Light from cosmic strings

Daniele A. Steer

APC* 10 rue Alice Domon et Leonie Duquet, 75205 Paris Cedex 13, France

Tanmay Vachaspati

Physics Department, Arizona State University, Tempe, Arizona 85287, USA(Received 2 January 2011; published 25 February 2011)

The time-dependent metric of a cosmic string leads to an effective interaction between the string and

photons—the ‘‘gravitational Aharonov-Bohm’’ effect—and causes cosmic strings to emit light. We

evaluate the radiation of pairs of photons from cosmic strings and find that the emission from cusps,

kinks and kink-kink collisions occurs with a flat spectrum at all frequencies up to the string scale. Further,

cusps emit a beam of photons, kinks emit along a curve, and the emission at a kink-kink collision is in all

directions. The emission of light from cosmic strings could provide an important new observational

signature of cosmic strings that is within reach of current experiments for a range of string tensions.

DOI: 10.1103/PhysRevD.83.043528 PACS numbers: 98.80.Cq

I. INTRODUCTION

Cosmic strings are possible remnants from the earlyuniverse (for a review see [1]) and there is significant effortto try and detect them. A positive detection of cosmicstrings will open up a window to very high energy funda-mental physics and can potentially have strong implica-tions for astrophysical processes. Hence it is of greatinterest to continue to discover new observational signa-tures of cosmic strings, as well as to refine features ofknown signatures. In this paper we address the radiationof photons by cosmic strings.

There is an extensive literature on gravitational radiationfrom cosmic strings, particularly motivated by upcomingand future gravitational wave detectors. More relevant tothe work presented here, however, is the analysis in [2,3] ofthe emission of particles due to the time-dependent metricof cosmic strings from the viewpoint of Aharonov-Bohmradiation. The case of photon emission—which we treat inthe present paper—was not explicitly discussed there. Acrucial feature which emerged in these calculations is thatcusps and kinks on cosmic strings emit radiation with a flatspectrum all the way up to the string scale. However, thoseresults were based on studying two rather specific loopconfigurations with cusps and kinks. As we show here inmore generality (namely for any loop configuration) theflat spectrum also applies to the emission of photons fromcusps and kinks, as well as kink-kink collisions. Thus lightemitted from cosmic strings in this way leads to a new andobservable signature of cosmic strings that is completelyindependent of the details of the underlying particle phys-ics model. As we shall see, the effect is small, however,being proportional to ðG�Þ2 where G is Newton’s constant

and � the string tension. Despite that, since photons arebeing emitted, it may be more easily measurable than,say, the gravitational wave bursts also emitted by cuspsand kinks.The total power emitted in scalar particles from cosmic

strings due to their gravitational coupling was first consid-ered in [4], using formalism developed in [5]. In this paperwe calculate the differential power emitted in photons fromcosmic strings due to the gravitational coupling. We callthis the ‘‘gravitational Aharonov-Bohm’’ effect becausethe metric is flat everywhere except at the location of thestring, and is closely analogous to the case of the electro-magnetic Aharonov-Bohm effect due to a thin solenoid. InSec. II we set up the calculation and evaluate the invariantmatrix element for the production of two photons. Theemission is dominant in three cases—from cusps, kinksand kink-kink collisions. Integrals relevant to these casesare evaluated in Sec. III. In Sec. IV we find the poweremitted from cusps, kinks and kink-kink collisions onstrings. Our results are summarized in Sec. V, where wealso consider observational signatures. Our numerical es-timate in Eq. (81) indicates that light from cosmic stringsmay potentially be detectable by current detectors for arange of string tensions.

II. GRAVITATIONAL AHARONOV-BOHM

The gravitational field of a cosmic string is characterizedby the parameter G� which is constrained to be less than�10�7. Hence it is sufficient to consider the case of aweak gravitational field and linearize the metric around aMinkowski background

g�� ¼ ��� þ h��: (1)

Then coupling between the gravitational field and thephoton becomes

*Universite Paris-Diderot, CNRS/IN2P3, CEA/IRFU andObservatoire de Paris

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1550-7998=2011=83(4)=043528(8) 043528-1 � 2011 American Physical Society

L int ¼ � 1

4

ffiffiffiffiffiffiffi�gp

F��F�� ¼ 1

2���F��F�

� þOðh2Þ;(2)

where

��� ¼ h�� � 14���h

��; (3)

and the electromagnetic field strength is F�� ¼ @�A� �@�A�. The coupling is quadratic in A� so that to lowest

order in h�� photons are created in pairs. The metric

perturbation, h��, due to the cosmic string energy-

momentum tensor, T��, follows from the Einstein equa-

tions: in Fourier space (denoted by tildes),

~��� ¼ ~h�� � 1

4���

~h�� ¼ � 16�G

k2

�~T�� � 1

4���

~T��

�:

(4)

From the Nambu-Goto action, and using the conformalgauge [1]

T��ðxÞ ¼ �Z

d2�ð _X�_X� � X0

�X0�Þ�4ðx� XÞ; (5)

where X�ð�; tÞ is the string world-sheet. A cosmic stringloop trajectory can be written in terms of left- and right-movers

