Ultrahigh energy neutrinos from superconducting cosmic strings

Ultrahigh energy neutrinos from superconducting cosmic strings

Veniamin Berezinsky,1,* Ken D. Olum,2,† Eray Sabancilar,2,‡ and Alexander Vilenkin2,x1INFN, Laboratori Nazionali del Gran Sasso, I–67010 Assergi (AQ), Italy

2Institute of Cosmology, Department of Physics and Astronomy, Tufts University, Medford, Massachusetts 02155, USA.(Received 6 January 2009; published 31 July 2009)

Superconducting cosmic strings naturally emit highly boosted charge carriers from cusps. This occurs

when a cosmic string or a loop moves through a magnetic field and develops an electric current. The

charge carriers and the products of their decay, including protons, photons, and neutrinos, are emitted as

narrow jets with opening angle �� 1=�c, where �c is the Lorentz factor of the cusp. The excitation of

electric currents in strings occurs mostly in clusters of galaxies, which are characterized by magnetic

fields B� 10�6 G and a filling factor fB � 10�3. Two string parameters determine the emission of the

particles: the symmetry breaking scale �, which for successful applications should be of order

109–1012 GeV, and the dimensionless parameter ic, which determines the maximum induced current as

Jmax ¼ ice� and the energy of emitted charge carriers as �X � ic�c�, where e is the electric charge of a

particle. For the parameters � and B mentioned above, the Lorentz factor reaches �c � 1012 and the

maximum particle energy can be as high as �c�� 1022 GeV. The diffuse fluxes of ultrahigh energy

neutrinos are close to the cascade upper limit, and can be detected by future neutrino observatories. The

signatures of this model are: very high energies of neutrinos, in excess of 1020 eV; correlation of neutrinos

with clusters of galaxies; simultaneous appearance of several neutrino-produced showers in the field of

view of very large detectors, such as JEM-EUSO; and 10 TeV gamma radiation from the Virgo cluster.

The flux of ultrahigh energy protons from cusps may account for a large fraction of the observed events at

the highest energies.

DOI: 10.1103/PhysRevD.80.023014 PACS numbers: 98.70.Sa, 11.27.+d, 98.80.Cq

I. INTRODUCTION

A. Neutrino astronomy

Ultrahigh energy (UHE) neutrino astronomy at energiesabove 1017 eV is based on new, very efficient methods ofneutrino detection and on exciting theories for neutrinoproduction. The most interesting range of this astronomycovers tremendously high energies above 1019–1020 eV. Infact, this energy scale gives only the low energy threshold,where the new observational methods, such as space-basedobservations of fluorescent light and radio and acousticmethods, start to operate. These methods allow observationof very large areas and so detection of tiny fluxes ofneutrinos. For example the exposure of the space detectorJEM-EUSO [1] is planned to reach �106 km2 yr sr. Theupper limits obtained by radio observations are presentedin Fig. 1.

The basic idea of detection by EUSO is similar to thefluorescence technique for observations of extensive airshowers (EAS) from the surface of the Earth. The UHEneutrino entering the Earth’s atmosphere produces an EAS.A known fraction of its energy, which reaches 90%, isradiated in the form of isotropic fluorescent light, whichcan be detected by an optical telescope in space. There is

little absorption of up-going photons, so the fraction of fluxdetected is known, and thus EUSO provides a calorimetricmeasurement of the primary energy. In the JEM-EUSOproject [1] a telescope with diameter 2.5 m will observe anarea �105 km2 and will have a threshold for EAS detec-tion Eth � 1� 1019 eV. The observations are planned tostart in 2012–2013.UHE neutrinos may also be very efficiently detected by

observations of radio emission by neutrino-induced show-ers in ice or lunar regolith. This method was originally

FIG. 1 (color online). The experimental upper limits on UHEneutrino fluxes in comparison with the electromagnetic cascadeupper limit in assumption of E�2 generation spectrum (labeled‘‘E�2 cascade’’) and with predictions for cosmogenic neutrinos.Neutrino fluxes are given for one neutrino flavor �i þ ��i.

*[email protected]†[email protected]‡[email protected]@cosmos.phy.tufts.edu

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http://dx.doi.org/10.1103/PhysRevD.80.023014

suggested by G. Askaryan in the 1960s [2]. Propagating inmatter the shower acquires excess negative electric chargedue to scattering of the matter electrons. The coherentCerenkov radiation of these electrons produces a radiopulse. Recently this method has been confirmed by labo-ratory measurements [3]. Experiments have searched forsuch radiation from neutrino-induced showers in theGreenland and Antarctic ice and in the lunar regolith. Inall cases the radio emission can be observed only forneutrinos of extremely high energies. Upper limits on theflux of these neutrinos have been obtained in the GLUEexperiment [4] by radiation from the moon, in the FORTEexperiment [5] by radiation from the Greenland ice, and inthe ANITA [6] and RICE [7] experiments from theAntarctic ice.

Probably the first proposal for detection of UHE neutri-nos with energies higher than 1017 eV was made in [8]. Itwas proposed there to use the horizontal EAS for neutrinodetection. Later this idea was transformed into the Earth-skimming effect [9] for � neutrinos. Recently the Augerdetector [10] put an upper limit on UHE neutrino flux usingthe Earth-skimming effect (see Fig. 1).

B. UHE neutrino sources

What might these new large-area UHE neutrino observ-atories detect? On the one hand, there are without doubtcosmogenic neutrinos, produced by ultrahigh energy cos-mic rays (UHECR) particles interacting with the cosmicmicrowave background (CMB) photons. On the otherhand, there may be neutrinos produced in decays or anni-hilation of superheavy particles; this is referred to as thetop-down scenario.

Cosmogenic neutrinos were first discussed in [11], soonafter the prediction of the Greisen-Zatsepin-Kuzmin(GZK) cutoff [12]. There, it was shown that UHE neutrinofluxes much higher than the observed UHECR flux can beproduced by protons interacting with CMB photons atlarge redshifts. The predicted flux depends on the cosmo-logical evolution of the sources of UHE protons and on theassumed acceleration mechanisms. Recent calculations ofcosmogenic neutrino fluxes (see e.g. [13–20]) are normal-ized to the observed UHECR flux, with different assump-tions about the sources.

The energies of cosmogenic neutrinos are limited by themaximum energy of acceleration, Emax

acc . To provide neu-trinos with energies above 1� 1020 eV, the energies ofaccelerated protons must exceed 2� 1021 eV. For non-relativistic shocks, the maximum energy of acceleration,Emaxp , can optimistically reach 1� 1021 eV. For relativistic

shocks this energy can be somewhat higher. Production ofcosmogenic neutrinos with still higher energies depends onless-developed ideas, such as acceleration in strong elec-tromagnetic waves, exotic plasma mechanisms of accel-eration, and unipolar induction.

