# Observing neutrinos from failed Supernovae at LNGS

Giulia Pagliaroli<EMAIL_ADDRESS>Istituto Nazionale di Fisica
Nucleare (INFN), Laboratori Nazionali del Gran Sasso, 67100 Assergi, L’Aquila
(AQ), Italy  Christoph A. Ternes<EMAIL_ADDRESS>Istituto
Nazionale di Fisica Nucleare (INFN), Laboratori Nazionali del Gran Sasso,
67100 Assergi, L’Aquila (AQ), Italy

###### Abstract

We discuss the possibility to observe neutrinos emitted from a failed core
collapse Supernova in the various experiments at Laboratori Nazionali del Gran
Sasso. We show that the veto regions of dark matter and neutrinoless double
beta decay experiments can be used as a network of small detectors to measure
Supernova neutrinos. In addition we show that this network can measure very
precisely the moment of black hole formation, which can be then used in the
nearby VIRGO detector and future Einstein Telescope to look for the
gravitational wave counterpart to the neutrino signal.

##  I Introduction

When stars with masses $M>8M_{\odot}$ burn all their fuel the radiation
pressure in the iron core region becomes not sufficient to balance the
gravitational force. The iron core collapses until the central density
overcomes the nuclear density. At this point the outer part of the core
bounces off the inner compact region. The bounce induces a shock wave that,
after a stalled phase, can be revived by neutrinos providing the final
Supernova explosion [1]. The Core Collapse Supernova (CCSN) releases an energy
of several $10^{53}$ ergs and neutrinos carry away most of this energy
($99\%$) over a time interval of about 10 s [2]. So far, only the Supernova
SN1987A has been observed in neutrinos, due to the low CCSN rate within the
Milky Way (viz. $R_{\text{MW}}=1.63\pm 0.46$ per century [3]). However, the
small data-set of 24 events observed by the Kamiokande-II [4, 5], Baksan [6]
and IMB detectors [7] testifies our capability to detect this emission and to
use neutrino data to constrain CCSN emission models [8]. Current and future
detectors would see thousands of events in neutrinos from a Galactic CCSN [9]
providing a fast alert to optical telescopes to catch the upcoming
electromagnetic emission and shock breakout [10].

In the case of more massive stars, with $M\gtrsim 20M_{\odot}$ [11], it is
possible that the explosion mechanism fails and the accretion of matter on the
central proto-neutron star generates a black hole (BH) very shortly after the
core bounce. In this scenario, called failed SN, neutrino and gravitational
wave (GW) emissions stop abruptly at the same time [12, 13]. This
astrophysical event is expected to emit only a very faint electromagnetic
signal from the disappearing progenitor star [14], so that the observation of
neutrinos and gravitational waves can be the only way to hunt for the BH
formation.

Neutrinos from Supernovae can be detected in different types of detectors with
different detection channels and neutrino flavor sensitivities [15]. They can
be looked for in Cherenkov detectors [16, 17, 18, 19, 20], liquid scintillator
detectors [21, 22, 23, 24, 25, 26, 27], in lead-based detectors [28], in noble
gas detectors [29] or also in Dark Matter detectors [30].

The Gran Sasso National Laboratory or Laboratori Nazionali del Gran Sasso
(LNGS) hosts at present the LVD experiment [23], aimed at CCSN neutrino
detection, and many other experiments dedicated to different physics topics,
nowadays with an important focus on neutrinoless double beta decay and direct
detection of dark matter. These experiments use large water tanks as veto
regions against different types of backgrounds, such as cosmic muons or
natural radioactivity in the surrounding cave material. These veto regions can
be used as a network of Water Cherenkov detectors to look for neutrinos from
CCSNe.

