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Neutrino Spectrum from SN 1987A andfrom CosmicSupernovae Hasan Yu¨ksel1,2 and John F. Beacom1,3,2 1Department of Physics, Ohio State University, Columbus, Ohio 43210 2CenterforCosmologyandAstro-ParticlePhysics, OhioStateUniversity, Columbus, Ohio43210 3Department of Astronomy, Ohio State University, Columbus, Ohio 43210 (Dated:18October2007) ThedetectionofneutrinosfromSN1987A bytheKamiokande-II andIrvine-Michigan-Brookhaven detectors provided the first glimpse of core collapse in a supernova, complementing the optical observations and confirmingourbasicunderstandingofthemechanismbehindtheexplosion. Onelong-standingpuzzleisthat, whenfittedwiththermalspectra,thetwoindependentdetectionsdonotseemtoagreewitheithereachotheror typicaltheoreticalexpectations. Weassessthecompatibilityofthetwodatasetsinamodel-independent way andshowthattheycanbereconciledifoneavoidsanybiasontheneutrinospectrumstemmingfromtheoretical conjecture.WereconstructtheneutrinospectrumfromSN1987Adirectlyfromthedatathroughnonparametric 8 inferentialstatisticalmethodsandpresentpredictionsfortheDiffuseSupernovaNeutrinoBackgroundbasedon 0 SN1987Adata. Weshowthatthispredictioncannotbetoosmall(especiallyinthe10-18MeVrange),since 0 themajorityofthedetectedeventsfromSN1987A wereabove18MeV(including6eventsabove35MeV), 2 suggestinganimminentdetectioninoperationalandplanneddetectors. n a PACSnumbers:97.60.Bw,98.70.Vc,95.85.Ry,14.60.Pq J 1 2 I. INTRODUCTION Theprincipalfocusofourstudyistoassessthecompatibil- ity of the Kam-IIand IMB data sets in a model-independent 3 Core-collapsesupernovaexplosionsarehowmassivestars fashion,andtostudytheimmediateimplicationsforneutrino v endtheirlives[1]. Oncetheirnuclearfuelisexhausted,they detection from Galactic and cosmic supernovae. We begin 3 producea neutronstar or blackhole, recycletheirashesinto with a brief formulationof neutrinodetection froma nearby 1 cosmic dust to seed subsequent starbursts and spread their supernova and summarize the main ingredients. Next, we 6 heavy elements[2, 3, 4], andflood the universewith a burst 2 determine the relative prospects of detecting neutrinos from ofneutrinos. Supernovaexplosionsareveryprolificneutrino 0 SN 1987A in either Kam-II or IMB, based on their reported producers,since the enormousamountof gravitationalbind- 7 energy-dependentefficiencies and the numbers of targets in 0 ing energy, liberated when the core of a massive star col- their fiducial volumes. We compare this relative probability / lapses, caneffectivelybedisseminatedonlybyneutrinos. In h of detection to the actual detected positrons for Kam-II and simulations, the rebounding collapse of the core often pro- p IMBandconcludethatthereisnosignificantdiscrepancybe- duces a shock that stalls, which may indicate the necessity - tweenthedatasetsunlessanapriorispectralshapeisforced o ofshockrevivalbyenergytransferfromtheintenseneutrino to fit the two data sets simultaneously. We establish that the r flux[5,6,7,8,9,10,11,12,13]. Whileneutrinoscanbecru- t Kam-IIandIMBdetectorsmainlyprobeddifferentenergydo- s cial for diagnosing a successful explosion and revealing the a mains,theformertestingthelowenergypartofthespectrum conditionsofthecore,theirelusivenaturemakestheirdetec- : and the latter testing the high energy part, and, in fact, only v tionchallenging. a propercombinationof Kam-IIorIMBdatasets providesa i X The detection of neutrinos from SN 1987A in the Large sensitiveprobeoverallenergyranges. r Magellanic Cloud (at a distance of D 50 kpc) by Using nonparametric(distribution-free)inferentialstatisti- a ∼ the Kamiokande-II (Kam-II) [14, 15] and Irvine-Michigan- calmethods[20],wedeterminethestructureoftheunderlying Brookhaven (IMB) [16, 17] detectors transformed super- spectrum directly from the data. Since these methods make nova physics from purely hypothesis- and simulation-driven noaprioriassumptionsanddonotrelyonparameterestima- to discovery-driven science. One puzzling feature of the tion, they allow for more efficient processing of small data SN 1987A data is that the neutrinos detected by the IMB samples. Then, we present the incoming neutrino spectrum detector were seemingly more energetic than those detected inferred directly from the data, which would be an effective bytheKam-IIdetector,whichwereclusteredatlowenergies. spectrumreceivedon Earth after effectsmodifyingthe over- Despitetheuncertaintiesassociatedwiththelowstatisticsand allshape are taken into account. Finally, we providepredic- the differencesof the detector properties(like efficiencyand tions for the Diffuse Supernova Neutrino Backgroundbased size),itseemshardtoaccommodatethetwodetectionswitha on the SN 1987A data (as Refs. [21, 22] have also recently