The top-down scenarios, on the other hand, naturallyprovide neutrinos with energies higher and much higher

than 1� 1020 eV [21]. The mechanism common to manymodels assumes the existence of superheavy particles withvery large masses up to the grand unified theory scale�1016 GeV. Such particles can be produced by topologicaldefects (TD) (see [22] for a general review). They thenrapidly decay and produce a parton cascade, which isterminated by production of pions and other hadrons.Neutrinos are produced in hadron decays.The production of unstable superheavy particles—the

constituent fields of TD—is a very common feature of theTD [23]. However, the dynamics of TD is highly nonlinearand complicated, the distance between TD is model-dependent, and the calculation of UHE particle fluxesrequires special consideration for different types of TD[24].Cosmic strings can release particles in the process of

self-interaction, and in the final evaporation of tiny loops,but only a few particles are produced by each such inter-action. Of more interest are cosmic string cusps, where thestring doubles back on itself and moves with a hugeLorentz factor [25]. Particles emitted by cusps have ener-gies much higher than their rest masses, because of theboost. However, the flux from such events is too low to beobserved [26,27].Monopole-antimonopole pairs connected by strings

[28–30] can release superheavy particles when the mono-pole and antimonopole finally annihilate. However, suchdefects, similar to superheavy dark matter (see below),would be accumulated inside galaxies, and, in particular,in the Milky Way. The resulting UHECR flux would bedominated by photons, which can reach us easily fromshort distances. Such photons are not observed [31] at thelevel that would be necessary if top-down production wereto account for the observed UHECR.If each monopole is attached to two strings, we have

necklaces. Necklaces are an attractive source for UHEneutrinos [32,33], but simple models of necklaces maylead to rapid annihilation of the monopoles [34]. In othermodels, however, the monopoles may survive for muchlonger, providing a detectable flux of UHE neutrinos.1

In a wide class of particle physics models, cosmic stringscan be superconducting, in which case they respond toexternal electromagnetic fields as thin superconductingwires [35]. String superconductivity arises when a conden-sate of charged particles (which can be either bosons orfermions) is bound to the string. These particles have zero

1The main point of Ref. [34] is that the relativistic motion ofstrings causes monopoles to develop large velocities along thestring. As a result monopoles frequently run into one another andannihilate. A possible way to avoid this is to consider lightstrings, which remain overdamped till very late times and there-fore move slowly. Another possibility is that the strings havezero modes, which act as a one-dimensional gas on the stringsand slow the monopoles down. These models need furtherinvestigation.

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mass in the bound state, whereas away from the string theyhave some mass mX. Loops of superconducting stringdevelop electric currents as they oscillate in cosmic mag-netic fields. Near a cusp, a section of string acquires a largeLorentz boost �c, and simultaneously the string current isincreased by a factor �c. If the current grows to a criticalvalue Jmax charge carriers rapidly scatter off each other andare ejected from the string. The decay products of theseparticles can then be observed as cosmic rays. This modelwill be the subject of the present paper.

Apart from TD, superheavy particles can naturally beproduced by thermal processes [36,37] and by time-varying gravitational fields [38,39] shortly after the endof inflation. These particles can survive until the presentand produce neutrinos in their decays. Protected by sym-metry (e.g. discrete gauge symmetry, in particular, R-parityin supersymmetric theories), these particles can have verylong lifetimes exceeding the age of the Universe. Theresulting neutrino flux may exceed the observed flux ofUHECR. However, like any other form of cold dark matter(CDM), superheavy particles accumulate in the MilkyWayhalo and produce a large flux of UHE photons. The non-observation of these photons puts an upper limit on theneutrino flux from intergalactic space.

C. The cascade bound

The neutrino fluxes are limited from above. The mostgeneral upper bound for UHE neutrinos, valid for bothcosmogenic neutrinos and neutrinos from top-down mod-els, is given by the cascade upper limit, first considered in[8,40]. The production of neutrinos in these scenarios isaccompanied by production of high energy photons andelectrons. Colliding with low energy target photons, aprimary photon or electron produces an electromagneticcascade due to the reactions �þ �target ! eþ þ e�, eþ�target ! e0 þ �0, etc. The cascade spectrum is very close

to the EGRET observations in the range 3 MeV–100 GeV[41]. The observed energy density in this range is!EGRET � ð2–3Þ � 10�6 eV=cm3. To be conservative, wewill use the lower end of this range. It provides the upperlimit for the cascade energy density. The upper limit onUHE neutrino flux J�ð>EÞ (sum of all flavors) is given bythe following chain of inequalities

!cas >4�

c

Z 1

EE0J�ðE0ÞdE0 >

4�

cEZ 1

EJ�ðE0ÞdE0

� 4�

cEJ�ð>EÞ: (1)

Here c is the speed of light, but will generally work in unitswhere c ¼ 1 and @ ¼ 1. In terms of the differential neu-trino spectrum, Eq. (1) gives

E2J�ðEÞ< c

4�!cas; with !cas <!EGRET: (2)

Equation (2) gives a rigorous upper limit on the neutrinoflux. It is valid for neutrinos produced by high energyprotons, by topological defects, by annihilation and decaysof superheavy particles, i.e., in all cases when neutrinos areproduced through decay of pions and kaons. It holds for anarbitrary neutrino spectrum decreasing with energy. If oneassumes some specific shape of neutrino spectrum, thecascade limit becomes stronger. For a generation spectrumproportional to E�2, which is used for analysis of obser-vational data, one obtains a stronger upper limit. Given forone neutrino flavor it reads [42]

E2JiðEÞ � 1

3

c

4�

!cas

lnðEmax=EminÞ ; (3)

where Emax and Emin give the range of neutrino energies towhich the E�2 spectrum extends, and i ¼ �� þ ��, or i ¼�e þ ��e, or i ¼ �� þ ��. This upper limit is shown inFig. 1. One can see that the observations almost reach thecascade upper limit and thus almost enter the region ofallowed fluxes.The most interesting energy range in Fig. 1 corresponds

to E� > 1021 eV, where acceleration cannot provide pro-tons with sufficient energy for production of these neutri-nos. At present the region of E� > 1021 eV, and especiallyE� � 1021 eV is considered as a signature of top-downmodels, which provide these energies quite naturally.

D. Model assumptions

In this paper we consider superconducting string loopsas a source of UHE neutrinos. We consider a simple modelin which a magnetic field of magnitude B, occupying afraction of space fB, is generated at some epoch zmax �2–3. The strings are characterized by two parameters: thefundamental symmetry breaking scale � and the criticalcurrent Jmax. We take the mass per unit length of string tobe � ¼ �2.The predicted flux of UHE neutrinos depends on the

typical length of loops produced by the string network.This issue has been a subject of much recent debate, withdifferent simulations [43–46] and analytic studies [47,48]yielding different answers. Here we shall adopt the picturesuggested by the largest and, in our view, the most accuratesimulations of string evolution performed to date [45,46].According to this picture, the characteristic length of loopsformed at cosmic time t is given by the scaling relation

l� t; (4)

with � 0:1.For simplicity and transparency of the formulas obtained

in this paper we use several simplifications. We assumecosmology without � term with �cdm þ�b ¼ 1, the ageof the Universe t0 ¼ ð2=3ÞH�1

0 ¼ 3� 1017 s, teq � 1�1012 s, and ð1þ zÞ3=2 ¼ t0=t for the connection of age tand redshift z in the matter era.

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We also assume the fragmentation function for the decayof superheavy X particle into hadrons is

dN=dE / E�2; (5)

while Monte Carlo simulation and the DGLAP methodgive closer to E�1:92 [49].

