Supernova Neutrino Burst Detection with the Deep Underground Neutrino Experiment

A core-collapse supernova¹¹1“Supernova” always refers to a “core-collapse supernova” in this paper, although we are aware that not all core collapses produce electromagnetically visible supernovae, and not all supernovae result from stellar core collapse. occurs when a massive star reaches the end of its life. As a result of nuclear burning throughout the star’s life, the central region of such a star gains an “onion” structure, with an iron core at the center surrounded by concentric shells of lighter elements (silicon, oxygen, neon, magnesium, carbon, etc). At temperatures of $T\sim 10^{10}$ K and densities of $\rho\sim 10^{10}$ g/cm³, the Fe core continuously loses energy by neutrino emission (through pair annihilation and plasmon decay Braaten:1993jw ). Since iron cannot be further burned, the lost energy cannot be replenished throughout the volume and the core continues to contract and heat up, while also growing in mass thanks to the shell burning. Eventually, the critical mass of about $1.4M_{\odot}$ of Fe is reached, at which point a stable configuration is no longer possible. As electrons are absorbed by the protons in nuclei and some iron is disintegrated by thermal photons, the degeneracy pressure support is suddenly removed and the core collapses essentially in free fall, reaching speeds of about a quarter of the speed of light²²2Other collapse mechanisms are possible: an “electron-capture” supernova does not reach the final burning phase before highly degenerate electrons break apart nuclei and trigger a collapse..

The collapse of the central region is suddenly halted after $\sim 10^{-2}$ seconds, as the density reaches nuclear (or super-nuclear) values. The central core rebounds and an outward-moving shock wave is formed. The extreme physical conditions of this core, in particular the densities of order $10^{12}-10^{14}$ g/cm³, create a medium that is opaque even for neutrinos. As a consequence, the core initially has a trapped lepton number. The gravitational energy of the collapse at this stage is stored mostly in the degenerate Fermi sea of electrons ($E_{F}\sim 200$ MeV) and electron neutrinos, which are in equilibrium with the former. The temperature of this core is not more than 30 MeV, which means the core is relatively cold.

At the next stage, the trapped energy and lepton number both escape from the core, carried by the least interacting particles, which in the standard model are neutrinos. Neutrinos and antineutrinos of all flavors are emitted in a time span of a few seconds (their diffusion time). The resulting central object then settles to a neutron star, or a black hole. A tremendous amount of energy, some $10^{53}$ ergs, is released in $10^{58}$ neutrinos with energies $\sim 10$ MeV. A fraction of this energy is absorbed by beta reactions into the material behind the shock wave that then blasts away the rest of the star, creating, in many cases, a spectacular explosion. Yet, from the energetics point of view, this visible explosion is but a tiny perturbation on the total event. Over 99% of all gravitational binding energy of the $1.4M_{\odot}$ collapsed core – some 10% of its rest mass – is emitted in neutrinos.

The core-collapse neutrino signal starts with a short, sharp “neutronization” (or “break-out”) burst primarily composed of $\nu_{e}$ from $e^{-}+p\rightarrow\nu_{e}+n$. These neutrinos are messengers of the shock front breaking through the neutrinosphere (the surface of neutrino trapping): when this happens, iron is disintegrated, the neutrino scattering rate drops and the lepton number trapped just below the original neutrinosphere is suddenly released. This quick and intense burst is followed by an “accretion” phase lasting some hundreds of milliseconds, depending on the progenitor star mass, as matter falls onto the collapsed core and the shock is stalled at the distance of $\sim 200$ km. The gravitational binding energy of the accreting material is powering the neutrino luminosity during this stage. The later “cooling” phase over $\sim$10 seconds represents the main part of the signal, over which the proto-neutron star sheds its trapped energy.

The flavor content and spectra of the neutrinos emitted from the neutrinosphere change throughout these phases, and the supernova’s evolution can be followed with the neutrino signal.

