated event number of neutrinos for several ${\sin }^2{\theta }_{13}$ values as a function of the nadir angle. The event number does not depend on the nadir angle very much and we will revisit later for the reason. It is also noted that the total event number is larger than that of the ordinary supernova ($\sim 10,000$) kotake06 because the total and averaged energies of emitted neutrinos are higher for the black hole progenitors. For instance, for the ordinary supernovae, the total and averaged energies of the neutrinos emitted as $\overline{{\nu }_e}$ are $4.7\times {10}^{52}$ ergs and 15.3 MeV, respectively totani98, while they are $1.42\times {10}^{53}$ ergs and 24.4 MeV in our model. Even the sum of total energies of emitted neutrinos over all species is lower in the ordinary supernova as $2.9\times {10}^{53}$ ergs.

In case of the inverted mass hierarchy, the event number gets smaller for the larger ${\sin }^2{\theta }_{13}$. This is because the survival probability of ${\overline{\nu }}_e$, which contributes to the event number most by the reaction (11a), is almost zero for ${\sin }^2{\theta }_{13}\gtrsim {10}^{-3}$ (Fig. 2). It is important to note that this tendency is different from the case of ordinary supernovae. For the ordinary supernova neutrinos, the flux of ${\overline{\nu }}_x$ before the oscillation is larger than that of ${\overline{\nu }}_e$ in the energy regime $E\gtrsim 25$ MeV and the neutrino flux is not so small for this regime (see Fig. 35 in Ref. kotake06). For the case of inverted mass hierarchy and larger ${\sin }^2{\theta }_{13}$, as seen in Fig. 2, half of ${\overline{\nu }}_x$ converts to ${\overline{\nu }}_e$ and the event number of neutrinos with $E\gtrsim 25$ MeV increases. Thus the resultant total event number of ordinary supernova neutrinos becomes larger. On the other hand, for failed supernova neutrinos, the neutrino flux is very small for the energy regime ($E\gtrsim 80$ MeV) where the flux of ${\overline{\nu }}_x$ before the oscillation becomes larger than that of ${\overline{\nu }}_e$ (see Fig. 5).

The event number gets larger for the larger ${\sin }^2{\theta }_{13}$ in the normal mass hierarchy while the difference is small. The difference owing to ${\sin }^2{\theta }_{13}$ is mainly induced by the reaction (11f). From Fig. 5, we can see that the differential number of ${\nu }_x$ before the oscillation is larger than that of ${\nu }_e$ for $E\gtrsim 80$ MeV. Thus the differential number of ${\nu }_e$ becomes larger by the oscillation for $E\gtrsim 80$ MeV while it does smaller for $E\lesssim 80$ MeV. Incidentally, the threshold of this reaction is high $\sim 15.4$ MeV and the cross section is roughly proportional to $E^2$. Therefore the difference arises crucially for the high energy regime and resultant total event number is slightly larger for the larger ${\sin }^2{\theta }_{13}$. It should be remarked, however, that the precise evaluation for the high energy tail of neutrino spectra is difficult in the numerical computations and our results are thought to have some quantitative ambiguities. In addition, this feature is consistent with the case of ordinary supernovae. It is also noted that the mass hierarchy does not make any difference for ${\sin }^2{\theta }_{13}\lesssim {10}^{-6}$, as is the case of ordinary supernovae.

In Fig. 7, we show the energy spectra of the time-integrated event number for several models. Results of the models with ${\sin }^2{\theta }_{13}={10}^{-5}$ for the normal mass hierarchy and ${\sin }^2{\theta }_{13}={10}^{-8}$ for the inverted mass hierarchy are not shown because they are very close to the result of the model with ${\sin }^2{\theta }_{13}={10}^{-8}$ for the normal mass hierarchy. The most outstanding case is the inverted mass hierarchy with ${\sin }^2{\theta }_{13}={10}^{-2}$. The event number is much smaller than those of the other cases especially for the energy regime, $E\lesssim 50$ MeV, and accordingly the spectrum becomes harder. The reason is again that the survival probability of ${\overline{\nu }}_e$ is almost zero and the original flux of ${\overline{\nu }}_e$ is larger than that of ${\overline{\nu }}_x$ especially for this energy regime. This spectral feature may be useful for the restrictions of the mixing parameters, as already studied for ordinary supernovae kotake06.