These simplifications give us a great advantage in under-standing the dependence of calculated physical quantitieson the basic parameters of our model, in particular, onfundamental string parameter �. Our aim in this paper is toobtain the order of magnitude of the flux of UHE neutrinosand to indicate the signatures of the model. We believe oursimplified model assumptions are justified, given the un-certainties of string evolution and of the evolution ofcosmic magnetic fields.

II. PARTICLE EMISSION FROMSUPERCONDUCTING STRINGS

A. Particle bursts from cusps

As first shown by Witten [35], cosmic strings are super-conducting in many elementary-particle models. As theyoscillate in cosmic magnetic fields, such strings developelectric currents. Assuming that the string loop size issmaller than the coherence length of the field l & lB �1 Mpc, the electric current can be estimated as [22,35]

J � 0:1e2Bl: (6)

where e� 0:1 is the elementary electric charge inGaussian units, renormalized to take into account self-inductance [22].

Particles are ejected from highly accelerated parts ofsuperconducting strings, called cusps, where large electriccurrents can be induced [50,51]. The current near a cuspregion is boosted as

Jcusp � �cJ; (7)

where J is the current away from the cusp region and �c isthe Lorentz factor of the corresponding string segment.Particles are ejected from portions of the string that de-velop Lorentz factors

�c � Jmax=J; (8)

where the current reaches the critical value Jmax. Thismaximum current is model-dependent, but is bounded byJmax & e�, where � is the symmetry breaking scale of thestring.

One may parametrize Jmax by introducing the parameteric < 1:

Jmax ¼ ice�: (9)

If the charge carrier is a superheavy particle X with massmX, the case which will be considered here, one may use�rX for the energy of an X particle in the rest system of thecusp and �X in the laboratory system. Then �rX ¼ �mX ¼

ic� and

�X � ic�c�; (10)

respectively, where � is the average Lorentz factor of the Xparticle in the rest system of the cusp. In Eq. (10) we tookinto account that the energy of the X particle in the labo-ratory system is boosted by the Lorentz factor of the cusp�c.The number of X particles per unit invariant length of the

string is �J=e, and the segment that develops Lorentzfactor �c includes a fraction 1=�c of the total invariantlength l of the loop. Hence, the number of X particlesejected in one cusp event (burst) is

NbX � ðJ=eÞðl=�cÞ � J2l=eJmax: (11)

The oscillation period of the loop is l=2, so assuming onecusp per oscillation, the average number of X particlesemitted per unit time is

_N X � 2J2=eJmax; (12)

and the luminosity of the loop is

Ltot � _NX�X: (13)

The X particles are short-lived. They decay producingthe parton cascade which is developed due to parton split-ting in the perturbative regime, until at the confinementradius the partons are converted into hadrons, mostly pionsand kaons, which then decay producing gamma rays, neu-trinos, and electrons. These particles together with lessnumerous nucleons give the observational signatures ofsuperconducting cusps.The neutrino spectrum at the present epoch, z ¼ 0,

produced by the decay of one X particle with energy �X �ic�c� at epoch z can be calculated using the fragmentationfunction (5) for an X particle at rest:

�ðEÞ � ic��c

2ð1þ zÞ lnðErestmax=E

restminÞ

1

E2; (14)

where Erestmax and Erest

min are the maximum and minimum

neutrino energies in the rest system of X particle.Particle emission from a cusp occurs within a narrow

cone of opening angle

�c � ��1c � J=Jmax: (15)

The duration of a cusp event is [51]

tburst � l��3c : (16)

B. Superconducting loops in the Universe

In any horizon-size volume of the Universe at arbitrarytime there are a few long strings crossing the volume and alarge number of small closed loops. As loops oscillateunder the force of string tension, they lose energy byemitting gravitational waves at the rate


023014-4

_E g � �G�2; (17)

where �� 2 is the string mass per unit length, G ¼1=m2

Pl is the gravitational constant, and �� 50 is a nu-

merical coefficient.The number density of loops with lengths in the interval

from l to lþ dl at time t can be expressed as nðl; tÞdl. Ofgreatest interest to us are the loops that formed during theradiation era t < teq and still survive at t > teq. The density

of such loops at time t is given by [22]

nðl; tÞdl� t1=2eq t�2l�5=2dl; (18)

in the range from the minimum length lmin to the maximumlength l� teq, where

lmin � �G�t� 3� 1011�210ð1þ zÞ�3=2 cm (19)

and �10 ¼ �=1010 GeV. Here and below we assume thatthe loop length parameter in (4) is � 0:1, as suggested bysimulations [45,46]. Loops of the minimum length are ofmost importance in our calculations because they are themost numerous.

For a loop of length l at redshift z, the Lorentz factor atthe cusp �c can be expressed as

�c ¼Jcusp

J¼ ice�

0:1e2Bl¼ �cðlminÞ lmin

l; (20)

where �cðlminÞ ¼ �0ð1þ zÞ3=2 and

�0 ¼ 10ic�

eBt0�G�¼ 1:1� 1012ic

B�6�10

; (21)

where B�6 is the magnetic field in microgauss.

C. Limits on �

The string motion is overdamped at early cosmic times,as a result of friction due to particle scattering on movingstrings. The friction-dominated epoch ends at

t� � ðG�Þ�2tp; (22)

where tp is the Planck time. In the above analysis we have

assumed that loops of interest to us are formed at t > t�.The corresponding condition

�G�t0= * t� (23)

yields

� * 109 GeV: (24)

For strings with �< 109 GeV, loops of the size given by(19) never form. Instead, the smallest loops are those thatform at time t� with length

lmin � t�; (25)

and then survive until the present day.We should also verify that energy losses due to particle

emission and to electromagnetic radiation in recent epochs

(after magnetic fields have been generated) are sufficientlysmall, so the lifetimes of the loops (which we estimatedassuming that gravitational radiation is the dominant en-ergy loss mechanism) are not significantly modified.The average rate of energy loss due to particle emission

is

_E part � fB _NX�X � 2fBJJmax=e2; (26)

where we have used Eqs. (10) and (12). The electromag-netic radiation power is smaller by a factor e2 � 10�2.The factor fB in Eq. (26) is the filling factor—the

fraction of space filled with the magnetic field. It givesthe fraction of time that cosmic string loops spend inmagnetized regions. We assume that loop velocities aresufficiently high that they do not get captured in magne-tized cosmic structures (such as galaxy clusters or large-scale structure filaments). To justify this assumption, wenote that particle emission can start only after the cosmicmagnetic fields are generated, that is, at z� 3 or so. Beforethat, gravitational radiation is the dominant energy lossmechanism, and the loops are accelerated to high speeds bythe gravitational rocket effect [52,53]. The smallest loopsof length (19) have velocities v� 0:1, certainly largeenough to avoid capture.The particle emission energy rate (26) should be com-

pared to the gravitational radiation rate (17). The ratio ofthe two rates is zero at z > zmax, where zmax � 2–3 is theredshift of magnetic field production. At z < zmax it isgiven by

_E part= _Eg � 50f�3B�6ic��110

�l

lmin

�ð1þ zÞ�3=2; (27)

where f�3 ¼ fB=10�3 and lmin is given by (19).

If particle emission is the dominant energy loss mecha-nism, then the lifetime of a loop is

�part � �l_Epart

� 5�

eicfBB� 0:025

t0�10

f�3B�6ic: (28)

Note that � is independent of l. This means that all loopssurviving from the radiation era decay at about the sametime.For the time being, we shall assume that particle radia-

tion is subdominant. We shall discuss the opposite regimein Sec. II G.

