A DECam Search for Explosive Optical Transients Associated with IceCube Neutrino Alerts

IceCube is a cubic-kilometer-scale neutrino detector located at the geographic South Pole and built into the Antarctic ice at depths ranging from 1450 to 2450 m (Aartsen et al. 2017b). Incident neutrinos are detected indirectly through Cerenkov radiation emitted by secondary charged particles produced in the interaction of a neutrino with a nucleus of the ice. Photomultipliers in 5160 optical modules sense the Cerenkov light. Events detected by IceCube are classified as "shower" events or "track" events. Showers are produced in charged-current interactions of electron and tau neutrinos and in neutral-current interactions of all neutrino flavors. A shower of secondary particles with a few meters of extension is produced, which is small compared to the distance between the optical modules. Therefore, the Cerenkov light produced by the shower particles appears almost spherical. Such events can be reconstructed with an angular resolution of the incoming neutrino on the order of ∼15 deg. Tracks are produced by charged-current interactions of muon neutrinos in or close to the detector, which produce a muon that travels along a long linear path through the ice, providing a good lever arm for the angular reconstruction. Specifically, for track neutrino events with energies above 100 TeV, IceCube achieves median angular resolutions for high-energy starting events (HESE) and extremely high energy events (EHE) of 0.4–1.6 deg and <1.0 deg, respectively (Aartsen et al. 2014, 2017a).

In an effort to locate the sources of these neutrinos and to facilitate follow-up studies, the IceCube Collaboration has created a real-time alert system for events of likely astrophysical origin (Aartsen et al. 2017a). The system has a median latency of 33 s in which neutrino events are processed and reconstructed. Within three minutes, alerts are published to streams such as the Astrophysical Multimessenger Observatory Network (AMON; Smith et al. 2013; Keivani et al. 2017) and the Gamma-ray Coordination Network (GCN; Barthelmy et al. 1998). Alerts are accompanied by an estimate of the probability that the neutrino is of astrophysical origin, referred to as the "signalness." The signalness describes how likely the event is of astrophysical origin relative to the total atmospheric background rate and is a function of the neutrino energy and the zenith angle. For more information on the calculation of the signalness, refer to the discussion in Aartsen et al. (2017a).

On 2017 September 22, IceCube detected a singular high-energy neutrino. Another singular high-energy neutrino was detected on 2017 November 6. Both events had good angular resolution and are likely to be of astrophysical origin. The details of the AMON alerts are presented in the top portion of Table 1. Both alerts were track-type singular neutrinos with energies on the 100–200 TeV scale. These alerts featured signalness scores of ∼0.57 and ∼0.75 and 90% confidence level containment region areas smaller than the FoV of DECam. The high energies and signalness scores suggest an astrophysical origin of the events and motivate the search for an optical counterpart in the form of CC SNe.

Of the 31 alerts issued thus far by the IceCube Collaboration, 21 have occurred after the start of this analysis. Of the 21 possible alerts, 16 were either deemed unobservable based on atmospheric, moon, or Sun conditions, or were retracted by the IceCube Collaboration. The two alerts presented here are two of only five alerts that have been observable from the location of the Blanco Telescope in Chile. The additional alerts (IC181023A, IC190331A, and IC190503A) were observed and will be discussed in a future work.

The Dark Energy Camera (Flaugher et al. 2015) is a 570-megapixel optical imager mounted on the 4 m Blanco Telescope at Cerro Tololo Inter-American Observatory (CTIO) in Chile. Blanco/DECam's location, FoV (∼3 deg²), and deep imaging capabilities make it the only southern hemisphere imager with a wide FoV matched to the IceCube angular resolution and a large enough aperture to efficiently detect explosive optical transients at the redshifts relevant for proposed neutrino sources.

The Dark Energy Survey is a wide-field optical survey (expected to reach a 10σ depth for point sources of grizY = 25.2, 24.8, 24.0, 23.4, 21.7 mag over 5000 deg²) with 577 full DECam observing nights (Mohr et al. 2012). Operating within a large survey program is ideal for our follow-up study because there is no need to interrupt community observers, and the interruptions are short (20 minutes). Specifically, we use triggered target-of-opportunity observations to promptly respond to IceCube alerts. Our observing strategy was to dedicate about one month to characterize the rise and peak of potentially associated CC SNe for each alert. During the one-month observing period, we took observations roughly every five to six nights depending on atmospheric and moon conditions. As well, we added an additional observing night to the start of the follow-up to look for rapidly evolving optical transients. Observations consisted of two 150 s exposures in each of the gri optical bands, and exposures within the same band were stacked to reduce noise and increase imaging depth. For the IC170922A follow-up, we also adjusted the pointing of the telescope by 0.01 deg in both R.A. and decl. between exposures of the same band to cover areas of the sky that mapped to gaps between adjacent CCDs, leading to a slightly larger effective DECam FoV area in Table 1.