In the case without the earth effects, spectra are not fluctuating while the cusps are artifacts due to the binning of original spectral data in Paper I and II. One exception is the model with ${\sin }^2{\theta }_{13}={10}^{-5}$ for the inverted mass hierarchy, whose spectrum is fluctuating from 20 MeV to 30 MeV. This is because, as already mentioned in Sec. II.3, the H resonance lies at the boundary of shells for the progenitor model WW95. The fluctuation originates from spikes in the upper left panel of Fig. 3. In the case with the earth effects for the nadir angle ${\theta }_{nad}=0^{\circ }$, ${30}^{\circ }$ or ${60}^{\circ }$, fluctuating spectra are from the earth effects. This is because the spectral shapes of survival probabilities for each neutrino species are deformed to wave-like as shown in Sec. II.4. In the case of the inverted mass hierarchy with ${\sin }^2{\theta }_{13}={10}^{-2}$, where there is no earth effect on the anti-neutrino sector, the deformation of spectra is not clear comparing with the other cases. Integrating over the neutrino energy, the fluctuation is smoothed out. Therefore the total event number does not depend on the nadir angle very much.

As found in Papers I and II, the luminosity and duration time of failed supernova neutrinos depend on EOS and progenitor model. In Fig. 8, the time-integrated event numbers of models W40S and W40L are shown for various parameter sets. The total event number of model W40S is at least 29,124 while that of model W40L is at most 16,779. This large difference arise from not only its long duration time but also its high average neutrino energy for model W40S. For all parameter sets, event number of model W40S is at least 2.5 times larger than that of model W40L. Estimating from the studies of postbounce evolutions ($<1$ sec) of the neutrino emission sumi05 and protoneutron star coolings suzuki06, the difference by the EOS is not so large for ordinary supernovae. Unfortunately, there is no previous investigation of an EOS dependence of supernova neutrinos from numerical computations of the successful explosion and the successive long-term ($\gtrsim 10$ sec after the bounce) emission. However, it is likely that the black hole formation is a better site to probe the EOS of nuclear matter.

In Table 2, the summary of event numbers for all models are shown. We can see that the event number of failed supernova neutrinos is mainly determined by properties of the EOS. Especially for the models by Shen-EOS (W40S and T50S), the difference is very minor. As discussed in Paper II, the duration time of the neutrino emission, which is mainly determined by the EOS, is also affected by the mass accretion rate after the bounce. The accretion rate depends on the density profile of a progenitor and T50S is reported to have lower accretion rate than W40S for late phase. If the accretion rate is lower, the neutrino duration time becomes longer but the neutrino luminosity is lower. This is because the neutrino luminosity is roughly proportional to the accretion luminosity. Therefore both effects are canceled out and the resultant event number becomes similar for models W40S and T50S. In case of LS-EOS (W40L, T50L and H40L), the event numbers differ by a factor of 1.8 at most and they are roughly proportional to the neutrino duration time. For these cases, the black hole formation occurs fast ($\lesssim 0.6$ s) after the bounce and the difference in neutrino luminosities induced by the accretion rate is not so large for this phase. Differences between the progenitor models are smaller than those between the EOS’s for the models investigated in this study.

We show in Fig. 9 the spectra of each model for several parameter sets. We can see the clear difference between two EOS’s. Moreover, In the case of the inverted mass hierarchy with ${\sin }^2{\theta }_{13}={10}^{-5}$, another interesting feature is seen in the spectra (lower left panel of Fig. 9). It is notable that peaks are mismatched in the comparison between the spectra of models W40S and T50S. This comes from the different dependence of the neutrino survival probabilities inside the stellar envelope as shown in Fig. 2. The density at H resonance for neutrinos with the energy which gives the peak of spectrum in Fig. 9 roughly corresponds to that of the boundary of layers, where the density profile becomes steep.

