Pinpointing astrophysical bursts of low-energy neutrinos embedded into the noise
C. Casentini G. Pagliaroli C. Vigorito V. Fafone
astro-ph.HE physics.data-an

Bayesian inference in Y-linked two-sex branching processes with mutations: ABC approach
    Miguel González Cristina Gutiérrez Rodrigo Martínez
    Received: date / Accepted: date
=======================================================================================





§ INTRODUCTION

Core-Collapse Supernovae (CCSNe) <cit.> represent the final explosive phase of massive stars and
the detection of a galactic event could be the unique opportunity for us to grasp the physical mechanism driving 
the final explosion of the structure. 
“Failed" Supernovae <cit.> are collapsing stars failing to explode and 
forming an inner black hole. The lack of the final explosion makes these sources optically silent and, 
at the present, have never been directly observed. 
Quark Novae <cit.> are expected when a neutron star suddenly converts into a quark star. Their 
existence is strongly related to the fundamental state of the matter  and their detection could provide 
the first clear evidence of the presence of strange matter in the universe. 

A common signature for all these catastrophic astrophysical phenomena is expected to be an impulsive 
(∼ 10 s) emission of low-energy, (∼ 10 MeV), neutrinos <cit.>. 
Despite the large amount of total energy (∼10^53ergs) released in neutrinos, when the source distance
increases and/or the average energy of emitted neutrinos decreases, the signal statistics drops and 
the identification of these astrophysical bursts embedded into the detector noise could be challenging. 

The search of these astrophysical signals is one of the main goals of several low-energy neutrinos  
detectors based on different techniques and characterised by different capabilities (see <cit.> for a Review).  
Moreover, the SuperNova Early Warning System (SNEWS) <cit.> provides an early warning of a galactic supernova
demanding the coincident observation of low-energy neutrinos bursts from several detectors. 
The detection strategy adopted in each neutrino detector, working alone or in the SNEWS system, 
relies on the competition between the background rate collected in a 
fixed time window and the expected signal rate in the same time interval. 
In order to separate real signals from background fluctuations, the standard procedure
is purely statistical, data are selected requiring that the Poisson probability that background 
fluctuation produces the observed number of events is very small. 
This means that only very strong signals are well identified whereas
small signals cannot be separated from statistical fluctuations and are unavoidably lost. 

In this paper, we improve the detectors capability to disentangle astrophysical bursts of low-energy 
neutrinos from background signals. This powerful method exploits the different temporal structure 
expected for an astrophysical burst with respect to background fluctuations that are near uniformly distributed in a time window.
This characteristic, described with a new parameter, can be used as an additional degree of freedom that, added to the 
statistical requirement, improves our capability to identify real signals allowing the detection of weaker/far away 
astrophysical sources.  



§ ASSUMPTIONS

For all the astrophysical sources we are interested in, we assume 
that the total energy radiated in neutrinos is ℰ=3· 10^53 erg.
Moreover, based on CCSN study, we consider that the total energy is 
partitioned in equal amount among the six types of neutrinos, 
that should be true within a factor of 2 <cit.>.

As highlighted, the novelty of the proposed method
is the introduction of a discrimination parameter based on the different temporal shapes
of background and signal. We consider a very general description of an astrophysical burst of low-energy neutrinos 
characterised by the following temporal evolution

    f(t)=[1-exp(-t/τ_1)]exp(-t/τ_2),

where τ_1 =(10-100)ms is the rising time and τ_2≥1 s 
represents the decaying time of the signal. 
This ansatz is very general, model independent and it fits all the expected neutrino 
emissions from CCSNe<cit.>, Failed Supernovae<cit.> and 
Quark Novae<cit.>. 
Moreover, SN1987A, the only CCSN detected 
so far by neutrinos telescopes, agrees with this simple temporal model. The complete set 
of events observed by Kamiokande-II<cit.>, IMB<cit.> and Baksan<cit.> fitted with 
such a model provides a best-fit time scale of τ_2∼ 1s <cit.>.   