The DECam observations were processed by the DES Difference Imaging Pipeline (Kessler et al. 2015, DiffImg), which subtracts template images from the observations and produces catalog-level photometric fluxes in each band for all detected transients in the FoV. For this analysis, since our observations did not overlap with the DES footprint where template images would be readily available, we use the first-night images as templates. Observations are also run through autoscan (Goldstein et al. 2015), which identifies potential image artifacts and poor image subtractions. This process injects fake artifacts into the images to train the algorithm to remove artifacts that may be present in the images. The probability that a DiffImg candidate is a real astrophysical object as opposed to an artifact is characterized by the autoscan machine learning (ML) score. This score is utilized in Section 4. From the DiffImg outputs, objects that are likely CC SNe and temporally coincident with the neutrino alert based on light curve properties are selected by an automated neutrino source candidate identification pipeline described in Section 4. The DiffImg products and pipeline candidates for both IC170922A and IC171106A are shown in Figure 1. The light curves of the pipeline candidates are shown in Figure 2.

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3.2.1. DECam Observations of TXS 0506+056

Selecting CC SNe associated with IceCube neutrino alerts requires extrapolating the explosion time of the SN from its light curve. To model the rise times of SNe, we employ the SuperNova ANAlysis software suite (SNANA; Kessler et al. 2009), which enables us to simulate SN light curves based on our observing cadence. Based on SNANA-simulated SN light curves for our observations, we develop selection criteria to filter out unassociated SNe. Then by applying the selection criteria to the same simulations, we determine the detection efficiency for an associated CC SN and the expected number of unassociated SNe found by NCIP for each follow-up.

We simulated SNe light curves for 500 DECam FoVs for each follow-up. The background population of simulated light curves consists of both type Ia and CC SNe that reach peak brightness in a range of MJDs extending from 30 nights before the observing window to 100 nights after the observing window has concluded. For the simulated type Ia SNe, we use templates from Guy et al. (2010) and rates from Dilday et al. (2008), and for the simulated CC SNe we use templates from Kessler et al. (2010) and rates from Bernstein et al. (2012). We do not include asteroids or AGNs in the background sample because those models are not available within SNANA, though it has been shown that they can be effectively removed with additional selection criteria (Bailey et al. 2007). The simulated signal population consists of only CC SNe with an explosion time set to be the date of the IceCube neutrino alert. Since explosion times are not observed, CC SNe models are more tightly constrained near peak brightness than during the rising phases, and we therefore use the flux relative to its peak value as a proxy for when the explosion began. Within SNANA, we define the explosion time to be the earliest MJD for which the flux in any band reaches 1% of the peak flux in the rest frame of the SN. To construct the signal sample, CC SN light curves are scanned for the MJD exhibiting a rise in flux above the 1% peak flux level, and then shifted in time so that the MJD aligns with the desired explosion time. By setting the neutrino trigger to the date of the start of the explosion, we are only considering prompt neutrino emission in this analysis, rather than neutrino emission that may occur during later stages of the collapse. Both sets of simulations account for the measured point-spread function, zero point, and sky noise in each exposure to yield a light curve quality that is representative of the actual DECam follow-up observations. The simulations do not account for uncertainty in the CC SN luminosity function, which is currently approximated using a Gaussian distribution with a mean and standard deviation determined from observed SN luminosity (Li et al. 2011).

The process of background removal consists of a series of selection criteria applied to the simulated light curves. We list the criteria here and discuss the motivation for each cut in the remainder of this section:

• 1.  
  Quality Cut. Light curves must have detections on two separate nights, irrespective of the band of the detection. If a light curve passes this cut, we refer to it as "optically detectable."
• 2.  
  Rising Cut. Light curves must not have a detection on the first post-template night following the trigger, or if there is a detection, there must be an increase in flux of at least one magnitude in at least one band over the first two post-template nights.
• 3.  
  Phase Cut. The peak MJDs predicted by the type Ibc and type II templates fit to the light curve must be at least six nights and 16 nights after the trigger, respectively.
• 4.  
  Classify Cut. Light curves must be classified as a CC SN by our Random Forest Classifier.

