High-resolution supernova neutrino spectra represented by a simple fit

0.22 MB·English

by  Irene Tamborra

MPP-2012-149
High-resolution supernova neutrino spectra represented by a simple fit
Irene Tamborra,1 Bernhard Mu¨ller,2 Lorenz Hu¨depohl,2 Hans-Thomas Janka,2 and Georg Raffelt1
1Max-Planck-Institut fu¨r Physik (Werner-Heisenberg-Institut), F¨ohringer Ring 6, 80805 Mu¨nchen, Germany
2Max-Planck-Institut fu¨r Astrophysik, Karl-Schwarzschild-Str. 1, 85748 Garching, Germany
(Dated: January 9, 2013)
To study the capabilities of supernova neutrino detectors, the instantaneous spectra are often
represented by a quasi-thermal distribution of the form f (E) ∝ Eαe−(α+1)E/Eav where E is
ν av
the average energy and α a numerical parameter. Based on a spherically symmetric supernova
model with full Boltzmann neutrino transport we have, at a few representative post-bounce times,
re-converged the models with vastly increased energy resolution to test the fit quality. For our
examples, the spectra are well represented by such a fit in the sense that the counting rates for
3
a broad range of target nuclei, sensitive to different parts of the spectrum, are reproduced very
1
0 well. Therefore, the mean energy and root-mean-square energy of numerical spectra hold enough
2 informationtoprovidethecorrectαandtoforecasttheresponseofmulti-channelsupernovaneutrino
detection.
n
a PACSnumbers: 14.60.Pq,97.60.Bw
J

8
I. INTRODUCTION has the form [3]

]
R f (E)∝Eαe−(α+1)E/Eav. (1)
ν
S Theneutrinosignalfromthenextnearbycore-collapse
. supernova(SN) remainsthe mostcovetedtargetforlow- Here, E is the average energy and α a numerical pa-
h av
energy neutrino astronomy. Galactic SNe are rare, per- rameter describing the amount of spectral pinching; the
p
- haps a few per century, and such an observation will be value α = 2 corresponds to a Maxwell-Boltzmann spec-
o a once-in-a-lifetime opportunity to look deeply inside a trum, and α = 2.30 to a Fermi-Dirac distribution with
r
t collapsing star and learn about its astrophysical work- zero chemical potential. In general the neutrino spectra
s
ings as well as about neutrino properties. Several de- are fitted by 2 ≤ α ≤ 4 [3] with higher α > 2 indicat-
a
[ tectors worldwide are in operation that will register a ing stronger pinching and α < 2 meaning anti-pinching
high-statistics signal while others are in preparation or relativetoaMaxwell-Boltzmanndistribution. Thethree
2
under discussion [1]. parameters E , α, and the overallnormalizationcan be
v av
determined, for example, if a numerical SN simulation
0 In order to assess the physics potential of various de-
2 tectors or detection principles one needs the expected provides the energy flux (luminosity) Lν in some flavor,
9 flavor-dependent SN neutrino flux spectra. In the ab- the average energy Eav = hEνi and some other energy
3 moment, for example hE2i or sometimes hE3i.
sence of self-consistent three-dimensional core-collapse ν ν
. While spectra of the form of Eq. (1) certainly pro-
1 simulations there is no standard SN neutrino flux model
1 and moreover, the flavor-dependent flux spectra depend videareasonableoverallrepresentation,itisnotobvious
2 how well the spectral tails are reproduced. For study-
onthepropertiesoftheincompletelyknownneutron-star
1 ing detector responses with target nuclei with a signifi-
equationofstate,onthepropertiesofthecollapsingstar,
: cantenergy threshold, it is imperative that the accuracy
v notably its mass, and, of course, on time for any given
i case. Inthissituationparametricstudies,takingaccount of the α-fit in the high-energy tail be checked against
X
solutions of the neutrino transport equations. This is
of the plausible range of predictions, are the preferred
r especially relevant for example for lead in the Halo de-
a course of action.
tector [5], argon in future large-scale liquid argon detec-
We here address one particular aspect of such studies,
tors [6], or subdominant detection channels on oxygen
i.e. the plausible spectral shape of SN neutrino fluxes.
in water Cherenkov detectors [7] or on carbon in liquid
On a rough level of approximation, the spectra follow a
scintillator detectors [8].
