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22 February 2001 Physics Letters B 500 (2001) 215–221 www.elsevier.nl/locate/npe Bursts of gravitational radiation from superconducting cosmic strings and the neutrino mass spectrum Herman J. Mosquera Cuesta a,b,c , Danays Morejón González b,d a Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, Miramare 34014, Trieste, Italy b Centro Brasileiro de Pesquisas Físicas, Laboratório de Cosmologia e Física, Experimental de Altas Energias, Rua Dr. Xavier Sigaud 150, Cep 22290-180, Urca, Rio de Janeiro, RJ, Brazil c Centro Latinoamericano de Física (CLAF), Avenida Wenceslau Bras 173, Botafogo, Rio de Janeiro, RJ, Brazil d Pontifícia Universidade Católica do Rio de Janeiro, Rua Marquês de São Vicente 225, Cep 22453-900, Gávea, Rio de Janeiro, RJ, Brazil Received 31 August 2000; received in revised form 8 November 2000; accepted 6 December 2000 Editor: J. Frieman Abstract Berezinsky, Hnatyk and Vilenkin showed that superconducting cosmic strings could be central engines for cosmological gamma-ray bursts and for producing the neutrino component of ultra-high energy cosmic rays. A consequence of this mechanism would be that a detectable cusp-triggered gravitational wave burst should be released simultaneously with the γ -ray surge. If contemporary measurements of both γ and ν radiation could be made for any particular source, then the cosmological time-delay between them might be useful for putting unprecedentedly tight bounds on the neutrino mass spectrum. Such measurements could consistently verify or rule out the model, since strictly correlated behaviour is expected for the duration of the event and for the time variability of the spectra. 2001 Published by Elsevier Science B.V. Cosmic strings (CSs) are topological defects formed during phase transitions in the early Universe in- duced by spontaneous symmetry breaking at GUT- scale energies [1]. The energy of the unbroken vac- uum phase is released as GUT quanta of the gauge and scalar fields, forming the CSs [2]. It has been suggested that ordinary CSs could be the cosmolog- ical sources of the biggest explosions in the Uni- verse [3]: the cosmological (classical) gamma-ray bursts (GRBs) [4], with energies E GRBs 10 5354 erg, timescales 10 2 T GRBs 10 3 s, as observed by BATSE [5]. CSs may also be the origin of the ul- tra high energy cosmic rays (UHECRs) with ener- gies above the Greisen–Kuzmin–Zatsepin (GZK) cut- E-mail address: [email protected] (H.J. Mosquera Cuesta). off: E × 10 19 eV [6], as well as the very high en- ergy neutrinos observed today [2,7,8]. Also, ordinary cosmic strings are potential sources of gravitational radiation. Emission of gravitational waves (GWs) is considered the main channel for CS loops to de- cay [1,9]. Turning attention to superconducting cosmic string (SCCS) loops, Babul, Paczy´ nski and Spergel [10], and most recently Berezinsky, Hnatyk and Vilenkin (BHV2000) [4], have proposed that such objects could be the central engines of most cosmological bursts of gamma-rays. In the first study, the cur- rents are thought of as being induced in the strings by primordial magnetic fields, and the source dis- tance scale is assumed to be (10 2 z 10 3 ) [10]. In the second one, on the other hand, the string cur- rents are seeded by an intergalactic magnetic field, 0370-2693/01/$ – see front matter 2001 Published by Elsevier Science B.V. PII:S0370-2693(01)00073-9

Bursts of gravitational radiation from superconducting cosmic strings and the neutrino mass spectrum

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22 February 2001

Physics Letters B 500 (2001) 215–221www.elsevier.nl/locate/npe

Bursts of gravitational radiation from superconducting cosmicstrings and the neutrino mass spectrum

Herman J. Mosquera Cuestaa,b,c, Danays Morejón Gonzálezb,d

a Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, Miramare 34014, Trieste, Italyb Centro Brasileiro de Pesquisas Físicas, Laboratório de Cosmologia e Física, Experimental de Altas Energias, Rua Dr. Xavier Sigaud 150,

Cep 22290-180, Urca, Rio de Janeiro, RJ, Brazilc Centro Latinoamericano de Física (CLAF), Avenida Wenceslau Bras 173, Botafogo, Rio de Janeiro, RJ, Brazil

d Pontifícia Universidade Católica do Rio de Janeiro, Rua Marquês de São Vicente 225, Cep 22453-900, Gávea, Rio de Janeiro, RJ, Brazil

