Cosmic rays from cosmic strings with condensates

Cosmic rays from cosmic strings with condensates

Tanmay Vachaspati

Institute for Advanced Study, Princeton, New Jersey 08540, USAand CERCA, Department of Physics, Case Western Reserve University, Cleveland, Ohio 44106-7079, USA

(Received 24 November 2009; published 18 February 2010)

We revisit the production of cosmic rays by cusps on cosmic strings. If a scalar field (‘‘Higgs’’) has a

linear interaction with the string world sheet, such as would occur if there is a bosonic condensate on the

string, cusps on string loops emit narrow beams of very high energy Higgses which then decay to give a

flux of ultrahigh energy cosmic rays. The ultrahigh energy flux and the gamma to proton ratio agree with

observations if the string scale is �1013 GeV. The diffuse gamma ray and proton fluxes are well below

current bounds. Strings that are lighter and have linear interactions with scalars produce an excess of

direct and diffuse cosmic rays and are ruled out by observations, while heavier strings (� 1015 GeV) are

constrained by their gravitational signatures. This leaves a narrow window of parameter space for the

existence of cosmic strings with bosonic condensates.

DOI: 10.1103/PhysRevD.81.043531 PACS numbers: 98.80.Cq

I. INTRODUCTION

The detection of cosmic topological defects can providea direct window to fundamental physics and the very earlyuniverse (for a review, see [1]). Hence it is no surprise thatthere has been a concerted effort to examine observationalconsequences of cosmic topological defects, includinggravitational wave emission, gravitational lensing, fluctua-tions in the cosmic microwave background, and ultrahighenergy cosmic rays. If cosmic strings are superconducting,they may also lead to electromagnetic phenomena such asgamma ray and radio bursts.

The gravitational effects of cosmic strings are strongerfor heavier strings and current observations of the cosmicmicrowave background rule out strings above the �few�1015 GeV energy scale. As lighter strings are considered,gravitational effects become less significant, and otherparticle physics signatures become relatively important.If the strings are superconducting, the currents on the stringcan lead to electromagnetic radiation that could be observ-able. However, if the strings are not superconducting, suchsignatures will be absent, and one must turn to particleemission from cosmic strings. If string dynamics forcesloop production at the smallest possible scales, as sug-gested in [2], particles can be copiously emitted, leadingto strong constraints. However, other studies indicate thatstring loops are large compared to microscopic lengthscales [1] and particle emission is suppressed [3]. Evenin this case, portions of a string loop may get boosted tovery high Lorentz factors, creating a ‘‘cusp’’ on the string(see Fig. 1), and this may potentially provide a burst ofparticles that could be seen in cosmic ray detectors.

Particle emission from cusps on cosmic strings has beenstudied by several authors. Srednicki and Theisen [4]considered a quadratic interaction of a scalar field withan idealized (zero thickness) string and came to the con-clusion that particle emission is insignificant for astrophys-

ical size string loops. Our analysis for the radiation issimilar to that of Refs. [4,5], though the particular linearinteraction of the scalar field (call it H) with the stringworld sheet, as would occur if H condenses on the string,1

has not been considered before. A linear interaction causesan enhancement of particle production by a factor ofM=mwhereM is the string scale and m is the mass of H. IfM isthe grand unification scale, while m is the electroweakscale, this factor can be as large as 1013.Particle emission from cusps on thick strings has been

considered in Refs. [7,8]. Now the cusp consists of over-lapping, oppositely oriented strings that can annihilate andgive off energetic particles. A careful study of this process,including numerical evolution of the field equations, showsthat the resulting flux of particles is too small to be ofinterest [9,10]. In contrast, the linear interaction mecha-nism we study is insensitive to the thickness of the string,and occurs over a string length that is much larger than thelength over which cusp annihilation occurs. Thus we canignore cusp annihilation and work in the zero thicknesslimit.There are important observational constraints on the so-

called ‘‘top down’’ models for production of ultrahighenergy cosmic rays (UHECR), in which a heavy particledecays to give ultrahigh energy protons and gamma rays.The constraint arises because there are bounds on thediffuse gamma ray flux in the EGRET window and alsoon the fraction of photons to protons in the observedUHECR. In previous studies, using cusp annihilation oncosmic strings as the source of Higgs injection, it wasfound that the EGRET constraint on diffuse gamma rayfluxes implies that the flux of UHECR is uninterestingly

1The condensate is similar to that for bosonic superconductingstrings [6], but dissimilar in that we do not require chargedmodes to propagate along the string.