D. Rate of cusp events

The rate of observable cusp bursts (i.e., the bursts whosespot hits the Earth) is given by

d _Nb ¼ fBd�

4��ðl; zÞdl dVðzÞ

1þ z; (29)

where, as before, fB is the fraction of space with magneticfield B, d� ¼ 2��d� is the solid angle element, with �limited by the angle of cusp emission �c � 1=�c, �ðl; zÞ ¼nðl; zÞ=ðl=2Þ is the frequency of the bursts with nðl; zÞ given


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by Eq. (18), and dVðzÞ is a proper volume of space limitedby redshifts z and zþ dz,

dVðzÞ ¼ 54�t30½ð1þ zÞ1=2 � 12ð1þ zÞ�11=2dz: (30)

Integrating Eq. (29) over �, l, and z, we obtain

_N b ¼ 54�ðteqt0Þ1=2ð�G�Þ�1=2ðe=10ic�Þ2

�Z zmax

0dz

½ð1þ zÞ1=2 � 12ð1þ zÞ11=4 fBðzÞB2ðzÞ; (31)

where zmax is the redshift at which the magnetic fields aregenerated. Since the Earth is opaque to neutrinos with theenergies we are considering, only half of these bursts canactually be detected by any given detector at the surface ofthe Earth or using the atmosphere.

The value of the integral in (31) depends on one’sassumptions about the evolution of the magnetic field Band of the volume fraction fB. This evolution is not wellunderstood. If we take these values out of the integral inEq. (31) as the average and characterize them by theeffective values of parameters B�6 and f�3 in the range0< z < zmax, then Eq. (31) reduces to

_N b ¼ 2:7� 102B2�6f�3

i2c�310

I

0:066yr�1; (32)

where the integral

I ¼Z z0

0dz

½ð1þ zÞ1=2 � 12ð1þ zÞ11=4

¼ 4

3½1� ð1þ z0Þ�3=4 � 8

5½1� ð1þ z0Þ�5=4

þ 4

7½1� ð1þ z0Þ�7=4 (33)

is equal to 0.015, 0.042, and 0.066 for z0 ¼ zmax ¼ 1, 2, and3, respectively.

The integrand in Eq. (31) includes the productfBðzÞB2ðzÞ. In the calculations of other physical quantitiesbelow, similar integrals will have different combinations offBðzÞ and BðzÞ. Nevertheless, we shall assume that theaverage values taken out of the integral are characterizedby approximately the same values of f�3 and B�6.

All cosmic structures—galaxies, clusters, and filamentsof the large-scale structure—are magnetized and contrib-ute to the rate of cusp bursts. In the recent epoch, z & 1, thedominant contribution is given by clusters of galaxies withB2�6f�3 � 1. The magnetic fields of galaxies have about

the same magnitude, but the corresponding filling factor fBis orders of magnitude smaller. We shall assume that thisholds in the entire interval 0< z < zmax. The sources in ourmodel are then essentially clusters of galaxies.

E. Diffuse flux of UHE neutrinos

The diffuse differential neutrino flux, summed over allproduced neutrino flavors, is given by the formula

J�ðEÞ ¼ 1

4�

Zd _NbN

bX�ðEÞ 1

�jetr2ðzÞ ; (34)

where d _Nb is the rate of cusp bursts (29), NbX is the number

of X particles produced per burst, given by Eq. (11), �ðEÞis the neutrino spectrum produced by the decay of one Xparticle, given by (14),

�jet ¼ ��2c ¼ �

�2c

; (35)

rðzÞ ¼ 3t0½1� ð1þ zÞ�1=2 (36)

is the distance between a source at redshift z and theobservation point at z ¼ 0, and �jetr

2 is the area of the

burst spot at the Earth from a source at redshift z.Using expressions (18) and (30), and assuming that the

product fBðzÞBðzÞ does not change much in the interval0< z < zmax, we obtain

2

E2J�ðEÞ ¼0:3icmplðteq=t0Þ1=2ðeBt20ÞfB

7�ð�Þ1=2t0ðct0Þ2 lnðErestmax=E

restminÞ

� ½1� ð1þ zmaxÞ�7=4: (37)

Numerically, this gives for the neutrino flux summed overneutrino flavors

E2J�ðEÞ ¼ 6:6� 10�8icB�6f�3 GeV cm�2 s�1 sr�1;

(38)

where we have set zmax ¼ 3 and estimated the logarithmicfactor as �30.For ic � 1, the flux (38) is close to the cascade upper

limit shown in Fig. 1. Notice that the diffuse neutrino flux(37) does not depend on �. The neutrino flux must corre-late with clusters of galaxies.To detect this flux, we need to monitor a target with

some large mass M. The effective cross section of thedetector is then

� ¼ ��NM=mN; (39)

where ��N � 3� 10�32 cm2 is the neutrino-nucleon crosssection at E * 1010 GeV and mN the mass of a nucleon.Because of the opacity of the Earth, the detector will seesolid angle about 2� sr. The detection rate of particles withenergy above E is

2�EJ�ðEÞ� � 23

�M

1018 g

��1010 GeV

E

�icB�6f�3 yr�1:

(40)

2We note that numerical simulations of the magnetic fieldevolution performed by Ryu et al. [54] do indicate that the spaceaverage of the magnetic field hBðzÞi ¼ fBðzÞBðzÞ remainsroughly constant at �10�9 G for 0< z & 3 and decreases atlarger values of z. The effective values B�6 and f�3 could bedifferent from those in Eq. (32) for the rate of bursts, but weneglect the possible difference.


023014-6

In the case of JEM-EUSO in tilt mode, M� 5� 1018 g,and thus we expect about 100ic detections per year, soevents can be expected for ic * 0:01.

F. Neutrino fluence and the number of neutrinosdetected from a burst

The fluence of neutrinos incident on the detector from aburst at redshift z can be calculated as

�ð>EÞ ¼ NbX�ð>EÞ�jetr

2ðzÞ : (41)

Consider a neutrino burst from a loop of length l atredshift z. Using Nb

X from (11), lmin from (19) and�ð>EÞ from (14), we obtain for a loop of any length l,

�ð>EÞ � 10i3c�3

18�eBt20E lnðErestmax=E

restminÞ½ð1þ zÞ1=2 � 12 ;

(42)

which numerically results in

�ð>EÞ � 1:2� 10�2 i3c�

310

B�6

�1010 GeV

E

�

� 1

½ð1þ zÞ1=2 � 12 km�2: (43)

The number of neutrinos detected in a burst is

Ndet� ��ð>EÞ�: (44)

With M� 5� 1018 g as above,

Ndet� ð>EÞ � 0:11

1010 GeV

E

i3c�310

B�6

1

½ð1þ zÞ1=2 � 12 :(45)

Therefore, for a certain range of ic�10 values and sourceredshifts z, multiple neutrinos can be detected as paralleltracks from a single burst. For example, for ic�10 � 3, andz� 1, Ndet

� � 17.For neutrino energies of interest, E� * 1� 1020 eV, the

neutrino Lorentz factor is so large that there is practicallyno arrival delay for neutrinos with smaller energies. Allneutrinos from a burst arrive simultaneously and produceatmospheric showers with parallel axes, separated by largedistances.