Received 31 August 2000; received in revised form 8 November 2000; accepted 6 December 2000Editor: J. Frieman

Abstract

Berezinsky, Hnatyk and Vilenkin showed that superconducting cosmic strings could be central engines for cosmologicalgamma-ray bursts and for producing the neutrino component of ultra-high energy cosmic rays. A consequence of thismechanism would be that a detectable cusp-triggered gravitational wave burst should be released simultaneously with theγ -raysurge. If contemporary measurements of bothγ andν radiation could be made for any particular source, then the cosmologicaltime-delay between them might be useful for putting unprecedentedly tight bounds on the neutrino mass spectrum. Suchmeasurements could consistently verify or rule out the model, since strictly correlated behaviour is expected for the duration ofthe event and for the time variability of the spectra. 2001 Published by Elsevier Science B.V.

Cosmic strings (CSs) are topological defects formedduring phase transitions in the early Universe in-duced by spontaneous symmetry breaking at GUT-scale energies [1]. The energy of the unbroken vac-uum phase is released as GUT quanta of the gaugeand scalar fields, forming the CSs [2]. It has beensuggested that ordinary CSs could be the cosmolog-ical sources of the biggest explosions in the Uni-verse [3]: the cosmological (classical) gamma-raybursts (GRBs) [4], with energiesEGRBs∼ 1053–54 erg,timescales 10−2 � TGRBs� 103 s, as observed byBATSE [5]. CSs may also be the origin of the ul-tra high energy cosmic rays (UHECRs) with ener-gies above the Greisen–Kuzmin–Zatsepin (GZK) cut-

E-mail address:[email protected] (H.J. Mosquera Cuesta).

off: E× ∼ 1019 eV [6], as well as the very high en-ergy neutrinos observed today [2,7,8]. Also, ordinarycosmic strings are potential sources of gravitationalradiation. Emission of gravitational waves (GWs) isconsidered the main channel for CS loops to de-cay [1,9].

Turning attention to superconducting cosmic string(SCCS) loops, Babul, Paczynski and Spergel [10],and most recently Berezinsky, Hnatyk and Vilenkin(BHV2000) [4], have proposed that such objectscould be the central engines of most cosmologicalbursts of gamma-rays. In the first study, the cur-rents are thought of as being induced in the stringsby primordial magnetic fields, and the source dis-tance scale is assumed to be(102 � z � 103) [10].In the second one, on the other hand, the string cur-rents are seeded by an intergalactic magnetic field,

0370-2693/01/$ – see front matter 2001 Published by Elsevier Science B.V.PII: S0370-2693(01)00073-9

216 H.J. Mosquera Cuesta, D. Morejón González / Physics Letters B 500 (2001) 215–221

with the GRB sources being located at distances char-acteristic of superclusters of galaxies, i.e.,z� 5 [4].In both of the models, a surface defect referredto as a cusp is the trigger of the bursts, whilethe electric currents are induced by oscillation ofstring loops in an external intergalactic magneticfield.

A further by-product of these pictures is thata beamed surge develops naturally at the place wherea superconducting string loop cusp annihilates. Be-cause of the very large Lorentz factor achieved by thecontracting cusp when it is nearly at the point where itwill trigger the GRB, this beamed radiation is a veryinteresting feature of the model. It fits quite well withthe current trend among GRB workers who claim thatrecent observations provide strong evidence for somedegree of beaming in GRBs [11].

Based on BATSE observations, we point out thatthe Berezinsky, Hnatyk and Vilenkin SCCS scenario[4] appears to be more well-motivated than the ear-lier one for explaining GRBs from CSs. Thus, weshall follow its main lines here in order to demon-strate that in such a view for the central engine ofGRBs, an accompanying burst of gravitational radi-ation should also be released. As shown in Figs. 1and 2, the characteristics of such GW bursts makethem potentially observable with the forthcoming gen-eration of Earth-based interferometric GW observato-ries LIGO, VIRGO, TAMA and GEO-600, and thespace-borne LISA, as well as by the resonant-massTIGAs.

A superconducting cosmic string loop, with en-ergy per unit lengthµ ∼ η2 (whereη is the stringsymmetry-breaking scale), oscillating in a magneticfield B, behaves like anac generator, and an electriccurrentI0 ∼ e2Bl flows in it. Herel ∼ αct , defines thestring loop invariant wavelength (l ≡ E/µ, whereEis the energy of the loop in the center-of-mass frame),andα ∼ κgGµ� 1 is a parameter determined by thegravitational back reaction [4].

