Analysis of matter suppression in collective neutrino oscillations duringthe supernova accretion phase

Sovan Chakraborty,1 Tobias Fischer,2,3 Alessandro Mirizzi,1 Ninetta Saviano,1 and Ricard Tomas1

1II Institut fur Theoretische Physik, Universitat Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany2GSI, Helmholtzzentrum fur Schwerionenforschung GmbH, Planckstraße 1 64291 Darmstadt, Germany

3Technische Universitat Darmstadt, Schlossgartenstraße 9, 64289 Darmstadt, Germany(Received 13 May 2011; published 5 July 2011)

The usual description of self-induced neutrino flavor conversions in core-collapse supernovae is based

on the dominance of the neutrino density n� over the net electron density ne. However, this condition is

not met during the post-bounce accretion phase, when the dense matter in a SN is piled up above the

neutrinosphere. As recently pointed out, a dominant matter term in the anisotropic SN environment would

dephase the flavor evolution for neutrinos traveling on different trajectories, challenging the occurrence of

the collective behavior in the dense neutrino gas. Using the results from recent long-term simulations of

core-collapse SN explosions, based on three-flavor Boltzmann neutrino transport in spherical symmetry,

we find that both the situations of complete matter suppression (when ne � n�) and matter-induced

decoherence (when ne * n�) of flavor conversions are realized during the accretion phase. The matter

suppression at high densities prevents any possible impact of the neutrino oscillations on the neutrino

heating and hence on the dynamics of the explosion. Furthermore, it changes the interpretation of the

Earth matter effect on the SN neutrino signal during the accretion phase, allowing the possibility of the

neutrino mass hierarchy discrimination at not too small values of the leptonic mixing angle �13 (i.e.,

sin2�13 * 10�3).

DOI: 10.1103/PhysRevD.84.025002 PACS numbers: 14.60.Pq, 97.60.Bw

I. INTRODUCTION

Massive star explosions are an active subject of research,in terms of core-collapse supernova (SN) models. Thesupernova problem is related to the revival of the stalledbounce shock, which forms when the collapsing stellarcore reaches nuclear matter density and bounces back. Itpropagates out of the core and stalls on its way out due tothe continuous energy loses via neutrino emission anddissociation of heavy nuclei. Independently from the ex-plosion mechanism (see, e.g., [1–7]) the total energy emit-ted in neutrinos (�) and antineutrinos ( ��) during a SN is ofthe order of several 1053 erg, which makes a SN the mostpowerful neutrino source in the Universe. The total dura-tion of the neutrino emission, known as the neutrino burst,can last up to 30 seconds. This � signal represents a power-ful tool to probe fundamental neutrino properties as well asthe dynamics of the explosion [8].

The role of astrophysical messengers played by neutri-nos during a stellar collapse is largely associated with thesignatures imprinted on the observable SN neutrino burstby weak processes and flavor conversions occurring deepinside the star. In the last few years, it has been understoodthat the description of neutrino flavor conversions in super-novae, based on the only Mikheyev-Smirnov-Wolfenstein(MSW) effect with the ordinary matter [9], was incom-plete. In particular, in the deepest supernova regions theneutrino density is so high that the neutrino-neutrino inter-actions dominate the flavor changes. Even if the importantrole played by the neutrino-neutrino interactions for the

flavor evolution in supernovae was pointed out since a longtime ago [10–14], only recently the first numerical experi-ments have been performed to characterize realisticallythese effects [15,16]. These seminal investigations havestimulated a still streaming torrent of activities (see, e.g.,[17], for a recent review). The general result of thesestudies is that neutrino-neutrino interactions create a largepotential for the neutrinos, which causes large and rapidconversions between different flavors. The transitions oc-cur collectively, i.e., in a coherent fashion, over the entireenergy range. The most important observational conse-quence of �-� interactions is a swap of the �e and ��e

spectra with the nonelectron �x and ��x spectra in certainenergy ranges [16–34].The basic idea behind the self-induced neutrino oscil-

lations in supernovae is that these can produce significantflavor conversions close to the � decoupling region (i.e., atsmall radii) even in the presence of a large matter density[15,35]. This would have inhibited any possible flavortransition, far from the MSW resonances (i.e., far outsidethe neutrino-matter decoupling region) at larger radii [36].Subsequently, it has been realized in [37], and then con-firmed in [38], that this original idea was only in part truesince the matter density cannot be arbitrarily large before itaffects collective flavor conversions after all. Indeed, forthe strongly anisotropic emission of neutrinos streamingoff the SN core, the matter effect is trajectory dependent,inducing different oscillation phases for neutrinos travelingin different directions. When the neutrino-neutrino inter-action is sufficiently strong, it forces neutrinos reaching the

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same position from different paths to have the same oscil-lation phase, overcoming the trajectory-dependent disper-sion induced by the matter. Conversely, when the electrondensity ne significantly exceeds the neutrino density n�, ithas been shown that the large phase dispersion induced bythe matter would suppress the collective phenomena.Depending on the electron density, the matter suppressioncan be complete, when ne � n�, or partial when the matterdomination is less pronounced. Finally, when ne * n�, theinterference of the two comparable effects would lead tothe decoherence of the collective neutrino flavor changes,producing an equal mixture between the oscillatingelectron and nonelectron neutrino species [37].