thermal distribution that can be described in terms of
Modern numerical SN codes treat neutrino transport
aneffective temperature. Indetailthe spectraformation
with Boltzmann solvers that are expected to produce
is a complicated process, different energy groups emerg-
physically accurate spectra. On the other hand, neu-
ingfromdifferentdepthsintheproto-neutronstaratmo-
trino transport is the most CPU-time consuming aspect
sphere [2, 3], and a more refined description is necessary
of SN simulations so that in practice the energy resolu-
for a more detailed understanding.
tionislimited. Forexample,intypicalsimulationsofthe
The nextlevelof sophisticationis to describe the non- Garchinggroup,17–21energybinsareused,correspond-
equilibrium spectra by a three-parameter fit that al- ing to an energy resolution ∆ǫ/ǫ ∼ 0.3. Whether such
lows for deviations from a strictly thermal spectrum [4]. a(seemingly)coarsezoning isadequate formodeling the
Oneparticularlysimplerealizationarespectrawherethe high-energy tail of the spectrum and hence for judging
energy-dependent neutrino number flux for each flavor the quality ofα-fits needs to be determinedatleastonce
2
by a resolution study. Spectra from standard resolution transport equations for a representative time during the
(SR) neutrino transport simulations are therefore of lit- accretionphase(261ms after bounce)andforthe proto-
tle value forassessingthe quality ofα-fits inthe spectral neutron star cooling phase (1016 ms and 1991 ms) using
tail, as are Monte Carlo studies [3, 4], which suffer from the matter background from the dynamical simulation.
limited sampling at high energies. Differentfromthe dynamicalrun,the originaltreatment
In this paper, we investigate whether α-fits and SR of[14]for the Doppler andredshiftterms is usedinstead
spectra from SN simulations are sufficiently accurate to ofthatof[15],andthethresholdvaluesforthemonochro-
predict detection rates that are sensitive to the high- maticneutrinoenergydensitybelowwhichaspectralex-
energy tail. To this end, we study the spectral shape trapolationis performed has been reduced by a factor of
and its impact on the relative detector response of dif- 100. We also transition continuously from the comoving
ferentmaterialsusinghigh-resolution (HR) spectracom- frame to the lab frame once neutrino interactions cease
puted for a few representative post-bounce times of a in order to eliminate artifacts due to the velocity jump
representative spherically symmetric simulation. These in the shock. These numerical changes accelerate con-
spectra have been obtained by re-converging models at vergenceandhelptoobtainsmootherandmoreaccurate
a given post-bounce time after refining the energy bin- spectra.
ning. With the HR spectra we can then test how well a The spacing of the logarithmic energy grid was much
global fit of the form of Eq. (1) reproduces the relative finer with 42 bins and δǫ/ǫ = 0.11 in the HR case. For
neutrino counting rates of different materials that probe better comparison, the stationary solutions for the SR
rather different parts of the spectrum. casewerere-computedwithpreciselythesamenumerics.
We always ignore the effect of neutrino flavor con- Figure 1 shows the obtained HR spectra for the species
versions. The ordinary Mikheyev-Smirnov-Wolfenstein ν , ν¯ , and ν (with ν meaning a single kind of ν , ν¯ ,
e e x x µ µ
(MSW) effect [9, 10] or self-induced flavor conver- ν , and ν¯ ).
τ τ
sions [11] strongly modify the flavor-dependent spectra In order to compare the numerical spectra (his-
seen by a detector on Earth [12]. For example, the de- tograms)with a quasi-thermalfit of the form of Eq. (1),
tectable ν¯e flux will be a certain linear combination of wedeterminetheparameterαintermsoftheenergymo-
the ν¯e and ν¯x spectra emitted at the source. These de- ments of the distribution. One way of fixing α is to use
tectionspectraneednotberepresentedbyasimplefit,in the energy moment of order k and notice that
particular if self-induced conversions cause features such
as “spectral splits.” Determining the detection spectra hEki k+α
ν = hE i. (2)
in terms of the source spectra is the task of oscillation hEk−1i 1+α ν
ν
studies, usually performed by post-processing the fluxes
provided by numerical models. Justifying simple repre-
We have checked that for our cases the implied value of
sentations of the source spectra for the purpose of flavor
α does not strongly depend on k, which already implies
conversionstudies is one prime motivation for our work.
thatthistypeoffitisareasonablerepresentation. Inthe
To this end we describe in Sec. II our numerical SN
following we will use k =2, i.e., the relation
modelandinSec.IIIthetargetmaterials. Acomparison
between counting rates based on the numerical spectra hE2i 2+α
and the fitted ones is performed in Sec. IV before con- ν = . (3)
hE i2 1+α
cluding in Sec. V. ν
Weshowexamplesofthefitsascontinuouscoloredcurves
alongside the HR spectra (step functions) on a logarith-
II. NUMERICAL SUPERNOVA MODEL
mic and on a linear scale in Figs. 1 and 2, respectively.