During brief time intervals, a noticeable augmenta-tion of the local current intensity can occur in domainsquite close to the cusp location, the point at whichthe string speed gets closest to the velocity of light.Several cusps may appear during a single loop oscil-lation period. Inside a cusp domainδlc (at maximumcontraction) the string shrinks by a large factor,l/δl,leading to a relativistic Lorentz factorΓ ∼ l/δl (in the

string rest frame),1 this condition being sustained fora timescaleδtc ∼ δlc/c, within a physical length scaleδlc ∼ δl/Γ ∼ l/Γ 2. Most of the huge cusp rest energyis converted into kinetic energy. Since, in general, thestring velocity is extremely high near to the cusp col-lapse (it spreads out inside a cone with opening an-gle θ ∼ 1/Γ oriented along the direction in which thecusp contracts), a quadrupole distribution of the localenergy density is expected to develop (see Ref. [4] forfurther details).

This scenario implies that a short burst of GW emis-sion should occur in the time leading up to cusp anni-hilation. In this brief time scale,δtc, the large asym-metric cusp shrinkage and energy reconversion meanthat a powerful GW burst would be emitted. As sug-gested above, we expect that the GW burst and theγ -ray burst should have exactly the same duration,and emphasize that long bursts (�t ∼ 200 s) mayalso be possible [4]. This last point may be real-ized if a special combination of intergalactic magneticstrengths, i.e.,B ∼ 10−7 G, and GRB fluences, e.g.,S ∼ 10−8 erg cm−2, is invoked. This possibility is il-lustrated in Table 1, where severalγ -ray fluences fromparticular events are combined with magnetic fieldstrengths thought to exist around the GRB sourcesin the context of the SCCS cusp annihilation mecha-nism.

The GW characteristics (amplitude and frequency)can be estimated using the typical dynamical timescalefor GRBs in this model. According to BHV2000 [4],theγ -ray timescale (GRB duration) is given by

T GRBsGWs � 0.24 ms

(B

10−8 G [13])2( α

10−8

)4

(1)

×(

10−4 erg cm−2

S

)(1+ z)−1

[(1+ z)−1/2 − 1]−2 ,

whereS is the GRB fluence in units of 10−4 erg cm−2,B is the intergalactic magnetic field in units of10−8 G [13], andz ∼ 4 is the source redshift. This

1 In the BHV2000 model this Lorentz factor may reach upto Γ ∼ 6.7 × 107. However, when the back-reaction effect ofthe electromagnetic radiation emitted in the process is taken intoconsideration the factor reduces toΓ � 104 [12], which is veryconsistent with the correspondings values inferred from BATSEobservations [5,11].

H.J. Mosquera Cuesta, D. Morejón González / Physics Letters B 500 (2001) 215–221 217

Fig. 1. Locus of the GW characteristics (−3/2 slope equally spaced lines) of the burst produced by the SCCS cusp annihilation (computed forη = 1016 GeV and�t = 2.4 ms) plotted against strain (burst) spectral densities and frequency bandwidth of the interferometers LIGO (I, II)and the futuretwin TIGAs: theMário Schönberg(MS, Brazil) and theMini-GRAIL (MG, Netherlands). Symbols *, XX and @ denote GWbursts with frequencies 150 Hz (Γ = 300), 420 Hz (Γ = 300) and 3200 Hz (Γ = 103, very short bursts (SB)), respectively. These GW signalsmay be triggered simultaneously with GRBs having the characteristics shown in Table 1 and Lorentz factors (LF)Γ as indicated here.

Table 1Inferred duration of the GRB (GW) emission phase (giving a large part of the total energy) as a function of characteristic values for the BATSEGRB fluence, and the assumed intergalactic magnetic field strength, with the CS parameterα ∼ 0.4× 10−8

TGRBsGWs [s] FluenceS [erg cm−2] Magnetic fieldB [G] EnergyIsotγ [erg] GRB [BATSE]

6.7× 10−6 3× 10−3 10−8 [13] ∼ 3× 1054 971214

3.2× 10−4 @ (Fig. 1) 1× 10−4 10−8 6.7× 1053 991216

2.5× 10−3 * (Fig. 1) 1× 10−5 10−7 6.7× 1053 991216

6.7× 10−3 XX (Fig. 1) 3× 10−4 10−7 � 3× 1054 990123

2.0× 10−2 1× 10−5 10−7

200 LISA target 1.5× 10−8 10−7 [4] � 5× 1051

value for the redshiftz at which the SCCS is locatedin this model has been taken in agreement with BATSEobservations of the most distant and most energeticGRBs ever detected: GRB000131 atz = 4.5, Ander-

sen et al. (VLT Team) [14]. It is also consistent withexisting models for ultrahigh-energy cosmic rays andthe observed shape of their spectrum.