For the energy differential fluence we assume quasi-thermal spectra described by

    Φ_i^0=ℰ_i/4π D^2×E^α e^-E/T_i/ T_i^α+2Γ(α+2)    i=ν_e,ν_μ,ν_τ,ν̅_e,ν̅_μ,ν̅_τ,

where E is the neutrino energy, ℰ_i=ℰ/6 is the energy radiated in each specie 
due to the equipartition hypothesis, 
and the `temperature' is T_i=⟨ E_i⟩/(α+1).
The average energy per flavour is ⟨ E_i⟩ and the parameter 
α=3 represents a mild deviation from a thermal distribution.
According to SN1987A data<cit.> and recent numerical simulations <cit.>,
we set ⟨ E_ν_e⟩=9 MeV,  ⟨ E_ν̅_e⟩=12 MeV and non-electronic temperature,
⟨ E_ν_x⟩, 30% higher than ⟨ E_ν̅_e⟩.   




Here, we consider the main interaction channel, namely 
the inverse beta decay (IBD) ν̅_e+p → n+e^+.
Due to neutrino oscillations, the ν̅_e fluence at the detector 
is an admixture of the unoscillated flavour fluences at the source: 
Φ_ν̅_e=P Φ^0_ν̅_e+ (1-P)Φ^0_x, where x indicates 
the non-electronic flavours and P is the survival probability for the ν̅_e. 
Depending on the neutrinos mass hierarchy, this probability can be P=0 for Inverted Hierarchy (IH) 
or P≃ 0.7 for Normal Hierarchy (NH). 
The expected number of IBD interactions is S(E_ν,D)= N_pσ_ν̅_e p(E_ν)
Φ_ν̅_e(E_ν,D)ϵ (E_vis), where D is
the source distance, N_p is the number of target protons within the
detector, σ_ν̅_e p<cit.> is the process cross section and
ϵ is the detector efficiency as a function of the visible energy E_vis. 
For an energy threshold E_thr∼ 1 MeV, the positron energy spectrum of the IBD channel 
is completely observed and, in the optimistic case of a total detection efficiency (ϵ=100%), 
a fixed number of IBD interactions S can be obtained in three different cases: 
1) Inverted neutrino mass hierarchy, average energy ⟨ E_ν̅_e⟩=12 MeV 
and source distance D; 
2) Normal neutrino mass hierarchy, average energy ⟨ E_ν̅_e⟩=12 MeV 
and source distance 0.896 D; 
3) NH, ⟨ E_ν̅_e⟩=15 MeV and source distance D.  

In other words, the effect on the number of events due to 
a change of the source distance or of the average energy of the spectrum 
or of the neutrino mass hierarchy is the same. 
By taking into account this degeneracy we show our results for the 
NH case with ⟨ E_ν̅_e⟩=15 MeV. 
By using previous considerations these results can be 
rescaled to the IH case or to a different neutrino average energy.  




§ METHOD

The aim of this paper is to show an efficient method to discriminate astrophysical burst of low-energy neutrinos
from fake burst of events induced by background. For this reason background knowledge and characterization
is fundamental to demonstrate the potential of this method. At the present, low-energy neutrinos detectors 
on data-taking, viz. Super Kamiokande<cit.>, LVD<cit.>, Borexino<cit.> and KamLAND<cit.> provide all the information needed to perform this study and we report results 
for these detectors considering both the situation where the detector operates alone and the case in which multiple detectors  
operate as a network exploiting the advantages of a combined coincident search.

In all the considered detectors the search of astrophysical bursts of low-energy ν is based  
on the definition of clusters of events.
Following <cit.> we define a cluster as the group of the events contained 
in consecutive time windows of w = 20 seconds. Each cluster is characterised by its multiplicity m_i,
i.e. number of events inside the time window, 
and its time duration Δ t_i, defined as the time difference among
the first and the last event detected.
In order to increase the detection probability this search is performed one more time 
by shifting the consecutive time windows of 10 seconds 
with respect to the first search (for more details see <cit.>).