The aim of this paper is twofold: On one hand, we suggest to exploit the veto
regions of detectors, generally dedicated to other physics topics, as a free-
of-charge network of small detectors for neutrinos from CCSNe; on the other
hand, we discuss the possibility of neutrino detectors to identify the time of
black hole formation in very massive stars providing an important timing to
look for the gravitational counterpart of such events. We show that, in case
of a failed Supernova the LNGS infrastructure can provide precise information
on the time when the BH forms. This timing could be crucial for the search for
the associated gravitational waves. In the absence of the electromagnetic
counterpart, LNGS is the only facility that can provide this timing in real
time without the need to know the sky position of the SN, thanks to the
negligible time of fly to the VIRGO gravitational wave interferometer [31] and
to the future Einstein Telescope [32], located in Pisa and in Sardinia [33],
respectively.

Our paper is structured as follows: We discuss in Section II the neutrino
emission models under consideration in this work. Section III contains
information on the current and future LNGS-detectors and the calculation of
the event rates used in our analysis. We discuss our results in Section IV and
we draw our conclusions in Section V.

##  II Neutrino emission from failed Supernovae

Figure 1: Neutrino luminosity $L_{\nu_{\alpha}}$ and average energy $\langle
E_{\nu_{\alpha}}\rangle$ as a function of time after the core bounce for Model
1 (solid lines) and Model 2 (dashed lines) discussed in the text. With a
dashed gray line we highlight the time of the black hole formation in both
cases.

In order to investigate the capability of the LNGS detectors to provide a good
timing measurement of the BH formation in failed SNe we consider the two
different models reported in Fig. 15 of Ref. [34]. In both cases, the
progenitor, a star of 40 $M_{\odot}$, fails to explode and forms a central BH.
The two models, both using the ls221 equation of state [35], differ with
respect to the duration of their evolution until BH formation. In Model 1
[36], the BH formation occurs $t_{\text{BH}}^{(1)}=0.568$ s after the bounce,
whereas for Model 2 [37] the gravitational instability occurs at a later time
of $t_{\text{BH}}^{(2)}=2.113$ s after the bounce thanks to a lower mass
accretion rate of the collapsing stellar core. In Fig. 1, we show the neutrino
luminosity $L_{\nu_{\alpha}}$ and average neutrino energy $\langle
E_{\nu_{\alpha}}\rangle$ as a function of time for both models. The origin
represents the time of the core bounce inside the CCSN. In both cases the
neutrino emission terminates abruptly at the time of BH formation
$t_{\text{BH}}$. Following Ref. [38], the differential neutrino flux per
flavor is defined as

$\frac{d\Phi_{\alpha}^{0}}{dE_{\nu_{\alpha}}dt}=\frac{1}{4\pi
d^{2}}\frac{L_{\nu_{\alpha}}(t)}{\langle
E_{\nu_{\alpha}}(t)\rangle}\phi_{\alpha}(E_{\nu_{\alpha}},t),$ (1)

where $d$ is the SN distance and the normalized energy spectrum is
parameterized as:

$\phi_{\alpha}(E_{\nu_{\alpha}},t)=\langle
E_{\nu_{\alpha}}(t)\rangle\frac{1+\xi_{\alpha}}{\Gamma(1+\xi_{\alpha})}\left(\frac{E_{\nu_{\alpha}}}{\langle
E_{\nu_{\alpha}}(t)\rangle}\right)^{\xi_{\alpha}}\exp\left(-\frac{(1+\xi_{\alpha})E_{\nu_{\alpha}}}{{\langle
E_{\nu_{\alpha}}(t)\rangle}}\right),$ (2)

where the ”pinching parameter” is set to $\xi_{\alpha}=3$. Due to neutrino
oscillations, the flux of electronic antineutrinos arriving at the detector is
a combination of initial $\bar{\nu}_{e}$ and $\bar{\nu}_{x}$ at the emission,
see e.g. Ref. [34] for a deep discussion. Assuming normal neutrino mass
ordering (currently moderately preferred at 2.7$\sigma$ over the inverted one
[39, 40]), neglecting collective effects and considering a static supernova
matter profile, the electronic antineutrino flux which arrives at the detector
is given by

$\Phi_{\bar{\nu}_{e}}=\cos^{2}\theta_{12}\Phi_{\bar{\nu}_{e}}^{0}+\sin^{2}\theta_{12}\Phi_{\bar{\nu}_{x}}^{0}.$
(3)

Here we will concentrate on the electronic antineutrinos being the inverse
beta decay the most relevant detection channel for the experiments under
consideration. Note, however, that with other channels also different neutrino
flavors might be observable.