For comparison we also show the SR spectra and their
We have performed simulations of the 15M⊙ progeni- correspondentfitsasthindashedblacklinesinFig.1,and
torofWoosleyandWeaver[13]withthe neutrino hydro- find that visually the fits work very well in both cases.
dynamics code Prometheus-Vertex [14], whichsolves Valuesforthefitparameterα,theluminosityL ,andthe
ν
the 0th and 1st neutrino moment equations with a vari- neutrino mean energy hE i for the three snapshots and
ν
able Eddington factor closure that is provided by the forthe differentneutrino species ν , ν¯ ,and ν arelisted
e e x
formal solution of a simplified (model) Boltzmann equa- in Table I both for the HR and the SR case. We note
tion. The explosion of the spherically symmetric model that despite the lower energy resolution, the SR spectra
was artificially initiated 500 ms after bounce. For the from the dynamical simulation typically give very simi-
dynamicalevolutionofthe modelincluding the feedback lar luminosities and mean energies, which are consistent
into the hydrodynamics, we used 21 energy bins with a withthe HRcasewithin∼5%orless. Thefitparameter
typicalresolutionδǫ/ǫ(ratioofzone widthto zone mean αagreessimilarlywellexceptforν inthepost-explosion
e
energy)of ∼0.29. We refer to this setup as the SR case. phasewitha deviationofupto 9%. However,this is due
Inordertoaccuratelypredicttheshapeoftheneutrino to a very steep dependence of α on the second energy
spectra,particularlyinthehigh-energytail,wecomputed moment, hE2i, when the energy ratio in Eq. (3) is close
ν
high-resolution(HR)stationarysolutionsoftheneutrino to unity, and does not affect the fit curves appreciably.
3
56
10
ν , 261ms ν , 261ms ν , 261ms
e e x
1) 55
-V 10
e
M
-1s 1054
(
ν
f
53
10
ν , 1016ms ν , 1016ms ν , 1016ms
1055 e e x
)
1
-V
e 54
M 10
1
-s
(ν 1053
f
52
10
1055 ν , 1991ms ν , 1991ms ν , 1991ms
e e x
)
1
-V 54
e 10
M
1
-s 53
( 10
ν
f
52
10
0 10 20 30 40 0 10 20 30 40 0 10 20 30 40 50 60
E (MeV) E (MeV) E (MeV)
FIG.1: Spectraforspeciesν (leftcolumn),ν¯ (middlecolumn)andν (rightcolumn)forpost-bouncetimesof261(toprow),
e e x
1016 (middlerow), and1991 ms(bottomrow). Thedatafrom thePrometheus-Vertexspectraareshownasstepfunctions,
and the continuous curves are quasi-thermal fits according to Eq. (1). Thick magenta, turquoise, and green lines are used for
the different neutrino species (ν , ν¯ and ν , respectively) in the HR case, while the SR spectra and fits are shown as thin
e e x
dashed black lines.
TABLE I: Fit parameters α, luminosities L , and neutrino mean energies hE i for different neutrino species and post-bounce
ν ν
times for HRand SR spectra.
t species α α L L hE i hE i
HR SR ν,HR ν,SR ν HR ν SR
[ms] [1052ergs−1] [1052ergs−1] [MeV] [MeV]
261 ν 2.65 2.73 2.80 2.68 13.05 12.87
e
261 ν¯ 3.13 3.23 2.79 2.71 15.23 15.05
e
261 ν 2.42 2.48 1.58 1.63 14.48 14.48
x
1016 ν 2.90 3.14 0.62 0.62 10.14 10.28
e
1016 ν¯ 2.78 2.88 0.67 0.70 12.69 12.87
e
1016 ν 2.39 2.52 0.75 0.77 12.89 13.00
x
1991 ν 2.92 3.17 0.48 0.47 10.01 10.01
e
1991 ν¯ 2.61 2.67 0.44 0.44 12.28 12.36
e
1991 ν 2.34 2.49 0.54 0.55 12.31 12.39
x
III. DETECTION CROSS SECTIONS (IBD) reactionν¯ +p→n+e+. The crosssectionvaries
e
roughly as E2 so that the overall event rate is well re-
ν
In existing detectors, SN neutrinos are primarily mea- produced, if the SN neutrino spectrum is approximated
suredintheν¯ channelbyvirtueoftheinversebetadecay
e
4
0.1 span a broad range of spectral responses to neutrino
fluxes and show the energy-dependent cross sections in
ν
0.08 e Fig. 3. The reaction sensitive to the lowest range of en-
ergies is elastic scattering on electrons via both charged-
current(CC)andneutral-current(NC)forν andν¯ and
e e
0.06 via NC interaction for ν [16].