Most of the numerical simulations of the self-inducedflavor conversions in SN explosions have been focusedon the late times cooling phase (at post-bounce timestpb * 1 s), when the matter potential is smaller than the

neutrino-neutrino potential. In such a situation, collectiveoscillations would develop without any matter hindrance[16,18,31,32]. During the earlier phase before the onset ofan explosion, which is determined by mass accretion(tpb & 0:5 s), the electron density is not negligible with

respect to the neutrino density, as one can infer from thefollowing back-of-the-envelope calculation. Given themass accretion rate

_m ¼ @m

@rv ¼ 4��r2v; (1)

in terms of the rest-mass density � at radius r, and of thematter velocity v, one gets

�� 108g

cm3

�_m

0:4M�=s

��102 km

r

�2�104 km=s

v

�; (2)

for typical values in the region ahead the shock-wave.Then, the net electron density at r� 102 km can be esti-mated as

ne ¼ �Ye=mB � 1032 cm�3; (3)

where the electron fraction Ye ’ 0:5 and mB is the nucleonmass. On the other hand, during the accretion phase, wherethe charged-current reactions dominate, the far-distanceelectron (anti)neutrino luminosities can be expressed viathe change of the gravitational potential at the neutrino-sphere1 with radius r� as follows:

L�e; ��e ¼Gm

r_m

��������r�e; ��e

�1052erg

s

�m

1:5M�

��_m

0:4M�=s

�

��102 km

r

�: (4)

Therefore, the neutrino particle flux F�e; ��e¼ L�e; ��e=hE�e; ��e

i � 1057 s�1, for hE�e; ��ei ¼ 11 MeV, implying

n�e; ��e¼ F�e; ��e

4�r2� 1031 cm�3; (5)

at r� 102 km. From this simple estimation, n� can besmaller or on the same order than ne and hence we cannota priori assume that ne � n� holds during the post-bounceaccretion phase. This result is confirmed by many differentSN simulations [39–42]. It is a generic feature that appliesgenerally to core-collapse SNe of massive iron-core pro-genitors. As an important consequence, it is not guaranteeda priori that the results of the neutrino flavor evolutionobtained during the cooling phase, could be directly ap-plied to the accretion.The post-bounce accretion phase, where the neutrino

fluxes are large, represents a particularly interesting sce-nario for detecting signatures of neutrino flavor transfor-mations. Indeed, also the flavor-dependent flux differencesare large with a robust hierarchy for the neutrino numberfluxes, F�e

> F ��e� F�x

, where �x indicates the nonelec-

tron flavors. In such a situation, if dense matter effects areneglected and the flavor asymmetries are enough to preventmulti-angle effects [43], the self-induced flavor oscilla-tions seemed to have a clear outcome, producing a com-plete exchange of the electron and nonelectron flavorsfor almost all antineutrinos in the inverted neutrino masshierarchy (IH, �m2

atm ¼ m23 �m2

1;2 < 0) and a spectral

split in the energy distributions of the neutrinos [18]. Inthe normal neutrino mass hierarchy (NH,�m2

atm > 0), theywould leave the neutrino spectra unaffected. How thispicture would change if the effect of the dense matter isincluded, is subject of investigation of the present article. Itdevelops and clarifies the results recently presented in ourpaper in [44].We take as benchmark the results of the recent long-term

SN simulations from Ref. [7], to characterize the SNneutrino signal as well as the matter density profile evolu-tion. We consider three cases corresponding to differentsupernova progenitor masses, the 8:8M� O-Ne-Mg-coreand the 10:8M� and 18:0M� iron-cores. For these threecases we find that the matter density is always larger thanor comparable to the neutrino density, during the accretionphase. This finding suggests the possible hindrance (ordecoherence) of collective neutrino flavor transitions formatter densities higher than (or comparable with) theneutrino densities during the accretion phase.The plan of our paper is the following. In Sec. II we

describe the original supernova neutrino emission duringthe accretion phase and in Sec. III we introduce the sche-matic supernova neutrino model we use to characterize the� emission geometry, the neutrino potentials and the differ-ent radial ranges where self-induced oscillations and mat-ter effects are relevant. The results of our analysis onmatter suppression for the three different SN progenitormasses are presented in Sec. IVand the impact of the densematter suppression on the interpretation of the Earth mattereffect on the SN neutrino burst is discussed in Sec. V.

1The neutrinospheres are the neutrino energy and flavor-dependent spheres of last scattering.

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Finally, we close with comments of our results and theconclude in Sec. VI.

II. NEUTRINO SIGNAL FROM CORE-COLLAPSESUPERNOVA EXPLOSIONS

We investigate core-collapse supernova simulations ofmassive progenitor stars of 8.8, 10.8 and 18M�. The firstone belongs to the class of O-Ne-Mg-core progenitors[45,46] and represents the threshold between thermonu-clear explosions and core-collapse supernovae [4,7]. Thelatter two are iron-core progenitors [47]. All models wereevolved consistently through core collapse, bounce andthe early post-bounce phase up to several seconds afterthe onset of explosion [7]. The core-collapse model isbased on general relativistic radiation hydrodynamicsthat employs three-flavor Boltzmann neutrino transport inspherical symmetry and a sophisticated equation of statefor hot and dense nuclear matter [48] (for details about thesupernova model, see [7,49] and references therein).Explosions of massive iron-core progenitors cannot beobtained in spherically symmetric supernova models. Inorder to trigger the explosions for the 10.8 and 18M�progenitor models, the heating rates are artificially en-hanced in the gain region where neutrinos deposit energyin order to revive the stalled bounce shock.

Figure 1 shows the evolution of the neutrino luminositiesL��

(left panels) and average energies hE��i (right panels),during the post-bounce accretion phase for all modelsunder investigation. Here, �� ¼ ð�e; ��e; �xÞ where �x in-dicates both ð�; �Þ-neutrinos and antineutrinos. The largeelectron flavor neutrino luminositiesOð1052Þ erg=s are dueto the continuous mass accretion at the neutrinospheres,where the electron flavor neutrino luminosities are domi-nated by the charged-current reactions. The slightly larger��e luminosity over the �e luminosity and the magnitude ofthe differences obtained in the models discussed here, is anactive subject of research and may change slightly apply-ing improved weak rates and multidimensional supernovamodels. The increasing and decreasing electron flavorluminosities reflect the propagating bounce shock (andthe mass accretion rate at the neutrinospheres), whichcontracts and expands accordingly during the accretionphase driven by neutrino heating/cooling. Since muoniccharged-current processes are suppressed, ��=�’s are pro-

duced only after bounce via pair-processes and the (�=�)-neutrino luminosities are generally smaller than theelectron flavor neutrino luminosities. For the average en-ergies, the following hierarchy holds hE�e

i< hE ��ei< hE�x

ifor all models under investigation during the accretionphase. Furthermore, the average energies rise continuouslyfor all neutrino flavors during the accretion phase illus-trated in Fig. 1 (right panels).