The physics of neutrino decoupling and spectra formation is far from trivial, owing to the energy dependence of the cross sections and the roles played by both CC and neutral-current (NC) reactions. Detailed transport calculations using methods such as MC or Boltzmann solvers have been employed. It has been observed that flux spectra coming out of such simulations can typically be parameterized at a given moment in time by the following ansatz (e.g., Minakata:2008nc ; Tamborra:2012ac ):

where $E_{\nu}$ is the neutrino energy, $\langle E_{\nu}\rangle$ is the mean neutrino energy, $\alpha$ is a “pinching parameter”, and $\mathcal{N}$ is a normalization constant related to the total luminosity. Large $\alpha$ corresponds to a more “pinched” spectrum (suppressed tails at high and low energy). This parameterization is referred to as a “pinched-thermal” form. The different $\nu_{e}$, $\overline{\nu}_{e}$ and $\nu_{x}~(x=\mu,\tau,\bar{\mu},\bar{\tau}$) flavors are expected to have different average energy and $\alpha$ parameters and to evolve differently in time.

The initial spectra get further processed by flavor transitions, and understanding these oscillations is very important for extracting physics from the detected signal (see Sec. 2.4.1).

In general, one can describe the neutrino flux as a function of time by specifying the three pinching parameters in successive time slices. Figure 1 gives an example of pinching parameters as a function of time for a specific model, and Fig. 2 shows the spectra for the three flavors as a function of time corresponding to this parameterized description. We have verified that the time-integrated spectrum for each flavor is expected to be reasonably well approximated by the pinched-thermal form as well.

A number of astrophysical phenomena associated with supernovae are expected to be observable in the supernova neutrino signal, providing a remarkable window into the event. In particular, the supernova explosion mechanism, which in the current paradigm involves energy deposition into the stellar envelope via neutrino interactions, is still not well understood, and the neutrinos themselves will bring the insight needed to confirm or refute the paradigm.

There are many other examples of astrophysical observables:

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  The initial burst, primarily composed of $\nu_{e}$ and called the “neutronization” or “breakout” burst, represents only a small component of the total signal. However, flavor transition effects can manifest themselves in an observable manner in this burst, and flavor transformations can be modified by the “halo” of neutrinos generated in the supernova envelope by scattering Cherry:2013mv .

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  The formation of a black hole would cause a sharp signal cutoff (e.g., Beacom:2000qy ; Fischer:2008rh ; Li:2020ujl ).

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  Shock wave effects (e.g., Schirato:2002tg ) would cause a time-dependent change in flavor and spectral composition as the shock wave propagates.

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  The standing accretion shock instability (SASI) Hanke:2011jf ; Hanke:2013ena , a “sloshing” mode predicted by three-dimensional neutrino-hydrodynamics simulations of supernova cores, would give an oscillatory flavor-dependent modulation of the flux.

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  Turbulence effects Friedland:2006ta ; Lund:2013uta would also cause flavor-dependent spectral modification as a function of time.

The supernova neutrino burst is prompt with respect to the electromagnetic supernova signal and therefore can be exploited to provide an early warning to astronomers Antonioli:2004zb ; Scholberg:2008fa . Note that not every core collapse will produce an observable supernova, and observation of a neutrino burst in the absence of an electromagnetic event would be very interesting.

Even non-observation of a burst, or non-observation of a $\nu_{e}$ component of a burst in the presence of supernovae (or other astrophysical events) observed in electromagnetic or gravitational wave channels, would still provide valuable information about the nature of the sources. Furthermore, a long-timescale, sensitive search yielding no bursts will also provide limits on the rate of core-collapse supernovae.

Observation of a supernova neutrino burst in coincidence with gravitational waves (which would also be prompt, and could indeed provide a time reference for a time-of-flight analysis) would be especially interesting Arnaud:2003zr ; Ott:2012jq ; Mueller:2012sv ; Nishizawa:2014zna .

The better one can understand the astrophysical nature of core-collapse supernovae, the easier it will be to extract information about particle physics.

A core-collapse supernova is essentially a gravity-powered neutrino bomb: the energy of the collapse is initially stored in the Fermi seas of electrons and neutrinos and then gradually leaked out by neutrino diffusion. The key property of neutrinos that makes them play such a dominant role in the supernova dynamics is the feebleness of their interactions. It then follows that should there be new light ($<100$ MeV) particles with even weaker interactions, these could alter the energy transport process and the resulting evolution of the nascent proto-neutron star. Moreover, additional interactions or properties of neutrinos could also be manifested in this way.