commonquasi-thermalneutrinospectrum [18]. This incom- donethroughmoreconventionalmethods),whichturnoutto patibility between the data sets and the theory can be allevi- benearlyaslargeastypicalpredictionsintheliteraturebased atedbyassumingthatatleastoneexperimenthadananoma- on supernova models. Our study follows a transparent ap- lousstatisticalfluctuationorunderestimatedsystematics[19]. proach to extract the targeted information directly from the Here,weassumethatbothexperimentsobservedstatistically SN1987Adata, skippingtheintermediatestagesofmultipa- probable outcomes, as they were exposed to the same spec- rameterfunctionfitting,commonlyusedinthepastliterature. trum,andacceptthereportedevents,uncertaintiesandoverall The methods we present avoid over-interpretationof limited efficienciesatfacevalue. data,whilestillallowingforasubstantialanalysis. 2 II. DETECTIONOFSUPERNOVANEUTRINOS threshold. Kam-II had a low threshold of E 7.5 MeV thr ∼ and its efficiency quickly rose, starting at 5 MeV, to its ∼ Weconcentrateontheν¯ fluxreceivedonEarthfromasu- asymptotic value. This is because the efficiency function e pernovasincethedetectorsareparticularlysensitivetothein- around the threshold energy is non-trivial due to the proba- versebetadecay,ν¯ +p n+e+(weignorepossiblesmall bilisticnatureofthedetectionprocess(i.e.,inprinciple,alow e contributionsdueν +e−→orν+16O). We parametrizethe ν¯ energyneutrinocouldtriggera greaternumberofphotomul- e numberspectrumasφ(E ) = [L /E ]ϕ(E ),whereϕ(E ) tipliers and be detected). At the time of SN 1987A, almost ν ν 0 ν ν is a normalized function ( ϕ(E )dE = 1) describing the a quarter of the photomultipliers in the IMB detector were ν ν overall spectral shape withR no prior assumptions. Here E offline due to a power failure, resulting in a distinct energy 0 is the average energy ( E = E ϕ(E )dE = E ) and dependenceduetothegeometricalmodifications. Thus,IMB ν ν ν ν 0 Lν isthetime-integratehdluiminoRsity(i.e.,thetotalnumberof hadahigherthresholdofEthr ∼19MeVandrosegradually, neutrinos is L /E ) in this flavor. In the energy regime of startingat 16MeV.Atlowenergies,theyieldswillbedom- ν 0 ∼ interest, the crosssection of inversebeta decayat the lowest inatedbyKam-II,duetoitsmuchgreaterefficiency,whileat ordersininversenucleonmass,1/M,is highenergies,theyieldswillbedominatedbyIMB,duetoits greatersize. σ(E ) 9.5 10−44cm2(1 6E /M)[(E ∆)/MeV]2, ν ν ν ≃ × − − (1) expressed in terms of the neutrino energy,E [23]. The de- 5 ν tected positron energy (visible energy), E , can be related + t∆oordt=ehrecMionrcnroe−mctiinMognpsnaiesruettrhmienuoncuhecnslemeroganyllemarsa:tshsEand+iuff=necreeErntνcaei−n(tit∆ehse,ahwsisghohecerie-r 3210 ] 43 Combined atedwithenergyresolution).Thepositronspectruminagiven ) [ IM B detector, as a functionof visible energyfor a supernovaat a E+2 distanceD,canbecastas N ( 1 Kam-II N(E ) Φ(E+)= 4πD+2 σ(Eν)φ(Eν), (2) 0 1 wheretheeffectivenumberoftargetsavailableinthedetector isN(E+) = Ntη(E+)asafunctionofvisible(positron)en- 0.8 K weregiyg,hitnedwwhiicthhNthtesutapn-tdimsfeofrrtahcetinounmobfethreofdferteeec-toprroatnodnηta(rEge+t)s ( E)+0.6 am-II IMB isthedetectorefficiencyfunction. f 0.4 0.2 III. COMPATIBILITYOFTHEDATASETS 0 12 In the top panel of Fig. 1, we present the effective number of targets available in the Kam-II and IMB detectors by 10 normalizingthe targetsin the fiducialvolumeof each detec- 8 ] tor by the up-time and reported energy-dependent efficien- [ s 6 cies[24, 25], N(E+) = Ntη(E+). The combinedeffective t 4 number of targets for both detectors together is also shown (in which they are treated as one big detector with energy- 2 dependentefficiency,coveringthefullenergyrangesampled 0 bythedata).Thenumberoffree-protontargetsinthefiducial 0 10 20 30 40 50 60 volume of each detector was Nt 1.43 1032 for Kam-II E [ MeV ] andN 4.55 1032forIMB[24≃].Desp×itebeing 3times + t ≃ × ∼ largerin size, IMBtypically had lower efficiency,especially FIG.1: Top Panel: Theeffectivenumber of targets(freeprotons) atlowerenergies.Moreover,theIMBdetectorwasbrieflyof- available in the Kam-II and IMB detectors and their combination flinewheneveramuonpassedthroughthedetector,reducing as a function of the visible (positron) energy E+ (i.e., protons in theaverageup-timeby 13%[18]. Weemphasizethatwhile thefiducialvolumeofeachdetectornormalizedbytheup-timeand ∼ all previousanalyses did take the separate energy-dependent energy-dependent efficiencies). MiddlePanel: Thefractionof to- efficienciesofKam-IIandIMBintoaccount,ourapproachof taldetectedeventsexpectedtobeassignedtoeitherKam-IIorIMB consideringaneffectivecombineddetectorwithamorecom- (independent of incoming neutrino spectrum) as a function of the plicatedenergy-dependentefficiencyisnew. visibleenergy. BottomPanel:PositronsdetectedduringSN1987A NotethattheefficiencyofKam-II(andmostmodernneu- byKam-II(11filled-diamonds)andIMB(8open-circles)withtheir trino detectors) is basically a step function over the detector recordedenergies(Ei)andexperimentaluncertainties(∆Ei). 