Apartes de la Conferencia de la SJG del 14 y 21 de Enero de 2012: Neutrino mass spectrum from gravitational waves generated by double neutrino spin flip in supernovae
The Astrophysical Journal, 689:371Y376, 2008 December 10 A# 2008. The American Astronomical Society. All rights reserved. Printed in U.S.A. NEUTRINO MASS SPECTRUM FROM GRAVITATIONAL WAVES GENERATED BY DOUBLE NEUTRINO SPIN-FLIP IN SUPERNOVAE Herman J. Mosquera Cuesta1 and Gaetano Lambiase 2 Received 2008 May 28; accepted 2008 August 4 ABSTRACT The supernova (SN ) neutronization phase produces mainly electron (e ) neutrinos, the oscillations of which must take place within a few mean free paths of their resonance surface located nearby their neutrinosphere. The latest research on the SN dynamics suggests that a signiﬁcant part of these e can convert into right-handed neutrinos by virtue of the interaction of the electrons and the protons ﬂowing with the SN outgoing plasma, whenever the Dirac neutrino magnetic moment is of strength 10À11 B , with B being the Bohr magneton. In the SN envelope, some of these neutrinos can ﬂip back to the left-handed ﬂavors due to the interaction of the neutrino magnetic moment with the magnetic ﬁeld in the SN expanding plasma (see the work by Kuznetsov Mikheev; Kuznetsov, Mikheev, Okrugin; Akhmedov Khlopov; Itoh Tsuneto; and Itoh et al.), a region where the ﬁeld strength is currently accepted to be B k1013 G. This type of oscillation was shown to generate powerful gravitational wave (GW ) bursts (see the work by Mosquera Cuesta; Mosquera Cuesta Fiuza; and Loveridge). If such a double spin-ﬂip mech- anism does run into action inside the SN core, then the release of both the oscillation-produced and particles and emi the GW pulse generated by the coherent spin-ﬂips provides a unique emission offset ÁTGW$ ¼ 0 for measuring the travel time to Earth. As massive particles get noticeably delayed on their journey to Earth with respect to the Einstein GW they generated during the reconversion transient, then the accurate measurement of this time-of-ﬂight delay by SNEWS + LIGO, VIRGO, BBO, DECIGO, etc., might readily assess the absolute mass spectrum. Subject headingg: elementary particles — gravitational waves — methods: data analysis — neutrinos — s stars: magnetic ﬁelds — supernovae: general Online material: color ﬁgure 1. INTRODUCTION electrons and the protons in the SN outﬂowing plasma. Speciﬁ- The determination of the absolute values of neutrino masses is cally, the neutrino chirality ﬂip is caused by the scattering via the intermediate photon (plasmon) off the plasma electromagnetic cur-certainly one of the most difﬁcult problems from the experimental rent presented by electrons, L eÀ À R eÀ ; protons, L pþ À R pþ ; ! !point of view (Bilenky et al. 2003). One of the main difﬁculties ofthe issue of determining the masses from solar or atmospheric etc. (2) A second signal exists by virtue of the reconversion pro- cess of these sterile particles back into actives some time later,experiments concerns the ability of detectors to be sensitive to at lower density, via the interaction of the neutrino magnetic mo-the species mass square difference instead of being sensitive to ment with the magnetic ﬁeld in the SN envelope (SNE). The GWthe mass itself. In this paper we introduce a model-independent characteristic amplitude, which depends directly on the luminositynovel nonpareil method to achieve this goal. We argue that a highly and the mass square difference of the species partaking in theaccurate and largely improved assessment of the mass scale can coherent transition (Pantaleone 1992), and the GW frequency ofbe directly achieved by measurements of the delay in time of ﬂight each of the bursts are computed. Finally, the time-of-ﬂight delaybetween the particles themselves and the gravitational wave $ GW that can be measured upon the arrival of both signals to(GW) burst generated by the asymmetric ﬂux of neutrinos under- Earth