u.) νe In nuclei, CC interactxions proceed via ν +(N,Z) →
a. e
(ν (N−1,Z+1)+e− andν¯e+(N,Z)→(N+1,Z−1)+e+,
f 0.04 ν
x respectively. The ν¯e interaction is typically suppressed
at a given energy with respect to the ν interaction due
e
0.02 to Pauli blocking. Neutral current interactions may also
produce observable signals via ejected nucleons or de-
excitation photons. As one example we consider NC
00 10 20 30 40 50 interactions with 16O [7, 17–19], which is relevant for
E (MeV) water-baseddetectors.
Interactions with heavier nuclei, such as lead, may
FIG. 2: Spectra for species νe, ν¯e and νx. Histograms are yield quite high rates of both CC and NC interactions.
the Prometheus-Vertex HR spectra of our model at time Observable signatures include leptons and ejected nucle-
1016 ms. The curves are quasi-thermal fits according to ons. Single and multiple neutron ejections are possi-
Eq. (1): dashed for ν , continuous for ν¯ , and dot-dashed
e e ble. The relevant interactions for lead-based detectors
for νx. are ν +APb→e−+ABi∗ and ν +APb→ν +APb∗.
e x x
ForbothCCandNCcases,theresultingnucleide-excite
via neutron emission. Antineutrino CC interactions are
ν 208Pb stronglysuppressed. Although naturallead containsiso-
104 e topes other than 208Pb, the neutron cross section for
40Ar (coh NC) A = 208 should be similar to other components. For
the Pb cross sections see Refs. [20–23].
2m)102 νe 208Pb (NC) Liquidargondetectorswillhaveexcellentsensitivityto
−42 c νe 40Ar 16O (NC) νaenviniatetrhaectCioCninfotrerwachtiicohntνhee+d4e0-Aexrc→itatei−on+γ40sKfr∗o.mTh40isKi∗s
0
σ (1100 can be observed. The reaction ν¯e +40Ar → e+ +40Cl∗
νe e ν e will also occur and can be tagged via the pattern of γs.
νe e νx e NC excitations are possible, although little information
10−2 x is available in the literature because of the low-energy
nuclearrecoilsthatmakeitpracticallydifficulttodetect.
Energy thresholds as low as few MeV may be possible.
5 10 15 20 25 30 35 40 45 50
E (MeV) Although the chances for detecting such a reaction are
ratherlow,weconsiderNCcoherentscatteringanywayas
FIG.3: Neutrinocrosssectionsonsomerepresentativetarget an example of a reaction scanning the low-energy region
nuclei (see text for details). of the neutrino spectra. The cross sections are reported
in Refs. [6, 24–27].
For the different channels that we consider, we use
by a quasi-thermal spectrum where the second energy the cross sections as in [1]. They have been used in the
moment is used to determine α. SNOWGLOBES software package [28].
However,other detection channels are also of interest.
Oneiselasticscatteringonelectronswherethecrosssec-
tion varies roughly linearly with E . Of greater interest IV. COUNTING RATES
ν
for our purpose of testing the goodness of the α-fit are
reactions with a higher energy threshold or steeper en- The detector counting rate as a function of incoming
ergy variation than IBD. One example is the recently neutrino energy is proportional to f (E)σ(E). Based on
ν
built HALO detector in SNOLAB where 79 tons of lead the numerical neutrino spectra for the 1016 ms model,
are used as target material. Other examples are reac- we show this quantity in Fig. 4 for our suite of interac-
tions on argon in future liquid argon detectors that are tion processes, normalized to the maximum rate of each
primarily consideredin the context of long-baseline neu- process. This plotdemonstratesthat the differentdetec-
trinooscillationstudiesandreactionsonoxygeninwater tionprocessesprobevastlydifferentpartsoftheneutrino
Cherenkov detectors and on carbon in liquid scintillator spectra.