For other sets of parameters Ndet� < 1, i.e., only one

neutrino from a burst (or no neutrino) is detectable. As �increases, the rate of bursts (32) diminishes while thenumber of neutrinos per burst increases, so that the totalneutrino flux remains unchanged. The rate of detectedneutrino bursts with the number of detected neutrinosNdet

� > � for each burst is given by Eqs. (32) and (33),with zmax determined by Ndet

� ð>E; zmaxÞ ¼ � . UsingEq. (45) we obtain for xmax � ð1þ zmaxÞ

xmaxð>E; �Þ ¼�1þ

�0:11

�

i3c�310

B�6

1010 GeV

E

�1=2

�2; (46)

if (46) is less than 4, and xmax ¼ 4 if (46) is larger than 4.Introducing in Eq. (32) coefficient 1=2 which approxi-mately takes into account the absorption of UHE neutrinoscrossing the Earth we obtain for the rate of detected burstswith Ndet

� �

_N detb ð �Þ ¼ 2:1� 103

f�3B2�6

i2c�310

IðzmaxÞ yr�1; (47)

where IðzmaxÞ is given by Eq. (33) with zmax from Eq. (46).In Fig. 2, we have shaded the region of the parameter

space ð�; icÞ corresponding to a detectable flux of neutri-nos. Curved lines in the figure mark the regions where weexpect a burst with a given multiplicity of neutrinos, � ¼ 2,3, or 10, detected simultaneously by a detector with theparameters of JEM-EUSO tilted. To the left of the 2-neutrino-burst line, only a diffuse flux of single neutrinoscan be observed. This flux depends only on ic, and thevertical left boundary of the shaded region marks the valueof ic at which it drops below one particle detected per year.Note that the regions shown for multiple events are those

where we expect at least one burst per year whose averagemultiplicity is the given � or more. But it is possible even ifthe parameters are to the left of the � ¼ 2 line that wewould happen to observe multiple neutrinos from a singleburst, which would give a clear signature of neutrino-jetemission from cusps.Another quantity of interest is the rate of detected neu-

trinos f�ð �Þ in the events with neutrino multiplicitygreater than � . It is given by

108

109

1010

1011

1012

1013

0.01 0.1 1

η (G

eV)

ic

>2>3

>10

dominated by friction

one burst per year

one

neut

rino

per

year

particle radiation dominant

FIG. 2. The region of parameter space where neutrinos can beseen by a detector with the parameters of JEM-EUSO. Thecurved lines show the left edges of the regions in which burstscontaining at least 2, 3, and 10 neutrinos can be expected at leastonce per year. Below the dotted line, particle radiation is thedominant channel of energy loss from loops.


023014-7

f�ð �Þ ¼ 1

2

Z fB2

1

�2c

nðl; zÞdll

dVðzÞ1þ z

Ndet� ð>E; z; lÞ:

(48)

The important feature of the calculations is the indepen-dence of Ndet

� ð>E; z; lÞ from l. This allows us to integrateover l in Eq. (48) to obtain

f�ð �Þ ¼ 2:1� 103f�3B

2�6

i2c�310

Z zmaxð�Þ

0dzð1þ zÞ�ð11=4Þ

� ½ð1þ zÞ1=2 � 12Ndet� ð>E; zÞ; (49)

where zmaxð�Þ is given by Eq. (46). Using Eq. (45) forNdet

� ð>E; zÞ results inf�ð �Þ ¼ 1:3� 102icf�3B�6½1� x�7=4

max ðic; �10Þ yr�1;

(50)

for E> 1� 1019 eV. The asymptotic expression at0:11i3c�

310=B�6� � 1 gives

f�ð �Þ ¼ 1:5� 102ffiffiffi�

p i5=2c �3=210 B1=2

�6 yr�1: (51)

G. Neutrino fluxes in the particle-emission dominatedregime

So far we have assumed that gravitational radiation isthe dominant energy loss mechanism of strings. In theopposite regime, where the particle emission energy lossesdominate, the loop’s lifetime �part is independent of its

length and is given by Eq. (28). We shall analyze thisregime in the present section.

As before, we shall adopt the idealized model where themagnetic field B is turned on at time t ¼ tB, correspondingto redshift zmax,

tB � t0ð1þ zmaxÞ�3=2: (52)

The loops decay at the time tdec � tB þ �part. The rate of

observable bursts _Nb is given by Eq. (32) with I fromEq. (33), where the integration is taken between zdec andzmax and zdec is the redshift corresponding to the time tdec.

If �part * tB, the redshift zdec is significantly different

from zmax, with�z ¼ zmax � zdec * 1, and the value of I isnot much different from that evaluated in Sec. II D. This isan intermediate regime, in which the results we obtained inSecs. II D and II E for the rate of bursts and for the diffuseflux can still be used as order of magnitude estimates.

For �part � tB, the loops lose all their energy to particle

emission in less than a Hubble time. The condition �part �tB can also be expressed as _Epart= _EgðzmaxÞ � 1. Using

Eq. (27) with zmax � 3, we find this condition is met forthe smallest loops when

�� 6� 1010icf�3B�6 GeV: (53)

It marks the boundary of the strong particle-emission

domination regime and is shown by the inclined dottedline in Fig. 2. Below this line, the results of the precedingsections do not apply even by order of magnitude, but aswe shall see, detectable neutrino fluxes can still beproduced.The redshift interval �z ¼ zmax � zdec for �part � tB

can be estimated as

�z � 2

3

�part

tBð1þ zmaxÞ � 1; (54)

and the integral I in Eq. (33) is given by

I � �z½ð1þ zmaxÞ1=2 � 12

ð1þ zmaxÞ11=4: (55)

With zmax � 3, we have tB � t0=8, and

�part

tB� 0:2

�10

f�3B�6ic: (56)

The rate of bursts that are actually detected _Ndetb can be

expressed as a product of _Nb and the probability pdet� that at

least one neutrino from the burst will be detected. Thisprobability is simply related to the average number ofdetected neutrinos per burst Ndet

� , given by Eq. (45),

pdet� ¼ 1� expð�Ndet

� Þ: (57)

For Ndet� � 1, we have

pdet� � Ndet

� (58)

and again taking E> 1� 1019 eV,

_N detb � _NbN

det� � 60�10

ð1þ zmaxÞ7=4yr�1 � 5�10 yr�1; (59)

where in the last step we have used zmax � 3. Requiringthat _Ndet

b * 1yr�1, we obtain the condition

� * 109 GeV: (60)

Note that at the boundary of detectability, where ��109 GeV, we always have Ndet

� � 1, and thus the approxi-mation (58) is justified. This boundary is the lower hori-zontal line bounding the observable parameter range inFig. 2. Note also that Eq. (60) coincides with the condition(24) for the burst-producing loops to be unaffected byfriction.It is interesting to note that the detection rate (59) in the

particle-emission dominated regime is independent of icand depends only on the symmetry breaking scale �. Thisis in contrast with Eq. (40) for the case of gravitationalradiation dominance, where the rate is proportional to icand independent of �.