Quality. The quality cut is designed to guarantee DECam detectability from photometric data quality. For this cut and the rising cut, we define a "detection" in a given band and epoch to mean the photometric data has the following three properties: (1) DiffImg finds the object and does not mark it as a bad subtraction; (2) the ML score is larger than 0.7, meaning observation was determined to have good image quality and is unlikely to be an artifact; and (3) the signal-to-noise ratio is larger than 10. The quality cut removes events that are photometrically of too poor quality to claim association with the neutrino by ensuring that the event is observable with DECam. In the DECam FoV, we expect 17.5 ± 0.3 SNe and observe 11 objects for IC170922A, and for IC171106A we expect 10.0 ± 0.3 SNe and observe 10 objects. The under-fluctuation for IC170922A is mediated by the application of further cuts.

Rising. The rising cut is designed to select light curves of recently exploded objects by removing SNe that reach peak brightness before the neutrino trigger. The effectiveness of the rising cut is shown in the left panel of Figure 3. For the two follow-ups presented here, the first observing night was used to make template images. Therefore, "post-template" nights refer to all nights after the first night. In future follow-ups, if it is not possible to observe the field right away, or if template images already exist, this criterion will need to be reformulated. After the rising cut, we expect 13.1 ± 0.3 SNe in the DECam FoV for IC170922A and observe 10 objects, and for IC171106A we expect 7.3 ± 0.3 SNe and observe eight objects.

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Phase. The most differentiating features between the signal and background populations that pass the first level of cuts are the type of SN and where in time the SN exploded relative to the neutrino trigger, which we refer to as the phase of the light curve. To exploit these features, we fit all light curves using the SNANA implementation of the Photometric SuperNova IDentification tool (Sako et al. 2011, PSNID), which offers not only best-fit phase information but also fit probabilities and χ² values for the type Ia, type Ibc, and type II templates fit to each light curve (henceforth ${\chi }_{\mathrm{Ia}}^{2}$, ${\chi }_{\mathrm{Ibc}}^{2}$, and ${\chi }_{\mathrm{II}}^{2}$). The phase cut is designed to remove light curves that exploded before the trigger and are still rising during the observing window. The cutoffs of six and 16 nights were empirically derived from analyzing simulations. The motivation for these values is displayed in the center panel of Figure 3, and the performance of the phase cut is also shown in the left panel of the same figure. After imposing the phase cut, we expect 5.6 ± 0.2 SNe in the DECam FoV for IC170922A and observe two objects, and for IC171106A we expect 2.74 ± 0.17 SNe and observe two objects.

Classify. Since PSNID was designed to make classifications with the goal of maximizing type Ia SNe purity, we opted to supplement the information used to make classifications with features that are tailored to the problem of selecting CC SNe. Specifically, we found that by including features representative of the probability a light curve was a CC SN, we were able to increase the fraction of known CC SNe classified as CC SNe (CC completeness) while simultaneously reducing the fraction of known type Ia SNe classified as CC SNe (Ia false-positive rate). PSNID returns χ² values for light curve fits to type Ia $({\chi }_{\mathrm{Ia}}^{2})$, type Ibc $({\chi }_{\mathrm{Ibc}}^{2})$, and type II $({\chi }_{\mathrm{II}}^{2})$ templates. Based on this information, we define the normalized Ia and CC χ² values ${\bar{\chi }}_{\mathrm{Ia}}^{2}$ and $1-{\bar{\chi }}_{\mathrm{CC}}^{2}$, where

$$\begin{eqnarray}&&{\bar{\chi }}_{\mathrm{Ia}}^{2}=\displaystyle \frac{{\chi }_{\mathrm{Ia}}^{2}}{{\chi }_{\mathrm{Ia}}^{2}+{\chi }_{\mathrm{Ibc}}^{2}+{\chi }_{\mathrm{II}}^{2}}\quad \mathrm{and}\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&{\bar{\chi }}_{\mathrm{CC}}^{2}=\displaystyle \frac{\min ({\chi }_{\mathrm{Ibc}}^{2},{\chi }_{\mathrm{II}}^{2})}{{\chi }_{\mathrm{Ia}}^{2}+{\chi }_{\mathrm{Ibc}}^{2}+{\chi }_{\mathrm{II}}^{2}}.\end{eqnarray} \tag{ 3 }$$