After the onset of explosion, when mass accretion van-ishes, the situation changes. The luminosities and averageenergies of all flavors decrease on a longer time scale on

the order of seconds. For the O-Ne-Mg-core, we considerthe neutrino signal only up to tpb ¼ 0:25 s (top panel of

Fig. 1). Note that mass accretion at the neutrinospheresvanishes already at about tpb ¼ 0:03 s. This determines the

onset of the explosion for this low-mass progenitor. For the10.8 and 18M� progenitor models discussed here, we showthe neutrino signal up to tpb ¼ 0:6 s. For these cases, the

onsets of explosion occur at about tpb ¼ 0:36 s, due to the

more massive envelopes surrounding the iron-core thatlead to a more extended accretion (and hence neutrinoheating) phase. Note the sharp drop of the luminositiesand average energies of all neutrino flavors in Fig. 1 afterthe onsets of explosion. These are due to the sudden flip ofmatter velocities from infall to expansion when the explo-sion shock passes through the distance of 500 km, wherethe observables are measured in a comoving frame ofreference. We note further that these results suggest loweraverage energies than often assumed in the literature [50]and a less pronounced spectral hierarchy, in particular,during the later proto-neutron star cooling phase. Theresults obtained for the low-mass O-Ne-Mg-core-collapsesupernova explosion and long-term simulation of theproto-neutron star cooling phase, are in qualitative andquantitative good agreement with recent simulations per-formed by the Garching group [51].

FIG. 1. Neutrino luminosities (left panels) and average ener-gies (right panels) for the core-collapse SN simulations consid-ered, where we distinguish �e (continuous line), ��e (dashed line)and �x (dotted line). The three cases are shown corresponding todifferent values of the supernova progenitor mass: 8:8M� (top),10:8M� (middle) and 18:0M� (bottom).

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III. SCHEMATIC SUPERNOVA MODEL

We present here our schematic assumptions to character-ize the flavor conversions during the accretion phase. Inparticular, we describe our model for the SN neutrinoemission geometry, the neutrino number fluxes of differentflavors, the neutrino-neutrino and matter potentials and thedifferent oscillation regimes.

A. Neutrino emission geometry

We assume a spherically symmetrical source (‘‘SNcore’’) that emits neutrinos and antineutrinos like a black-body surface, from a neutrinosphere at radius r ¼ r�. Wedefine r� to be the radius where the neutrino radiation fieldis assumed to be ‘‘half-isotropic,’’ i.e., all outward-movingangular modes are equally occupied [43]. We remind thatthis conventional definition of the neutrinosphere, used inthe context of the neutrino flavor conversions, is onlyintended to fix a boundary condition for the subsequentflavor evolution. We assume that weak processes take placeat higher densities inside the neutrino-matter decouplingsphere and are negligible outside the decoupling spherewhere neutrino oscillations can occur. The half-isotropicdefinition for the decoupling spheres does not necessarilycoincide with the definition of the neutrinosphere as neu-trino last scattering surface, defined as the radius where theoptical depth becomes 2=3. However, in the context ofcollective oscillations which start at radii much largerthan the assumed boundary, neutrinos are safely in afree-streaming regime [31]. Therefore, we do not have toworry about the details of the � decoupling.

Instead of choosing arbitrarily the neutrinosphere radius,we fix it consistently with the SN simulations which pro-vide the angular distributions of the different neutrinospecies as a function of time and energy at different radii.For definiteness, we have taken the �e’s distribution asrepresentative for all the different flavors, since electronneutrinos reach the free-streaming regime at larger radiicompared to the other neutrino species during the accretionphase. The angular distributions are functions of the en-ergy. For definiteness, we consider as reference the rootmean square (rms) energy relevant for the averageneutrino-nucleon interaction rates, that determine the neu-trino angular distributions. As an example, in Fig. 2 we plotthe �e angular distribution fðcos�rÞ ¼ dn�=d cos�r, where�r is the zenith angle of a given mode relative to the radialdirection at distance r. We consider Erms ¼ 11:5 MeV atthree different radii at the post-bounce time tpb ¼ 0:4 s,

taken form the 10:8M� model. As expected, at small radii(r ¼ 20 km in the Fig. 2) the angular distribution is iso-tropic since neutrinos are in a trapping regime. They areisotropically emitted in all directions. At large radii(r ¼ 70 km in the Fig. 2), neutrinos are free-streamingand their angular distribution becomes forward peaked.We schematically assume as neutrinosphere radius, theone at which the �e’s angular distribution has no longer

significant backward flux, i.e., a few % of the total one (atr ¼ 50 km in the Fig. 2). Even if there the real angulardistribution is not half-isotropic, we assume it to character-ize the further flavor evolution. For our purpose of evalu-ating the impact of the matter effects on the self-inducedflavor conversions, this simplified choice is conservative.Indeed, the real angular distributions, which are moreforward peaked than what we are assuming, would reducethe real strength of the neutrino-neutrino potential withrespect to what we are assuming.We stress that this procedure for fixing the neutrino-

sphere has to be taken only as an empirical prescription.However, we performed several neutrino flavor oscillationanalysis changing by a factor of 3 the neutrinosphereradius. We find that the results on matter effects remainedbasically unchanged, indicating a weak dependence of theoscillation analysis from the decoupling region.

B. Neutrino number fluxes for different flavors

The neutrino particle flux for the different flavors isdefined as follows F�� ¼ L��

=hE��i. In order to character-

ize the flux asymmetries among the different � species weintroduce the parameter [43]

� ¼ F�e� F ��e

F ��e� F ��x

; (6)

where we assume F�x¼ F ��x

.

In Fig. 3 we show the neutrino number flux F��(left

panels) and asymmetry parameter � (right panels) forthe three benchmark SN simulations based on the differ-ent progenitor models. The particle fluxes present theexpected hierarchy, F�e

> F ��e� F�x

, during the accretion

phase. The first part of the hierarchy is caused by the

FIG. 2. 10:8M� progenitor mass. Angular distributions for �e

at tpb ¼ 0:4 s for Erms ¼ 11:5 MeV at different radii: r ¼ 20 km

(dotted line), r ¼ 50 km (continuous line), r ¼ 70 km (dashedline).