##  III The LNGS detectors

At present the LNGS facility hosts the Large Volume Detector (LVD) [23] that
is a 1 kton liquid scintillator detector dedicated to search SN neutrinos and
several detectors which are already taking data or still under construction
and are dedicated to direct detection of dark matter or $0\nu\beta\beta$ decay
search. Remarkably, several of these detectors have a veto region sensitive to
SN neutrinos. In particular, the veto regions of Cryogenic Observatory for
SIgnals seen in Next-generation Underground Searches (COSINUS) [41], the Large
Enriched Germanium Experiment for Neutrinoless double beta Decay (LEGEND-200)
[42] and XENONnT [43] are water tanks with different shapes containing about
$270$ tons, $590$ tons and $700$ tons of water, respectively. These are the
experiments which are currently present at LNGS. We will also consider the
future experiments LEGEND-1000 [44] and DARWIN [45] (which might be hosted at
LNGS) containing $980$ tons and $2140$ tons of water, respectively. These are
the successor experiments to LEGEND-200 and XENONnT.

Figure 2: Expected rate in Hz of IBD events in $1$ ton of water for different
SN distances in kpc for emission Model 1 (solid lines) and Model 2 (dashed
lines).

The main interaction process for SN neutrinos in water and in liquid
scintillators is the inverse beta decay (IBD) interaction of electronic
antineutrinos with free protons. The IBD event rate is given by

$R_{\text{\tiny IBD}}(t)=N_{p}\int_{E_{\text{\tiny
thr}}}dE_{\nu}\sigma_{\text{\tiny
IBD}}(E_{\nu})\Phi_{\bar{\nu}_{e}}(E_{\nu},t)\,,$ (4)

where $N_{p}$ is the number of free protons inside the detector,
$\sigma_{\text{\tiny IBD}}$ is the IBD cross section taken from Ref. [46] and
$\Phi_{\bar{\nu}_{e}}(E_{\nu},t)$ has been discussed in Sec. II. As shown in
Fig. 2, this rate can be very large (several kHz for the LNGS detectors under
consideration) for a nearby failed SN, especially in the case of fast BH
formation. This information should be considered in future experiments or the
ones which are still under construction, in order to define the best data-
acquisition rate enabling the goal proposed in this work. The total number of
IBD events expected as a function of the SN distance is reported in Fig. 3 for
the different detectors, each represented with a different color, and for the
combined LNGS sensitivity in black. In the left panel we show the sensitivity
of the currently present experiments, whereas in the right panel we show the
one for future experiments. In each panel the solid (dashed) lines correspond
to emission Model 1 (Model 2), see Sec. II. The lines labeled as ”LNGS” show
the expectations for the combined configuration where all the detectors are
considered as different modules of a single big detector. We also show over-
imposed the probability density distribution (PDF) of core-collapse SNe as a
function of distance. This blue shadowed curve is obtained by considering the
normalized Milky Way distribution of supernova remnants [47] and by adding two
Gaussian contributions for the Large and Small Magellanic Clouds centered at
$d=49.5$ kpc and $62.8$ kpc, respectively. The width of each Gaussian accounts
for the radial and angular size of both Galaxies111Taken from the NASA/IPAC
Extragalactic Database (NED). while the normalization is $11\%$ of the Milky
Way contribution [48]. As can be easily observed the PDF shows a broad peak
around $d=10$ kpc. For this reason we select this SN distance to discuss
subsequent results. The number of expected IBD events for each detector and
for a SN at $d=10$ kpc is reported in the first column of Tab. 1 where we
assumed an energy threshold of $4$ MeV for LVD [49] and $7$ MeV for the veto
regions. From Fig. 3 it is evident that both the present configuration of LNGS
detectors and the future one will be sensitive to measure SN neutrinos coming
even from the Small Magellanic Cloud. As a final remark, it should be noted
that the intrinsic modular configuration of all veto regions (and LVD) assures
a very high duty cycle. Even if a single experiment was in maintenance during
a SN, the other experiments could observe neutrinos individually.