Both of the features range from zero to one, and they are expected to be closer to zero for a type Ia SN while closer to one for a CC SN. Using these features, we reclassified the sample using a Random Forest Classifier (Breiman 2001) implemented with Sci-Kit Learn (Pedregosa et al. 2011). To construct the classifier, we first used principal component analysis (PCA; Tipping & Bishop 1999) with eight components to project light curve fit information to the space of largest variance, and then employed an ensemble of 1000 decision-tree classifiers with a restriction of 50 for the maximum tree depth. The number of principal components, number of decision trees, and depth restriction on the decision trees were determined via an extensive search of the hyperparameter space using a separate validation data set.

The classifier was trained using a SNANA-simulated sample of CC light curves passing the quality, rising, and phase cuts that had an explosion time set to the neutrino trigger and a low redshift (z < 0.3) as the target class. The background class was composed of CC and type Ia light curves determined that passed the quality, rising, and phase cuts with no set restriction on the true explosion time of the SNe. We also include spectroscopically confirmed CC SNe and Ia SNe from the DES 3 Year SNe sample (Macaulay et al. 2019) in the training set signal and background samples, respectively. The ratio of simulated SNe to real SNe in the training set was found to strongly correlate to classifier accuracy, and we determined the optimum ratio for each follow-up using independent validation sets. We find that classification accuracy is optimized for both simulated and real SNe when 36% (40%) of the training set is simulated for IC170922A (IC171106A), and the determination of these ratios is shown in Figure 4. The dependence in classification accuracy on the inclusion of real SNe is likely due to the fact that the real SNe were observed under better conditions than our follow-ups. Since our classifier uses best-fit properties to make predictions, it is sensitive to how well the light curve fitting reflects the true properties of the explosion, so the better observing conditions of the real SNe could add new classification information that a simulated training set lacked.

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In the process of training, we used stratified fivefold cross validation to limit overfitting. This procedure trains on four-fifths of the training data, tests on the remaining one-fifth, and then repeats the process so that each fifth of the training data is used once for testing; the purpose is to introduce small variations in the training set so that the classifier does not effectively memorize the training set. The classifier operates at 83.7% purity, 68.1% completeness, and with a 7.3% false-positive rate on the training set, and at 81.8% purity, 66.7% completeness, and with a 8.1% false-positive rate on the testing set. The similarity of the performance of the classifier on familiar and unseen data is a good indicator that the classification is not suffering from overfitting and will be able to generalize to other data sets. An ROC curve for our classifier is displayed in the right panel of Figure 3 and shows the classifier operating with a mean area under curve (AUC) of 0.88. The standard deviation of the AUC over the five folds of the training data is 0.02, indicating classifications are relatively insensitive to the representation of exact members of the training set. After this final cut, we expect 0.74 ± 0.07 SNe in the DECam FoV for IC170922A and observe one object, and for IC171106A we expect 0.32 ± 0.06 SNe and observe one object. These two remaining objects are our candidates from these first two follow-up efforts.

We applied the selection criteria to our SNANA simulations of the signal and background samples. For the signal samples, we report the fraction of events passing each successive cut, as well as the fraction of optically detectable events passing each successive cut. These fractions are multiplied by the CC SN rate to determine the expected number of signal-like events per IceCube alert as a function of redshift and scaled to the IceCube 90% confidence level containment region (IC90). The number of background events passing each cut was normalized based on the number of FoVs generated to accurately reflect the expected number of background-like events per follow-up. These results are presented for each of the two events in Table 2.

Table 2.  Application of the Candidate Selection Pipeline to Simulations and Data for IC170922A and IC171106A