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deleptonization of the collapsed stellar core whereas thesecond is due to the absence of charged-current interac-tions for neutrino species other than �e and ��e. As aconsequence of this hierarchy, the electron and nonelectronneutrino particle fluxes differ by almost a factor of 2 duringthe accretion phase. Conversely, flux differences tend tobecome much smaller during the cooling of the SNremnant.

Passing to the asymmetry parameter � [Eq. (6)], thestrong excess of �e’s during the early post-bounce dele-ptonization (tpb & 0:05 s), increases � � 1. Then, � drops

and reaches values between 0.3–0.5 during the accretionphase (see Fig. 3). After that, since the flux differenceF ��e

� F ��x drops more rapidly than F�e� F�x

, � rises again

and becomes larger than 1 for the iron-core SNe. We willsee that this behavior will have important consequences onthe development of the self-induced flavor conversions.

C. Nonlinear neutrino flavor mixing

Our description of the nonlinear neutrino flavor conver-sions is based on the two-flavor oscillation scenario,driven by the atmospheric mass-square difference�m2

atm ’2:6� 0�3 eV2 and by a small (matter suppressed) in-medium mixing �eff ¼ 10�3 [52,53]. Three-flavor effects,associated with the solar sector, are small for the neutrinoflux ordering expected during the accretion phase [32,54].

In the normal mass hierarchy and for the spectralordering of the accretion phase, no self-induced flavor

conversion will occur and thus dense matter cannot pro-duce any sizable new effect. Therefore, in the following wewill always refer to the inverted mass hierarchy case.The impact of the nonisotropic nature of neutrino emis-

sion on the self-induced flavor conversions is taken intoaccount by ‘‘multi-angle’’ simulations [16], where onefollows a large number ½Oð103Þ� of neutrino trajectories.The � emitted from a SN core naturally have a broadenergy distribution. However, this is largely irrelevant forour purposes, since large matter effects would lock to-gether the different neutrino energy modes, both in thecase of suppression [37] and of decoherence [43] of col-lective oscillations. Therefore, in order to simplify thenumerical complexity of this problem, we followRefs. [37,43] and assume that all the neutrinos are repre-sented by a single energy E ¼ 15 MeV. This approxima-tion is reasonable since our main purpose is to determine ifthe dense matter affects the development of the self-induced transformations. This choice results in the neu-trino vacuum oscillation frequency

! ¼��m2

atm

2E

�¼ 0:4 km�1: (7)

The strength of the neutrino-neutrino interaction, nor-malized at the neutrinosphere, is parametrized by [43]

�r ¼ffiffiffi2

pGF½n ��e

ðrÞ � n ��xðrÞ�

¼ 7� 105 km�1

�L ��e

hE ��ei �

L ��x

hE ��xi�15 MeV

1052 erg

�10 km

r

�2

(8)

where n��ðrÞ is the flux of the neutrino species �� at radius

r [for a definition of n��, see Eq. (5)]. The matter potential

is represented by [36]

r ¼ffiffiffi2

pGFneðrÞ

¼ 1:9� 106 km�1 ��Ye

0:5

���

1010 g=cm3

�(9)

encoding the net ne � ne� � neþ electron density, whereYe ¼ Ye� � Yeþ is the net electron fraction and � is thematter density. The neutrino term �r declines always asr�2 due to the geometric dilution outside the decouplingspheres, whereas r is given by the detailed matter profilefrom the SN simulations.

D. Oscillation regimes

In the absence of matter suppression, collective neutrinoflavor transformations will start outside the synchroniza-tion radius, given by [43]

rsyncr�

¼� ffiffiffiffiffiffiffiffiffiffiffiffi

1þ �p � 1

2

�1=2

��rjr¼r�

!

�1=4

; (10)

FIG. 3. Neutrino particle fluxes (left panels) and flavor asym-metry parameter � (right panels) for �e (continuous line), ��e

(dashed line), �x (dotted line).

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and are expected to develop at least until radii when theneutrino-neutrino interaction strength becomes compa-rable to the vacuum term, i.e., at [18]

rendr�

¼��rjr¼r�

2!

�1=4

: (11)

We remind that outside rsync the multi-angle nature of the

neutrino trajectories in a supernova can lead to self-induced flavor decoherence between different angularmodes. However, it has been shown in [43] that if theasymmetry parameter � is significantly large, multi-angleeffects are suppressed and self-induced neutrino oscilla-tions exhibit a collective behavior. We will come back tothis point in more details when commenting our numericalresults.

Using the above definitions, in Fig. 4 we present theneutrinosphere radius r�, the synchronization radius rsyncand the radius rend at which collective effects saturate,for the case of the 10:8M� SN simulation. We see thatr� � 102 km during the accretion phase and drops to�30 km at the beginning of the cooling phase. This con-traction of the neutrinosphere radius reflects the gradualshift of the � opacities. These change from being absorp-tion dominated during the accretion, to being scatteringdominated during the cooling [55,56].

Collective oscillations are expected in the range r 2½rsync; rend�. This range is at r� ½600; 1500� km at tpb ¼0:1 s when the neutrinosphere radius r� > 102 km, push-ing to larger radii the flavor conversions. Then, followingthe contraction of the neutrinosphere radius, the rangeof conversions shifts at smaller radii, moving to r�½200; 500� km at tpb ’ 0:4 s. Finally at tpb ’ 0:6 s, the

lower neutrino luminosity and the larger asymmetryparameter � both conspire to push rsync towards rend [see

Eqs. (10) and (11)].

In the range ½rsync; rend�, the self-induced neutrino flavorconversions can be affected by the dense matter when [37]

ne * n ��e� n ��x

: (12)

In particular, when the two densities are comparable mattereffects induce multi-angle decoherence among differentneutrino modes, leading to a flavor equilibration amongthe different species. When the net electron density issignificantly larger than the neutrino density, collectiveoscillations are suppressed at all [37].We remark that previous numerical studies of self-

induced flavor conversions typically assumed smaller neu-trinosphere radii, namely r� ¼ Oð10 kmÞ (see, e.g., [43]).This choice is more appropriate for the cooling rather thanfor the accretion phase. As a consequence of this differentinput, in the current work flavor conversions start at largerradii than typically assumed in previous studies.