PHYSICAL REVIEW D 81, 043531 (2010)

1550-7998=2010=81(4)=043531(9) 043531-1 � 2010 The American Physical Society

http://dx.doi.org/10.1103/PhysRevD.81.043531

low. However, the constraint on diffuse gamma rays issensitive to the spectral features of the injected Higgsparticles and to the particular interaction of the string.The Higgs particle emission that we consider yields adiffuse flux of gamma rays and protons that is belowEGRET bounds if the string scale is >106 GeV and anUHECR flux consistent with observations for string scale�1013 GeV. The ratio of photons to protons in the UHECRflux is also consistent with current bounds. Hence ourmodel can explain the observed UHECR and not run intotrouble with the EGRET bounds.

It is to be noted that cosmic ray production by stringsincreases as the string scale decreases. Hence strings onscales less than �1013 GeV, and with these interactions,are excluded by the observed flux of UHECR. This isespecially interesting since heavier strings are constrainedby their gravitational signatures. Hence there is a narrowwindow between, say, 1013–1015 GeV for the mass scale ofcosmic strings having linear interactions with a scalar field.

To summarize, the novelty in the present work is that weare considering a new, generic, interaction of cosmicstrings with scalar fields that leads to high energy particleemission. This interaction seems to have been missed in theliterature. Also, we have been careful to take the beamednature of the emission into account. We have focused onderiving approximate analytical estimates so as to keep thephysical aspects of the problem as apparent as possible.More detailed predictions will require numerical evalu-ation of the production and propagation.

We begin in Sec. II by describing the field theory inter-actions under consideration. We then evaluate the rate atwhich a single cusp emits Higgs particles in Sec. III. InSec. IV we use the results of Sec. III to determine thecosmological Higgs injection function. The diffuse gamma

ray and proton fluxes are calculated in Secs. V and VIrespectively. These cosmic rays originate in strings thatare relatively far away from the Earth. Higgs emissionfrom strings that are closer to us and pointed at us cangive UHECR. The direct flux of UHECR is calculated inSec. VII where we also discuss the ratio of photons toprotons in UHECR. We conclude in Sec. VIII. In theAppendix we summarize some known facts about cosmicstring loop dynamics.

II. FIELD THEORY

The interaction we consider is

Sint ¼ ��MZ

d2�ffiffiffiffiffiffiffiffi��

ph; (1)

where � is a coupling constant assumed to be�1,M is thestring energy scale assumed to be at the grand unified scale,�ab is the string world-sheet metric, and h is a scalar field.A linear interaction can be considered quite generally. It

can also arise if there is a bosonic condensate on the string.To see this explicitly, consider the model

S ¼ S0½�; H; . . .� þ �Z

d4xð�y��M2ÞHyH; (2)

where S0 is a field theory action that yields cosmic stringsolutions when� gets a vacuum expectation value (VEV).H is a scalar field that we can take to be the electroweakHiggs for concreteness.At energy scales above the grand unified scale, the

VEVs of � and H are both zero. At lower energy scales,but still above the electroweak scale,� gets a VEV so thatjh�ij2 ¼ M2 but jhHij2 ¼ 0. At this stage, we also havestring solutions whose tension is ��M2 and width is�M�1. Inside the core of the string, where �y� canbecome small, it may be favorable for H not to vanishsince the coefficient of the HyH term in Eq. (2) becomesnegative. As in the case of bosonic superconducting strings[6], there can be an H condensate in the core of the string.SinceM is the only mass scale in the problem at this stage,the magnitude of H is of order M within the core of thestring. At a lower energy scale, H too gets a VEV. Forexample, if H is the electroweak Higgs, this scale is m�100 GeV. We will assume m � M and hence the VEVofH within the string is unaffected by the lower scale (elec-troweak) symmetry breaking.The interaction term in Eq. (2) can now be written as

Sint ¼ �Z

d2�Z

d2x?ffiffiffiffiffiffiffiffi��

p ð�y��M2ÞHyH

¼ �Z

d2�ffiffiffiffiffiffiffiffi��

p Zd2x?ð�y��M2ÞðhHiin þ hÞy

� ðhHiin þ hÞ� ��M

Zd2�

ffiffiffiffiffiffiffiffi��p

hþ � � � : (3)

θ0

L

d

z

O

FIG. 1 (color online). String segment at 3 different times, witha cusp at O, with velocity along z. The size of the curved sectionof string is L. The observer is at an angle �0 from the z axis at adistance d. There is strong emission from the cusp in a beamalong the z axis.