Thus, a core-collapse supernova can be thought of as an extremely hermetic system, which can be used to search for numerous types of new physics (e.g., Schramm:1990pf ; Raffelt:1999tx ). The list includes various Goldstone bosons (e.g., Majorons), neutrino magnetic moments, new gauge bosons (“dark photons”), “unparticles”, and extra-dimensional gauge bosons. The existing data from SN1987A already provide significant constraints on these scenarios by confirming the basic energy balance of the explosion. At the same time, more precision is highly desirable and will be provided with the next galactic supernova.

Such energy-loss-based analysis will make use of two types of information. First, the total energy of the emitted neutrinos should be compared with the expected release in the gravitational collapse. Note that measurements of all flavors, including $\nu_{e}$, are needed for the best estimate of the energy release. Second, the rate of cooling of the protoneutron state should be measured and compared with what is expected from diffusion of the standard neutrinos. The detection of a supernova neutrino burst also allows the exploration of corrections to the neutrino velocity that could arise due to violations of Lorentz invariance km2012 .

The flavor transition physics and its signatures are a major part of the physics program. Compared to the well-understood case of solar neutrinos, in a supernova, neutrino flavor transformations are much more involved. For supernovae, there are both neutrinos and antineutrinos, and the density profile is such that both mass splittings—“solar” and “atmospheric” — have an effect on the neutrino propagation. While flavor transitions can be reasonably well understood during early periods of the neutrino emission as standard Mikheyev-Smirnov-Wolfenstein (MSW) transitions in the varying density profile of the overlying material, during later periods the physics of the transformations is significantly richer. For example, several seconds after the onset of the explosion, the flavor conversion probability is affected by the expanding shock front and the turbulent region behind it. The conversion process in such a stochastic profile is qualitatively different from the adiabatic MSW effect in the smooth, fixed density profile of the Sun Mirizzi:2015eza .

Even more complexity is brought about by the coherent scattering of neutrinos off each other. This neutrino “self-refraction” results in highly nontrivial flavor transformations close to the neutrinosphere, typically within a few hundred kilometers from the center, where the density of streaming neutrinos is very high. Since the evolving flavor composition of the neutrino flux feeds back into the oscillation Hamiltonian, the problem is nonlinear. Furthermore, as the interactions couple neutrinos and antineutrinos of different flavors and energies, the oscillations are characterized by “collective” modes. This complexity leads to very rich physics that has been the subject of intense interest over the last decade and a voluminous literature exists exploring these collective phenomena, e.g., Duan:2005cp ; Fogli:2007bk ; Raffelt:2007cb ; Raffelt:2007xt ; EstebanPretel:2008ni ; Duan:2009cd ; Dasgupta:2009mg ; Duan:2010bg ; Duan:2010bf ; Wu:2014kaa . This is an active theoretical field and the effects are not yet fully understood. A supernova burst is the only opportunity to study neutrino-neutrino interactions experimentally.

Active-sterile neutrino transitions may also have observable effects Peres:2000ic ; Esmaili:2014gya ; Tang:2020pkp .

The new effects can imprint information about the inner workings of the explosion on the signal. The flavor transitions can modulate the characteristics of the signal (both event rates and spectra as a function of time). In particular, the flavor transitions can imprint distinctive non-thermal features on the energy spectra, potentially making it possible to disentangle the effects of flavor transformations and the physics of neutrino spectra formation. This in turn should help us learn about the development of the explosion during the crucial first 10 seconds.

In the world’s current supernova neutrino flavor sensitivity portfolio Scholberg:2012id ; Mirizzi:2015eza , the sensitivity is primarily to $\bar{\nu}_{e}$ flavor, via inverse beta decay on free protons in water and scintillator detectors worldwide (Ikeda:2007sa ; Abe:2016waf ; Abbasi:2011ss ; Eguchi:2002dm ; Agafonova:2014leu ; Monzani:2006jg ; NOvA:2020dll ; Wei:2015qga .) There is only minor sensitivity to the $\nu_{e}$ component of the flux, via elastic scattering on electrons (ES) and subdominant interaction channels on nuclei, as well as in small detectors (HALO Duba:2008zz and MicroBooNE Abratenko:2020hfy .) However $\nu_{e}$ statistics are small and and it can be difficult to disentangle the flavor content. The $\nu_{e}$ component carries with it particularly interesting information (e.g., neutronization burst neutrinos are created primarily as $\nu_{e}$.)