In order to claim the detection of an astrophysical burst two different requirements should be satisfied:
the cluster of expected events for a specific source distance should be populated enough (at least two neutrinos m_i≥ 2) 
and this cluster should be discriminated by the others due to standard background events. 
    
In the following, the first requirement will be discussed in term of detection efficiency, η, and the second one will 
be related to the misidentification probability ζ, i.e. the probability to confuse a background cluster for
a signal cluster. Both these quantities are strongly related to the background characteristics of each neutrino 
detector. In order to reproduce the background fluctuations of each detector and according to the parameters 
(frequencies and energy thresholds) reported in Tab.<ref>, a Monte Carlo simulation of 10 years of data-taking has been performed .
For the observed background clusters of events we calculate its imitation 
frequency f^im_i (day^-1), i.e. how many times in a day 
background events produce a cluster with the same multiplicity. 
This quantity is defined as:

    f^im_i=N·∑_k=m_i^∞(f_bkg w)^k e^-f_bkg w/k! day^-1,

where N=8640 is the number of windows of duration w=20 seconds overlapped 
every 10 seconds in a day, m_i is the cluster multiplicity and f_bkg is the background 
frequency of the experiment. In order to reduce the background, standard search 
procedures assume a selective cut on f^im_i (or equivalently on m_i)<cit.>.
Lower is the f^im_i threshold value used higher is the probability that survived clusters are due to 
real astrophysical signals. For example, to reduce the background fluctuations to a negligible value,
the SNEWS threshold is f^im≤ 1/100years <cit.>. 
On the other hand, higher is the allowed value for the f^im_i 
larger is the distance reach of our search or the sensitivity to weaker signals, 
since, here, the real signals with small statistics can enter in our analysis. 
To test the new method to discriminate signal from background, we set 
f^im≤ 1/day as working threshold for this statistical cut. 

As discussed before, this background reduction is purely statistical and no physical characteristics 
of the signal are used in order to separate background clusters from signal clusters.
To perform this step forward, we simulate the signals expected in each neutrino detector by 
considering different source distances D in the range 8.5-500 kpc. 
Simulated signals are randomly injected inside the background. 
Once clusters are obtained following the previous procedure, we only select clusters 
with f^im< 1/day. 

For each cluster we define the parameter, ξ_i as the ratio between the cluster multiplicity and the cluster duration:

    ξ_i=m_iΔ t_i
 
and we study the ξ_i distributions of pure background clusters 
and background plus signal clusters in term of normalised Probability Density Functions (PDF). 
In Fig <ref>(a) we show the result obtained for SuperK detector. 

All the PDFs are well described by a 4 parameters Gamma distribution. 
The distribution of clusters due to pure background events is reported with a black solid line and 
is characterised by very small values of the ξ_i parameter as expected for events with a temporal 
uniform distribution, i.e. Δ t_i→ w seconds. 
On the other hand, clusters where also an astrophysical signal is present show a PDF shifted at higher values of ξ_i, 
as expected for clusters with events that cumulate faster in time Δ t_i < 20 seconds. 
For any fixed source distance, the PDF is different and in Fig.<ref>(a) we show with different color
the case of D=65, 140, 300 and 400 kpc as expected in SuperK. Obviously, for closer source distances the 
expected multiplicity in presence of signal increases, allowing a better separation of the PDFs where astrophysical events are present. 
This separation becomes less and less as the source distance increases and the statistics decreases. 