TANMAY VACHASPATI PHYSICAL REVIEW D 81, 043531 (2010)

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In the first line we have split the integral into world-sheetand transverse integrals and included the Jacobian factor(

ffiffiffiffiffiffiffiffi��p

) where �ab (a, b ¼ 0, 1) denotes the induced metric

on the string world sheet. The subscript ‘‘in’’ on theangular brackets denotes that the relevant value is theVEV within the core of the string and we take hHiin �M. Note that h in the last line denotes the radial componentof H evaluated on the world sheet.

In the next section, we will estimate the radiation ofHiggses from cusps on cosmic string loops.

III. HIGGS EMISSION

The equation of motion for the Higgs field is

ðhþm2Þh ¼ j; (4)

where

jðxÞ ¼ ��MZ

d�d�ffiffiffiffiffiffiffiffi��

p�4ðx� Xð�; �ÞÞ: (5)

Then the number of Higgs particles with momentum kproduced due to a source is

dNk ¼ j~jð!k;kÞj2 d3k

2!k

; (6)

where the Fourier transform of the source jðxÞ is given by

~jð!k;kÞ ¼ ��MZ

d�d�ffiffiffiffiffiffiffiffi��

pe�i½!k��k�Xð�;�Þ�; (7)

where X�ð�; tÞ denotes the string world sheet and !k ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 þm2

p.

The dominant contribution to ~j comes from the regionaround the cusp where j!k�� k �Xð�; �Þj< 1. Choosingworld-sheet coordinates so that the cusp occurs at � ¼ 0

and � ¼ 0, this occurs for a range j�j, j�j< L=ðkLÞ1=3provided k > m

ffiffiffiffiffiffiffiffimL

p(see the Appendix). Also,

ffiffiffiffiffiffiffiffi��p ¼ 1� _x2 � j�j2

L2� ðkLÞ�2=3: (8)

Therefore

~jð!k;kÞ � �Mj�j4L2

� �ML2

ðkLÞ4=3 : (9)

The angular width of the beam of particles emitted fromthe cusp can also be estimated by evaluating the integral inEq. (7) in the stationary phase approximation, or as inRef. [11]. The result is

�� 1

ðkLÞ1=3 : (10)

Then Eq. (6) gives

dNk � j~jj2�2kdk� �2M2L2=3 dk

k7=3: (11)

The estimate applies for

k 2 ðm ffiffiffiffiffiffiffiffimL

p;M

ffiffiffiffiffiffiffiffiML

p Þ: (12)

The upper cutoff on k arises because the wavelength of theemitted particles should be larger than the string width inthe rest frame of the cusp: � >M�1. Boosting to the rest

frame of the loop, this yields k <M=ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� _x2

pand together

with the estimate in Eq. (8) gives the upper cutoff. Notethat k can be much larger than the string scale because ofthe enormous boost factors at the cusp. The lower cutoffcomes from the requirement that jk � Xj< 1 (see the

Appendix). For k < mffiffiffiffiffiffiffiffimL

p, j�j � k=m2 and the spectrum

is a rapidly increasing function of k

dNk � �2 M2k25=3

m16L14=3dk: (13)

Hence dNk goes to zero very fast as k ! 0. In the followingsections, we will ignore this part of the spectrum and only

consider k > mffiffiffiffiffiffiffiffimL

p.

At this stage we can also compare Higgs emission fromcusps to the process of cusp annihilation (see Fig. 2). Higgs

emission at momentum k occurs over a length L=ðkLÞ1=3.With the upper bound, k ¼ kmax ¼ M

ffiffiffiffiffiffiffiffiML

p, this length isffiffiffiffiffiffiffiffiffiffiffi

L=Mp

and coincides with the cusp annihilation length[12]. Hence, cusp annihilation does not affect our estimatesof Higgs emission for k < kmax.A caveat to this statement is that the presence of the

condensate should not significantly change the string dy-namics. Also, we have been considering cusp annihilation,but the condensate itself has some width which is largerthan the string width and there could, in principle, be‘‘condensate annihilation’’ even where there is no cuspannihilation [13].

IV. HIGGS INJECTION FUNCTION

Let d�Hðk; tÞ be the number of Higgs produced by stringcusps with energy ðk; kþ dkÞ at time t per unit volume perunit time. d�H is called the ‘‘Higgs injection function.’’

cusp annihilation

Higgs emission

FIG. 2. Higgs emission with momentum k occurs from asection of the string with length �L=ðkLÞ1=3. Cusp annihilation

occurs over a length � ffiffiffiffiffiffiffiffiffiffiffiL=M

p[12] and is smaller than the region

for Higgs emission for k <MffiffiffiffiffiffiffiffiML

p.