218 H.J. Mosquera Cuesta, D. Morejón González / Physics Letters B 500 (2001) 215–221

Fig. 2. Locus of the GW characteristics (computed forη= 1010 GeV and�t = 1 ms) compared with the strain (burst) spectral density and thefrequency bandwidth of LISA. The confusion limit produced by the background of white dwarf binaries is also plotted. The symbol @ indicatesa GW burst with frequency 10−2 Hz andh= 4× 10−23, in units of (Hz)−1/2, for a GRB withΓ = 6.5× 106.

This is a timescale that could be observed byBATSE (time resolution�t ∼ 100 µs) in very shortGRBs from this sort of cosmological source (seeTable 1), were it not for its low efficiency for suchbursts (see for example Trigger Number 01453,�t =0.006± 0.0002 in Table 1 in Cline, Matthey andOtwinowski [15]). In what follows we concentrate onthis kind of GRB.

One can use the general relativity (GR) quadru-pole formula to make an estimate of the charac-teristic GW amplitude: the dimensionless spacetimestrain (h), generated by the non-spherical dynam-ics of thecuspkinetic energy, is related to the dis-

tance of the CS (D) by hij = 2Gc4D

d2Qij

dt2, where

Qij is the second moment of the mass distribu-tion (quadrupole mass-tensor) in the transverse trace-less (TT) gauge and is given byQij ≡ ∫

ρ(xixj −1/3x2)d3x, with ρ being the mass-energy density

of the source. This expression can be rewrittenas [16]

(2)h= 2G

c4

ENon-sym

DL,

where DL is the luminosity distance defined byDL = rz(1+ z), with z being the source redshift(inferred from the spectrum of the GRB host), andrz is the comoving distance given byrz = 2 c

H0×

(1− [1+ z]−1/2). HereH0 is the present-day valueof the Hubble constant. For the case which we arestudying, we can approximate the non-symmetric partof the kinetic energy of the annihilating cusp asENon-sym ∼ Mc(dl/dt)2, whereMc ∼ µδlc ∼ µcδtcis the total mass of the cusp. At the time when thegamma-ray outburst occurs, we can express the timederivative of the cusp characteristic linear dimensionas (δl/δt)2 ∼ (Γ δlc/δtc)2 ∼ Γ 2c2, with δt ∼ δtc , as

H.J. Mosquera Cuesta, D. Morejón González / Physics Letters B 500 (2001) 215–221 219

discussed by BHV2000. We also identify the GWpulse duration asδtc ∼ T GRBs

GWs , given by Eq. (1).Using these expressions, we can writeENon-sym ∼µc3Γ 2δtc, and Eq. (2) becomes

h∼[G

cDL

]µΓ 2T GRBs

GWB ∼ 1.9× 10−22Γ 2(T GRBsGWB

)

(3)×(

1028 cm

DL

)(µ

(1016GeV)2

),

where we have used as the GUT symmetry breakingscale for the SCCS:η ∼ 1016 GeV, and have assumedthat the source is at a distance equal to the Hubbleradius. Consistently with current observations, andfollowing BHV2000, we will consider sources atlow z ∼ 5 [14] for which Γ = 300 andΓ = 103

are plausible Lorentz factors according to BATSEobservations and the fireball model [5]. Recall thathigh Γ values are needed in order for the fireballto avoid overproduction of electron–positron pairs[17]. From Eq. (3) we then havehΓ=300

6.7 ms ∼ 9.4 ×10−21 Hz−1/2, hΓ=300

2.4 ms ∼ 2.1 × 10−21 Hz−1/2 and

hΓ=103

0.32 ms∼ 1.5 × 10−21 Hz−1/2, for the burst durationand Lorentz factor, as indicated.