observatories is then estimated, and the prospective of ob-going coherent (Pantaleone 1992) helicity (spin-ﬂip) transitionsduring either the neutronization phase or the relaxation (diffusion) taining the mass spectrum from such measurements is discussed.phase in the core of a Type II supernova (SN) explosion. Because 2. DOUBLE RESONANT CONVERSIONspecial relativistic effects do preclude massive particles from trav- OF NEUTRINOS IN SUPERNOVAEeling at the speed of light, while massless particles are not (thegraviton in this case), the measurement of this time lag leads to 2.1. Interaction of L Dirac Magnetic Momenta direct accounting of its mass. We posit from the start that two with SN Virtual Plasmonbursts of GWs can be generated during the protoYneutron star The neutrino chirality conversion process L $ R in a SN has(PNS) neutronization phase through spin-ﬂip oscillations: (1) one been investigated in many papers (see, for instance, Voloshin 1988;signal from the early conversion of active particles into right- Peltoniemi 1992; Akhmedov et al. 1993; Dighe Smirnov 2000).handed partners, at density $ few ; 1012 g cmÀ3, via the inter- Next, we follow the reanalysis of the double spin-ﬂip in SNeaction of the Dirac neutrino magnetic moment [of strength recently revisited by Kuznetsov Mikheev (2007) and Kuznetsov(0:7Y1:5) ; 10À12 B , with B being the Bohr magneton] with the et al. (2008), who obtained a more stringent limit on the neutrino magnetic moment, , after demanding compatibility with the 1 Instituto de Cosmologia, Relatividade e Astrof ´sica ( ICRA-BR), Centro ı SN 1987A luminosity. The process becomes feasible in virtueBrasileiro de Pesquisas Fısicas (CBPF), Rua Dr. Xavier Sigaud 150, 22290-180, ´ of the interaction of the Dirac magnetic moment with a virtualRio de Janeiro, Brazil; and ICRANet Coordinating Centre, Piazzalle dellaRepubblica 10, 065100, Pescara, Italy. plasmon, which can be produced, L À R þ ? , and absorbed, ! 2 ´ Dipartimento di Fisica ‘‘E. R. Caianiello,’’ Universita di Salerno, 84081 L þ ? À R , inside a SN. Our main goal here is to estimate the !Baronissi (Sa), Italy; and INFN, Sezione di Napoli, Italy. R luminosity after the ﬁrst resonant conversion inside the SN. 371
372 MOSQUERA CUESTA LAMBIASE Vol. 689This quantity is one of the important parameters for estimating form Ye ¼ 1/3. ( Typical values of Ye in SNE are Ye $ 0:4Y0:5,the GW amplitude of the signal generated at the transition (see x 3 which are rather similar to those of the collapsing matter). How-below). The calculation of the spin-ﬂip rate of creation of the R ever, the shock wave causes the nuclei dissociation and makesin the SN core is given by (Kuznetsov Mikheev 2007) the SNE material more transparent to particles. This leads to Z 1 the proliferation of matter deleptonization in this region and, con- dER dnR 0 0 sequently, to the so-called short outburst. According to the latest L R ¼V E dE dt 0 dE 0 research on SNe, a typical gap appears along the radial distribu- Z 1 tion of the parameter Ye where it can achieve values as low as V ¼ E 03 ÀðE 0 ÞdE 0 ; ð1Þ Ye $ 0:1 (see Mezzacappa et al. 2001 and also Fig. 2 in Kuznetsov 2 2 0 et al. 2008, and references therein). Thus, a transition region un-where dnR /dE 0 deﬁnes the number of right-handed particles avoidably exists where Ye takes the value of 1/3. It is remarkableemitted in the 1 MeV energy band of the energy spectrum, and that only one such point appears where the Ye radial gradient isper unit time, À(E 0 ) deﬁnes the spectral density of the right-handed positive, i.e., dYe /dr 0. Nonetheless, the condition Ye ¼ 1/3 is luminosity, and V is the plasma volume. Thus, by using the SN the necessary but yet not the sufﬁcient one for the resonant con-core conditions that are currently admitted (see, for instance, Janka version R ! L to occur. It is also required to satisfy the so-calledet al. 2007), plasma volume V ’ 4 ; 1018 cm3, temperature adiabatic condition. This