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Then

d�Hðk; tÞ ¼ dkZ Lmax

Lmin

dLdNH

dk

dnLdL

dNc

dt; (14)

where the first factor in the integrand is the number ofHiggses with energy k produced by a cusp on a loop oflength L. The second factor is the number density of loopsof length L at time t. The third factor is the number of cuspsper unit time on a loop of length L. The integration is overall loops of length from Lmin to Lmax. The smallest loop canhave length�M�1 but most of the ultrahigh energy cosmicray signal will come from longer loops. Lmax is clearlybounded by the cosmic horizon size �t but, for a fixedvalue of k, it is also bounded due to the constraints inEq. (12).

Let us deal with the last factor in (14) first. Since themotion of the loop is periodic or quasiperiodic,

dNc

dt¼ fc

L; (15)

where fc is a parameter that gives the average number ofcusps on a loop per oscillation period. For loops that are nottoo complicated, we expect fc � 1.

The rate of Higgs production from a single cusp—thefirst factor in the integral in (14)—is given in Eq. (11). Nowwe need to determine the second factor—the number den-sity of loops—in the integrand in Eq. (14).

The number distribution of cosmic string loops is cur-rently under discussion [2,14–18]. In [2] the authors findthat loops will only be of microscopic size, in which caseloops decay very quickly. The conventional scenario,though, is where there is a distribution of loops of all sizesat all times. Simulations of (nonradiating) string networksin an expanding universe give

dnLdL

¼ A

L2i t

2; (16)

where A � 10 and Li is the initial length of the loop. Inthese simulations, the loops do not shrink due to radiationand L ¼ Li at all times. To include the effects of radiationfrom string loops we will express the initial length, Li, interms of the length at time t. This is given by the differen-tial equation for the rate of energy loss

�dL

dt¼ ��gG�

2 � �h

�ffiffiffiffiffiffiffiffimL

p ; (17)

where �g and �h are numerical coefficients characterizing

the gravitational and Higgs radiation. The energy lost toHiggs radiation is found by integrating Eq. (11) aftermultiplication by k. In Eq. (17), the energy lost in onecusp event is averaged over one oscillation period of theloop to get a rate of energy loss.

Equation (17) has to be solved with the initial conditionLðt ¼ 0Þ ¼ Li, which is equivalent to assuming that all thestring loops are effectively laid down at t ¼ 0. Strictly,

there will be loops of size Li that are produced at some latercosmic time ti, and these will then evolve. However, for thepurpose of finding the distribution function for loops, onecan view these as larger loops that were produced at t ¼ 0which then shrunk to Li at the time ti. This is the usualscheme for determining the loop distribution function withthe inclusion of loop evaporation (e.g. see [1]). A moresophisticated analysis that attempts to take radiation back-reaction into account [18] gives a similar loop distributionfunction though with a different dependence on L.However, the effects of backreaction on the loop produc-tion function, which feeds into the loop distribution func-tion, have not yet been included [18] and hence we havestayed with the conventional loop distribution function.With the rescalings

y ¼��gG�

�h

�2mL; x ¼

��gG�

�h

�3�hmt: (18)

Equation (17) becomes

dy

dx¼ �1� 1ffiffiffi

yp ; yðx ¼ 0Þ ¼ yi: (19)

The differential equation can be solved but it is simpler toapproximate it as

dy

dx¼ �1; y > 1;

dy

dx¼ � 1ffiffiffi

yp ; y < 1:

(20)

The Higgs injection function in Eq. (14) can now bewritten as

d�Hðk; tÞ ¼ �Afct2

4=3 dk

k7=3IðY; xÞ; (21)

where Y � Lmax with

��gG�

�h

�2m (22)

and

IðY; xÞ ¼Z Y

0

dy

y1=31

y2i: (23)

Note that we have set the minimum length to 0 since theintegration will be dominated by the upper limit. Themaximum length (needed to determine Y) cannot bemore than the Hubble size but it is also restricted by the

value of k since k > kmin ¼ mffiffiffiffiffiffiffiffimL

p. Therefore

Y ¼ min

�k2

m3; t

�: (24)

To determine IðY; xÞwe need towrite yi in terms of y andx in Eq. (23). For this we need to solve Eq. (20) for all yi toobtain yðx; yiÞ. Then this function should be inverted to getyiðyðxÞ; xÞ which can then be inserted in Eq. (23) (see


043531-4

Fig. 3). We shall do these steps approximately butanalytically.