Figure 3: Number of IBD events expected in the veto regions of current (left)
and future (right) LNGS detectors as a function of the Supernova distance in
kpc for Model 1 (solid lines) and Model 2 (dashed lines). The number of IBD
events expected in LVD is also reported with a blue line in the left panel for
comparison. The black lines provide the global LNGS sensitivity in this
detection channel. We also show the probability density distribution (blue
shadowed area) for a supernova explosion in the Milky Way and in the
Magellanic Clouds, which has to be read with respect to the right (blue) axis.

##  IV Results

Detector | $N_{\text{\tiny IDB}}$ |  $t^{1}\pm\delta t^{1}$ [s] |  $t^{\text{\tiny last}}\pm\delta t^{\text{\tiny last}}$ [s] |  $1/\xi$ [s]
---|---|---|---|---
LVD |  $303$ (527) |  $0.018\pm 0.007$ ($0.019\pm 0.009$) |  $0.566\pm 0.002$ ($2.109\pm 0.003$) | 0.002 (0.004)
COSINUS-veto | 66 (116) |  $0.04\pm 0.02$ ($0.04\pm 0.02$) |  $0.563\pm 0.005$ ($2.10\pm 0.01$) | 0.008 (0.018)
Legend200-veto | 145 (253) |  $0.02\pm 0.01$ ($0.03\pm 0.01$) |  $0.565\pm 0.003$ ($2.107\pm 0.007$) | 0.004 (0.008)
XENONnT-veto | 173 (301) |  $0.02\pm 0.01$ ($0.02\pm 0.01$) |  $0.565\pm 0.003$ ($2.107\pm 0.005$) | 0.003 (0.007)
Legend1000-veto | 242 (530) |  $0.019\pm 0.008$ ($0.021\pm 0.009$) |  $0.566\pm 0.002$ ($2.108\pm 0.005$) | 0.002 (0.005)
DARWIN-veto | 530 (922) |  $0.014\pm 0.006$ ($0.013\pm 0.006$) |  $0.5672\pm 0.0009$ ($2.111\pm 0.003$) | 0.001 (0.002)
Table 1: The table shows part of the results obtained in our analysis. The
numbers outside (inside) the parenthesis correspond to emission Model 1 (Model
2). In the first column we show the number of expected IBD events in the each
detector. The quantity $t^{1}$ is the relative time between the time of the
bounce and the time of the first neutrino detection averaged over 100 MC
simulations. The statistical fluctuation of this quantity is labeled $\delta
t^{1}$. The quantity $t^{\text{\tiny last}}$ is the average relative time
between the last event and the time of the bounce and $\delta t^{\text{\tiny
last}}$ is its statistical fluctuation. Finally the last column shows the
inverse of the observed event rate, which is equivalent to the uncertainty of
the time of BH formation.

As discussed in the Introduction and shown in Fig. 1, the neutrino emission in
failed SNe starts at the time of the bounce of the outer core off the inner
one and ends sharply at the time of the BH formation. This is true also for
the corresponding GW signal, see e.g. Fig. 3 of Ref. [50]. While the
observation of the neutrino signal is guaranteed with current neutrino
detectors up to the Magellanic Clouds, the observation of the GW counterpart
is still challenging [51]. As demonstrated in previous works [52, 53], the use
of a fixed temporal window to look for a GW burst and the combination of
several detectors in temporal coincidence to decrease the false alarm rate
probability, are powerful multi-messenger strategies to increase the GW
detection efficiency.