To disentangle as well as possible the signal from the noise we define for each detector X the function

    Ξ[ξ]_X=∫_0^ξ_XPDF^bkg_X  dξ+∫_ξ_X^∞PDF^sig+bkg_X(D)  dξ

and we look for the ξ maximising this function. 
This value defines the best separation, for each detector X, between pure background distribution and signal plus background distribution. For any fixed source distance the PDF is different and a different optimal cut value for the ξ parameter
can be defined. By performing several simulations, we define the function ξ(D), reported in Fig.<ref>(b) for the case of SuperK. 
As expected, we found a descendent behavior of the cut value with the increasing of the distance and this is observed for all the considered detectors. 
If the source distance is known, the optimal cut value of the new parameter is determined by this curve, however when a blind search of astrophysical signals is performed on real data the distance of the source is unknown. In this case we believe that the search 
should be optimised to the larger distance achievable. So that we define as optimal value of the cut parameter ξ_X the smaller one allowing a clear separation between background and signal PDFs. Finally, in the last column of Tab.<ref>, we report the optimal cut parameters found for each detector considered. 

As a consequence, we add as a new cut, on the statistically selected clusters, 
the condition ξ_i ≥ξ_X and we investigate 
its impact on the detection probability η and on the misidentification probability ζ. 
The detection probability is defined as the ratio between the number of signal clusters 
surviving after the cuts and the number of signal clusters initially injected into the background. 
In a similar manner the misidentification probability ζ is obtained as the fraction of background 
clusters that survive all the cuts over the total number of clusters observed. 



As a leading example, we apply our search procedure to SuperK detector and 
we compare the detection probability obtained with the new procedure with  
the one achieved with the standard method. 
The selected statistical threshold f^im≤ 1/day for SuperK 
is equivalent to a multiplicity cut of m_i≥ 4: this implies that an 
astrophysical burst, described as in our model, can be observed 
on average till a distance of D_sk≃400 kpc. This distance represents
the expected horizon for the detector operating at this statistical threshold. 

To show the improvement provided by our method we plot in 
Fig.<ref>(a) the detection efficiencies η with red lines 
and the misidentification probabilities ζ with blue lines. 
In particular, dashed lines are obtained by using the standard procedure 
only based on statistical cut as described above, 
whereas the solid lines are obtained by applying our additional selection criterium. 
It is evident from the figure that the efficiency is unchanged 
till a distance of ∼ 200 kpc, whereas the misidentification, ≃ 23% by 
using the standard procedure, drops to very small value  
≃ 3% by applying the ξ̅ cut.   

 

A gain factor of the order of 10 implies that SuperK can work at a f^im threshold  
10 times higher then the one based on the standard method
obtaining the same background reduction thanks to the new ξ̅ cut. 
A detector working at an higher statistical threshold 
is sensitive to larger distances, so that our result can be also expressed in term of an increased horizon. 
Let us consider, for example, the threshold assumed by SuperK for normal warning 
in the online search for Supernova bursts<cit.>: m_i ≥ 25, 
that corresponds to f^im≤ (3.5·10^-10)/year. This normal warning is sent to
SNEWS and it corresponds to an horizon of 147 kpc. By considering the new additional cut, based
on the ξ parameter, the same background reduction can be obtained by operating at a threshold
f^im≤ (3.5·10^-9)/year corresponding to m_i≥ 11 and increasing the horizon up to 221 kpc.


The improvement of SuperK is representative of the new search method, on the other hand, 
for completeness, we also investigate the others detectors. In particular, we show in the last two columns 
of Tab.<ref> the maximal distance D̅(kpc) for which the additional ξ̅ cut provides no efficiency loss  
and the corresponding gain factor, G=ζ/ζ', calculated for such a distance as the ratio 
between the misidentification probability before and after the ξ̅ cut. 
This gain factor as a function of the source distance is reported in Fig.<ref>(b) for SuperK.






§ NETWORK EXTENSION.
  