IV. MATTER EFFECTS FOR DIFFERENTPROGENITOR MASSES

In this section we will present our results concerning theeffects of dense matter on of the self-induced neutrinooscillations for the three representative SN simulationsbased on the different progenitor masses.

A. 10:8M� progenitor mass

We start our investigation with the case of the 10:8M�iron-core SN. In Fig. 5 we show the net electron densityne (left panel) and the difference of neutrino densitiesn ��e

�n ��x(right panel) entering in the potential �r

[Eq. (8)] for different post-bounce times. While the declineof the neutrino density is r�2, the electron density presentsa more complicated behavior. In particular, one can recog-nize the abrupt discontinuity in ne associated with thesupernova shock front that propagates in time. Theshock-wave initially dissipates its energy due to heavynuclei dissociation and eventually stalls at tpb ¼ 0:05 s.

The standing accretion shock is revived via neutrino heat-ing, on a time scale on the order of 0.1 seconds, andexpands accordingly as well as contracts via neutrinocooling. After 0.35 seconds post-bounce, neutrino heating

FIG. 4 (color online). 10:8M� progenitor mass. Time evolu-tion of r� (continuous line), rsync (dashed curve) and rend (dotted

curve).

FIG. 5. 10:8M� progenitor mass. Radial evolution of the netelectron density ne (left panel) and of the neutrino densitydifference n ��e

� n ��x(right panel) at different post-bounce times.

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succeeds and the standing accretion shock turns into adynamic shock with positive matter velocities. It initiatesthe onset of explosion at about 0.4 seconds post bounce.

For this SN model we find that during the accretionphase the matter density in the post-shock region declinesslower than the neutrino density, typically as�r�1:5. Fromthe comparison between the electron and the neutrinodensities, we realize that at the different post-bounce timesconsidered, ne is always larger than or comparable ton ��e

� n ��x. It suggests that one cannot ignore matter effects

on self-induced flavor transformations during the accretionphase.

In order to quantify the relative strength of the electronand neutrino densities, in Fig. 6 we show the ratio

R ¼ nen ��e � n ��x

(13)

as a function of the radial coordinate r at different post-bounce times for tpb 2 ½0:1; 0:6� s. The range rsync < r <

rend is delimited with two vertical dashed lines. The valuesof rsync and rend determine the possible range for the self-

induced flavor conversions and the shock radius rsh denotesthe abrupt drop in the electron density. Therefore, theirrelative position is crucial to assess the impact of mattereffects. In the expected oscillation range, R � 1will implya strong matter dominance in the flavor conversions andthus complete suppression of the self-induced effects.

Instead, when electron and neutrino densities are compa-rable (R * 1), decoherence will occur for the collectiveoscillations.The ratio R, being very large behind the shock front,

prevents flavor conversions in this region. However, theratio can go down to R * 1 for r > rsh, leading to matter-induced decoherence and thus partial flavor changes.Let us discuss in more detail what occurs at different

post-bounce time snapshots in Fig. 6. At very early times(tpb ¼ 0:1 s) the matter term is strongly dominant also

behind the shock-front (R � 1). Under these conditionsoscillations are always blocked. Then, at intermediatetimes (tpb ¼ 0:225, 0.3 s) the matter density in the post-

shock region, where flavor conversions are possible, isdropping faster than the neutrino one. Therefore, the ratioR drops at 1–2 in this range and matter-induced decoher-ence is possible in this case. Subsequently, at tpb¼0:325 s,

oscillations are suppressed behind the shock front, but thendecoherence will develop at larger radii (r * 300 km)when R * 1. Eventually at later times (tpb ¼ 0:4, 0.6 s),

since the shock has resumed its forward motion, the regionrelevant for the oscillations is at r < rsh, where R � 1. Inthis situation self-induced oscillations will be suppressed.From these different snapshots we realize that R has apeculiar nonmonotonic behavior as a function of time. Itsuggests a time-dependent pattern for the matter effects onthe self-induced transitions during the accretion phase,namely, complete-partial-complete suppression.In order to confirm these expectations, we have per-

formed a multi-angle numerical study of the equationsfor the neutrino flavor evolution in the schematic modeldescribed in Sec. III. Our treatment closely follows the onepresented in Ref. [37] to which we address the interestedreader for further details. We only mention here that inorder to achieve convergence in our simulations we had tosimulate 103 neutrino angular modes. In Fig. 7 we show theradial evolution of the ��e survival probability Pee fordifferent post-bounce times, obtained taking into accountthe effects of the SN matter profile (continuous curve). Forcomparison, we show also the results obtained settingne ¼ 0 (dashed curve).In the case with ne ¼ 0, for the given flavor asymmetry

� * 0:3 we would have expected the ‘‘quasisingle angle’’behavior described in Ref. [43], where after the onset of theconversions at r ¼ rsync, the survival probability Pee de-

clines smoothly approaching zero at large radii. However,in the situation we are studying flavor conversions developat radii larger than what is typically shown in previousworks (see, e.g., [43]). Therefore, the evolution is moreadiabatic (i.e., the evolution length scale l� � r [30]). As a

consequence, effects of self-induced multi-angle decoher-ence have more chances to develop in this case, producingsome small disturbance in the smooth decline of the sur-vival probability at large radii (visible at tpb ¼ 0:1, 0.3,

0.325 s for r * 700 km). This finding is potentially

FIG. 6 (color online). 10:8M� progenitor mass. Radial evolu-tion of the ratio R between electron and neutrino densities atdifferent post-bounce times. The two dashed vertical stripsdelimit the position of rsync (left line) and rend (right line).

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interesting, however, since matter effects will anyhowdramatically alter this picture we have not performed asystematic study on this (subleading) self-induceddecoherence.