3 When both the efficiencies and numbers of target protons Considering the low statistics and large experimental un- inthefiducialvolumesofthetwodetectorsaretakenintoac- certaintiesinvolved,itseemsthatthereisnoobviousconflict countsimultaneously,itbecomesapparentwhyKam-IIregis- between the distribution of detected positronsin the two ex- ters all the positrons below 16 MeV, while IMB is expected periments,suggestingthepossibilityofreconcilingtheKam- todominateover30MeV.Aninformativeperspectiveincom- IIandIMBdatasetswithasuitableincomingneutrinospec- paring the effectiveness of Kam-II and IMB is presented in trum. It is evident that while each experiment may suggest the middle panel of Fig. 1. This shows the fraction of total aradicallydifferentincomingneutrinospectrumindividually, positronsexpectedtoberecordedbyeitheroftheexperiments since the signal will be sampled distinctly by the detectors, (i.e., the ratio of effectivenumberoftargetsin each detector onlytheirpropercombinationcouldprovideasensitiveprobe to the combined number of targets) as a function of visible over all energy ranges. In order to minimize the impact of energy. Note that this is independentof the shape of the in- statisticsduetothesmallsample, wewillpresentoursubse- coming neutrino spectrum or the cross section, since it only quentresultswithanemphasisonthecombineddata. compares the effectiveness of the detectors relative to each other. One can directly compare this information to the ac- tualpositronsdetectedduringSN1987Ainordertoassessthe IV. EXPECTEDNEUTRINOSPECTRUM compatibilityofthedetectors,withouteverreferringtotheac- tualreceivedspectrumofneutrinos. Asexplainedbelow,we Neutrinosareproducedthermallyfromtheplasmaandre- areassumingacommonincomingneutrinospectrum. main trapped inside the collapsed core of a massive star un- We summarize the visible energies (E ) and experimen- til their optical depth becomes unity around the correspond- i tal uncertainties (∆E ) of positrons detected at the time of ingneutrino-sphereofeachflavor. Thentheystreamthrough i SN 1987A in Kam-II (11 data points marked with filled- theenvelope,preservingtheirinitialenergydistribution. The diamonds)and IMB (8 data pointmarkedwith open-circles) average energy at free streaming is dictated by the temper- in the bottom panelof Fig. 1. We exclude the Baksan Scin- ature at the neutrino-sphere, which is in turn determined by tillatorTelescopedata[26,27],whichhasmuchlargeruncer- the strength of the coupling of neutrinos to the matter. One tainties associated with backgrounds, from our main results expects a hierarchy of energies, E < E < E , h νei h ν¯ei h νµ,µ¯i butcommentonpossibleimpactlater. Datasetsaresynchro- sincetheelectronneutrinoflavorenjoysbothchargedandneu- nized based on the arrival time of the first events (note that tralcurrentinteractions(andcouplestomattermoststrongly), we limit our study to time-integrated neutrino flux). While while the non-electron flavors feel a weaker attachment to Kam-IIoriginallyreported12signalevents,oneofthem,be- the plasma. The time-integrated luminosity and the hierar- ing attributed to background, is excluded from our analysis, chy of energiesamong neutrinoflavors from a core-collapse asinpreviousstudies.Thevisibleenergyassignedtoeachde- supernovaarestillnotwellknownanddiversevaluesarere- tectedeventisbasedonthenumberoftriggeredphotomulti- portedby modelers[5, 6, 7, 8, 9, 10, 11, 12, 13]. While the pliersfromtheCˇerenkovlightofthepositrons.Thestatistical supernovaneutrinospectraare expectedto be quasi-thermal, uncertaintyonthisvisibleenergyisproportionaltothesquare modificationsdueto non-standardeffects, like neutrinomix- rootofthe totalnumberofhitsandis shownbya horizontal ing among various flavors [28, 29, 30, 31, 32, 33, 34, 35], bar. neutrinodecay[38,39,40,41,42],neutrino-neutrinointerac- Interestingly,thedistributionsofthedetectedpositronsbe- tions[43,44,45,46],additionalchannelsofenergyexchange tweenKam-IIandIMB,asshowninthebottompanel,quan- betweenflavors[5,47,48,49],and/oranyothernovelmecha- titativelyagreesquitewellwithourexpectationsbasedonthe nismduetounknownphysics,mayproduceatime-integrated middle panel of Fig. 1. While Kam-II dominated the detec- spectrumreceivedonEarththatdeviatessignificantlyfroma tionsbelow16MeV,theeventsbetween16–30MeVineach quasi-thermalshape. detectorarecomparableasonewoulddeducefromthemiddle Manypreviousstudies dealingwith the SN 1987Aneutri- panel.Atevenhigherenergies(wherethedirectcomparisonis