1. 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 of
the 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 GW
the mass itself. In this paper we introduce a model-independent
characteristic amplitude, which depends directly on the luminosity
novel nonpareil method to achieve this goal. We argue that a highly
and the mass square difference of the species partaking in the
accurate and largely improved assessment of the mass scale can
coherent transition (Pantaleone 1992), and the GW frequency of
be directly achieved by measurements of the delay in time of flight
each of the bursts are computed. Finally, the time-of-flight delay
between 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) transitions
during 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 CONVERSION
special relativistic effects do preclude massive particles from trav-
OF NEUTRINOS IN SUPERNOVAE
eling at the speed of light, while massless particles are not (the
graviton in this case), the measurement of this time lag leads to 2.1. Interaction of L Dirac Magnetic Moment
a direct accounting of its mass. We posit from the start that two with SN Virtual Plasmon
bursts 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 SNe
action 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 virtue
Brasileiro de Pesquisas Fısicas (CBPF), Rua Dr. Xavier Sigaud 150, 22290-180,
´ of the interaction of the Dirac magnetic moment with a virtual
Rio de Janeiro, Brazil; and ICRANet Coordinating Centre, Piazzalle della
Repubblica 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

2. 372 MOSQUERA CUESTA LAMBIASE Vol. 689
This 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 makes
in 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 remarkable
emitted in the 1 MeV energy band of the energy spectrum, and that only one such point appears where the Ye radial gradient is
per 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-called
et al. 2007), plasma volume V ’ 4 ; 1018 cm3, temperature adiabatic condition. This means that the diagonal element CL in
range T ¼ 30Y60 MeV, electron chemical potential range e ¼ ˜ equation (3), at least, should not exceed the nondiagonal element
280Y307 MeV, neutrino chemical potential ¼ 160 MeV,3 one
˜ B? , when the shift is made from the resonance point at the
obtains 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 dr
which 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 condition
minosities estimated in our earlier papers (Mosquera Cuesta 2000, reads as
2002; Mosquera Cuesta Fiuza 2004) to compute the GW am- À12
10 B
plitude from flavor conversions, which were different from the B? k 2:6 ; 1014 G
one 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 8
may 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 neutrino
the additional energy CL, which the left-handed electron-type neu- helicity (Dar 1987), L ! R ! L , can be realized whenever
trino 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 reaches
Voloshin 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 ¼
np /nB , Ye ¼ ne /nB , and ne; p; e are the densities of the electrons, 3. OSCILLATIONYDRIVEN GW
protons, and neutrinos, respectively, B? is the transverse compo- DURING SN NEUTRONIZATION
nent of the magnetic field with respect to the propagation direc- The characteristic GW amplitude of the signal produced by the
ˆ
tion, 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- Z
ergy 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 ej
is remarkable that the possibility exists for this value to be zero c D À1
just in the region of the SNE we are interested in. And, in turn, 4G
this 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 Ye
in 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 collapse
the core bounce. of newborn neutron stars, etc.

3. 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; Burrows
where D is the source distance, L (t) is the total luminosity, Hayes 1996; Mezzacappa et al. 2001; van Putten 2002; Arnaud
ei 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 expected
taneous 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 ( Mayle
through 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 taking
condition 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 GW
Thus, the PNS magnetic field vector B in equation (7) distorts the signal from the collective motion of neutron matter in a just-born
surface 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 its
onance can be parameterized as r(

4. ) ¼ r þ % cos

5. , where % (¯ )
¯ r peak. Meanwhile, according to our probability discussion above,
is the radial deformation and cos

6. 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 away
the 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 to
be ¼ (Sþ À SÀ )/(Sþ þ SÀ ), where SÆ is the area of the upper/ the upper limit computed in Peltoniemi [1992], L ¼ (2/10) ;
lower hemisphere, whence one obtains ’ %/ r.5 The anisotropy
¯ 1053 ½e /(10À12 B )Š 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-changing
emitted 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 2c2
To 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 profile
time 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. ¯