H. Cascade upper limit on neutrino fluxin the superconducting string model

In Sec. I C, we gave a very general upper limit for UHEneutrino flux. The presence of such a limit does not contra-


023014-8

dict the existence of stronger upper limits in some particu-lar models with additional assumptions.

In this section, we calculate the energy density of thecascade radiation in our model and compare it with!cas ¼2� 10�6 eV cm�3 allowed by EGRET measurements.

The cascade energy density can be calculated as

!cas ¼Z zmax

0

dz

ð1þ zÞ4Z lmaxðzÞ

lminðzÞdlfBnðl; tÞLemðl; tÞ; (61)

where Lemðl; tÞ � 12Ltotðl; tÞ is the loop luminosity in the

form of UHE electrons and photons produced by piondecays. The standard calculation (for zmax ¼ 3) results in

!cas �1:2icðeBt20Þðteq=t0Þ1=2fB�

7ð�G�Þ1=2t30½1� ð1þ zmaxÞ�7=4

� 8:3� 10�7icf�3B�6 eV cm�3: (62)

The energy density (62) does not depend on � and since!cas <!EGRET, it respects the general upper limit (3). Foric � 1, the predicted neutrino flux (38) is close to the upperlimit shown in Fig. 1.

III. GAMMA-RAY JETS AND SINGLE GAMMARAYS FROM THE CUSPS

A. Bursts from loops in the Milky Way

In each galaxy, including the Milky Way, there areapproximately Nl loops with l * lmin,

Nl � nð>lminÞVg � 2:5� 105��310 Vg=10

3 kpc3; (63)

where Vg is the volume of the magnetized part of the

galaxy. A narrow jet of particles emanating from a cuspon such a loop can in principle hit the Earth. The proba-bility of such a catastrophic event is very small because ofthe smallness of solid angle �jet of jet emission. The

number of jets hitting an area S on the Earth per unittime does not depend on S if S � �jetr

2, where r is the

distance from the source. This rate is given by

_N b ¼ PVg

Zdl

2nðlÞl

; (64)

where again we assume one cusp per oscillation, and P ¼�jet=4� ¼ 1=ð4�cÞ2 is the probability to hit the detector.

After the standard calculations, we obtain for Vg � 1�103 kpc3,

_N b ¼ 1� 10�13B2�6�

�310 i

�2c yr�1: (65)

Thus, for particles propagating rectilinearly, the jetsfrom cusps in our Galaxy are unobservable.

The most important components of the galactic jets arephotons and neutrinos. A photon jet at the highest energiesundergoes widening of the jet angle due to photon absorp-tion in the galactic magnetic field [55]. Absorption ofphotons �þ B ! Bþ eþ þ e� is followed by energy

loss by electrons and positrons in the magnetic field, withthe emission of synchrotron photons in directions differentfrom that of the primary photon. This results in the widen-ing of the solid angle �jet [55].

The widening of photon jets in the Milky Way is negli-gible. This can be illustrated by a numerical example. Thehighest energy of a photon in a jet is Emax

� � �c��1031ic=B�6 eV. Photons with E� 1025 eV are absorbed

in galactic magnetic fields. The produced electrons andpositrons with Ee � 1025 eV have lifetime �� 103 s forsynchrotron energy losses and attenuation length latt � 3�1013 cm. Since the Larmor radius of such electrons is rL �3� 1028 cm, the deflection angle �� latt=rL � 10�15 is ofno consequence.

B. Cascade gamma radiation from Virgo cluster

As was discussed above, the photon jets from the galac-tic cusps are not widening and thus are invisible. For cuspsat large distances, the widening of the photon jet efficientlyoccurs in the cascading on CMB photons, �þ �CMB !eþ þ e�, eþ �CMB ! e0 þ �0, etc., and the source can beseen in gamma radiation. As in the case of diffuse cascaderadiation (see Sec. I C), all primary photons with energyhigher than the absorption energy �a are absorbed on CMBradiation and converted into low energy cascade photons.Thus, cusps can be seen in 100 GeV–100 TeV gammaradiation, similar to the sources of UHE protons whichcan be seen in TeV gamma radiation [56].The nearest source from which this radiation can be

expected is the Virgo cluster. It is located at distance r ¼18 Mpc, and the number of loops within the core of radiusRc � 3 Mpc, where a magnetic field B� 10�6 G can bereliably assumed, can reach nlR

3c � 7� 1012��3

10 , with the

luminosity in �eþe� component for each loop L�loop � 2�

1029ic�310B�6 erg s�1. Half of this energy goes into cas-

cade radiation Lcas � 0:5L�loop.

The spectrum of the cascade photons at distance r�20 Mpc has two characteristic energies [40]: the absorp-tion energy �a � 100 TeV and the energy �x. The latter isthe energy of a photon produced by an electron born in �þ�CMB ! eþ þ e� scattering by a photon with E� ¼ �a.

The energy of this electron is Ee � 0:5�a � 50 TeV andthe second characteristic energy is �x � 20 TeV for r�20 Mpc.The spectrum of cascade photons at observation is cal-

culated in [40] as

J�ðE�Þ ¼�KðE�=�xÞ�3=2; E� � �xKðE�=�xÞ�2:0; �x � E� � �a

: (66)

The spectrum constant K in (66) can be expressed interms of the cascade luminosity Lcas and the distance to thesource r as


023014-9

K ¼ Lcas

�effr2

1

�2xð2þ lnð�a=�xÞÞ; (67)

where �eff is the effective solid angle produced by scat-tering of cascade electron in extragalactic magnetic field.In case of full isotropization �eff � 4�. Cascade luminos-ity can be estimated as 1=4 of the total luminosity of cuspsin a cluster Lcas � 1

4LloopNloop. Using Lloop ¼ 4:4�1029ic�

310B�6 erg s�1 and Nloop � 2:5� 1011��3

10 , valid

for a cluster core with Rc � 3 Mpc, one obtains for the flux

J�ð>�xÞ ¼Z �a

�x

dE�J�ðE�Þ � 1

� 10�13icB�6ðRc=3 MpcÞ3 cm�2 s�1 (68)

which is marginally detectable by present telescopes. Notethat Lcas and the flux J� do not depend on �. We consider

the estimate (68) as a very rough indication of detectabilityof the gamma-ray flux from the Virgo cluster. Much moreaccurate calculations are needed for a reliable prediction ofthis flux.

IV. UHE PROTONS FROM SUPERCONDUCTINGSTRINGS

The cusps of superconducting strings in clusters ofgalaxies produce UHE nucleons at fragmentation of partonjets with a fraction of nucleons �N ¼ 0:12 [33] relative tothe total number of hadrons. The generation rateQpð�pÞ ofUHE protons with Lorentz factor �p per unit comoving

volume and unit time can be expressed through emissivity,

L 0 ¼Z �max

p

�minp

d�pmN�pQpð�pÞ; (69)

where the emissivity L0 is the energy released in UHEprotons at z ¼ 0 per unit comoving volume per unit time,�maxp and �min

p � 1 are the maximum and minimum Lorentz

factors of the protons, respectively, and mN is the nucleonmass. For a power-law generation spectrum Qpð�pÞ ��2p , we have

Qpð�pÞ ¼ L0

mN ln�maxp

��2p : (70)

The emissivity is calculated as

L 0 ¼ �NfBZ lmax

lmin

dlnðlÞLcusptot ðlÞ; (71)

where lmin is given by (19), while nðlÞ and Lcusp are given

by (13) and (18) respectively. For Lcusptot one readily obtains

Lcusptot ¼ J2l

eJc

ic�c�

l=2¼ 0:2iceBl�; (72)

and after a simple calculation we have

L 0 � 0:4ic�NfBðteq=t0Þ1=2eBt20ð�G�Þ1=2

�

t0

1

ðt0Þ3� 1:4� 1045icf�3B�6 ergMpc�3 yr�1: (73)

One more parameter relevant for the calculation ofQpð�pÞ is �max

p ¼ Emaxp =mN . It can be estimated using

Emaxp � 0:1�X, where �X ¼ ic�c� is the energy of the

boosted X particles in the laboratory system, which beingestimated for loops of length lmin gives

�maxp ¼ 1� 1010�10i

2c

1

�G�

�

eBt0

�1 GeV

mN

�: (74)

Notice that �maxp does not depend on � and that it enters

Qpð�pÞ through ln�maxp .