The next generation of supernova neutrino detectors will be dominated by Hyper-Kamiokande Abe:2018uyc , JUNO An:2015jdp and DUNE. Hyper-K and JUNO are sensitive primarily to $\bar{\nu}_{e}$, and Hyper-K in particular will have potentially enormous statistics. The next-generation long-string water detectors, IceCube Gen-2 Aartsen:2014njl and KM3NeT Adrian-Martinez:2016fdl , will bring improved burst timing. New tens-of-ton scale noble liquid detectors such as DARWIN Aalbers:2016jon will bring new all-flavor sensitivity via NC coherent elastic neutrino-nucleus scattering. To this landscape, DUNE will bring unique $\nu_{e}$ sensitivity via $\nu_{e}$ charged-current ($\nu_{e}$CC) interactions on argon nuclei. It will offer a new opportunity to measure the $\nu_{e}$ content of the burst with high statistics and good event reconstruction.

The past decade has also brought rapid evolution of multi-messenger astronomy. With the advent of the detection of gravitational waves as well as high-energy extragalactic neutrino detection in IceCube, a broad community of physicists and astronomers are now collaborating to extract maximum information from observation in a huge range of electromagnetic wavelengths, neutrinos, charged particles and gravitational waves. This collaboration resulted in the spectacular multi-messenger observation of a kilonova kilonova . The next core-collapse supernova will be potentially an even more spectacular multi-messenger observation. Worldwide neutrino detectors are currently participants in SNEWS, the SuperNova Early Warning System snews , which will be upgraded to have enhanced capabilities over the next few years Kharusi:2020ovw . Information from DUNE will enhance the SNEWS network’s reach.

Neutrino pointing information is vital for prompt multi-messenger capabilities. Only some supernova neutrino detectors have the ability to point back to the source of neutrinos. Imaging water Cherenkov detectors like Super-K can do well at this task via directional reconstruction of neutrino-electron ES events. However, other detectors lack pointing ability due to intrinsic quasi-isotropy of the neutrino interactions, combined with lack of detector sensitivity to final-state directionality. Like Super-K, DUNE is capable of pointing to the supernova via the good tracking ability of its time projection chamber (TPC.)

Supernova neutrino detection is more of a collaborative than a competitive game. The more information gathered by detectors worldwide, the more extensive the knowledge to be gained; the whole is more than the sum of the parts. The flavor sensitivity of DUNE is highly complementary to that of the other detectors and will bring critical information for reconstruction of the entire burst’s flavor and spectral content as a function of time Ankowski:2016lab .

While a core-collapse burst is a known source of low-energy ($<$100 MeV) neutrinos, there are other potential interesting sources of neutrinos in this energy range. Nearby Type Ia  Wright:2016gar ; Wright:2016xma or pair instability supernova Wright:2017zyq events may create bursts as well, although they are expected to be fainter in neutrinos than core-collapse supernovae. Mergers of binary neutron stars and of neutron stars and black holes will be low-energy neutrino sources Caballero:2009ww ; Kyutoku:2017wnb , although the rate of these close enough to detect (i.e., within the Galaxy) will be small. There are also interesting steady-state sources of low-energy neutrinos — in particular, there may still be useful oscillation and solar physics information to extract via measurement of the solar neutrino flux. DUNE will have the unique capability of measuring solar neutrino energies event by event with $\nu_{e}$CC interactions with large statistics, in contrast to other detectors, which primarily make use of recoil spectra Capozzi:2018dat ; Ioannisian:2017dkx . The technical challenge for solar neutrinos is overcoming radiological and cosmogenic backgrounds, although preliminary studies are promising. The diffuse supernova background neutrinos Moller:2018kpn are another interesting target; these have higher energy than solar neutrinos, but are very challenging due to very low event rate. There may also be surprises in store, both from burst and steady-state signals, enabled by unique DUNE liquid argon tracking technology.

Liquid argon has a particular sensitivity to the $\nu_{e}$ component of a supernova neutrino burst, via the dominant interaction, CC absorption of $\nu_{e}$ on ⁴⁰Ar,

for which the observable is the $e^{-}$ plus deexcitation products from the excited ⁴⁰K^∗ final state. Additional channels include a $\bar{\nu}_{e}$ CC interaction and ES on electrons. Cross sections for the most relevant interactions are shown in Fig. 3. It is worth noting that none of the neutrino-⁴⁰Ar cross sections in this energy range have been experimentally measured, although several theoretical calculations exist marley ; marley2 ; snowglobes . The uncertainties on the theoretical calculations are not generally quantified, and they may be large.