In this work we quantify the capability of the veto regions of LNGS detectors
to provide the universal time of the BH formation for GW search
($T^{\text{\tiny GW}}_{\text{\tiny BH}}$). We perform 100 Monte Carlo (MC)
simulations for a failed SN at $10$ kpc with both emission models discussed
above. For each detector we estimate the relative time of the first event
$t^{1}=T^{1}-T^{\nu}_{\text{\tiny bounce}}$, where $T^{1}$ is the time of the
first non-isolated event, the relative time of the last event $t^{\text{\tiny
last}}=T^{\text{\tiny last}}-T^{\nu}_{\text{\tiny bounce}}$, where
$T^{\text{\tiny last}}$ is the time of the last event, and the average rate of
observed events $\xi=N_{\text{\tiny IBD}}/(t^{\text{\tiny last}}-t^{1})$. In
Tab. 1 we show the average values for $t^{1}$ and $t^{\text{\tiny last}}$ and
their statistical fluctuations obtained from the MC simulations.

We are interested to estimate the time of BH formation at a GW detector
related to the time of BH formation at the neutrino detector
$T^{\nu}_{\text{\tiny BH}}$ through the equation $T^{\text{\tiny
GW}}_{\text{\tiny BH}}=T^{\nu}_{\text{\tiny BH}}\pm t_{\text{\tiny fly}}$,
where $t_{\text{\tiny fly}}$ is the time of fly between the neutrino and the
GW detectors. In addition, we are also interested to provide the uncertainty
$\delta T^{\text{\tiny GW}}_{\text{\tiny BH}}$ on this quantity.

The first proxy of the quantity $T^{\nu}_{\text{\tiny BH}}$ will be the time
of the last neutrino event $T^{\text{\tiny last}}$. However, we find, in
agreement with the results from Ref. [54], that this time is systematically
smaller than the BH formation time by an interval time that is well described
by the data-driven quantity $1/\xi$, reported in the last column of Tab. 1. In
addition, the same quantity can be assumed as a conservative estimation of the
associated uncertainty $\delta T^{\nu}_{\text{\tiny BH}}$. In other words, for
each neutrino detector the best proxy for the BH formation time is

$\displaystyle T^{\nu}_{\text{\tiny BH}}$ $\displaystyle=$ $\displaystyle
T^{\text{\tiny last}}+1/\xi\,,$ (5) $\displaystyle\delta T^{\nu}_{\text{\tiny
BH}}$ $\displaystyle=$ $\displaystyle 1/\xi,$ (6)

where each term can be obtained by the observed events without any theoretical
assumption on the emission model. Note that the data-driven quantity $\xi$ is
also a powerful parameter to discriminate real SN signals from background and
to increase the detection capability of neutrino detectors for small
statistics signals [55].

In Fig. 4 we show the obtained values of $T^{\nu}_{\text{\tiny BH}}\pm\delta
T^{\nu}_{\text{\tiny BH}}$ for each neutrino detector considered in this work.
Evidently, the real BH formation time is always within the reconstructed
interval provided by these estimations. The veto regions of the LNGS detectors
can also be combined as modules of a bigger detector. In this case, the best
proxy for the BH formation time will be