Passing now to the matter case, we see that at tpb ¼ 0:1,

0.4, 0.6 s the flavor oscillations are completely blocked,since R � 1 in the conversion range. For the other threeintermediate times (tpb ¼ 0:225, 0.3, 0.325 s), the presence

of a large matter term at rsync will significantly delay the

onset of the flavor conversions with respect to the case withne ¼ 0. Then, at larger radii (r > 700 km) when R * 1–2,matter effects produce a partial suppression of the flavorconversions, with a tendency toward flavor decoherence.When the latter is complete, it leads to Pee ! 1=2 (attpb ¼ 0:3, 0.325 s), implying a complete mixture between

��e and ��x.Finally, we would summarize our results with a figure of

merit that captures in a compact way the interplay betweenmatter and self-induced effects as a function of time.Therefore, in Fig. 8 we show the ratio R (upper panel) atits minimum value

Rmin ¼ min

�ne

n ��e� n ��x

�r2½rsync;rend�

; (14)

in the range r 2 ½rsync; rend� relevant for collective conver-sions. This value provides a conservative estimation for the

impact of matter effects. Indeed, if Rmin � 1 one shouldexpect a complete matter suppression of the flavor changes.Conversely, when Rmin * 1 the matter suppression will bepartial and will lead to flavor decoherence. For comparisonwe also show the values of R at the two radii rsync and rend.

Moreover, in the lower panel we represent the time evolu-tion of the radial positions rsync, rend, rmin and rsh.

One realizes that R presents a nontrivial behavior in therange between rsync and rend. In particular, the position of

the minimum value Rmin is at

rmin ¼�rsh if rsync < rsh < rendrsync otherwise.

(15)

For tpb & 0:3 s, since the neutrino density grows and the

electron density decreases in time, Rmin decreases mono-tonically from �20 at tpb ¼ 0:05 s to �1–2 at tpb ’ 0:3 s

post bounce, changing from complete to partial mattersuppression. Then, since at the end of the accretion phasethe neutrino density decreases faster than the electrondensity, Rmin rises again to �20 at tpb ’ 0:4–0:6 s,

suppressing flavor conversions. The behavior of Rmin

FIG. 7 (color online). 10:8M� progenitor mass. Radial evolu-tion of the survival probability Pee for electron antineutrinos atdifferent post-bounce times for the multi-angle evolution inpresence of matter effects (continuous curve) and for ne ¼ 0(dashed curve).

FIG. 8 (color online). 10:8M� progenitor mass. Upper panel:Time evolution of the ratio Rsync (dashed curve), Rmin (continu-

ous curve) and Rend (dotted curve). Lower panel: Time evolutionof the radial position rsync (dashed curve), rmin (continuous

curve), rend (dotted curve) and rshock (dot-dashed curve).

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shows in a compact way the succession of phases withcomplete-partial-complete matter suppression in the timeevolution of the neutrino self-induced conversions duringthe accretion phase.

A similar analysis to the one presented above based onFig. 8, may allow to obtain a first answer about the role ofthe matter on the self-induced effects. This schematicapproach does not require a detailed numerical study ofthe neutrino flavor evolution. For this reason, we find ituseful to compare the role of matter in different SN simu-lations based on different progenitor models. In the follow-ing, we will apply such analysis to the other SNsimulations from Ref. [7] based on different progenitormasses.

B. 18:0M� progenitor mass

As further example, we consider the SN simulation ofthe more massive 18M� iron-core progenitor. In Fig. 9 weshow the net electron density ne (left panel) and the dif-ference of neutrino densities n ��e

� n ��xfor different post-

bounce times. From the comparison with Fig. 5 we see thatthe time evolution of the electron and neutrino densityprofiles is similar to the case 10:8M� SN simulation.Therefore, we expect a similar impact for the matter effectson the neutrino flavor evolution. In Fig. 10 we show thetime evolution of Rmin in the same format as in Fig. 8. Wesee that the behavior of Rmin is also very similar to theprevious case described above. Therefore, one expect ananalogous pattern of complete 0:05 & tpb & 0:2 s), partial

(0:2 & tpb & 0:35 s) and complete (0:35 & tpb & 0:6 s)

matter suppression in the self-induced flavor conversions.We have repeated the same analysis also for a 15:0M�

iron-core progenitor mass, giving a not-exploding super-nova. In the time range before the recollapse of the star(tpb & 0:3 s) the results obtained are similar to the one

shown for the two exploding cases. Therefore, for thesake of the brevity, we will not show them here.

C. 8:8M� progenitor mass

Finally, we pass to analyze the case of the SN of thelow-mass 8:8M� O-Ne-Mg core. In Fig. 11 we present the

net electron density ne (left panel) and the difference ofneutrino densities n ��e

� n ��x(right panel) for different post-

bounce times. We realize that since in this case the matterdensity of the envelope is very low compared to theiron-core progenitors, the electron density profile abovethe core is very steep, declining as �r�2:5, faster than theneutrino density. Moreover, also the neutrino densities fortpb & 0:2 s are roughly a factor�3 smaller than in the case

of the iron-core supernovae. This reflects the practicalabsence of an extended accretion phase for this low-massstar. In this case the explosion succeeds very shortly afterthe core bounce. Therefore, the shock-front is alreadybeyond the radial range interesting for the flavorconversions.Figure 12 shows the time evolution of Rmin in the range

r 2 ½rsync; rend� relevant for collective conversions. We

realize that the behavior of Rmin is different with respectto the previously considered iron-core supernovae. Since inthis case the matter does not present an abrupt discontinu-ity, but it monotonically decreases, the position of Rmin willalways coincide with rend. Moreover, since the electron

FIG. 9. 18:0M� progenitor mass. Radial evolution of the netelectron density ne (left panel) and of the neutrino densitydifference n ��e

� n ��x(right panel) at different post-bounce times.

FIG. 10 (color online). 18:0M� progenitor mass. Upper panel:Time evolution of the ratio Rsync (dashed curve), Rmin (continu-

ous curve) and Rend (dotted curve). Lower panel: Time evolutionof the radial position rsync (dashed curve), rmin (continuous

curve), rend (dotted curve) and rshock (dash-dotted curve).

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density is never much larger than the neutrino density inthe conversion region, it results in Rmin & 3.