If Y is very small compared to x, the integration over y isfor y � x and then yi � x up to some numerical factor oforder 1. Therefore

IðY; xÞ � Y2=3

x2; Y � x: (25)

In the opposite limit, Y x, the integral needs to be splitinto two pieces, one for y from 0 to x and the other from xto Y. The first piece is approximated as for the Y � x case.The second piece is different since here yi � y is moreappropriate. More explicitly, for Y x,

IðY; xÞ ¼Z Y

0

dy

y1=31

y2i� x2=3

x2þ

Z Y

x

dy

y7=3� 1

x4=3: (26)

Note that IðY; xÞ does not depend on Y and, since Y is kdependent [Eq. (24)], neither does IðY; xÞ depend on k inthis limit.

Therefore

d�Hðk; tÞ � �Afct2

4=3 dk

k7=3Y2=3

x2; Y � x (27)

and

d�Hðk; tÞ � �Afct2

4=3 dk

k7=31

x4=3; Y x: (28)

A more rigorous derivation will lead to a smooth interpo-lation between these asymptotic forms. For the purpose ofour estimates, it is sufficient to extend the above asymp-totic forms so that they connect continuously at Y ¼ x.

These equations can be written more neatly in terms of

k � mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�gG�mt

q: (29)

Then


m4

k4

dk

k; k � k; (30)


m4

k8=3

dk

k7=3; K > k � k; (31)

where K � mffiffiffiffiffiffimt

pand the bound k < K ensures that Y ¼

k2=m3 i.e. the loops are less than the horizon size.

However, the k�7=3 falloff is also valid for k > K becausethen IðY; xÞ in Eq. (21) is independent of k. So the restric-tion K > k in Eq. (31) may be dropped. Equation (31) isour estimate for the Higgs injection function.An important feature of the injection function in Eq. (31)

is that it is inversely proportional to the string scale. Thuslighter strings inject more Higgses than heavier strings.This feature was also noted in Ref. [8] and can be explainedby the greater longevity of light string loops.

V. DIFFUSE PHOTON FLUX

Once Higgses are injected into the cosmological me-dium, they will decay, lose energy, and eventually cascadeinto gammas in the EGRET energy range. The energy ingammas in the EGRETwindow is estimated as a fraction ofthe total injected energy using [19]

!cas � f2

Z t0

0dt

Zdkk

d�H

dk

1

ð1þ zÞ4 ; (32)

where f is the fraction of energy of the Higgses that goesinto pions and z is the cosmological redshift. The 1=2accounts for how much energy goes into gammas. Thediffuse gamma ray background measured by EGRET ison the order of!cas;obs � 10�6 eV=cm3 in the energy range

10 MeV–100 GeV [20].With the Higgs injection function found in the previous

section, we have

Zdkk

d�H

dk�

Z k

0dk

�Afct2

m4

k4� �Afcm

ð�gG�mÞ3=21

t7=2:

(33)

The integration is dominated by the upper limit (k). Theintegration for k 2 ðk;1Þ will be dominated by its lower

limit because of the faster (k�4=3) falloff and will give acomparable result.Now with 1þ z ¼ aðt0Þ=aðtÞ where aðtÞ is the scale

factor in a matter dominated universe the time integrationcan be done and gives

!cas � 3Affc

ð�gG�Þ3=2�

m2ðmt0Þ1=2 m

t30: (34)

The time integral is dominated by the present cosmic timeand Higgs production during the radiation dominated erahas been ignored. As noted at the end of the previous

y

yi

1

0 x0 x

Y

FIG. 3. The integration in IðY; xÞ is over y (dashed verticalline) at a definite value of x (say x0). The different looptrajectories are labeled by the initial length, yi, at x ¼ 0. Thedotted line at y ¼ 1 is where the differential equation changesbehavior [see Eq. (20)]. For small values of yðx0Þ, yi � x up tofactors of order 1. For large yðx0Þ, yi � y.


043531-5

section, the energy cascade into gamma rays is larger forlighter strings.

To get a feel for the numbers, we use m ¼ 102m2 GeV,� ¼ ð1013�13 GeVÞ2 (therefore G� ¼ 10�12�2

13), t0 ¼1028 cm. Also we take A� 10, �g � 100, and we assume

f � 1. Then

!cas � 10�13��113 m

�1=22

eV

cm3: (35)

This is quite a bit smaller than the EGRET observationunless the string scale is less than �106 GeV. Lighterstrings with the interactions in Eq. (2) are ruled out, thoughwith the caveat that the dynamics of much lighter strings(M< 1 TeV) can be dominated by friction until thepresent cosmic epoch and so the network properties andloop dynamics can be quite different [1].