means that the diagonal element CL inrange T ¼ 30Y60 MeV, electron chemical potential range e ¼ ˜ equation (3), at least, should not exceed the nondiagonal element280Y307 MeV, neutrino chemical potential ¼ 160 MeV,3 one ˜ B? , when the shift is made from the resonance point at theobtains distance of the order of the oscillation length. This leads to the condition (Voloshin 1988) 2 LR ’ (0:4Y2) ; 1077 erg sÀ1 ; ð2Þ dCL 1=2 3GF dYe 1=2 B B? k ’ pﬃﬃﬃ : ð4Þ dr 2 mN drwhich for a ¼ 3 ; 10À12 B compatible with SN 1987A neu-trino observations and preserving causality with respect to the left- And values of these typical parameters inside the considered re-handed diffusion luminosity LR LL P1053 erg sÀ1, renders gion are dYe /dr $ 10À8 cmÀ1 and $ 1010 g cmÀ3 . Therefore,LR ¼ 4 ; 1053 erg sÀ1. This constraint is on the order of the lu- the magnetic ﬁeld strength that realizes the resonance conditionminosities estimated in our earlier papers (Mosquera Cuesta 2000, reads as2002; Mosquera Cuesta Fiuza 2004) to compute the GW am- À12 10 Bplitude from ﬂavor conversions, which were different from the B? k 2:6 ; 1014 Gone estimated by ( Loveridge 2004). More remarkable, this anal- ysis means that only $1%Y2% of the total number of L particles 1=2 1=2 dYe 8may resonantly convert into R particles. ; 10 cm : ð5Þ 1010 g cmÀ3 dr 2.2. Conversion of R À L in the SN Magnetic Field ! Thus, one can conclude that the analysis performed above shows Kuznetsov et al. (2008) have shown that by taking into account that the Dar scenario of the double conversion of the neutrinothe additional energy CL, which the left-handed electron-type neu- helicity (Dar 1987), L ! R ! L , can be realized whenevertrino e acquires in the medium, the equation of the helicity evo- the neutrino magnetic moment is in the interval 10À13 B lution can be written in the form ( Voloshin Vysotsky 1986; 10À12 B and when the strength of the magnetic ﬁeld reachesVoloshin et al. 1986a, 1986b; Okun 1986, 1988) k1014 G ( Kusenko 2004) in a region R between the neutrino- sphere R and the shock wave stagnation radius Rs , where R @ R 0 B? R i ¼ E ˆ0 þ ; R Rs .4 Thus, the L luminosity during this stagnation time, @t L B? CL L ÁTs ’ 0:2Y0:4 s, is LL ’ 3 ; 1053 erg sÀ1, as the conservation 3GF 4 1 law allows one to expect 10À12 B . Once one has all these pa- *CL ¼ pﬃﬃﬃ Ye þ Ye À ; ð3Þ rameters in hand, one can then proceed to compute the correspond- 2 mN 3 3 ing GW signal from each of the resonant spin-ﬂip transitions.where the ratio /mN ¼ nB is the nucleon density, Ye ¼ ne /nB ¼np /nB , Ye ¼ ne /nB , and ne; p; e are the densities of the electrons, 3. OSCILLATIONYDRIVEN GWprotons, and neutrinos, respectively, B? is the transverse compo- DURING SN NEUTRONIZATIONnent of the magnetic ﬁeld with respect to the propagation direc- The characteristic GW amplitude of the signal produced by the ˆtion, and the term E0 is proportional to the unit matrix, however, outﬂow can be estimated by using the general relativistic quad-it is not crucial for the analysis below. rupole formula (Burrows Hayes 1996) As pointed out by Kuznetsov et al. (2008), the additional en- Zergy CL of left-handed particles deserves a special analysis. It 4G t hTT (t) ¼ 4 ij ðt 0 ÞL ðt 0 Þdt 0 ei ejis remarkable that the possibility exists for this value to be zero c D À1just in the region of the SNE we are interested in. And, in turn, 4Gthis is the condition of the resonant transition R ! L . When the À!h ’ 4 Á L ÁTfL !fR ; ð6Þ c D density in the SNE is low enough, one can neglect the value Yein the term CL , which gives the condition for the resonance in the 4 These kinds of magnetic ﬁeld strengths have been extensively said to be reached after the SN core collapse forms just-born pulsars (magnetars), in the central 3 These conditions could exist in the time interval before the ﬁrst second after engines of gamma-ray burst outﬂows, and during the quantum-magnetic collapsethe core bounce. of newborn neutron stars, etc.