In order to construct a list of candidate signal clusters, 
when two or more detectors operate together as a network, a further step is required, i.e. the temporal 
coincidence among clusters of different detectors within a time window 
that we assume to be w_c=10 seconds <cit.>. 
The requirement to observe a coincident signal naturally decreases the background 
contamination, therefore increases the detection efficiency. 
In this case the statistical concept of false imitation frequency 
is substituted by the false alarm rate  

    FAR=2w_c^N_d-1∏_X=1^N_df^im_X,

where N_d is the number of detectors in the network and f^im_X is the imitation frequency 
of each detector. 
Once that a threshold for the global FAR is defined, the corresponding threshold in f^im_X 
for the different detectors depends on the network configuration.
Assuming for example as global FAR of the network ≤ 1/day, the required 
single detector threshold is f^im_X≤ 66/day for a 2-fold coincidence (i.e. two detectors network), 
whereas becomes f^im_X≤ 265/day for the 3-fold coincidence and so on.

Simulated astrophysical signals are injected inside the detectors background 
by taking into account the time of flight between the detectors.
The procedure for the clusters definition is the same as described in 
the previous section and, once that clusters are defined, only clusters coincident 
in the time window w_c are selected.
The new additional cut based on the ξ parameter is now applied to coincidences,  
we require that the product of the ξ_X of coincident clusters is greater then the global cut value:

    ξ^*=∏_X=1^N_dξ_X.

The sensitivity of the neutrino network can be obtained by using 
an extended definition of the detection efficiency η^*, i.e. the number 
of astrophysical clusters surviving the statistical cut on f_im^X that are found 
in temporal coincidence and are also characterised 
by a global ξ^* greater than the cut value defined in Eq.<ref> 
over the total injected signals.
In a similar manner the network definition 
of the misidentification probability ζ^* becomes 
the ratio among background coincidences and the total number of found coincidences.

As a leading example, we show the case of LVD & Kamland working together at a FAR≤ 1/day. 
The detection efficiency and the misidentification probability of this network are showed 
in Fig.<ref>(a). As in the previous plot dashed lines represent 
the old method based on statistical cut plus temporal coincidence search, whereas 
solid lines show the same quantities obtained by adding the ξ^* cut as described above.









In particular, the misidentification probability is nearly constant until 75 kpc around a value of 4% with the standard procedure and decreases 
to a value around 0.2% with the new cut. The gain factor obtained in this distance range is around ∼ 20 as reported in Fig.<ref>(b).
This reduction of the misidentification can be also converted in term of a reduction of the FAR for the network. In other words, the network 
LVD & Kamland operating at a FAR of 0.001/day  (0.365/year) with the inclusion of our method, based on the ξ^* cut, 
can reach the same background level of LVD &Kamland working at the SNEWS threshold of 1/1000years where only the statistical selection is applied. 


We apply this extended procedure to all the possible sub-networks of detectors 
including LVD, Borexino, Kamland and SK. In any case the improvement obtained 
is of the same order, indeed also for the cases of combined search between 
LVD & Borexino or Kamland & Borexino the gain factor obtained is of ∼19, 
however with a reduced distance D̅∼ 50 kpc due to the lower 
sensitivity of Borexino. 




§ DISCUSSION

In conclusion, we propose a novel search method 
for astrophysical bursts of low-energy neutrinos. 
This method allows us a powerful discrimination between 
background and signal by exploiting their 
different cumulative rate. The achieved results show a 
decrease of the misidentification probability of a factor 10-20 
without loosing on detection efficiency.       

The proposed method can be applied both on single detector search and
in combined search among different detectors and can be easily implemented 
in the SNEWS online search for enhancing its detection potential and horizon.
Finally, we stress that the proposed method works 
for any low-energy neutrinos detector, water Cherenkov, liquid scintillator 
or argon based, that will be on data-taking in the next future, viz.
HyperK<cit.>, JUNO<cit.> and DUNE<cit.>.

Moreover combined search of core collapse supernovae 
with low-energy neutrinos and gravitational waves 
that are on going<cit.>, can profit 
from this new procedure, being already based both on a  
complete data-sharing and on a combined search with different detectors.
This will also increase the detection probability 
for gravitational wave bursts expected from 
the here considered astrophysical sources.    


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