detectors. We now compare the total counting rates for each of
Inourstudyweconsiderseveralselectedreactionsthat our reactions using as a neutrino spectrum either the
5

1 moments of the numericaldistribution, is sufficiently ac-
ν e
0.9 e
ν 16O (NC) TABLE II: Ratio of counting rate based on our spectral fit
0.8 e and thedirect numerical HR spectrum, N /N .
fit num
u.) 00..67 νx e νe 208Pb (NC) t[ms] νe40Ar νe208Pb νee νxe νe16O
a. CC CC NC
σ ( 0.5 261 1.01 1.02 1.00 1.00 1.01
fν 0.4 ν 40Ar (cohNC) 1016 1.02 1.06 1.01 1.00 0.91
e
0.3 1991 1.02 1.07 1.01 1.00 0.90
0.2
0.1 t[ms] ν¯ 40Ar ν¯ 208Pb ν¯ e ν¯ e ν¯ 16O
e e e x e
00 5 10 15 20 25 30 35 40 45 50 NC NC NC
E (MeV) 261 1.00 1.02 1.00 1.00 1.05
1016 1.01 1.03 1.00 1.00 1.16

1 1991 1.00 1.03 1.00 1.00 1.18
0.9 ν e ν 16O (NC)
e e
0.8
curate to predict the detector response for target nu-
0.7
clei with relatively high threshold energies. In order to
u.) 0.6 ν 208Pb (CC) answer this question reliably, we have produced high-
a. e
σ ( 0.5 resolution (HR) neutrino spectra at several postbounce
fν0.4 times of a spherically symmetric SN simulation. In ad-
dition, these spectra have allowed us to check for the
0.3
first time whether the coarser standard resolution (SR)
0.2 of typicalsupernova models (which is dictated by strong
0.1 νe 40Ar (CC) ν e CPU-time limitations) already allows robust signal pre-
x
dictions.
00 5 10 15 20 25 30 35 40 45 50
E (MeV) WehavecomparedSRandHRspectraandfittedboth
withthe analyticfitofEq.(1),basedonthefirsttwoen-
FIG. 4: Detection rate as a function of neutrino energy for ergymomentsofthenumericaldistribution. Thesefitted
different reactions, based on our 1016 ms model. Each curve spectra account well for the detection rates in SN neu-
isnormalizedtoitsmaximum. Top: Anti-neutrinos. Bottom: trino detectors with vastly different target nuclei. We
Neutrinos. conclude that for the purpose of signalforecast in differ-
ent detectors reasonably good accuracy can be achieved
byconsideringthe lowesttwoenergymomentsofnumer-
numerical output directly (rate N ) compared with
num ically produced spectra. Moreover,we have verified that
the rate inferred if the spectrum is represented by our
amodeststandardenergyresolutionasaffordableintyp-
fit function (rate N ). In Table II, we report the ra-
fit icalsupernovasimulationsalreadygivesenergymoments
tio N /N for neutrinos (top part) and antineutrinos
fit num surprisingly close to the HR case. This insight simplifies
(bottompart). The agreementbetweenN andN is
num fit parametricstudies ofdetectionforecastsfortheneutrino
very good and typically the error is only a few percent.
signalfromthe nextnearbySN becauseit is sufficientto
Even for the oxygen reaction, which probes the highest-
representthe non-equilibrium spectra by a simple three-
energy tail of all examples, the difference is only some
parameter quasi-thermal fit.
10–20%. Assuming complete flavor conversions due to
neutrinooscillations(ν →ν ,andthe sameforantineu-
e x
trinos), we tested that the ratio N /N differs from
fit num
theonesreportedinTableIIonlyontheleveloftenthsof Acknowledgments
a percent (results not shownhere). Such a result further
confirms the good quality of the alpha-fit. We acknowledge partial support by the Deutsche
Forschungsgemeinschaft under grant TR-7 “Gravita-
tional Wave Astronomy” and the Cluster of Excellence
V. CONCLUSIONS EXC-153“Originand Structure of the Universe” andby
the European Union Initial Training Network Invisibles
In this paper, we have investigated whether the sim- PITN-GA-2011-289442. I.T. acknowledges support by
ple analytic fit of Eq. (1), based on the first two energy the Alexander von Humboldt Foundation.
6
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