Now we can calculate the space density of UHE protonsusing the generation rate Qpð�pÞ given by (70) and taking

into account propagation through CMB radiation with thehelp of the kinetic equation [57,58]

@

@tnpð�p; tÞ � @

@�p

½bð�p; tÞnpð�p; tÞ ¼ Qpð�p; tÞ; (75)

where bð�p; tÞ ¼ �d�=dt describes energy losses of UHE

protons interacting with CMB photons. For � 3� 1010,the proton energy losses become large and one can neglectthe first term in the left-hand side of Eq. (75). Then Eq. (75)becomes stationary and its solution for t ¼ t0 reads

npð�pÞ ¼ 1

bð�pÞZ �max

p

�p

Qpð�pÞd�p � L0

mN�pbð�pÞ ln�maxp

:

(76)

In terms of the proton energy E ¼ mN�p and the diffuse

flux JpðEÞ ¼ ðc=4�ÞnpðEÞ, we have, in the standard form

of presentation,

E3JpðEÞ � c

4�

L0

ln�maxp

E2

bðEÞ ; (77)

where bðEÞ ¼ dE=dt. With bðEÞ taken from [58] a nu-merical estimate at E ¼ 3� 1019 eV gives

E3JpðEÞ � 1:3� 1024icf�3B�6 eV2 m�2 s�1 sr�1: (78)

With ic � 1, the calculated flux (78) coincides well withthe measurements at the same energy, e.g. with the HiRes[59] flux E3JpðEÞ ¼ 2:0� 1024 eV2 m�2 s�1 sr�1, so the

cusp emission may account for the observed events at thehighest energies. For ic & 0:1 the UHE proton flux fromsuperconducting strings is subdominant.The UHE proton spectrum from superconducting strings

has a sharper GZK cutoff than the standard spectrum forhomogeneously distributed sources. This is due to theabsence of clusters of galaxies in the vicinity of ourGalaxy. The nearest cluster, Virgo, is located at 18 Mpcfrom the Milky Way; other clusters are located at muchlarger distances. Nearby sources affect the spectrum at


023014-10

E 1� 1020 eV, where the proton spectrum from super-conducting strings is predicted to be steeper than thestandard one. The experimental data at present have toolow statistics to distinguish the two cases.

In contrast, homogeneously distributed sources such asnecklaces [32] give the dominant contribution at E ð7–8Þ � 1019 eV in the form of UHE photons, comingfrom nearby sources. In the case of superconducting stringssuch component is absent. The UHE photon componentfrom superconducting strings is not dominant at energylower than 5� 1019 eV, because absorption of photons atthese energies is stronger than for protons.

V. CONCLUSIONS

Superconducting cosmic strings produce high energyparticles in the decay of charge carriers, X particles,ejected from the string cusps. The large Lorentz factor�c of the cusp boosts the energies of these particles andcollimates them in a narrow beam with opening angle ��1=�c. The basic string parameter is �, the scale of sym-metry breaking, which we parametrize as � ¼�1010

10 GeV. Another free parameter ic & 1 determinesthe critical electric current in the cusp, Jmax ¼ ice�, andthe mean energy of the charge carriers X escaping from thestring, �X ¼ ic�c�.

The astrophysical parameter which determines the elec-tric current induced in the string is the magnitude of themagnetic field B in the relevant cosmic structures. Thefraction fB of the Universe occupied by magnetic field Bdetermines the flux of high energy particles produced bysuperconducting strings. The most favorable values of Band fB for the generation of a large flux of UHE neutrinosare B� 10�6 G and fB � 10�3. They correspond to clus-ters of galaxies.

The main uncertainties of our model are related to theuncertainties in our understanding of the evolution ofcosmic strings and of the origin and evolution of cosmicmagnetic fields. On the cosmic string side, the key un-known quantity is the parameter which sets the charac-teristic length of string loops in Eq. (4). Here, we used thevalue of � 0:1, as suggested by numerical simulations inRefs. [45,46]. We have also disregarded the effects of loopfragmentation. Toward the end of its life, the loop’s con-figuration may be sufficiently modified by radiation back-reaction that the loop will self-intersect and break up intosmaller pieces. These smaller loops will have more fre-quent cusps, shorter lifetimes, higher velocities, andsmaller induced currents. The effect of such loops on theneutrino fluxes is hard to assess without a quantitativemodel of loop fragmentation. This will have to await thenext generation of string evolution simulations.

On the astrophysical side, basically unknown is thecosmological evolution of the magnetic field parametersfBðzÞ and BðzÞ in the redshift interval 0< z < zmax, wherezmax � 2–3 is the redshift when the magnetic field was

generated. For the space average value hfBðzÞBðzÞi weuse the numerical simulation by Ryu et al. [54], accordingto which this value remains roughly constant at 0< z < 3.Some important quantities, such as the diffuse neutrinoflux J�ðEÞ, the cascade energy density !cas, and the UHEproton emissivity are determined by the evolution of theproduct fBðzÞBðzÞ. However, some other quantities, such asthe rate of neutrino bursts and fluence depend on theevolution of fBðzÞ and BðzÞ in other combinations. In thesecases we consider the parameters f�3 and B�6 as effectivevalues, using f�3 � B�6 � 1.In addition, we adopted the following simplifying as-

sumptions. The Lorentz factor of the cusp is characterizedby a single fixed value �c, while in reality there is adistribution of Lorentz factors along the cusp. The spec-trum of particles in a jet is approximated as E�2, while aQCD calculation [49] gives a spectrum which is not apower law, with the best power-law fit as E�1:92. We usecosmology with � ¼ 0. The spectrum of photons fromVirgo cluster and the diffuse spectrum of UHE protonsare calculated using very rough approximations. Given theuncertainties of string and magnetic field evolution, thesesimplifications are rather benign. On the other hand, theyhave the advantage of yielding analytic formulas, whichallow us to clearly see the dependence of the results on theparameters involved in the problem. In particular, with theassumed particle spectrum �E�2, the diffuse flux of neu-trinos, the cascade upper limit, the flux of TeV photonsfrom Virgo cluster, and the diffuse flux of UHE protons donot depend on �. Since the realistic spectrum is very closeto E�2, this means that the quantities listed above dependon � very weakly.We summarize the results obtained in this work as

follows.As our calculations show, among different sources, such

as galaxies, group of galaxies, filaments, etc., the largestdiffuse flux is produced by clusters of galaxies with B�10�6 G in a cluster core and fB � 10�3. The calculateddiffuse neutrino flux for three neutrino flavors and forzmax ¼ 3 is

E2J�ðEÞ � 6:6� 10�8icf�3B�6 GeV cm�2 s�1 sr�1:

(79)

This flux respects the cascade upper limit, provided bythe energy density of electrons, positrons, and photons,which initiate electromagnetic cascades in collisions withCMB photons. The cascade energy density is calculatedfrom Eq. (79) as

!cas � 8:3� 10�7icf�3B�6 eV cm�3 (80)

and is close to the cascade limit for ic � 1. It is the same asgiven by Eq. (62).At energies E & 1022 eV, the flux (79) is detectable by

future detectors JEM-EUSO and Auger (South and North).The signature of the superconducting string model is the


023014-11

correlation of neutrinos with clusters of galaxies. We note,however, that the neutrino flux from the nearest cluster,Virgo, is undetectable by the above-mentioned detectors.