No. 1, 2008 NEUTRINO MASS SPECTRUM FROM GRAVITATIONAL WAVES 373 tion ( Barkovich et al. 2002). As a result, one gets Ájpj/jpj ¼ R R 6 ( 0 Fs = u dS )/( 0 Fs = n dS ) ’ 2%/9¯ (n is6 a unit vector nor- 1 r ˆ ˆ mal to the resonance surface and u ¼ B/jBj). An anisotropy of $1% would sufﬁce to account for the observed pulsar kicks ´ ( Kusenko Segre 1996; Loveridge 2004; Mosquera Cuesta 2000, 2002); hence, ’ 0:045 $ O(0:01)YO(0:1), which is con- sistent with numerical results ( Burrows Hayes 1996; Muller ¨ Janka 1997). Finally, the conversion probability is PeL ! R ¼ ˜ ˜ ˜ 1/2 À 1/2 cos 2i cos 2f (Okun 1986, 1988), where is deﬁned as tan 2(r) ¼ 2 B? =(B = p þ Ve À 2c2 ): ˜ ˆ ð8Þ ˜ ˜ ˜ ˜ The quantities i ¼ (ri ) and f ¼ (rf ) are the values of the mix- ing angle at the initial point ri and the ﬁnal point rf of the neutrino path.7 Meanwhile, the average timescale of this ﬁrst spin-ﬂip con- version is (Dar 1987; Voloshin 1988) 2 2 B me mp ÁTfL !fR ¼ 2 (1 þ hZi)Y ; ð9Þ fsc e where hZi $ O(1Y30) is the average electric charge of the nuclei and fsc is the ﬁne-structure constant. Using the current bounds Fig. 1.— Illustration of the combined effect of the spin coupling to the star on the neutrino magnetic moment P 3 ; 10À12 B , Ye ’ 1/3,magnetic ﬁeld and rotation. This ﬁgure was taken from Mosquera Cuesta Fiuza hZi $ 10, $ 2 ; 1012 g cmÀ3, and $ 0:04, it follows that(2004). ÁTf L !f R ’ (1 À 10) ; 10À2 s (parameters have been chosen from SN simulations evolving the PNS on timescales of $3 ms around core bounce; Mayle et al. 1987; Walker Schramm 1987; Burrowswhere D is the source distance, L (t) is the total luminosity, Hayes 1996; Mezzacappa et al. 2001; van Putten 2002; Arnaudei ej is the GW polarization tensor, the superscript TT stands et al. 2002; Beacom et al. 2001). In such a case, the above time-for the transverse-traceless part, and ﬁnally, (t) is the instan- scale suggests that the GW burst would be as long as the expectedtaneous quadrupole anisotropy. Above, we estimated the R lu- duration of the pure neutronization phase itself, i.e., ÁTneut $minosity; next, we estimate the degree of asymmetry of the PNS 10Y100 ms, according to most SN analyses and models ( Maylethrough the anisotropic parameter and the timescale ÁTfL !fR et al. 1987; Walker Schramm 1987; Burrows Hayes 1996;for the resonant transition to take place, as discussed above. Mezzacappa et al. 2001; van Putten 2002; Arnaud et al. 2002; To estimate the star asymmetry, let us recall that the resonance Beacom et al. 2001), with the maximum GW emission takingcondition for the transition eL ! R is given by (at the reso- max place around ÁTneut $ 3 ms (van Putten 2002; Arnaud et al. 2002; ¯nance r ) Mosquera Cuesta 2000, 2002; Mosquera Cuesta Fiuza 2004). Hence, the outcoming GW signal will be the evolute ( linear Ve ( r ) þ B( r ) = p À 2c2 ¼ 0: ¯ ¯ ˆ ð7Þ superposition) of all the coherent eL ! ;R oscillations taking place over the neutronization