VI. DIFFUSE PROTON FLUX

We shall follow Berezinsky et al. [21] and use a power-law fragmentation function to obtain the diffuse protonflux, IpðEÞ, as [see Eq. (A13) of [21]]

IpðEÞ ¼ ð2� pÞfN4p

_nHEH

�E

EH

��pRpðEÞ: (36)

Here p ¼ 1:9 [22] and fN is the fraction of energy trans-ferred to nucleons when a Higgs decays which, for order ofmagnitude estimates, we will take to be 1. _nH is the rate atwhich Higgses are being produced per unit volume and canbe found from our result for d�H above. EH is the energyat which the Higgses are produced and RpðEÞ is the protonattenuation length at energy E due to scattering off theCMB. At energies of 1019 eV i.e. below the Greisen-Zatsepin-Kuzmin cutoff, Rp � t0 � 1028 cm.

To estimate IpðEÞ we takeEH ¼ k � 1019�13m

3=22 GeV; (37)

and

_n Hðk; t0Þ ¼ �Afct20

m4

k4� 10�58m�2

2 ��213 cm�3 s�1 sr�1:

(38)

This gives

E3IpðEÞ � 1018�E

E19

�1:1m�0:65

2 ��1:113

eV2

m2 � s� sr; (39)

where E19 ¼ 1019 eV. This is to be compared with theobserved flux [23]

½E319IpðE19Þ�obs � 1024

eV2

m2 � s� sr: (40)

Hence the proton flux is much less than the observed fluxunless �13 � 10�6, in which case both the diffuse gamma

flux and the diffuse proton flux are comparable toobservations.

VII. DIRECT FLUX

String loops that are relatively close by to the MilkyWaymay beam Higgses directly at the Earth and these would beseen as ultrahigh energy cosmic rays. However, as theparticles propagate from the cusp to the Earth, they loseenergy due to scattering with various components of thecosmological medium (see Fig. 4). The energy of a particledrops exponentially with distance from the cusp. If k0 is theenergy of a particle at the cusp, k is the energy at Earth, andr is the distance from the loop to the Earth. We have

k ¼ k0e�r=R: (41)

As the initially emitted Higgs particle decays and losesenergy due to interactions, the products spread out over awider beam angle, �b. The total energy in all the particles isproportional to k�2b and we assume that this remains

roughly constant along the length of the beam. Thereforethe angular spread of the beam at Earth is

�Eb ¼ �lbeþr=2R; (42)

where the superscripts denote Earth (E) and the loop (l).The number of particles at Earth with energy k producedfrom a loop of length L, at distance r follows from Eq. (21)

d�Eðk; L; rÞ ¼ �Afc4=3

t2dk0

k07=3dy

y1=31

y2idVd�b: (43)

Note that the left-hand side is the flux at Earth and hence isat energy k, while the right-hand side contains the injectedflux at the location of the loop and hence is at k0.The beaming solid angle is

d�b � ð�Eb Þ2 ¼ ð�lbÞ2eþr=R ¼ er=R

ðk0LÞ2=3 ; (44)

where we have used Eq. (10). Using Eq. (41) and therescalings in (18) and integrating over loop lengths andspatial volume, we get

Earth

Loop

r

FIG. 4 (color online). A cusp on a loop close to Earth emitsparticles that lose energy as they propagate and also spread out.


043531-6

d�EðkÞ ¼ 42�Afc2

t2dk

k3

Z 1

0drr2e�r=R

Z ymax

ymin

dy

y

1

y2i:

(45)

The limits of integration over loop lengths (y) alsodepend on r since [see Eq. (12)]

ymin ¼ k02

M3¼

k2

M3e2r=R; (46)

ymax ¼ min

�k02

m3; t

�¼ min

�k2

m3e2r=R; t

�; (47)

where is defined in Eq. (22).In the ðr; yÞ plane of integration in Eq. (45), there are

four values of r which are significant (see Fig. 5). The first,r1, is where ymax becomes equal to x [see Eq. (18)], and isgiven by

e�r1=R ¼ k

k: (48)

The second, r2, is where ymax becomes equal to the Hubblesize loop

e�r2=R ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi�hG�

p k

k: (49)