$\displaystyle T^{\nu}_{\text{\tiny BH}}$ $\displaystyle=$
$\displaystyle\text{Max}[T^{\text{\tiny last}}_{i}]+1/\xi_{\text{Max}}$ (7)
$\displaystyle\delta T^{\nu}_{\text{\tiny BH}}$ $\displaystyle=$
$\displaystyle\sqrt{1/\sum_{i}\left(\xi_{i}^{2}\right)}$ (8)

where $i$ runs over all the considered detectors and $\xi_{\text{Max}}$ is the
$\xi$ associated to $\text{Max}[T^{\text{\tiny last}}_{i}]$. In this case the
uncertainty profits from the network configuration. The uncertainties on the
black hole formation time obtained for the individual detectors are summarized
in the last column of Tab. 1 and range from 1 ms to 8 ms for emission Model 1,
and from 2 ms to 18 ms for emission Model 2. Using the combined network of
current veto regions the uncertainty can be improved to $\delta T^{\text{\tiny
LNGS-veto}}_{\text{\tiny BH}}=2.3$ ms ($4.8$ ms) for Model 1 (Model 2). These
results can be compared with the sensitivity of LVD, namely $\delta
T^{\text{\tiny LVD}}_{\text{\tiny BH}}=1.8$ ms ($3.9$ ms) showing that the
veto region network could provide a similar precision as a dedicated Supernova
experiment. Of course, at the present, all these detectors are in operation at
LNGS and the overall sensitivity, ”LNGS-present” in Fig.4, can reach even
$\delta T^{\text{\tiny LNGS}}_{\text{\tiny BH}}=1.4$ ms ($3$ ms). In the
future, bigger detectors are expected to be hosted at LNGS. LEGEND-1000 can
determine the time of BH formation within $\delta T^{\text{\tiny
Legend-1000}}_{\text{\tiny BH}}=2$ ms ($5$ ms) while DARWIN could shrink the
uncertainty to $\delta T^{\text{\tiny Darwin}}_{\text{\tiny BH}}=1.0$ ms
($2.2$ ms). When the possibility to combine these two detectors is considered
we obtain the best sensitivity of $\delta T^{\text{\tiny LNGS}}_{\text{\tiny
BH}}=0.9$ ms ($2.0$ ms).

Figure 4: The capability to identify the BH formation time of the different
detectors considered in this paper for emission Model 1 (left panel) and
emission Model 2 (right panel).

Note that these detectors are not competitive with other observatories, e.g.
Super-Kamiokande, in terms of providing the most precise measurement of
$T^{\nu}_{\text{\tiny BH}}$, which can be up to one order of magnitude more
precise than the LNGS measurement, see the discussion below. However, LNGS
will provide important temporal information to the nearby GW detector Virgo
and the future Einstein Telescope. Indeed, as stated before, the time of BH
formation for the GW detector is given by $T^{\text{\tiny GW}}_{\text{\tiny
BH}}=T^{\nu}_{\text{\tiny BH}}\pm t_{\text{\tiny fly}}$, where $t_{\text{\tiny
fly}}$ is the time of fly between the detectors, $t_{\text{\tiny
fly}}=\vec{r}\cdot\hat{n}$, where $\vec{r}$ is the distance-vector between the
detectors and $\hat{n}$ is the direction pointing to the Supernova. The
associated uncertainty is $\delta T^{\text{\tiny GW}}_{\text{\tiny BH}}=\delta
T^{\nu}_{\text{\tiny BH}}+\delta t_{\text{\tiny fly}}$. The error $\delta
t_{\text{\tiny fly}}$ is negligible for an astronomically identified SN.
However, it can be equal to $t_{\text{\tiny fly}}$ for an unknown SN position.
For a failed SN the electromagnetic counterpart is missing and the
identification of the SN position in the sky is more challenging. The distance
between LNGS and Virgo or the Einstein telescope is very small so that
$t_{\text{\tiny fly}}\leq 1$ ms. This means that, even in the worse scenario
of slow BH formation (emission Model 2), the temporal uncertainty that the
current network of LNGS detectors can provide to Virgo is

$\delta T^{\text{\tiny GW}}_{\text{\tiny BH}}=\delta T^{\text{\tiny
LNGS}}_{\text{\tiny BH}}+\delta t_{\text{\tiny fly}}=4~{}\text{ms}\,.$ (9)