Another process of interest for supernova detection in liquid argon detectors, not yet fully studied, is NC scattering on Ar nuclei by any type of neutrino: $\nu_{X}+{\rm Ar}\rightarrow\nu_{X}+{\rm Ar}^{*}$, for which the observable is the cascade of deexcitation gammas from the final state Ar nucleus. A dominant 9.8-MeV Ar^∗ decay line has been recently identified as a spin-flip M1 transition Hayes . At this energy the probability of $e^{+}e^{-}$ pair production is relatively high, offering a potentially interesting NC tag. Other transitions are under investigation. NC interactions are not included in the studies presented here, although they represent a topic of future investigation.

The predicted event rate from a supernova burst may be calculated by folding expected neutrino flux differential energy spectra with cross sections for the relevant channels, and with detector response; this is done using SNOwGLoBES snowglobes  (see Sec. 5.3.1.)

Supernova neutrino events, due to their low energies, will manifest themselves primarily as spatially small events, perhaps up to a few tens of cm scale, with stub-like tracks from electrons (or positrons from the rarer $\bar{\nu}_{e}$ interactions). Events from $\nu_{e}$CC, $\nu_{e}+{}^{40}{\rm Ar}\rightarrow e^{-}+{}^{40}{\rm K}^{*}$, are likely to be accompanied by de-excitation products — gamma rays and/or ejected nucleons. Gamma rays are in principle observable via energy deposition from Compton scattering, which will show up as small charge blips in the time projection chamber. Gamma rays can also be produced by bremsstrahlung energy loss of electrons or positrons. The critical energy for bremsstrahlung energy loss for electrons in argon is about 45 MeV. Ejected nucleons may result in loss of observed energy for the event, although some may interact to produce observable deexcitations via inelastic scatters on argon. Such MeV-scale activity associated with neutrino interactions has been observed in the ArgoNeuT LArTPC PhysRevD.99.012002 . ES on electrons will result in single scattered electron tracks, and single or cascades of gamma rays may result from NC excitations of the argon nucleus. Each interaction category has, in principle, a distinctive signature. Figure 4 shows examples of simulated $\nu_{e}$CC and neutrino-electron ES interactions in DUNE.

The canonical event reconstruction task is to identify the interaction channel, the neutrino flavor for CC events, and to determine the four-momentum of the incoming neutrino; this overall task is the same for low-energy events as for high-energy ones. The challenge is to reconstruct the properties of the lepton (if present), and to the extent possible, to tag the interaction channel by the pattern of final-state particles. LArSoft larsoft.org open-source event simulation and reconstruction software tools for low-energy events is employed; a full description of the algorithms is beyond the scope of this work. Performance is described in Sec. 5.2.2. Enhanced tools are under development, for example for interaction channel tagging; however, standard tools already provide reasonable capability for energy reconstruction and tracking of low-energy events. Event reconstruction in this energy range has been demonstrated by MicroBooNE for Michel electrons Acciarri:2017sjy .

Given the rarity of a supernova neutrino burst in our galactic neighbourhood and the importance of its detection, it is essential to develop a redundant and highly efficient triggering scheme in DUNE. In DUNE, the trigger on a supernova neutrino burst can be done using either TPC or photon detection system information. In both cases, the trigger scheme exploits the time coincidence of multiple signals over a timescale matching the supernova luminosity evolution. Development of such a data acquisition and triggering scheme is a major activity within DUNE and will be the topic of future dedicated publications. Both TPC and PD information can be used for triggering, for both SP and DP. Here are described two concrete examples of preliminary trigger design studies. Note that the general strategy will be to record data from all channels over a 30-100 second period around every trigger Abi:2020loh , so that the individual event reconstruction efficiency as described in Sec. 5.2.2 will apply for physics performance.