nosadoptatemplateneutrinospectrumandtrytoextractpa- possible),IMBdetected5eventswhileKam-IIdetectedonly rametersdescribingthisspectrumfromtheKam-IIandIMB 1ti.caIfllwyepraocbcaebpltethoauttcthoemIeM,sBinocbestehrevaetfifoenctoivfe5neuvmenbtesriosfatsatragteists- danatda.FeBromthi-DMiraaxcwseplle-cBtroaltz[m51a]n,nϕ[(5E0ν],)ϕ∝(EνE)ν2∝/[eEEνν2/eT−E+ν/1T],, inIMBisalmosttwicetothatofKam-IIattheseenergies,one require only a single temperature parameter, T. An addi- wouldexpecttosee 2.5eventsinKam-II.Withthisexpec- tional degeneracy parameter, η, is used to describe spec- tation,theprobability∼ofseeing<2eventsisover50%,sothat tral pinching in a Fermi-Dirac distribution [52], ϕ(E ) ν theobservationofonlyasingle∼eventintheKam-IIsampleat E2/[eEν/T−η + 1]. More recently, a quasi-thermal distr∝i- ν highenergiesisprobablysimplyadownwardfluctuation,and bution (based on a gamma distribution), is suggested [5], notindisagreementwiththe IMBsample. Instead,if we as- ϕ(Eν) ∝ Eνγe−(γ+1)Eν/E0, which can handle anti-pinched sume that the Kam-II observationof 1 event is correct, then (broader) spectra, as well as pinched ones, through the pa- onewouldexpecttosee 2eventsinIMB.Withsuchalow rameterγ. IthasbeenshownthatSN1987Adatamayfavor ∼ expectation,seeing5 eventsinIMBis quiteunlikely(witha a highly anti-pinchedspectrum [53]. However, all such pre- 15%probabilityofgetting4ormoreevents,anda 4%prob- definedfunctionsdescribeonlyaverylimitedclassofpossi- ability of getting 5 or more events), suggesting instead that ble shapes, whichmay notnecessarily fitthe actualreceived the truespectrumwasindeedmostfaithfullysampledbythe spectrumofneutrinos. largerIMBdetector. More complex composite spectral shapes are also consid- 4 eredintheliterature. Today,itiswellestablishedthatneutri- usefulforstudieswhichmayrequiresuchinput,regardlessof nos may change flavor while they travel through matter, so- theunderlyingphysicsandassumptions.Asnoted,weassume called MSW effect [54, 55, 56, 57, 58, 59, 60]. The den- thatthetwodetectorswereexposedtothesameincomingneu- sityprofileofthesupernovaenvelope,whichneutrinostravel trinospectrum. Thetheoreticalrelationbetweenthereceived through, spans many orders of magnitude and thus enables neutrinospectrumφ(E )andthecorrespondingpositronde- ν manyresonancesandopportunitiestoswapflavors. TheIMB tection spectrum Φ(E ) is given in Eq. (2). The actual de- + andKam-IIdetectorsareexpectedtoobservealmostidentical tected positron spectrum as recorded by an experiment can spectrasincetheEartheffects[61,62,63]areunlikelytoin- beexpressedasasumof“bumps”placedattheobservations, troducesignificantdifferences(usingrecentdeterminationsof Φ(E ) = δ(E E ), where the index is over the set + i + − i themixingparameters). Asuperpositionoflow-energy/high- ofeventsunPderconsideration.Thiswouldbeafaithfulrepre- luminosity and high-energy/low-luminosity thermal spectra sentationofthetheoretically-expectedpositronspectrum(as- duetoneutrinomixingfitsthedatabetterthanasinglequasi- sumingthatthedatawasaprobableoutcomeandnotastatis- thermal spectrum [31]. A bimodalneutrino distribution the- ticalanomaly),onlyifthedetectednumberofpositronswere oretically arising from the accretion and the cooling phases verylarge(i.e.,inthecaseofafutureGalacticsupernova,op- ofasupernova[25],isanotherexampleofacompositespec- erational detectors are expected to observe many thousands trum. However, due to limited statistics, inferring multiple of events) and the uncertainties of the visible energies were theoreticalparametersisachallengingtask,duetoseverede- small. Then,onecaninferthereceivedneutrinospectrumby generacies among them. While both approaches effectively invertingEq.(2)as increase the number of parameters describing the spectrum, naturallyimprovingthefittothedata,onecanonlyprobean φ(E )=4πD2 δ(E+−Ei) , (3) effective ν¯ spectrum after any physical mechanisms modi- ν N(E )σ(E +∆) e X i i i fying the spectrum, including oscillations, are taken into to account. whichcanbeverifiedbysubstitutingbackintoEq.(2). When the statistics are limited, as here, we must regulate the dis- cretebumpsinthespectrumbyanappropriatesmoothing;the mostphysicallymotivatedmethodistousetheapproximately V. NEUTRINOSFROMSN1987A Gaussian uncertaintieson the measured energies. Hence we replaceeachδfunctionbyaGaussianwithawidthofε , i We resort to the data directly and attempt to find the sim- plest description of the effective ν¯e spectrum that could be δ(E E ) exp[ (E E )2/(2ε2)]/(√2πε ). (4) +− i → − +− i i i SinceverysharplypeakedGaussianswouldrevealthespuri- 10. ousfinestructureofthedata,wechooseagenerouswidthof ε = 1.5∆E (where∆E istheuncertaintyontheassigned i i i energy of each detected positron), which enables us both to take into account uncertainties