The maximum frequency of the GW burst in thereference frame of the loop can be approximated asfGW ∼ t−1

dyn, with tdyn ∼ T GRBsGWs being the dynamical

timescale for annihilation of the cusp. This thenimpliesh∼ f−3/2

GW and so we can write

(4)

fGW ≡ (T GRBs

GWs

)−1 ∼(δlc

c

)−1

∼ 150,420,3200 Hz,

for GRBs with the durations given in Table 1. Suchfrequencies fall just within the range of highest sensi-tivity for the LIGO (I, II), VIRGO and GEO-600 in-terferometers, and for the Brazilian Mário Schönbergand Dutch Mini-GRAIL TIGAs, as shown in Fig. 1.Thus, in the high frequency regime, the bursts maybe detectable for higherΓ and much lowerη, fora given�t , than is the case at lower frequencies.

For the very low GW frequency band 10−4–100 Hz,the LISA antenna could observe these signals even forextremely low GUT energy scales but largeΓ factors,as shown in Fig. 2 (of course, it could also observe theGW pulses for higherη and lowerΓ ).

In the context of the BHV2000 SCCS model forGRBs it is also possible for an outburst of ultra

high energy cosmic rays to be released simultaneouslywith the GRB surge. Assuming that a packet of suchparticles comes in the form of a neutrino burst, wecan estimate the neutrino time-of-flight delay withrespect to both the GRB and GW signals. Thismeasurement can provide an indication of the neutrinomass eigenstates. Next we make a rough estimate ofthe overall characteristics of such aν-spectrum, anduse it to constrain the predicted time lag. (A moreconsistent calculation of the actual UHECR spectrumin this picture, in the light of their detectability by theAUGER experiment, is now under way [18].)

Topological defects or unstable relic particles pro-duce ultra high energy photons at a ratenp =np,0(t/t0)

−m, wherem = 0,3,4 for decaying parti-cles, ordinary CSs and necklaces, and SCCS, respec-tively [1,2]. In the GRB fireball picture [17], the de-tectedγ -rays are produced via synchrotron radiationcoming from ultrarelativistic electrons boosted by in-ternal shocks in an expanding relativistic blast wave(wind) consisting of electron–positron pairs, somebaryons and a huge number of photons. The typicalsynchrotron frequency is constrained by the character-istic energy of the accelerated electrons and also bythe intensity of the magnetic field in the emitting re-gion. Since the electron synchrotron cooling time isshort compared with the wind expansion time, elec-trons lose their energy radiatively. The standard energyof the observed synchrotron photons (see Refs. [19,20]for a more complete review of this mechanism) isEbγ = Γ hγ 2

e eB/(mec) which is then given by

Ebγ � 45 MeVξ1/2B ξ

3/2e

(Lγ

1054 erg s−1

)1/2(103

Γ

)2

(5)×(

0.25 ms

�t

),

whereLγ = 1054 erg s−1 is the power released inthe most energetic GRBs observed by BATSE. ThisLγ may imply Lorentz expansion factorsΓ ∼ 103,and we assume this in the following.�t = 0.25 msis the inferred timescale for the shorterγ -ray burstsfrom the BHV2000 SCCSs,ξB is the fraction of theenergy carried by the magnetic field: 4πr2

d cΓ2B2 =

8πξBL, whereL is the total wind luminosity, andξeis the luminosity fraction carried away by electrons.No theory is available to provide specific values forξB andξe. However, for values near to equipartition,

220 H.J. Mosquera Cuesta, D. Morejón González / Physics Letters B 500 (2001) 215–221

the break energyEbν for photons in this model is inagreement with the observed one forΓ ∼ 103 and�t = 0.25 ms, as discussed below. More precisely,the hardness of the GRB spectra, which extend up to18 GeV, constrains the wind to have Lorentz factorΓ ∼ 103, while the observed variability of the GRBflux on a timescale�t � 1 ms implies that the internalcollisions occur at a distance from the center ofrd ∼Γ 2�t , due to variability of the central engine on thesame timescale. Since most of the BATSE observedGRBs show variability with�t � 10 ms and thereis rapid variability with �t � 1 ms, the impliedcharacteristic size of the emitting region isrem ∼ 107

cm which means that it must be a compact domain.The SCCS loop cusp clearly satisfies this constraint.