IC170922A Follow-up
Sample	Signal Efficiency		Signal Efficiency		Background Eventsa in IC90		Datab	Datab
	All SNea		Optically Detectable SNea				in FoV	in IC90
Cut	z < 0.3	z > 0.3	z < 0.3	z > 0.3	z < 0.3	z > 0.3	⋯	⋯
Total	1.0000−0.0001	1.00000−0.00001	⋯	⋯	9.26 ± 0.14	414.2 ± 0.9	617	240
Quality	${0.199}_{-0.004}^{+0.004}$	${2.47}_{-0.15}^{+0.16}\times {10}^{-3}$	${1.0000}_{-0.0006}$	${1.000}_{-0.005}$	1.48 ± 0.05	3.96 ± 0.09	11	2
Rising	${0.198}_{-0.004}^{+0.004}$	${2.47}_{-0.15}^{+0.15}\times {10}^{-3}$	${0.9984}_{-0.0011}^{+0.0007}$	${1.000}_{-0.005}$	1.00 ± 0.04	3.07 ± 0.08	10	2
Phase	${0.179}_{-0.004}^{+0.004}$	${2.47}_{-0.15}^{+0.15}\times {10}^{-3}$	${0.9964}_{-0.0015}^{+0.0012}$	${1.000}_{-0.005}$	0.51 ± 0.03	1.25 ± 0.05	2	0
Classify	${0.121}_{-0.003}^{+0.003}$	${1.26}_{-0.11}^{+0.11}\times {10}^{-3}$	${0.608}_{-0.011}^{+0.011}$	${0.51}_{-0.03}^{+0.03}$	0.116 ± 0.015	0.113 ± 0.015	1	0

IC171106A Follow-up
Sample	Signal Efficiency		Signal Efficiency		Background Eventsa in IC90		Datab	Datab
	All SNea		Optically Detectable SNea				in FoV	in IC90
Cut	z > 0.3	z > 0.3	z < 0.3	z > 0.3	z < 0.3	z > 0.3	⋯	⋯
Total	1.0000−0.0001	1.00000−0.00001	⋯	⋯	5.13 ± 0.10	236.4 ± 0.7	1868	646
Quality	${0.134}_{-0.003}^{+0.003}$	${7.1}_{-0.8}^{+0.9}\times {10}^{-4}$	${1.0000}_{-0.0008}$	${1.000}_{-0.016}$	0.65 ± 0.04	1.46 ± 0.05	10	0
Rising	${0.134}_{-0.003}^{+0.003}$	${7.1}_{-0.8}^{+0.9}\times {10}^{-4}$	${0.9978}_{-0.0016}^{+0.0010}$	1.000−0.016	0.43 ± 0.03	1.11 ± 0.05	8	0
Phase	${0.130}_{-0.003}^{+0.003}$	${7.1}_{-0.8}^{+0.9}\times {10}^{-4}$	${0.970}_{-0.005}^{+0.004}$	${0.986}_{-0.02}^{+0.010}$	0.187 ± 0.019	0.39 ± 0.03	2	0
Classify	${0.095}_{-0.003}^{+0.003}$	${3.3}_{-0.4}^{+0.4}\times {10}^{-4}$	${0.713}_{-0.012}^{+0.012}$	${0.46}_{-0.06}^{+0.06}$	0.047 ± 0.010	0.020 ± 0.006	1	0

Notes. The term "optically detectable" is equivalent to the light curve passing the quality cut. Background event values have a statistical uncertainty of $\pm \sqrt{N/500}$ on the mean number of events generated, corresponding to the number of DECam FoVs simulated. The uncertainties reported for the signal efficiencies give a 68% confidence interval. The numbers of background events reported are scaled to reflect just the events within the IceCube 90% CL containment region. The data in the total row are not expected to match the simulations because they can contain non-SNe transients or bad image subtractions that are removed by the quality cut.

^aResults presented are based on SNANA simulations. ^bResults presented are the number of candidates remaining from DECam observations.

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For events with observing conditions of quality similar to the IC170922A follow-up, we expect to detect roughly 12.1% of nearby (z < 0.3), neutrino-emitting CC SNe using DECam and NCIP, while limiting the background to 0.23 unassociated SNe up to redshift z = 1 within IC90. For the events similar to the IC171106A follow-up observations, we expect to detect 9.5% of nearby signal events and 0.067 background events within IC90 per follow-up. The low signal detection efficiency is a result of the faintness of CC SNe and the magnitude limit of DECam, rather than the strictness of our selection criteria, evidenced by roughly 90% of optically detectable, low-redshift signal events passing all cuts across both events. The remaining SNe background sample's magnitude, temporal, and redshift distributions are displayed in Figure 5.

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Next, we consider the signal detection efficiency and number of remaining background events per set of follow-up observations as functions of redshift in order to calculate the maximum redshift to which we will be able to detect the potential CC SN counterpart of an IceCube alert. These relationships are shown for the IC170922A follow-up in Figure 6. Based on Figure 6, the fraction of signal sample events remaining after all cuts quickly diminishes as redshift increases; however, this behavior is largely a result of the low optical detectability of high-redshift CC SNe.