associated with the detection 1-1 processadequatelyandtopresentasufficientlysmoothspec- ] V trum.Anylargerwidthwouldintroduceexcessivesmoothing, e Model whichwouldobscuredetailsofthedistributionandspuriously M 560 / 10-1 ennehuatrninceotshpeetcatirlusm. Niontethtihsatwianyo,ridteirstocrduicrieacltltyharetctohnesntreuucttritnhoe 1 andpositronenergiesarerelatedinaone-to-onewayasinin- [ φ ( E) ν10-2 Kam-CIIombineIdMB eIvIxeraFosmriegpIubMlreeet)Ba.2dddeaicstapaylasey(utssn,thliaekseinaffoefrurrnnecedtuisotprniencootf-reanlfeercuottmrroinneoisthceaentreterthgrieyn.Kg,Tafmhoer- differencesbetweentheinferredspectrafromKam-IIorIMB stems from a few facts. Since the IMB detector has no sen- sitivityatlowerenergies,wherewecanrelyonKam-IIonly, -3 thetwoleastenergeticeventsofIMB,forwhichtheefficiency 10 0 10 20 30 40 50 60 isverylow,thenbecomedisproportionatelysignificant. Also Eν [ MeV ] thefactthatonlyasinglepositronwasdetectedbyKam-IIat higherenergiesrequiresa significantsuppressionoftheneu- FIG.2:TheinferredneutrinoemissionspectrafromeithertheKam- trinospectrum. However,the effectivenumberoftargetsfor II or IMB data setsaloneor their combination, as discussed inthe Kam-II is low compared to IMB (as shown in the top panel text(takingintoaccountthecorrespondingeffectivenumberoftar- oftheFig.1)andtheexpectednumberofeventscouldeasily gets and cross section). The shaded shape is a Fermi-Dirac spec- fluctuatedown. trum with canonical neutrino emission parameters (average energy We have earlier established that Kam-II and IMB probed E0=15MeVandtime-integratedluminosityL=5×1052erg). different energy domains and also showed that the detected 5 positrons showed no obvious conflict with this expectation. VI. DSNBDIRECTLYFROMSN1987A Soratherthanfocusingonthedifferencesbetweenthespectra suggestedby the Kam-IIor IMBdata sets, we will focuson Neutrinos from past core-collapse supernovae have been what can be learned from the combined data set, since both studied extensively [21, 22, 64, 65, 66, 67, 68, 69, 70, 71, experimentswere measuringthe identicalincomingneutrino 72, 73, 74, 75, 76, 77, 78, 79, 80] and experimental limits spectrumusinganidentical(water-Cˇerenkov)technique. are already suggesting an impending detection [81, 82, 83]. Bycombiningthetwodatasetsandthecorresponding(en- The Diffuse Supernova Neutrino Background (DSNB) flux ergy dependent)effective number of targets (as displayed in dependsnotonlyonthetypicalsupernovaneutrinoemission the top panel of Fig. 1), we avoid over-emphasizingthe dif- spectrum,butalsotheexpansionrateoftheuniverse(redshift- ferences due to low statistics. The combined result is also timerelationh(z)= Ω (1+z)3+Ω ,whereΩ =0.3, M Λ M less susceptible to Poisson fluctuations associated with such Ω =0.7)andthecopre-collapsesupernovarate(SNR),which Λ limitedstatistics,yetcoversthewholeenergydomain,andis presumablytracksthehistoryofstarformationrate(SFR)(see more conservative. The neutrino spectra based on this com- e.g. [84] and referencestherein). The neutrino emission per bineddatasetisshowninFig.2(solidline). Forcomparison, supernovaisthemostuncertainquantity[77,84]andishence wealsoshowaFermi-Diracspectrumwithcanonicalneutrino our focushere. The currentprecisionof the data shows that emissionparameters(anaverageenergyE = 15MeVanda 0 theevolutionoftheSFRcanbeparametrizedwithapiecewise integratedluminosityL =5 1052erg). ν linearfit: × Since we construct the spectra directly from the data, it is not meaningful to make goodness-of-fit tests between the R (z) = R0 (1+z)α for z <z SF SF p dataandtheseconstructedspectra(weightedwiththeenergy- R0 (1+z)β dependentproportionalityfactors). If we use delta functions = SF for z <z <z , (5) (1+z )β−α p max to represent the data, then the cumulative distributions of p the measured and ‘predicted’ spectra would be identical, so where α 3.44, β 0, z 1 and z 5. p max thataKolmogorov-Smirnovtestwouldindicateperfectagree- The overal∼l normalizatio∼n in the lo∼cal universe is R0∼ = SF ment. When we use Gaussians to represent the data, the 0.0095M⊙/(yr Mpc3). Thecore-collapsesupernovarate as Kolmogorov-Smirnovtest,whilenotstrictlymeaningful,pro- afunctionofredshiftisR (z) = ζR (z),whilethefrac- SN SF videsconfirmationthatourchosenwidthisnottoolarge. tion of stellar mass ending as supernovae, ζ = 0.0132/M⊙ Whileapinchedtemplatespectrumputsmoreweighttothe (fortheBaldry–GlazebrookIMF[85]),dependsonthestellar peak, and an anti-pinchedspectrum puts more weight to the massfunction. Wenotethatthisdependenceontheassumed tail, the shapefromthecombineddatacanbebestexplained stellarmassfunctionissmall,asexplainedinRef.