In the acceleration region, protons (the fireballbaryon load) are also expected to be shocked. Theirphoto–mesoninteraction with observedγ -burst pho-tons should produce a surge of neutrinos almost si-multaneously with the GRB via the decayp+n0

γ →π+ ↔ µ+ + νµ ↔ e+ + νe + νµ + νµ. The neutrinospectrum for a fireball driven explosion is expectedto follow the observedγ -ray spectrum, which is ap-proximately a broken power-lawdNγ /dEγ ∝ E−β

γ ,with β ∼ 1 for low energies andβ ∼ 2 for high en-ergies as compared with the observedbreak energyEβγ ∼ 45 MeV, whereβ changes. The interaction of

protons from the surrounding medium, accelerated toa power-law distributiondNp/dEp ∝ E−2

p , with thefireball photons, leads to a broken power-law neutrinospectrumdNν/dEν ∝ E−β

ν , with β = 1 forEν < Ebν ,andβ = 2 for Eν > Ebν . Thus the neutrino break en-ergyEbν is fixed by the threshold energy of photonsfrom photo-productioninteracting with the dominant∼ 45 MeV fireball photons (in our case), and is

(6)Ebν � 1.3× 1015(Γ

103

)2(45 MeV

Ebγ

)eV.

Thus, forνs produced with the above energy a furtherFermi cycle in the ultrarelativistic blast wave mayamplify the UHECR energy by a factor ofΓ 2 which,for the case of protons, may push them over the GZKlimit [6] ( νs do not have a GZK cut-off). The part ofthe total fireball luminosity that escapes as theν-flux isdetermined by the efficiency of pion production. Theenergy fraction lost via pion production by protons

producing νs above the break energyis essentiallyindependent of the energy and can be expressed as

fπ = 0.23

(Lγ

1054 erg s−1

)(45 MeV

Ebγ

)(103

Γ

)4

(7)×(

0.25 ms

�t

).

Thus, an important part of the total wind energy isgiven to these very high energyνs.

The time-of-flight delay of theνs with respect tothe GWs (and the GRB) may be computed by usingthe spectrum Eq. (6) and Table 2, above. This gives[21]

�T GWs–GRBsν

(8)

∼ 1.545 s

(D

3 Gpc

)(m2ν

100 eV2

)(100 GeV2

E2ν

).

This equation was originally derived as a way of es-timating thetime-of-flight lagbetween massive neu-trinos and massless ones, which should travel at thespeed of light [21]. However, it can also be applied tothe problem which we are studying here, since we as-sume that the GWs propagate at the velocity of light,as in GR. It turns out that the detection of such a neu-trino pulse with a delay of approximately 1.5 seconds(for Eν ∼ 1010 eV) after the GRB and GW outburstsfrom the same source on the sky would make it pos-sible to impose tighter bounds on the neutrino massspectrum, since the source distance may be estimatedfrom its redshift and the GWs detected. Of course,there are some uncertainties involved in the derivationof Eq. (8). However, it is foreseeable that if atomicclocks were installed in both the GW andν observa-tories, a very precise measurement of the arrival timesmight be obtained, making this determination a plau-sible one in the near future.

To summarize, since theν-spectrum ranges overfourteen orders of magnitude (MeVνs from SN1987A,above GZKνs detected by AGASA, Fly’s Eye, etc.,see Table 2), and theν-energy can be measured di-rectly at the detector, the detection of any species ofneutrino in near spatial and temporal coincidence withobserved GW+ GRB signals might yield a very accu-rate estimate of the time-delay between them. Throughthe analysis of such a time lag, one may verify or ruleout the BHV2000 SCCS model, and also clarify the

H.J. Mosquera Cuesta, D. Morejón González / Physics Letters B 500 (2001) 215–221 221

Table 2Time delay between GWs (GRBs) andν-bursts as a function of theν-energy in the BHV2000 SCCS model

�TGRBs–GWsνs [s] ν energy [eV]

1.545 1010

1.545× 10−8 1014

1.545× 10−20 1020

value of the GW propagation velocity which is a quan-tity of great interest for discriminating between differ-ent relativistic theories of gravity [22,23].

Acknowledgements

We are indebted to Prof. John C. Miller (SISSAand OXFORD) for his patience in reviewing our pa-per and for his important suggestions for making thismanuscript comprehensible. H.J.M.C. thanks the Cen-tro Latino-Americano de Física (Rio de Janeiro) andConselho Nacional de Desenvolvimento e Pesquisa(CNPq, Brazil) for financial support. D.M.G. alsothanks CNPq for a Graduate Scholarship.

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