The third, r3, is where ymin becomes equal to x

e�r3=R ¼�m

M

�3=2 k

k; (50)

and the last, r4, where ymin equals t

e�r4=R ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi�hG�

p �m

M

�3=2 k

k: (51)

The integration in the ry plane in Eq. (45) can be split upinto four integrations over the ranges ð0; r1Þ, ðr1; r2Þ,ðr2; r3Þ, and ðr3; r4Þ. For y values less than x we can useyi � x, as argued in Sec. IV. For y values greater than x, weuse yi � y. While the full integration can be done, it isunnecessary because the dominant contribution comesfrom the interval ð0; r1Þ i.e. the closest strings. The otherintegrals are suppressed by powers of k=k [see Eq. (29)].The integration over ð0; r1Þ is

I1 ¼Z r1

0drr2e�r=R

Z ymax

ymin

dy

y

1

y2i¼ 6

x2ln

�M

m

�R3; (52)

where we have used yi � x, ymax=ymin ¼ ðM=mÞ3, andassumed k=k � 1.This leads to

kd�E

dk¼ 242Afc ln

�M

m

�M2

ð�hG�Þ2k2R3

t4: (53)

Using t ¼ 104 Mpc ¼ 1017 s, and dividing by theEarth’s surface area, 109 km2, we get the flux per unit area

kd�E

dAdk� 10�3

�1020 eV

k

�2�107 GeV

M

�2�

R

5Mpc

�3m�2 s�1:

(54)

ForM� 1013 GeV, this is comparable to the observed fluxof ultrahigh energy cosmic rays, 10�36 m�2 s�1 sr�1 eV�1

at 1020 eV [23]. Note that m only enters through thelogarithm and its precise value does not make much dif-ference to the overall estimate. The E�3 spectrum inEq. (54) is harder than the observations above 4�1019 eV which suggest an exponent �� 4:2 [23].However, the spectral exponent depends on our treatmentof R as an energy-independent constant.The direct photon flux will also be given by Eq. (53), but

the attenuation length R will be specific to photons whichis less than that for protons at energy 1019 eV, and thebranching ratio for Higgs decay into protons differs fromthat to photons. Hence the photon to proton ratio is

�

p¼ N�

Np

�R�

Rp

�3; (55)

where N� and Np are the number of gammas and the

number of protons produced by a Higgs. From Figs. 9and 11 of Ref. [24], and also Fig. 2 of [21], we findR�=Rp � 10�2 at 1019 eV. If the heavy particle emitted

by the cusp is the electroweak Higgs, the decay productscontain a pion fraction of �0:75 and a nucleon fraction�0:15. The pions then decay into gamma photons. As aconservative estimate of the number of photons to protonsin the decay products, we take N�=Np � 102. Then the

gamma to proton fraction at 1019 eV is �10�4, which isconsistent with the observed AUGER bound �=p < 0:02[25]. As pointed out in Ref. [21], at higher energies, sinceRp falls quite rapidly, we expect the gamma to proton ratio

tα

maxy

ymin

r1 r2 3r 4r

y

x

0 r

FIG. 5. The integration region in (45) in the ry plane falls inthe region bounded by the ymin and ymax curves from r ¼ 0 tor ¼ r4, while the integrand depends on whether y < x or y > x.The dominant contribution to the integral comes from the region0< r < r1.


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to be larger. For example, at 1020 eV, R�=Rp � 10�1 and

�=p� 10�1, whereas the observed AGASA and Yakutskbound is 0.3 [26].

We would like to point out that we have performed ouranalysis by taking R to be a constant, as in Eq. (41). This isonly an approximation since R depends on the energy ofthe particle and changes as the particle propagates. A morecomplete analysis would take the energy loss to be givenby

dk

dr¼ � k

RðkÞ ; (56)

where RðkÞ follows by considering the various interactionsthat a cosmic ray may encounter en route. Our results arevalid only if

dR

dk

��k� RðkÞ

k: (57)

An improved analysis should take the detailed form of RðkÞinto account.

VIII. CONCLUSIONS

It is generally difficult to construct astrophysical scenar-ios that can accelerate protons to high energies sufficient toarrive as ultrahigh energy cosmic rays. Top down modelsinvolving topological defects can naturally give the requi-site energies. Depending on the precise variety of defect,however, they are likely to suffer from an excess of diffusegamma ray production.