We can compare this quantity with the equivalent one obtained for Super-
Kamiokande with $32$ kton of water and an energy threshold of $E_{\text{\tiny
thr}}=7$ MeV. Super-Kamiokande will collect a very big number of events
(13776) with a corresponding very small uncertainty $\delta t^{\text{\tiny
SK}}_{\text{\tiny BH}}=0.15$ ms, in agreement with a previous estimation in
Ref. [56]. However, the distance between Super-Kamiokande and Virgo or the
Einstein Telescope is quite large so that $t_{\text{\tiny fly}}$ could be as
large as $28$ ms providing a final uncertainty window of

$\delta T^{\text{\tiny GW}}_{\text{\tiny BH}}=\delta T^{\text{\tiny
SK}}_{\text{\tiny BH}}+\delta t_{\text{\tiny fly}}=28.3~{}\text{ms}$ (10)

for the same emission Model 2. Of course the uncertainty on the time of fly
could be reduced by exploiting the information provided by directional
neutrino interactions as the elastic scattering on electrons [57] or by
triangulation between different neutrino detectors [58] as discussed also for
this specific case in Ref. [59]. However, all these techniques require deep
analysis of neutrino data in order to tag the IBD channel [60] or to combine
the neutrino data streams of different detectors around the world [61], while
our information can be provided immediately to the GW detector as soon as the
neutrino data are collected.

##  V Conclusions

In conclusion, we have investigated the role of veto regions of detectors
mainly dedicated to other physics topics, to detect supernova neutrinos. We
concentrate on the peculiar case of the LNGS infrastructure where three
different detectors (COSINUS, LEGEND-200 and XENONnT) use water tanks of
different shapes and volumes as veto regions, that can be easily combined to
work together at a network of small detectors. We show that this combination
of small free-of-charge Cherenkov detectors is sensitive to neutrinos from
CCSNe up to the Magellanic Clouds. In particular, we quantify the capability
of these detectors to identify the time of BH formation for a failed SN. This
information can be very relevant to look for the gravitational wave
counterpart to the neutrino emission. We show that the sensitivity of current
LNGS detectors working as a network could provide the time of BH formation
with the smallest uncertainty thanks to the very small distance between the
LNGS facility and the Virgo gravitational wave interferometer. We also provide
a projection of this sensitivity with future detectors expected in the same
laboratory. We want to stress that the approach described in this work is
completely model independent and can be applied to any type of BH forming SN,
also the ones that collapse to a BH after the explosion [62]. This statement
is not true, however, if we were interested to identify the starting time of
the neutrino signal, i.e. the time of the bounce. In this case the estimation
of the $T^{\nu}_{\text{\tiny bounce}}$ will be affected by a model dependent
uncertainty, that is quantified as the absolute value of $t^{1}$ in our Tab.
1. This term, usually called response time of the detector [63] cannot be
deduced by the neutrino data-stream without any assumption on the emission
model.

Finally, as a last remark we point out that our analysis is very conservative
and timely. It is conservative because it provides a lower limit of the total
sensitivity of the LNGS detectors. Indeed we are considering only the IBD
detection channel and only the veto regions of the detectors. The sensitivity
could be improved by including other detection channels both in water and in
liquid scintillator and by considering also the inner regions of the dark
matter or $0\nu\beta\beta$ detectors that are also sensitive to SN neutrinos
[30]. The corresponding increase of statistics can be easily converted in a
reduced temporal uncertainty on the time of BH formation. The analysis is
timely because the COSINUS and LEGEND-200 detectors are now still defining the
best configuration for their veto regions.

###### Acknowledgements.

The authors thank N. Di Marco for useful discussions. The work of G.P. is
partially supported by grant number 2022E2J4RK ”PANTHEON: Perspectives in
Astroparticle and Neutrino THEory with Old and New messengers” under the
program PRIN 2022 funded by the Italian Ministero dell’Università e della
Ricerca (MUR) and by the European Union – Next Generation EU.

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