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The distribution of remaining background events per set of follow-up observations based on the observing conditions of the IC170922A follow-up is shown as a function of redshift in Figure 7. We note that the type Ia SNe background has been suppressed in the redshift range of highest sensitivity (approximately z < 0.2). Therefore, only events that are similar to the signal population in both SN type and phase remain in the final SNe background population. The background sample of this analysis was normalized within SNANA such that the number of events generated was equal to the expected number of SNe within a DECam FoV. The signal sample is normalized to one CC SNe, since only one or zero signal CC SNe is possible per follow-up. The redshift distribution is derived by folding the detection efficiency as a function of redshift with the cosmic star formation rate as described in Section 2. We also multiply the signal distribution by the IceCube signalness value, which is the probability that the neutrino has an astrophysical origin.

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From the signal and background redshift distributions, we integrate over all redshifts to obtain the total number of expected signal and background events for a given follow-up. These numbers are displayed on Figure 7 as the AUC values and become important when redshift information for pipeline candidates is unavailable, requiring us to account for all events along the line of sight. Next, we compare the expected number of detected CC SNe associated with the IceCube alert to the number of unassociated SNe expected to pass our selection criteria in a single follow-up observation. Figure 7 displays this comparison for the IC170922A follow-up. The signal clearly dominates the background at low redshifts, and the cumulative number of signal events dominates the cumulative number of background events until roughly z = 0.1. Beyond this redshift, uncertainties in the modeling of SN rise times weaken the effect of the timing-based cuts (rising and phase) on reducing the background. Additionally, as the redshift increases, CC SNe exploding at the time of the neutrino alert quickly become too faint to detect. Together, both of these effects lead to background domination at high redshifts.

Previous efforts have been made to associate CC SNe with neutrino telescope alerts. ROTSE and TAROT were used in optical follow-ups of 42 ANTARES neutrino alerts, reaching a maximum r-band limiting magnitude of 18.6 with an FoV diameter of 2 deg (Adrián-Martínez et al. 2016). One neutrino doublet IceCube alert has been followed up with ROTSE and PTF, and while a CC SN was found in the FoV, it was determined to be old and type IIn via spectroscopy from Keck I LRIS and Gemini/GMOS-N and therefore unassociated with the neutrinos (Aartsen et al. 2015b). The optical instruments in this effort reached a peak r-band limiting magnitude of 19.5 with an FoV diameter of 2.0 deg. Only a small fraction of cosmic neutrinos will be detected as doublets, while the large majority will be singlets. Accordingly, the follow-up of singlets is a promising approach. However, as shown by Figure 5, significantly deeper observations are required in the singlet case. Specifically, our SNANA simulations, with peak r-band magnitudes displayed in Figure 5, show the importance of DECam's imaging depth in searches for CC SNe since the fainter SNe in the 21–23.5 mag range become detectable. A complementary version of Figure 5 where the limiting magnitude was set to 21 mag is available in Figure 10, illustrating the small fraction of all occurring SNe detectable at this optical depth. Because CC SNe are expected to follow the cosmic massive star formation rate, they are expected to be mostly distributed at higher redshifts, and hence faint in the optical bands. Therefore, it is not surprising that deep imaging offers a significant advantage to follow-up campaigns.

Even with the deep DECam observations presented in this work, detecting a CC SN as a teraelectronvolt to petaelectronvolt neutrino counterpart is still difficult, as our sensitivity analysis suggests. Our simulations and NCIP analysis show that only ∼9.5%–12% of associated CC SNe with z ≤ 0.3 would be bright enough to be detectable. For z > 0.3, less than 1% of associated CC SNe can be detected. When we fold this low detection efficiency with the cosmic massive star formation rate as described in Section 2, unassociated SNe are expected to overwhelm the signal population. For example, based on the expected signal (0.038) and background (0.229) candidates per follow-up from Figure 7 for IC170922A, we would detect approximately four associated CC SNe and 23 unassociated SNe out to a redshift of z = 1.0 within the IC90 regions over the course of 100 IceCube follow-up experiments using DECam.

The redshift distribution and SN type composition of the expected background after our candidate selection pipeline are displayed in Figure 7 and are particularly useful in understanding the difficulty of claiming association between a neutrino and a CC SN. In Figure 7, the area under the dashed black curve represents the total expected number of signal events for a given follow-up.