The peculiar time evolution of Rmin in this case willdepend on the small differences between the matter and theneutrino term. At tpb & 0:13 s, the electron density is

smaller than the neutrino density in the range of collectiveconversions, giving Rmin & 1. Then, the ratio Rmin growsup to �2–3 at tpb ’ 0:2 s since the electron density in the

conversion region is larger than in the previous case. As aresult, the time evolution of the neutrino survival proba-bilities presents a change between a regime dominated by

the �-� effects at very early times and one of matter-suppressed oscillations at later times.In Fig. 13 we explicitly show this behavior, representing

the radial evolution of the ��e survival probability Pee forthe same post-bounce times snapshots of Fig. 11, with(continuous curves) and without (dashed curves) matter.We find that at tpb ¼ 0:08 s where R & 1, the matter

suppression is relatively small and the neutrino-neutrinointeractions produce a quite almost complete swap be-tween ��e and ��x spectra (Pee ¼ 0:15 at the end of theevolution). Conversely, for the later times considered, theflavor conversions are strongly suppressed with a finalPee ’ 0:7–0:9, since R * 1. We want to remark that thetime evolution of the neutrino oscillation probability dur-ing the accretion phase is significantly different with re-spect to the case of the iron-core supernovae we studied.This will allow the distinction of a O-Ne-Mg-core SN froma iron-core SN, in the case of the detection of a futuregalactic event.

V. EARTH MATTER EFFECT DURING THEACCRETION PHASE

It is plausible that a high-statistics neutrino detectionfrom a future galactic SN [57,58] would faithfully monitorthe abrupt time variations imprinted on the neutrino oscil-lation probabilities by the transitions between stages ofcomplete and partial matter suppression during the accre-tion phase. We plan to perform a dedicated investigation ofthe matter suppression during the accretion phase on theobservable SN neutrino burst in a future work.Here we discuss another consequence of the SN dense

matter effects, namely, the change of the interpretationof the Earth matter effect on the supernova neutrino signal.

FIG. 11. 8:8M� progenitor mass. Radial evolution of the netelectron density ne (left panel) and of the neutrino densitydifference n ��e

� n ��x(right panel) at different post-bounce times.

FIG. 12 (color online). 8:8M� progenitor mass. Upper panel:Time evolution of the ratio Rsync (dashed curve), Rmin (continu-

ous curve) and Rend (dotted curve). Lower panel: Time evolutionof the radial position rsync (dashed curve), rmin (continuous

curve), rend (dotted curve).

FIG. 13 (color online). 8:8M� progenitor mass. Radial evolu-tion of the survival probability Pee for electron antineutrinos atdifferent post-bounce times for the multi-angle simulations inpresence of matter effects (continuous curve) and for the casewith ne ¼ 0 (dash-dotted curve).

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It is widely known that if SN neutrinos arrive at thedetector from ‘‘below,’’ the Earth crossing will induce anenergy-dependent modulation in the neutrino survivalprobability [59]. The appearance of the Earth effect de-pends on the neutrino fluxes and on the mixing scenario.The accretion phase is particularly promising to detectEarth crossing signatures because the absolute SN � fluxis large and the flavor-dependent flux differences are alsolarge.

Before the inclusion of the collective effects in the SNneutrino flavor transitions, the detection of Earth mattermodulations has been proposed as a tool to distinguish theneutrino mass hierarchy at ‘‘large’’ values of the mixingangle �13 (i.e., sin2�13 * 10�3) [60]. Then, when collec-tive oscillations were taken into account, it was speculatedthat due to the self-induced flavor transformations, thepresence or absence of Earth matter effects will break thedegeneracy between normal and inverted mass hierarchyfor ‘‘small’’ values of the mixing angle �13 (i.e., sin

2�13 &10�5) while no hierarchy discrimination is possible forlarger values of �13 [61]. Now, due to the matter suppres-sion effect this conclusion should be revisited for theaccretion phase.

To determine the observable SN neutrino fluxes at Earth,we refer to [54]. We consider the modified flavor basis,

ð�e; �x; �yÞ, which is defined such that ð�e; �x; �yÞ ¼Ry23ð�23Þð�e; ��; ��Þ, where R23 is the 2–3 rotation matrix.

This will allow for collective oscillations between e and ystates. For definiteness, here we concentrate on the ��e

spectra observable through inverse beta decay reactions,��e þ p ! nþ eþ, at water/ice Cherenkov or scintillationdetectors [57,62].

In normal hierarchy, the ��e flux at Earth, FD��e, (without

collective effect and no MSW effect associated to �13) forany value of the mixing angle �13 is given by [60]

FD��e¼ cos2�12F ��e

þ sin2�12F ��x; (16)

where �12 is the 1–2 mixing angle, with sin2�12 ’ 0:3[52,53].

In the inverted mass hierarchy, MSW matter effects inthe SN envelope are characterized in terms of the level-crossing probability PH of antineutrinos, which is ingeneral a function of the neutrino energy and �13 [36].In the following, we consider two extreme limits, namelyPH ! 0 when sin2�13 * 10�3 (‘‘large’’), and PH ! 1when sin2�13 & 10�5 (‘‘small’’). In this situation and forcomplete matter suppression of collective oscillations forlarge �13, F

D��eis given by

FD��e¼ F ��x

; (17)

while small �13 results in the same case as for normalhierarchy [Eq. (16)].

In the case of matter-induced decoherence between ��e

and ��y in inverted hierarchy, leading to a complete mixture

of ��e and ��y, the detectable ��e flux for any value of the

mixing angle �13 would be given by

FD��e¼ cos2�12

F ��eþ F ��y

2þ sin2�12F ��x

: (18)

Earth effect can be taken into account by just mappingcos2�12 ! Pð ��1 ! ��eÞ and sin2�12 ! 1� Pð ��1 ! ��eÞ.where Pð ��1 ! ��eÞ is the probability that a state enteringthe Earth as mass eigenstate ��1 is detected as ��e at thedetector [59].Considering for definiteness the case of iron-core SNe,

the presence or absence of Earth matter effects at earlytimes (tpb & 0:2 s) will allow to distinguish the neutrino

mass hierarchy at large value of the mixing angle �13 [seeEqs. (16) and (17)]. Moreover, the Earth effects maybecome observable via the interesting time pattern intro-duced above, in the case of inverted mass hierarchy at large�13: (1) no effects at early times [Eq. (17)], (2) the appear-ance at intermediate times when decoherence occurs[Eq. (18)], and (3) disappearance again when the accretionrate decreases significantly.