[84].While by a spectral shape that is enhanced both at the peak (to ac- thedirectlymeasuredcorecollapsesupernovarateisslightly commodateeventsfromKam-IIwhereIMBwasnotsensitive) smaller[86, 87], recentstudiessuggestthismaybemislead- andhighenergytailofthe spectrum(tobetteraccommodate ing. A large fraction of supernova exploding in very dusty energeticeventsfromIMB),anddepressedinbetween,com- starburstenvironmentsmaygoundetected,causinga30-60% pared to a thermal Fermi-Dirac spectrum. This basic shape underestimate of the true core-collapse supernova rate [88]. of the underlying spectrum (see our Fig. 2), in agreement Thus,whilethesupernovaratedataaregenerallyconfirming, withthetwo-componentcompositespectrumofLunardiniand weusethemorereliablestarformationratedatafornow. Ul- Smirnov [22, 31], could reconcile the Kam-II and IMB data timately, it will be possible to use the supernovarate data to with each other. The correspondingluminosity for the com- more directly predict the DSNB flux, as first pointed out by bined spectrum, L 6 1052 erg, is quite similar to that ν Ref. [76]. The uncertainty on the flux due to astronomical ∼ × ofthe model,with an averageenergy,E 12 MeV,thatis 0 inputs is small and will be further reduced with anticipated ∼ somewhatlower. improvementsin the data. In our analysis, we assumed that Since we consider a data set of 19 detected positrons, the the normalizationof the SFR and its evolution can be deter- overalluncertaintyduetothePoissonnatureofthedetection minedindependentlybyastronomers,andusedasafixedin- will not be too large, 1/√19 25%. In the peak region, puttoextractmoreaccurateinformationontypicalsupernova ∼ our reconstructionis based on 12 events with a nominalun- properties. certaintyof1/√12 30%, sothattheexcessrelativetothe The neutrino emission per supernova times the supernova ∼ modelissignificant. Inthetailregion,thereare7eventswith rate, when integrated over redshift and convolved with the a nominal uncertainty of 1/√7 40%, which is also illus- cross section, yields the detected spectrum of DSNB neutri- ∼ tratedbythedifferencesbetweenthespectrainthisregionof nos, joint sensitivity. We emphasize that the tail region is not so uncertain,astherewere6eventsabove35MeV;thisstrongly c zmax RSN(z) ψ(E )= σ(E )N φ(E [1+z]) dz, precludesanyhypothesizedsuppressionofthetail. InFig.2, + H ν tZ ν h(z) 0 0 thecombinedspectrumabove40MeVdependsonthewidth (6) chosenfortheGaussians,astherewerenoeventsattheseen- where c is the light speed and H is the Hubble constant. 0 ergies(however,therewere6eventsbetween35and40MeV, For a modern detector like Super-Kamiokande (SK), the ef- each with relativelylargeenergyuncertainties, and theirsta- ficiencycansafely be takento beunity. We willpresentour tisticalweightsmustgosomewhere). Thisistheleastcertain resultsforaSuper-Kamiokandesizeddetectoroffiducialvol- partofthespectrum,duetothelowfluxthere. ume22.5kton,correspondingtoN =1.5 1033. t × 6 1.. energytail, both the smooth method (as presented in Fig. 3) and the δ-function method agree quite well over all energy ] ranges.InTableI,wesummarizetheDSNBeventratesinvar- ) n iousrangesofvisibleenergyforbothsmoothandδ-function o kt prescriptions (all values are quoted per 22.5 kton per year). 5 Thevaluesinthelastcolumn,reportingtheeventratesinthe 2. 10-1 energy range of 18–26 MeV can be compared to the event 2 V rate limit of 2 at 95% C.L. (as inferred in Ref. [77] from Me I theν¯e fluxlim∼itof 1.2cm−2 s−1 at95%C.L.reportedby r Model MB SK [81] throughno∼n-detectionofexcesscountsaboveback- ( yea Combined gtarloizuinndglflyucctluoasteiotnos)d,estuegcgtieosnt.ingInthtahtethreanDgSeN1B0–is1a8lrMeaedVy,tathne- ) [ 1/ 10-2 Kam-II eovnensutpreartensovaaremnoedarellys,assulgagrgeestiansgthaenuimsumalinpernedtidcitsiocnosvebrayseodf E+ DSNBiswellwithinthereachofcurrentdetectors,andespe- ( cially promisingif thresholdsare reduced by the addition of ψ gadolinium[89]. There are various advantages to predicting the DSNB di- -3 10 0 10 20 30 40 rectly from the data rather than through the intermediate E [ MeV ] stagesoffittingfunctions. TheDSNBmainlyconsistsofthe + higherenergypartoftheneutrinospectrumandismuchless susceptibletouncertainties/backgroundsaroundthethreshold FIG. 3: The DSNB detection spectra based on the neutrino spec- trainferredfromeither theKam-II or IMB datasetsalone or their ofdetectors. While theoreticallymotivatedformulasattempt combinationasinFig.2,comparedtoamodel(shadedshape)with to explain the bulk of the detected events, they tend to put canonical neutrinoemissionparameters(theassumedcore-collapse more weight in the low energy part at the expense of intro- SNhistoryisdescribedinthetext). ducing distortions at the high energy parts of distributions relevant for the DSNB. For a high threshold of 18 MeV, almost all of the statistical weight comes from the IMB data. In Fig. 3, the presented DSNB spectra are calculated by Whentheenergythresholdisdecreased,theKam-IIdatawill substituting the smoothneutrinospectra inferredfromeither theKam-IIorIMBdatasetsaloneortheircombinationasin Fig.2, intoEq.