The first example is a trigger based on the photon detection system of the DP module. A real-time algorithm should provide trigger primitives by searching for photomultiplier hits and optical clusters, where the latter combines several hits together based on their time/spatial information. According to simulations, the optimal cluster reconstruction parameters yield a 0.05 Hz radiological background cluster rate for a supernova $\nu_{\text{e}}$CC signal cluster efficiency of 11.8%. Once the optimal cluster parameters are found, the computation of the supernova neutrino burst trigger efficiency is performed using the minimum cluster multiplicity. This value, set by the radiological background cluster rate and the maximum fake trigger rate (one per month), is $\geq$3 in a 2-second window (time in which about half of the events are expected). Approximately 3/0.118$\simeq$25 interactions must occur in the active volume to obtain approximately 45% trigger efficiency while maintaining a fake trigger rate of one per month.

The triggering efficiency as a function of the number of supernova neutrino interactions is shown in Fig. 13. At 20 kpc, the edge of the Galaxy, about 80 supernova neutrino interactions in the 12.1-kton active mass (assumed supernova-burst-sensitive mass for a single DP module) are expected (see Fig. 12). Therefore, the DP photon detection system should yield a highly efficient trigger for a supernova neutrino burst occurring anywhere in the Milky Way.

The second example considered is a TPC-based supernova neutrino burst trigger in a SP module (SP photon-based triggering will be considered in a future study). Such a trigger considering the time coincidence of multiple neutrino interactions over a period of up to 10 seconds yields roughly comparable efficiencies. Figure 14 shows efficiencies for supernova bursts obtained in this way for a DUNE SP module and for supernova bursts with an energy and time evolution as shown in Fig. 1. Triggering using TPC information is facilitated by a multi-level data selection chain whereby ionization charge deposits are first selected on a per wire basis, using a threshold-based hit finding scheme. This results in low-level trigger primitives (hit summaries) which can be correlated in time and channel space to construct higher-level trigger candidate objects. Low-energy trigger candidates, each consistent with the ionization deposition due to a single supernova neutrino interaction, subsequently serve as input to the supernova burst trigger. Simulations demonstrate that the trigger candidate efficiency for any individual supernova burst neutrino interaction is on the order of 20-30%; see Fig. 14. However, a multiplicity-based supernova burst trigger that integrates low-energy trigger candidates over $\sim$10 s integration window yields high trigger efficiency out to the galactic edge while keeping fake supernova burst trigger rates due to noise and radiological backgrounds to the required level of one per month or less.

An energy-weighted multiplicity count scheme further increases efficiency and minimizes fake triggers due to noise and/or radiological backgrounds. This effect is illustrated in Fig. 14, where a nearly 100% efficiency is possible out to the edge of the galaxy, and 70% efficiency is possible for a burst at the Large Magellanic Cloud (or for any supernova burst creating $\sim$10 events). This performance is obtained by considering the summed-waveform digitized-charge distribution of trigger candidates over 10 s and comparing to a background-only vs. background-plus-burst hypothesis. The efficiency gain compared to a simpler, trigger candidate counting-based approach is significant; using only counting information, the efficiency for a supernova burst at the Large Magellanic Cloud is only 6.5%. These algorithms are being refined to further improve supernova burst trigger efficiency for more distant supernova bursts. Alternative data selection and triggering schemes are also being investigated, involving, e.g., deep neural networks implemented for real-time or online data processing in the DAQ bib:docdb11311 .

Timing for supernova neutrino events is provided by both the TPC and the photon detector system. Basic timing requirements are set by event vertexing and fiducialization needs. Here we note a few supernova-specific design considerations. During the first 50 ms of a 10-kpc-distant supernova, the mean interval between successive neutrino interactions is $0.5-1.7~\rm{ms}$ depending on the model. The TPC alone provides a time resolution of 0.6 ms (corresponding to the drift time at 500 V/cm), commensurate with the fundamental statistical limitations at this distance. However nearly half of galactic supernova candidates lie closer to Earth than this, so the rate can be tens or (less likely) hundreds of times higher. A resolution of $\mathord{<}1~\mu\rm{s}$, as already provided by the photon detector system, ensures that DUNE’s measurement of the neutrino burst time profile is always limited by rate and not detector resolution. The hypothesized oscillations of the neutrino flux due to standing accretion shock instabilities would lead to features with a characteristic time of $\sim$10 ms, comfortably greater than the time resolution. The possible neutrino “trapping notch” (dip in luminosity due to trapping of neutrinos in the stellar core) right before the start of the neutronization burst has a width of $1-2~\rm{ms}$. Identifying the trapping notch could be possible for the closest supernovae (few kpc).