VI. DISCUSSION AND CONCLUSIONS

Different simulations of core-collapse SNe [39–42],show independently that the matter density in the stellarinterior is large during the post-bounce accretion phasebefore the onset of an explosion. This implies that self-induced neutrino flavor transformations are affected by thehigh matter density, through trajectory-dependent phe-nomena [37].In order to characterize the SN � flavor evolution quan-

titatively, we performed a dedicated study, where we tookas benchmark for the SN neutrino emissivity and the matterprofiles the recent long-term core-collapse SN simulationsfrom Ref. [7]. We considered three different cases for thesupernova progenitor mass, the low mass 8:8M� O-Ne-Mg-core and the two more massive iron-core progenitorsof 10.8 and 18M�. In all three cases the electron density neis never negligible in comparison to the neutrino densityn� during the accretion phase. As a consequence, thetrajectory-dependent matter effects always influence thedevelopment of the self-induced transformations. We real-ized that during the accretion phase both the condition ofmatter suppression (ne � n�) and matter-induced deco-herence (n� � ne) are realized. This implies that interest-ing time-dependent features in the neutrino signal canoccur at early times. Moreover, the different patterns ofcomplete/partial matter suppression would allow, at least inprinciple, to distinguish from the flavor evolution betweeniron-core and O-Ne-Mg core SNe.In early schematic investigations, the accretion phase

seemed a particularly clear setup to probe the developmentof the collective neutrino transformations, because theexpected flux hierarchy is large and robust. In such asituation the pattern of the self-induced spectral swaps/

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splits were thought to be unambiguous [32]. In this sce-nario, the presence or absence of Earth matter effects onthe SN neutrino burst were proposed to break the degen-eracy between normal and inverted mass hierarchy for‘‘small’’ values of the mixing angle �13 [61]. Our findingsseem to change this picture once more. In particular, thedense matter suppression of collective oscillations insupernovae implies that Earth matter effects may allow todistinguish the neutrino mass hierarchy in the likely case oflarge �13 [63], as originally expected in the analysis per-formed without the inclusion of the collective phenomena.This intriguing possibility makes supernova neutrino de-tection a competitive way to get this yet unknown propertyof the neutrino mass spectrum in addition to terrestrialexperiments [64].

Furthermore, in the past [16,39] it has been speculatedthat swaps of neutrino fluxes between electron and non-electron flavors in the deepest SN regions may increase theneutrino energy deposition and hence the neutrino heatingin the gain region behind the standing bounce shock. It wasargued that it could possibly help to revive the shock and totrigger neutrino-driven explosions. We have shown that thepresence of the matter suppression of flavor conversions athigh densities, behind the shock front (see, e.g., Fig. 6),implies that these cannot play any significant role inchanging the neutrino energy deposition rate behind thestalled shock wave. As an important consequence, theproblem of the � flavor mixing in SNe can be decoupledfrom the � transport and impact on the matter heating/cooling. This result is confirmed by the analysis recentlyperformed in [65] using two-dimensional supernova mod-els. In that work, even if matter effects are not included inthe simulation of neutrino flavor oscillations, a negligibleimpact of the self-induced conversions is found on theenergy deposition behind the shock-wave. Indeed, wealso find that event neglecting matter effects, the possiblerange where self-induced oscillations would have signifi-cantly developed is always after the shock-wave positionðrsh & rsyncÞ at the times relevant for the shock revival (see

our Figs. 6–8).Our results have been obtained considering a spherically

symmetric neutrino emission. All the previous analysis inthe field have relied on this assumption to make the flavorevolution equations numerically tractable. It remains to beinvestigated if the removal of a perfect spherical symmetry

can provide a different behavior in the flavor evolution[66]. Moreover, in multidimensional SN models densityfluctuations are expected behind the standing bounceshock, due to the presence of convection and hydro insta-bilities. These can range at most between 10% to a factor of2–3 (see, e.g., [40,67]). Therefore, even in the case afraction of the neutrino beam crosses underdenses regions,the matter suppression of the collective oscillations willstill remain relevant. This claim is supported by a recentanalysis of the matter suppression, performed with two-dimensional SN simulations [65].Self-induced effects in the supernova neutrinos still

remain crucial during the later cooling phase, when thematter density decreases continuously (tpb * 1 s) and

becoming subdominant with respect to the neutrino den-sity. In this situation and for the flux ordering provided bythe simulations from Ref. [7], we confirm that self-induced oscillations in the cooling phase produce multiplesplits and swaps, like the one described in [26–32].Moreover, for low-mass O-Ne-Mg-core SN a peculiarinterplay of MSW and self-induced effects would produceinteresting signatures during the �e prompt neutronizationburst [68–70].In conclusion, the time evolution of the SN neutrino

signal provides a large amount of information on the flavortransformations of neutrinos in the deepest stellar regions.The early post-bounce deleptonization, accretion and cool-ing phases represent three different ‘‘experiments’’ thatallow us to probe different behaviors of the dense SNneutrino gas. In order to learn the most from a futureneutrino observation from a galactic event, improved theo-retical and experimental studies are necessary to decipherthe supernova neutrino enigma.

ACKNOWLEDGMENTS

We thank B. Dasgupta, T. Janka, G. Raffelt, C. Ott, andG. Sigl for useful discussions and suggestions during thedevelopment of this project. We also acknowledge E. Lisi,G. Raffelt, S. Sarikas, and I. Tamborra for helpful com-ments on the manuscript. The work of S. C., A.M., N. S.was supported by the German Science Foundation (DFG)within the Collaborative Research Center 676 ‘‘Particles,Strings and the Early Universe.’’ T. F. acknowledges sup-port from HIC for FAIR Project No. 62800075.

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