transient, in analogy with the GWThus, the PNS magnetic ﬁeld vector B in equation (7) distorts the signal from the collective motion of neutron matter in a just-bornsurface of resonance due to the relative orientation of p with re- pulsar. This implies a GW frequency of fGW $ 1/ÁTneut $ 100 Hz,spect to B (see vector B in Fig. 1). The deformed surface of res- max for the overall GWemission, and fGW $ 1/ÁTneut $ 330 Hz at itsonance can be parameterized as r(
, where % (¯ ) ¯ r peak. Meanwhile, according to our probability discussion above,is the radial deformation and cos
¼ B = p. The deformation en- ˆ ˆ about 1%Y2% of the total particles released during the SN neu-forces a nonsymmetrical outgoing neutrino ﬂux, i.e., the net ﬂux tronization phase may oscillate (Voloshin 1988; Peltoniemi 1992;of neutrinos emitted from the upper hemisphere is different from Akhmedov et al. 1993; Dighe Smirnov 2000), carrying awaythe one emitted from the lower hemisphere (see Fig. 1). There- an effective power L ¼ 3 ; 1054 Y1053 erg sÀ1, i.e., 0:01 ; 3 ;fore, a geometrical deﬁnition of the quadrupole anisotropy can 1053 erg, emitted during ÁTneut $ 10Y100 ms (this is similar tobe ¼ (Sþ À SÀ )/(Sþ þ SÀ ), where SÆ is the area of the upper/ the upper limit computed in Peltoniemi , L ¼ (2/10) ;lower hemisphere, whence one obtains ’ %/ r.5 The anisotropy ¯ 1053 ½e /(10À12 B ) erg sÀ1). Moreover, as is evident from equa-of the outgoing neutrinos is also related to the energy ﬂux Fs tion (6), the GW amplitude is a function of the helicity-changingemitted by the PNS and, in turn, to the fractional momentum eL ! luminosity, i.e., h ¼ h(Lmax ;R ). The luminosity itself depends ´asymmetry Ájpj/jpj ( Kusenko Segre 1996; Barkovich et al.2002; Lambiase 2005a, 2005b; Mosquera Cuesta Fiuza 2004). 6 To compute Ájpj/jpj one uses the standard resonance condition V ¼ 2c2To compute Fs , one has to take into account the structure of the (see Barkovich et al. 2002 for details). According to Mezzacappa et al. (2001),ﬂux at the resonant surface, which acts as an effective emis- during the ﬁrst 10Y200 ms, Ye may assume values ’1/3 so that Ve $ (3Ye À 1)sion surface, and the distribution in the diffusive approxima- is suppressed by several orders of magnitude. At $10 ms, $ 1012 g cmÀ3, r $ 50 km, and jpj $ 10 MeV, the resonance condition leads to a range for Ám 2 cos 2 consistent with solar (or atmospheric) neutrino data. 5 A detailed analysis of the asymmetry parameter requires one to study its 7 By using the typical values B k1010 G, P 9 ; 10À11 B , and the proﬁletime evolution during the SN collapse. Such a task goes beyond the aim of this ’ core (rc /r) 3 for r k rc (rc $ 10 km is the core radius and core $ 1014 g cmÀ3),paper. Working in the stationary regime, we may assume constant (see Burrows one can easily verify that the adiabatic parameter 2( B? ) 2/ðj 0/jÞ 1 at Hayes 1996; Burrows et al. 1995; Zwerger Muller 1997; van Putten 2002). ¨ the resonance r. ¯