We have multiplied the redshift probability distribution function described by the cosmic massive star formation rate by the IceCube signalness for the alert, which was ∼0.56 for IC170922A. This curve also follows the assumption that the entire IceCube teraelectronvolt to petaelectronvolt energy neutrino flux is caused by CC SNe, which we refer to as the λ = 1.0 case in Appendix A. We note that for low redshifts (z ≲ 0.2), the redshifted neutrino energy will be very close to its rest-frame energy. Therefore, we find it sufficient to leave out the contribution of the spectral energy distribution of neutrinos from the calculation. The red curve then folds the black curve with the signal detection efficiency from Figure 6 to obtain the expected number of detected CC SN teraelectronvolt to petaelectronvolt neutrino counterparts per follow-up as a function of redshift. In the redshift range where the red curve is nonvanishing, we note that the most dominant background is CC SNe that are present in the FoV but are not associated with the neutrino. These CC SNe have also passed the rising and phase cuts of our pipeline, meaning they appear to be temporally coincident with a neutrino as well. Therefore, the largest background in this analysis is CC SNe that are temporally coincident with the neutrino, which unfortunately are the exact characteristics of our signal population.

Given this background, our main tool for determining if a given candidate is associated with the neutrino alert is the proximity of the candidate to the alert centroid. Additional differentiating features between signal and background samples are the candidate redshift and spectroscopic characterization. The redshift can likely be determined to sufficient accuracy based on a photometric redshift of the host galaxy, but if spectra can be obtained, then it would also be possible to exclude the type Ia SNe background.

An individual follow-up with deep optical imaging, triggered observations, and redshift information for all candidates is still unlikely to reach a meaningful level of significance due to a signal-like background of unassociated CC SNe that are coincident in time with neutrino alerts and a low signal detection efficiency. Therefore, we frame this analysis as seeking to detect an excess of CC SNe located near IceCube neutrino alerts in space and time over the course of multiple follow-ups. We present a full calculation later in this section, but we outline the basis of it here and restrict our focus to the IC90 region for clarity; the full calculation utilizes the entire DECam FoV. In the IC90 region, from Figure 7 we expect 0.038 signal events and 0.229 background events for a follow-up similar to IC170922A, and from Figure 14 we expect 0.047 signal events and 0.067 background events for a follow-up similar to IC171106A. In the full calculation, if redshift information is available, the contribution to the total significance is weighted by the expected number of signal and background events at the particular redshift. Overall, this process just reduces the contribution of background events to the total significance, so we can account for the effect of having redshift information for a candidate by dividing the expected background by a factor of ∼3 to simulate a smaller redshift range. Applying this factor and averaging the results lead to a mean expected 0.043 signal events and a mean expected 0.049 background events. We then estimate the significance after N follow-ups using a Poisson distribution:

$$\begin{eqnarray}&&p \mbox{-} \mathrm{value}=1-\mathrm{PoissonCDF}(\mathrm{Observed}| \mathrm{Expected}).\end{eqnarray} \tag{ 4 }$$

Setting the "Expected" number of events to be N × 0.049 for the background-only null hypothesis and setting the "Observed" number of events to be N × (0.043 + 0.049) enable us to evaluate the significance as a function of the number of follow-ups. As well, one can see that increasing the number of follow-ups will increase the global significance. In this brief calculation, for 100 follow-ups, we would expect to reach a p value of ∼0.029, which is between 2σ and 3σ. The addition of positional information and a more accurate accounting of redshift information in the full calculation both increase the significance beyond this simplified example.

To make this calculation more rigorous, we simulate pipeline candidate events with frequencies matching the mean expected event rates from the two follow-ups. To determine the probability of an NCIP candidate being the source of the neutrino, we evaluate the candidate's position and redshift in the context of the expected signal and background population distributions of these quantities. For the signal sample, we fit independent 2D asymmetric Gaussian distributions to the 90% confidence level error bounds on the R.A. and decl. We multiply our detection efficiency by the fraction of events that would fall on a DECam CCD to account for chip gaps and the few events that fall outside the DECam FoV. For the background distribution, we adopt a uniform distribution of events on the individual DECam CCDs. The redshift distributions for the signal and background samples are taken from Figure 7. Using these distributions, we assess whether CC SNe contribute to the teraelectronvolt to petaelectronvolt neutrino flux by searching for an excess of temporally coincident CC SNe in our observations compared to the expected background contamination rate. Appendix A explains this process in greater detail and provides an overview of the maximum-likelihood formalism used to quantify the likelihood of association between our pipeline candidates and IceCube neutrino alerts.