(6). Acanonicalneutrinoemissionspectrum 1.0 described by Fermi-Dirac distribution (E = 15 MeV and 0 L=5 1052ergs[77])isalsopresented. Note×that we have replaced each detected positron by a ) ] Gaussian to obtain a sufficiently smooth neutrino spectrum. n o Thiscouldinadvertentlyintroducesomearbitrarinessanddis- kt tort the results. Since the calculation of the DSNB already 2.5 -1 involves an integration over redshift, we can instead predict 2 10 V theDSNBdirectlyfromthedata,whichwecalltheδ-function e M prescription. Whenredshiftingistakenintoaccount, Eq.(3) canbecastas: ear Model IMB y φ(Eν[1+z]) = 4πD2Xi δδ((z[1+(zEN]((EE+iE)+σ(∆)E/)Ei−+)(∆E)i+∆)) ) [ 1/ ( 10-2 KCaomm-IbIined = 4πD2 − i− + ν (7) E+ X EνN(Ei)σ(Ei+∆) ( i ψ wherewehaveusedthedeltafunctionidentity;δ(ax b) = − δ(x b/a)/a. SubstitutionofthisrelationintoEq.(6)elimi- -3 − 10 natestheintegrationoverredshift,sothatthedetectedpositron 0 10 20 30 40 spectrumis E [ MeV ] + c σ(E )N 4πD2 R ((E E )/E ) ψ(E )= ν t SN i− + ν .FIG. 4: The DSNB detection spectra based on the neutrino spec- + H0 XEνN(Ei)σ(Ei+∆) h((Ei E+)/Eν) trainferredfromeither the Kam-IIor IMB datasetsalone or their i − (8) combination,throughtheδ-functionprescriptionwhichenablesusto Figure4displaystheresultantDSNBdetectionspectra,as deduce theDSNBspectrawithout applyinganysmoothing (asdis- deducedfromtheKam-IIorIMBdatasetsaloneortheircom- cussedinthetext),comparedtoamodel(shadedshape)withcanon- icalneutrinoemissionparameters. bination,andcomparestothemodel.Apartfromtheveryhigh 7 largerDSNBflux.PartofourpointisthattheSN1987Adata, TABLEI:TheDSNBeventratesinvariousrangesofvisibleenergy when consideredwithouttheoreticalpriors, do notsupporta fromthespectradisplayedinFig.3andFig.4.Allquotedvaluesare verylowDSNBflux. per22.5ktonperyear. Wenowcommentontheeffectsofconsideringalsothedata Range(MeV) 4-10 10-18 18-26 fromtheBaksandetector[26,27]. Therelevantpropertiesof δ smooth δ smooth δ smooth this detector were very similar to that of Kam-II, except for being 10timessmaller. ThususingtheKam-IIyield, 1 Kam-II 1.8 1.7 0.7 1.0 0.2 0.3 ∼ ∼ event would be expected in Baksan, while 5 were observed, IMB 0.6 0.8 1.4 1.3 0.6 0.9 andwithasomewhathigheraverageenergythaninKam-II.If Combined 1.8 1.8 1.0 1.2 0.4 0.5 theBaksandatawereafaithfulrepresentationofthetruespec- Model – 1.5 – 1.8 – 0.8 trum,thenthecorrespondingDSNBfluxwouldbeenhanced bya factor> 5, which islikely alreadyexcluded. The most likelyresolu∼tionisthatBaksansawanupwardfluctuationof be given more weight. Note also that we do not attempt to their large backgroundrate, perhapsalong with some events deconvolve any poorly-understoodphysical effects from the fromSN1987A. observed spectrum to deduce the original spectra at formation. We are directly relating the observed spectrum of SN 1987A to the prediction for the observable DSNB spectrum VII. DISCUSSIONANDCONCLUSIONS frommanysupernovae. The idea of normalizing the DSNB flux prediction to the Neutrinosplayacrucialroleinthelifeanddeathofmassive SN 1987Adata is appealinglyempirical, and was first intro- stars and so far, they are the only messengersthat enable us duced by Fukugita and Kawasaki [21], and later developed to probe the inner workings of a core collapse. Thus, two in greater detail by Lunardini[22]. The disadvantage of re- decadesafterthedetectionofneutrinosfromSN1987A,they lying on the SN 1987A data is the uncertainties and appar- continuetoattractmuchattention. ent disagreements of the sparse data. While Fukugita and We assess the compatibility of the Kam-II and IMB data Kawasaki[21] used only the IMB data, obtaininga neutrino setsinamodel-independentwayandturntothedatadirectly, emissionpersupernovainagreementwiththeoreticalmodels, skipping any intermediate stages and assumptions about the LunardinishowedthatincludingtheKam-IIdatainthespec- incoming neutrino spectrum. Our main conclusion is that if tralfitleadstoamoreuncertainDSNBfluxprediction,includ- one drops theoretical prejudices on the spectral shape, the ingthepossibilityofitsbeingsignificantlylowerthanfound Kam-IIandIMBdetectionscanbereconciled. Thetwodata by other authors. Our predictions for the combined Kam-II setsareprimarilysensitivetodifferentenergyregimesofthe and IMB data are consistent with those of Lunardini when neutrino spectrum (as seen in Fig. 1), and only their proper differencesin the star formation rate are taken into account. combinationprobesthewholeenergyrange. The actualten- However, as we show, it is the IMB data that are presently sionisbetweentheadoptedtheoryandtheobservations,since