We have revisited cosmic ray production from cosmicstrings, taking into account the possibility of a linearinteraction between a scalar field and the string worldsheet. Such an interaction arises, for example, if a scalarfield acquires a VEV (‘‘condenses’’) within the string.Then beams of Higgs particles are emitted from cusps oncosmic string loops. Our analysis shows that such eventscan be responsible for the production of UHECR withinreasonable parameters and also be consistent with mea-surements of the diffuse backgrounds and the photon toproton fraction in UHECR. The cosmic ray events due tocosmic strings will be correlated with the positions of theloops. Since the loops have relativistic velocities, they donot cluster in the galaxy and are expected to be homoge-neosly distributed, and the resulting cosmic ray events willalso be isotropically distributed. The bursts from loops willbe repeated since the cusps are repetitive events. However,the beaming direction is sensitive to the backreaction of theemission on the loop dynamics. Without studying the back-reaction in detail, it is not possible to predict if subsequentcusp events will be seen as repeated events on Earth.

Our results are also interesting because they show thatstrings with bosonic condensates are allowed in a narrowwindow of energy scales. If they are much lighter than�1013 GeV, they will produce an excess of UHECR anddiffuse fluxes, while if they are much heavier, they will

cause conflicts with other cosmological constraints thatdepend on their gravitational effects. However, if thestrings are around the grand unified scale, they can producethe observed flux of ultrahigh energy cosmic rays and alsobe heavy enough to be detected by their gravitationalsignatures in the near future.We have focused on deriving analytical estimates of

cosmic ray fluxes under various simplifying assumptions.Further work is needed to obtain more detailed estimatesthat can be compared to observations.

APPENDIX: LOOP DYNAMICS AND CUSPS

Here we summarize some known features of cosmicstring loops that are used to obtain the radiation estimatesin Sec. III.A cosmic string loop oscillates under its own tension

such that its world sheet can be written as

X ð�; tÞ ¼ 12½að�� tÞ þ bð�þ tÞ�; (A1)

where a and b satisfy

ja0j ¼ 1 ¼ jb0j; (A2)

where primes denote derivatives with respect to the argu-ment. Also, since the loop is a closed string,

Z L

0a0d� ¼ 0 ¼

Z L

0b0d�: (A3)

Therefore �a0 and b0 are two closed curves on a unitsphere whose centers of mass coincide with the center ofthe sphere. These curves generally intersect leading to

a 0ð�� tÞ ¼ �b0ð�þ tÞ (A4)

for one or more values of � and t. Since the velocity of apoint on the loop is

v ð�; tÞ ¼ 12½�a0 þ b0�; (A5)

intersection of the�a0 and b0 curves implies a point on theloop that reaches the speed of light at one instant peroscillation. Such a point on the string is called a cusp.The ultrahigh boost factors at the cusp can be responsiblefor burstlike events from cosmic string loops.We choose our string parametrization so that the cusp

occurs at � ¼ 0 ¼ t, and then expand the functions a andb around a cusp

a ð��Þ ¼ a00�� þ 12a

000�

2� þ 16a

0000 �

3� þ � � � ; (A6)

b ð�þÞ ¼ b00�þ þ 1

2b000�

2þ þ 16b

0000 �

3þ þ � � � ; (A7)

where � ¼ � t. The expansion coefficients are con-strained by

a 00 ¼ �b0

0; a00 � a000 ¼ b00 � b00

0 ¼ 0;

ja00j ¼ jb00j ¼ 1:

(A8)


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These expansions give

!kt� k �X� ðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 þm2

p� kÞ� þ �kL�2�3 þ � � � ;

(A9)

where we have taken �þ � �� and denoted them collec-tively by � . � is a dimensionless constant that depends onthe shape of the cusp. We will take �� 1. To have j!kt�k �Xj< 1, we require that both terms in Eq. (A9) be less

than 1. This gives � <minðk=m2; LðkLÞ�1=3Þ. For k >

mffiffiffiffiffiffiffiffimL

p, we get � < LðkLÞ�1=3 and for k < m

ffiffiffiffiffiffiffiffimL

p, we

get � < k=m2.

ACKNOWLEDGMENTS

I am very grateful to Nima Arkani-Hamed, CliffordCheung, and Jared Kaplan for early collaboration and forcontinued support and encouragement. I am also grateful toVenyamin Berezinsky, Francesc Ferrer, Uri Keshet,Shmuel Nussinov, Ken Olum, Grisha Rubtsov, AlexVilenkin, Jay Wacker, and Edward Witten for their com-ments and advice. This work was supported by the U.S.Department of Energy at CaseWestern Reserve University.

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Cosmic rays from cosmic strings with condensates