374 MOSQUERA CUESTA LAMBIASE Vol. 689 TABLE 1 Time Delay between GW and (jpj ¼ 10 MeV ) Bursts from a SN Neutronization, as a Function of Mass and Distance arr ÁTGW$ (s) Distance Source ( kpc) 1 2 3GC ......................... 10 5:15 ; 10À9 5:15 ; 10À3 0.32LMC...................... 55 2:83 ; 10À8 2:83 ; 10À2 1.7M31 ....................... 2:2 ; 10 3 1:13 ; 10À6 1.13 68.8Source.................... 1:1 ; 10 4 5:66 ; 10À6 5.66 344.0 Note.— The masses in eV are 10À3, 1.0, and 2.5, for ﬂavors 1 , 2 , and 3 ,respectively.on the probability of conversion ( Peltoniemi 1992; MosqueraCuesta 2000, 2002; Mosquera Cuesta Fiuza 2004; Loveridge eL !2004), i.e., Lmax ;R ¼ (PeL !;R )L . tot The characteristic GW strain [per (Hz)1 2 ] from the outgoing = Fig. 2.— Characteristics (h(fL !f 0 R ) , fGW ) of the GW burst generated via the ﬂux of spin-ﬂipping (ﬁrst transition) particles is spin-ﬂip oscillation mechanism vs. detector noise spectral density. For sources at either the GC or LMC, the pulses will be detectable by LIGO-I and VIRGO. To P ! 0 distances $10 Mpc (farther out than the Andromeda galaxy), such radiation would be detectable by Advanced LIGO and VIRGO. Resonant GWantennas, tuned at the hf L !f 0 R h ’ 1:1 ; 10À23 HzÀ1=2 fL f R 0:01 frequency interval indicated, could also detect such events. Highlighted is the GW signal of a SN neutronization phase at Andromeda, which would have a frequency Ltot 2:2 Mpc ÁT fGW $ 100 Hz. [See the electronic edition of the Journal for a color version of this ; ; ð10Þ 3 ; 10 54 erg sÀ1 D 10À1 s 0:1 ﬁgure.]for a SN exploding at a ﬁducial distance of 2.2 Mpc, e.g., at the the amplitude of the coherent weak interaction of L with the PNSAndromeda galaxy (see Table 18). The GW strain in this mech- matter (Ve ) can cross smoothly enough to ensure adiabatic res-anism (see Fig. 2) is several orders of magnitude larger than in the onant conversion of f R into f L .9 Following Mezzacappa et al.SN diffusive escape (Burrows Hayes 1996; Muller Janka ¨ (2001), the region where Ve ¼ 0 as Ye ¼ 1/3 corresponds to a1997; Arnaud et al. 2002; Loveridge 2004) because of the huge postbounce timescale $100 ms and radius $150 km at whichluminosity the oscillations provide by virtue of being a highly the luminosity is L $ 3 ; 1052 erg sÀ1, and the matter densitycoherent process (Pantaleone 1992; Mosquera Cuesta 2000, 2002; is $ 1010 g cmÀ3. There, the adiabaticity condition demandsMosquera Cuesta Fiuza 2004). This makes it detectable from B? k 1010 G for the quoted above (such a ﬁeld is characteristicvery far distances. These GW signals are right in the bandwidth of young pulsars). This reverse transition (rt) should resonantlyof the highest sensitivity (10Y300 Hz) of most ground-based produce an important set of ordinary (muon and tau) L particles,interferometers. which could be found far from their own neutrinosphere and, hence, Spin ﬂavor oscillations eL ! R , which according to the latest can stream away from the PNS. Whence a second GW burst withresearch on SN dynamics do take place during the neutronization the characteristics h ’ 1 ; 10À23 HzÀ1/2 for D ¼ 2:2 Mpc andphase of core-collapse SNe (Mayle et al. 1987; Walker Schramm ÁTrt ’ 1:4 s is released in this region. Notice that this h is1987; Voloshin 1988; Dighe Smirnov 2000; Kuznetsov similar to the one for the ﬁrst transition despite the luminosityMikheev 2007), allow powerful GW bursts to be released from being lower. This feature makes it similar to the GW memory prop-one side (according to eq. ) and a stream of R particles to be erty of the -driven signal, i.e., time-dependent strain amplitudegenerated from the other side, over a timescale given by equation (9). with average value nearly constant ( Burrows Hayes 1996).The latter would in principle escape from the PNS were it not for To obtain this result, equations (9) and (10) were used. Where-the appearance of several resonances that catch up with them fore, the GW frequency fGW $ 1/ÁTrt $ 0:7 Hz falls in the low-(Voloshin 1988; Peltoniemi 1992; Akhmedov et al. 1993). If there frequency band and could be detected by the planned BBO andwere no such resonance, the fL ! f 0 R oscillation process would DECIGO GW interferometric observatories. Notice also that theleak away all the binding energy of the star, leaving no energy at time lag for the event at LIGO, VIRGO, etc., and the one at BBOall for the left-handed L particles that are said to drive the actual and DECIGO is then about 100 ms. It is this transition whichSN explosion and for us to have observed them during SN 1987A. deﬁnes the offset to measure the time-of-ﬂight delay, since both ¯A new resonance may occur at r k100 km from the center, which ; and GW free-stream away from the PNS at this point.converts $90%Y99% of the spin-ﬂipYproduced R particles backinto L ones (Voloshin 1988; Akhmedov 1988; Akhmedov Khlopov 1988a, 1988b; Itoh Tsuneto 1972; Itoh et al. 1996; 4. TIME-OF-FLIGHT DELAY $ GWPeltoniemi 1992; Akhmedov et al. 1993; Athar et al. 1995). As dis- The measurement of the $ GW time delay from oscil-cussed in these papers, in fact, in the outer layer of the SN core lations in SNe promises to be an innovative procedure to obtain the mass spectrum. Provided that Einstein’s GWs do propagate 8 The mass eigenstates listed are masses supposed to be estimated throughoutthe detection in a future SN event, not the mass constraints already establishedfrom solar and atmospheric neutrinos, the expected time delay of which is com- 9 The cross level condition once again involves the terms B = p. Nevertheless, ˆputable straightaway. If a nonstandard mass eigenstate is detected, then one can at that point the deformation of the resonance surface may be neglected, whenceuse the seesaw mechanism to infer the remaining part of the spectrum. no relevant GW burst is expected ( yet is quite low).