Another signature of the model is the possibility ofmultiple events, when several showers appear simulta-neously in the field of view of the detector, e.g. JEM-EUSO. They are produced by neutrinos from the samejet. The time delay in arrival of neutrinos with differentenergies is negligibly small. Such multiple events are ex-pected to appear for a certain range of parameters, asindicated in Fig. 2.

As an illustration, in Table I we show, for a representa-tive value � ¼ 5� 1010 GeV, the diffuse neutrino flux, inunits of the cascade upper limit Jmax

� , the rate of bursts, andthe average shower multiplicity for several values of ic.Note that the bottom row in the table is the average multi-plicity, that is, the average number of neutrinos detectedper burst. For example, the low multiplicity at ic ¼ 0:1indicates that only a small number (about 5) out of the 220bursts per year will actually be detected. For ic ¼ 1=3, theaverage multiplicity is below 1, but Fig. 2 shows that wecan expect at least one 2-neutrino burst per year.

A photon jet from the cusp initially propagates togetherwith the neutrino jet, within the same solid angle. However,at large enough distance, photons from the jet can beabsorbed in collisions with CMB photon (�þ �CMB !eþ þ e�), the produced electrons (positrons) emit highenergy photons in inverse-Compton scattering (eþ�CMB ! e0 þ �0), and thus an electromagnetic cascadedevelops. Electrons are deflected in magnetic fields, andphoton radiation is isotropized. Because of this effect, 10–100 TeV gamma radiation from the nearby cluster ofgalaxies, Virgo, can be marginally detectable. The corre-sponding photon flux is given by

J�ð>�xÞ � 1� 10�13icB�6 cm�2 s�1; (81)

where �x � 20 TeV.In the Milky Way, there may be a large number of loops

with radiating cusps, but because of the very small jetopening angle, the probability to observe UHE particlejets coming from these loops is extremely small.

The diffuse flux of UHE protons is suppressed by thesmall fraction of nucleons produced at decay of X particles

(the factor �N ¼ 0:12 is obtained in Monte Carlo andDGLAP calculations [49]), and by energy losses of protonsinteracting with the CMB during propagation. The calcu-lated flux at energy E 3� 1019 eV is given by theapproximate formula

E3JpðEÞ � c

4�

L0

ln�maxp

E2

bðEÞ ; (82)

where bðEÞ ¼ �dE=dt is the energy loss rate of protons,�maxp is the maximum Lorentz factor of a proton at produc-

tion, and L0 is the emissivity (energy in the form ofprotons emitted per unit comoving volume per unit time),given by

L 0 � 1:4� 1045icf�3B�6 ergMpc�3 yr�1: (83)

For ic � 1 and E� 3� 1019 eV, the proton fluxcan reach the value 1:3� 1024 eV2 m�2 s�1 sr�1, whichcan be compared, for example, with 2�1024 eV2 m�2 s�1 sr�1 measured by the HiRes experiment[59]. Thus, radiation from cusps may account for observedevents at the highest energies. The predicted spectrum atE> 8� 1019 eV is steeper than the standard UHECRspectrum with homogeneous distribution of sources. Theaccompanying UHE gamma radiation is very low, due tolarge distances between the sources (clusters of galaxies).As already mentioned, practically all predicted quanti-

ties, such as the diffuse neutrino flux (79), the cascadeenergy density (80), the UHE gamma-ray flux from Virgocluster (81), the diffuse flux of UHE protons (82), and theproton emissivity (83), do not depend on the basic stringparameter �. There are only two observable quantities thatdo: the rate of neutrino bursts _Nb and the neutrino fluence�ð>EÞ,

_N b � 3� 102B2�6f�3

i2c�310

yr�1; (84)

�ð>EÞ � 1� 10�2 i3c�

310

B�6

�1010 GeV

E

�

� 1

½ð1þ zÞ1=2 � 12 km�2: (85)

As � decreases (at a fixed ic), the rate of neutrino burstsgoes up and the number of neutrinos detected in a burst

Ndet� ð>EÞ � 0:11

1010 GeV

E

i3c�310

B�6

1

½ð1þ zÞ1=2 � 12 (86)

goes down, while the product _NbNdet� remains

�-independent.We have considered here only ‘‘regular,’’ field theory

cosmic strings. Recent developments in superstring theorysuggest [60–62] that the role of cosmic strings can also beplayed by fundamental (F) strings and by D branes. Suchstrings may be superconducting, in which case they will

TABLE I. The diffuse flux J�ðEÞ in units of the cascade upperlimit Jmax

� for 3 neutrino flavors, found from (3), the rate ofneutrino bursts, and the shower multiplicity (the average numberof neutrinos detected in one burst), for � ¼ 5� 1010 GeV,zmax ¼ 3, and different values of ic. The multiplicity is shownfor neutrinos with E * 1010 GeV from a burst at z ¼ 2.

ic 1.0 1=2 1=3 0.1

J�=Jmax� 0.42 0.21 0.14 0.042

Rate of bursts 2:2 yr�1 8:7 yr�1 19:6 yr�1 220 yr�1

Multiplicity 26 3.2 0.95 0.026


023014-12

also emit bursts of relativistic particles from their cusps.The main difference from the case of ordinary strings isthat the probability for two intersecting strings to recon-nect, which is p ¼ 1 for ordinary strings, can be p < 1 andeven p � 1 for F or D strings. A low reconnection proba-bility results in an enhanced density of loops; the particleproduction by loops is increased correspondingly.

UHE neutrinos from superconducting strings may havethree important signatures: correlation with clusters ofgalaxies; multiple neutrino-induced showers observed si-multaneously in the field of view of a detector, e.g. JEM-

EUSO; and detection of �10 TeV gamma radiation fromVirgo, the nearest cluster of galaxies.

ACKNOWLEDGMENTS

We would like to thank J. J. Blanco-Pillado for usefuldiscussions, and A. Gazizov for preparing Fig. 1 and dis-cussions. This work was supported in part by the NationalScience Foundation under Grant Nos. 0353314 and0457456 (USA), and by the contract ASI-INAF I/088/06/0 for theoretical studies in High Energy Astrophysics(Italy).

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Ultrahigh energy neutrinos from superconducting cosmic strings