Using the maximum-likelihood framework and the angular and redshift distributions for the signal and background samples, we perform 1000 realizations of optical follow-up campaigns of IceCube alerts. For each alert follow-up, we determine a list of candidates by sampling the signal and background event distributions. The list of candidates has a probability of containing a signal event of 0.038 × 0.89 = 0.0338 for the IC170922A follow-up, where 0.038 is the expected number of detected signal events per follow-up from Figure 7 and 0.89 is the fraction of the signal that falls on the DECam CCDs. The number of expected background events per follow-up (0.717) is determined from the background AUC in Figure 7 by scaling the value to the full DECam FoV. The number of background events to simulate is then determined by a Poisson probability distribution with a mean of 0.717. Once the number of candidates has been determined, we sample the angular and redshift distributions to obtain a realistic representation of the follow-up. We also introduce a signal normalization parameter, λ, which represents the fraction of the teraelectronvolt to petaelectronvolt neutrino flux caused by CC SNe, and we perform the realizations under different values of λ.

The test statistic distributions for 1000 realizations of follow-up campaigns of 60 alerts based on the conditions of both alerts are displayed in Figure 11. We determine the significance level at which we would be able to claim CC SNe contribute to the teraelectronvolt to petaelectronvolt IceCube flux by taking the median value of our alternative-hypothesis $(\lambda \ne 0.0)$ distributions and calculating the fraction of the null-hypothesis distribution (λ = 0.0) greater than the median test statistic. This fraction can be interpreted as the frequentist p value. Figure 11 shows that our sensitivity for detecting an excess of CC SNe temporally and spatially coincident with IceCube alerts is greater when redshift information is available and if CC SNe make up a large fraction of the teraelectronvolt to petaelectronvolt neutrino flux.

We then further the analysis of the viability of this type of study by calculating the discovery potential for follow-up campaigns with increasing numbers of alerts. The result of this calculation is displayed for the same three values of λ in Figure 8. The cases of every candidate having available redshift information and no candidates having available redshift information are taken as the boundary cases of the best- and worst-case scenarios. We note that a few alerts are required before the p value is expected to differ from 1.0 in all cases, but that this behavior is explained by the fact that we use the median test statistic to calculate the p value. The p value is the fraction of null-hypothesis test statistics strictly greater than the median of the signal-hypothesis test statistic distributions, so the median must be larger than zero in order to produce a p value less than one. Due to the low detection efficiency of signal events, follow-up campaigns of few alerts are very likely to be statistically consistent with the background and hence produce test statistic distributions that are predominantly composed of test statistics of zero. Based on this calculation, optical follow-up campaigns of this nature would require approximately 60 follow-ups before an excess of temporally and spatially coincident CC SNe could be detected at the 3σ confidence level in the case that photometric redshift information is available for each candidate and that CC SNe account for 100% of the teraelectronvolt to petaelectronvolt neutrino flux. In the less optimistic case where CC SNe account for a smaller percentage of the teraelectronvolt to petaelectronvolt neutrinos, or redshift information is unable to be obtained, the required number of follow-ups can easily exceed 200. The IceCube event rate is approximately one alert per month, and typically one in every three alerts is observable from CTIO based on solar and atmospheric conditions.

This analysis highlights several vital components for successful optical follow-ups of IceCube neutrino alerts searching for CC SNe. First, deep imaging is needed to efficiently detect CC SNe counterparts to teraelectronvolt to petaelectronvolt neutrinos at z ∼ 0.1, as shown by the comparison between Figures 5 and 10. A sufficiently large FoV is also required such that the IceCube localization region can be covered. DECam's ∼23.5 r-band limiting magnitude in 90 second exposures and ∼3 deg² FoV make it the current imager of choice in the southern hemisphere for this task. Even with DECam, we anticipate a large background of CC SNe that are unassociated with the teraelectronvolt to petaelectronvolt neutrinos and difficult to distinguish from neutrino-causing CC SNe. Photometric redshifts and spectroscopic characterization can help with the differentiation, but with the low IceCube event rate and optical detection efficiency of CC SNe, a follow-up campaign would take several years to be able to observe a significant excess of CC SNe near IceCube alerts. It is also possible that tightened constraints on the CC SNe luminosity function and SNe rise times will improve the accuracy of the simulations used in this analysis and in the significance forecasting. It is anticipated that an expansion of the IceCube detector or improved real-time alert selection could increase the event rate and improve the angular resolution of neutrino reconstruction.

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