In order to study the impact of the mass-loss of a progenitor, we compare models T50S, Tz50S, T50L and Tz50L in Fig. 10. Results for models T50S and Tz50S are the same except the case of the inverted mass hierarchy with ${\sin }^2{\theta }_{13}={10}^{-5}$. This is also the case for the comparison of models T50L and Tz50L. For the regime of a neutrino energy $E=30$-40 MeV, the spectrum of model T50S has a bump, which does not appear for model Tz50S. This bump comes again from a steep density profile due to the onion-skin-like structure of the progenitor. The envelope of progenitor model TUN07Z is stripped by the effects of mass-loss and corresponding boundary is absent. Thus, only when the mass hierarchy is inverted and ${\sin }^2{\theta }_{13}\sim {10}^{-5}$-${10}^{-4}$, we have chances of probing the structure of a stellar envelope and effects of mass-loss. Incidentally, mass-loss may affect also the structure of more inner region. If this is the case, the dynamics and neutrino emission themselves may be changed.

In conclusion, the event number of failed supernova neutrinos depends primarily on EOS. Ambiguities on the mixing parameters also affect the result and the spectra of detected neutrinos may be useful for the restrictions of them, while they can also be determined from other experiments. The event number reflects the properties of progenitor models though its dependence is weaker than that of EOS. For the limited case of the mixing parameters, we may be able to probe the structure of a stellar envelope. There is weak dependence on the nadir angle, which should be determined from the identification of the progenitor astronomically. Incidentally, Papers I and II discussed mainly about the time evolutions and duration times of the neutrino emission. Here, we have focused on the features of time-integrated event numbers and spectra, which are complementary to the previous studies.

We remark that the conclusion given above is based on the results of some restrictive models and more detailed investigations are necessary to examine how general our conclusion is. Incidentally, EOS including the new degree of freedom such as hyperons takochu or quarks self08 are shown to have different features at hot and dense regime recently. A new EOS of the nuclear matter for astrophysical simulations is also proposed to construct based on the many body theory, which is a different approach from Shen-EOS and LS-EOS kann07. However, we stress that the ambiguities and model dependences which can be taken into account under the current status have been fully involved and we feel that the essential point has been clarified in our investigations.

Finally, we speculate the observational feasibility. The event number of models computed under Shen-EOS is at least 29,004 while that of LS-EOS is at most 16,779. Let us assume that the signal of failed supernova neutrinos is discovered from the data of the detector and its progenitor is identified in future. These facts indicate that the difference of EOS can be distinguished even if the ambiguities on the mixing parameters and progenitor models are not constrained well. If the mixing parameters are determined till that time by another way, the discrimination of EOS is more promising. In this analysis, the determination of the distance from the progenitor is a key factor. If we are lucky, the progenitor will have already been monitored by the surveys as presented in Ref. kocha08. Even if there are no ex-ante surveys, we can determine the direction of the progenitor to some extent by the neutrino detection itself ando02. Then the progenitor may be identified from old photographs. In addition, the nadir angle of the progenitor can also be determined so that we can remove the ambiguities of the nadir angle.

The rate of these black hole forming collapses is highly uncertain but will be less than that of ordinary supernovae. Assuming the Salpeter’s initial mass function, the black hole formation rate is $\sim 25$% that of ordinary supernovae kocha08. Since the supernova rate of our galaxy is $\sim 0.03$ /yr ando05, we have to wait $\sim 130$ years for the black hole formation. One solution to this rate problem is to observe the nearby galaxies by the future detectors with larger fiducial volume. Andromeda galaxy (M31) lies 780 kpc away. In the most optimistic case of Shen-EOS, the event number of neutrinos emitted from it is roughly estimated as $\sim 8.2$ for SK. If we consider a future detector with a scale of 5 Mton (e.g. the proposed Deep-TITAND detector ysuzu01; kist08), the event number becomes $\sim 1900$. This will be enough number to compare the LS-EOS case with $\sim 500$ events. Since the cumulative supernova rate within the distance to M31 is $\sim 0.08$ /yr ando05, we should wait $\sim 50$ years. If we extend further out to 4 Mpc, the event number becomes 70 for the most optimistic case. Since the supernova rate within 4 Mpc is $\sim 0.3$ /yr, we should wait $\sim 13$ years for the black hole formation.