7. 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 3
GC ......................... 10 5:15 ; 10À9 5:15 ; 10À3 0.32
LMC...................... 55 2:83 ; 10À8 2:83 ; 10À2 1.7
M31 ....................... 2:2 ; 10 3 1:13 ; 10À6 1.13 68.8
Source.................... 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; Mosquera
Cuesta 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 PNS
Andromeda 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 a
1997; Arnaud et al. 2002; Loveridge 2004) because of the huge postbounce timescale $100 ms and radius $150 km at which
luminosity the oscillations provide by virtue of being a highly the luminosity is L $ 3 ; 1052 erg sÀ1, and the matter density
coherent process (Pantaleone 1992; Mosquera Cuesta 2000, 2002; is $ 1010 g cmÀ3. There, the adiabaticity condition demands
Mosquera Cuesta Fiuza 2004). This makes it detectable from B? k 1010 G for the quoted above (such a field is characteristic
very far distances. These GW signals are right in the bandwidth of young pulsars). This reverse transition (rt) should resonantly
of 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 with
research on SN dynamics do take place during the neutronization the characteristics h ’ 1 ; 10À23 HzÀ1/2 for D ¼ 2:2 Mpc and
phase of core-collapse SNe (Mayle et al. 1987; Walker Schramm ÁTrt ’ 1:4 s is released in this region. Notice that this h is
1987; Voloshin 1988; Dighe Smirnov 2000; Kuznetsov similar to the one for the first transition despite the luminosity
Mikheev 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. [6]) and a stream of R particles to be erty of the -driven signal, i.e., time-dependent strain amplitude
generated 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 and
were no such resonance, the fL ! f 0 R oscillation process would DECIGO GW interferometric observatories. Notice also that the
leak 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 BBO
all for the left-handed L particles that are said to drive the actual and DECIGO is then about 100 ms. It is this transition which
SN 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 back
into L ones (Voloshin 1988; Akhmedov 1988; Akhmedov
Khlopov 1988a, 1988b; Itoh Tsuneto 1972; Itoh et al. 1996; 4. TIME-OF-FLIGHT DELAY $ GW
Peltoniemi 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 throughout
the detection in a future SN event, not the mass constraints already established
from 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,
ˆ
putable straightaway. If a nonstandard mass eigenstate is detected, then one can at that point the deformation of the resonance surface may be neglected, whence
use the seesaw mechanism to infer the remaining part of the spectrum. no relevant GW burst is expected ( yet is quite low).

8. No. 1, 2008 NEUTRINO MASS SPECTRUM FROM GRAVITATIONAL WAVES 375
at 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 statistics
tions 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
2
second 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.
emi
by 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$, will be measured with a very high accuracy.
mass spectrum by ‘‘following’’ the GW and neutrino propaga- With these quantities, a very precise and stringent assessment of
tion to Earth observatories. The time lag in arrival is ( Beacom the absolute mass eigenstate spectrum will be readily set out by
et al. 2001) means not explored earlier in astroparticle physics: an innovative
technique involving not only particle but also GW astronomy. For
arr D m 10 MeV 2 instance, at a 10 kpc distance, e.g., to the Galactic center (GC in
ÁTGW$ ’ 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 eigenstates
2001; Beacom et al. 2001; van Putten 2002), the neutroniza- ( Beacom et al. 2001). Farther out, events are less promising
tion burst is a well-characterized process of intrinsic duration in this perspective, but we stress that one event collected by
ÁT ’ 10 ms, with its maximum occurring within 3:5 Æ 0:5 ms the planned megaton detector, from a large-distance source, may
after 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 relates
to the detectors’ approximate sensitivity to masses beyond the 6. SUMMARY
mass 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 recently
This 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, one
trinos,
-rays, X-rays, visible, infrared, or radio signals will be might conclusively assess the mass spectrum. In particular, sorting
detected. Therefore, their position on the sky and distance (D) may out the neutronization phase signal from both the light curve
be determined quite accurately, including—if far from the Milky and the second peak in the GW waveform (with its memorylike
Way—their host galaxy (Ando et al. 2005). Besides, the Universal feature; Burrows Hayes 1996) might allow one to achieve this
Time of arrival of the GW burst to three or more gravitational goal in a nonpareil fashion.
radiation interferometric observatories or resonant detectors will
be precisely established (Schutz 1986; Arnaud et al. 2002). The
uncertainty 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 and
1 ms being the rms width of the main GW peak (Arnaud et al. ICRANet Coordinating Centre, Pescara, Italy for hospitality
2002). Meanwhile, the type of and its energy and Universal during the early stages of this work. G. L. acknowledges support
Time of arrival to telescopes of the SNEWS network will be to this work provided by MIUR through PRIN Astroparticle
highly accurately measured (Antonioli et al. 2004; Beacom ´
Physics 2007 and by research funds of the Universita di Salerno.
Vogel 1999). The timing uncertainty is ÁTmax ¼ Cash (N )À1 2 ,
=
He also acknowledges ASI for financial support.
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