No. 1, 2008 NEUTRINO MASS SPECTRUM FROM GRAVITATIONAL WAVES 375at the speed of light, the GW burst produced by spin-ﬂip oscilla- with Cash $ (2:3 Æ 0:3) ms and N being the event statisticstions during the neutronization phase will arrive to GW observa- ( proportional to D). This leads to the SN distance-dependent un-tories earlier than its source (the massive particles from the certainty in the mass, m / ÁTmax /D $ 0:5Y0:6 eV 2 (Arnaud 2second conversion) will get to telescopes. et al. 2002), which implies m $ 7 ; 10À1 eV, which is consistent As pointed out above, the mechanism to generate GWs at the with our previous estimate from equation (12). Hence, those instant at which the second transition f 0 R ! fL takes place can particles and their spin-ﬂip conversion signals must be detected. emiby itself deﬁne a unique emission offset, ÁTGW$ ¼ 0, which Therefore, the left-hand side of equation (11), i.e., the time-of-makes possible a cleaner and highly accurate determination of the ﬂight delay ÁTGW$, will be measured with a very high accuracy. mass spectrum by ‘‘following’’ the GW and neutrino propaga- With these quantities, a very precise and stringent assessment oftion to Earth observatories. The time lag in arrival is ( Beacom the absolute mass eigenstate spectrum will be readily set out byet al. 2001) means not explored earlier in astroparticle physics: an innovative technique involving not only particle but also GW astronomy. For arr D m 10 MeV 2 instance, at a 10 kpc distance, e.g., to the Galactic center (GC in ÁTGW$ ’ 0:12 s : ð11Þ Fig. 2), the resulting time delay should approximate ÁTGW ! ¼ 2:2 Mpc 0:2 eV jpj 5:2 ; 10À3 s, for a ﬂavor of mass m 1 eV and jpj $ 10 MeV. A SN event from the GC or Large Magellanic Cloud ( LMC) 5. DISCUSSION would provide enough statistics in SNO, SK, etc., $5000Y8000 In most SN models (Burrows Hayes 1996; Mezzacappa et al. events, so as to allow for the deﬁnition of the mass eigenstates2001; Beacom et al. 2001; van Putten 2002), the neutroniza- ( Beacom et al. 2001). Farther out, events are less promisingtion burst is a well-characterized process of intrinsic duration in this perspective, but we stress that one event collected byÁT ’ 10 ms, with its maximum occurring within 3:5 Æ 0:5 ms the planned megaton detector, from a large-distance source, mayafter core collapse (Mayle et al. 1987; Walker Schramm 1987; prove sufﬁcient (see further arguments in Ando et al. 2005).van Putten 2002; Burrows Hayes 1996). This timescale relatesto the detectors’ approximate sensitivity to masses beyond the 6. SUMMARYmass limit In this paper, it has been emphasized that knowing the ab- solute mass scale with enough accuracy would turn out to be a 2:2 Mpc ÁT 1=2 jpj fundamental test of the physics beyond the standard model of fun-m 6:7 ; 10 eVÀ2 : ð12Þ D 10 ms 10 MeV damental interactions. By virtue of the very important two-step mechanism of spin-ﬂavor conversions in SNe, very recentlyThis threshold is in agreement with the current bounds on masses revisited by Kuznetsov et al. (2008), we suggest that by combining(Fukuda et al. 1998). the detection of the GW signals generated by those oscillations and Nearby SNe will somehow be seen. Apart from GWs and neu- the signals collected by SNEWS from the same SN event, onetrinos, -rays, X-rays, visible, infrared, or radio signals will be might conclusively assess the mass spectrum. In particular, sortingdetected. Therefore, their position on the sky and distance (D) may out the neutronization phase signal from both the light curvebe determined quite accurately, including—if far from the Milky and the second peak in the GW waveform (with its memorylikeWay—their host galaxy (Ando et al. 2005). Besides, the Universal feature; Burrows Hayes 1996) might allow one to achieve thisTime of arrival of the GW burst to three or more gravitational goal in a nonpareil fashion.radiation interferometric observatories or resonant detectors willbe precisely established (Schutz 1986; Arnaud et al. 2002). Theuncertainty in the GW timing depends on the signal-to-noise ratio(S/N ) as ÁT (GWjD¼10 kpc ) $ 1:45 /(S/N ) $ 0:15 ms, with $ H. J. M. C. thanks FAPERJ, Brazil for ﬁnancial support and1 ms being the rms width of the main GW peak (Arnaud et al. ICRANet Coordinating Centre, Pescara, Italy for hospitality2002). Meanwhile, the type of and its energy and Universal during the early stages of this work. G. L. acknowledges supportTime of arrival to telescopes of the SNEWS network will be to this work provided by MIUR through PRIN Astroparticlehighly accurately measured (Antonioli et al. 2004; Beacom ´ Physics 2007 and by research funds of the Universita di Salerno.Vogel 1999). 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