morerelevantforthedetectableDSNBflux,andaccordingly, assumedtheoreticalshapesarealwayslimitedtoacertainsub- pessimistic DSNB detection rate predictions are disfavored. setofallmathematicalpossibilitieswhichdonotnecessarily Thefactthat6 ofthe SN 1987Aeventswere detectedabove describe the true underlying physics. When using the data 35MeVstronglyprecludesasupernovaspectrumthatiseither directly, we consider the combined spectrum, based on both toosoftortoolow. datasets,tobethemostreliable. Aswithotherformsofstatisticalinference,weareassum- Using parameter-free inferential statistical methods, we ingthattheobserveddatawererepresentativeofthetruth(i.e., have shown that the combinedKam-II and IMB data can be which is the maximally likely outcome), while considering best explained by a spectral shape that is enhanced both at how uncertain our subsequent results are. We are not using the peak and the tail of the spectrum and depressed in be- theoryorotherconsiderationstojudgethatthedataweresub- tween,comparedtoawell-knownFermi-Diracspectrum(e.g., ject to any particularstatistical fluctuation. In particular, we Fig.2). Suchadistributionmaybearisingfrommanydiffer- found no evidence from the data alone that the Kam-II and entphysicalprocesses, includingthe details of core collapse IMB data sets were incompatible. A second point to con- orneutrinomixing,whicharebeyondthescopeofthisstudy. sider is whether SN 1987A was a representative supernova. Oncetheeffectivereceivedneutrinospectrumisdetermined, Despite the large range of progenitors, and the wide vari- onecanthenworkbackwardsanduncoverthescenariosthat ety of optical supernova properties, the time-integrated neu- will yield this measured spectrum. Considering that super- trinosignalsfromcore-collapsesupernovaeareexpectedtobe nova models still fail to robustly explode despite increased mostly uniform, as they are well-connectedto the properties sophisticationin modeling, it is an alluringpossibility thata of the newly-producedneutron stars. We note that both the keyelementmaystillbemissing[90]. Themodeltheoretical SN 1987A data and the present SK limit on the DSNB flux spectrawithcanonicalemissionparametersadoptedinmany both depend on statistical uncertainties at the 30% level. studiescouldprovideandcaptureessentialsofthesupernova ∼ It may well be that SN 1987A is not a representative super- neutrinospectrum,andbeadequateformostpurposes.While nova,andindeed,thisispartofwhatwewanttotestwiththe theoreticalmodelsstillprovideessentialguidance,theneces- DSNB[77].AsshowninFigs.3and4,andsummarizedinTa- sityoffreshdataonsupernovaneutrinosisobvious.Therarity bleI,moretypicalresultsfromtheoreticalexpectationsgivea ofgalacticsupernovae,withthemostoptimisticrateestimates 8 ofatmostafewpercentury,makesthisachallenge. andespeciallypromisingifthresholdsarereducedbythead- Apartfromtheproposaltodetectindividualneutrinosfrom dition of gadolinium [89]. A gadolinium-enhanced Super- galaxieswithin the 10 Mpc neighborhoodof the Milky Way Kamiokande should also be able to provide a measurement with future Mton-scale detectors [91], the DSNB presents of the spectral shape [77, 92], providing further clues. The the only sensible alternative to gain informationon neutrino DSNB,whichmayevenonedayserveasatoolforextracting emissionfromsupernovae.Weuseanonparametricapproach cosmological evolution parameters [93], is a leading candi- in orderto determineoneobservable, the Diffuse Supernova datealongwiththeothercontenders,likecosmogenicneutri- Neutrino Background, directly from another observable, the nos[94],toopennewdoorstothecosmosandprovidethefirst SN 1987A data on supernova neutrino emission, rather then neutrinodetectionoriginatingfromcosmologicaldistances. proceeding through intermediate stages of fitting functions. We show thatthis predictioncannotbe too small (especially in the 10-18 MeV range), since the majority of the detected ACKNOWLEDGMENTS events from SN 1987A were above 18 MeV (with 6 above 35 MeV). We emphasize that our DSNB predictionsare not very dependent on the details of our analysis procedure, as WethankArnonDarandGeorgRaffeltforhelpfuldiscus- can be seen by comparing Figs. 3 and 4, and especially by sions,andMattKistlerandCeciliaLunardiniforthesameand comparingthe“δ”and“smooth”casesinTableI. also a careful reading of the manuscript. This work is sup- These results are nearly as large as the usual predictions ported by the National Science Foundation under CAREER basedonsupernovamodels, suggestingan imminentdiscov- GrantNo.PHY-0547102toJB,andCCAPPatTheOhioState ery of DSNB is well within the reach of current detectors, University. [1] A.Heger,C.L.Fryer,S.E.Woosley,N.LangerandD.H.Hart- [20] B. W. 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