X�ð�; tÞ ¼ 12½a�ð��Þ þ b�ð�þÞ�; (6)

where �� ¼ �� t, and we will adopt world-sheet coor-dinates such that

a0 ¼ ���; b0 ¼ �þ; ja0j ¼ 1 ¼ jb0j; (7)

where primes denote derivatives with respect to the argu-ment. Substituting (6) into (5) yields

T��ðxÞ ¼ ��

4

Zd�þd��ða0�b0� þ a0�b0�Þ�4ðx� XÞ

(8)

which, when Fourier transformed, is

~T ��ðkÞ ¼ ��

4ðIþ;�I�;� þ Iþ;�I�;�Þ: (9)

Here, for the periodic oscillations of a loop of length L

I�þ ¼ X1n¼1

�

�k0L

4�� n

�Z L

0d�þb0�e�ik�b=2; (10)

I�� ¼ X1n¼1

�

�k0L

4�� n

�Z L

0d��a0�e�ik�a=2: (11)

It will be important in the following to notice that as aresult of the periodicity of the loop,

k � I� ¼ 0: (12)

We can now calculate the amplitude for the pair creationof two outgoing photons of momentum p and p0, andpolarization and 0, respectively, where

p2 ¼ 0 ¼ p02; p � ¼ 0 ¼ p0 � 0: (13)

This is given by the tree level process shown in Fig. 1, andon using Eqs. (2), (4), and (9) we find

M ðp; p0Þ ¼ � 4�G�

k2I�þðkÞI��ðkÞQ��ðp; ;p0; 0Þ; (14)

where momentum conservation imposes that

k ¼ pþ p0; (15)

while

Q�� ¼ P�� þ P�� � 12���P

��; (16)

P�� ¼ ðp��� � p�

��Þðp0

�0�� � p0�0�� Þ: (17)

The number of photon pairs produced in a phase spacevolume, the ‘‘pair production rate,’’ is given by (e.g.Sec. 4.5 of [6])

dN ¼ X;0

d3p

ð2�Þ31

2!

d3p0

ð2�Þ31

2!0 jMj2; (18)

where ! and !0 are the energies of the two outgoingphotons, while the energy emitted in the pairs is

dE ¼ X;0

d3p

ð2�Þ31

2!

d3p0

ð2�Þ31

2!0 ð!þ!0ÞjMj2: (19)

Clearly the crucial relevant quantity is

jMj2tot �X;0

jMj2 ¼�4�G�

k2

�2X;0

I�þI��þ I�I��� Q�Q

���:

(20)

Note that in this paper we only study the total radiation ratefrom a cosmic string loops: a discussion of any polarizationsignatures is left for subsequent work.After substitution ofQ�� given in Eq. (16), the sum over

photon polarizations in (20) can be simplified through thereplacement

X;0

��� ! ����; (21)

provided certain conditions hold [6]. More specifically, letus define M���� via

ε

’

p

p’ε

FIG. 1 (color online). Feynman diagram showing two photonproduction from a classical string.

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Q�� ¼ ��0��M����; (22)

so that

M ���� ¼ N���� þ N���� � 12���N��

��; (23)

N���� � ðp���� � p����Þðp0��

�� � p0����Þ: (24)

Then the required condition [6] is that

p�M���� ¼ 0 ¼ p0�M����: (25)

However, it is straightforward to check that this conditionis satisfied, since it is an immediate consequence of thedefinition of M���� in (23) and (24).

Then, after quite a bit of algebra and on using (21), wefind that jMj2tot is given by

jMj2tot ¼�2�G�

p �p0

�2½8jp � Iþj2jp � I�j2

þ 4p �p0fjp � Iþj2jI�j2 þ jp � I�j2jIþj2þp � I�þp � I�Iþ � I�� þp � Iþp � I��I�þ � I��p � Iþp � I�I�þ � I�� �p � I�þp � I��Iþ � I�gþ 2ðp �p0Þ2fjIþj2jI�j2 þ jI�þ � I�j2 � jIþ � I�j2g�;

(26)

where jI�j2 � I��� I��, and in order to simplify the result

we have made extensive use of p � I� ¼ �p0 � I� whichfollows (12) since k ¼ pþ p0. Finally, we have expressedthe answer in powers of k2 ¼ 2p � p0: this will be impor-tant later when we will see that the k2 ! 0 limit plays acrucial role.

III. EVALUATION OF jMj2totIn order to calculate the energy radiation in photon pairs,

we need to evaluate jMj2tot, where the dynamics of cosmicstring loops enters Eq. (26) through the integrals I�ðkÞ.These integrals, defined in (10) and (11), also occur in thecalculation of other forms of radiation from strings andhave been discussed in the past (e.g. [7,8]). There is,however, a key difference between Aharonov-Bohm(AB) radiation and other forms of radiation from stringsthat are commonly studied, namely, AB radiation involvestwo particle final states. As a result, the kinematics of theproblem is potentially more complicated.

Generally, however, it is well known that I�ðkÞ decayexponentially with k0L, where L is the length of the loop,unless either the phase in these integrals has a saddle pointon the real line, or there is a discontinuity in the integranddue to kinks in b0� and/or a0� (see e.g. Ch. 6 of Ref. [9]).Below we study these cases in turn, and we will see thatdespite the two-particle nature of the final state, a saddlepoint in both I�þ and I�� corresponds to a cusp on thestring—namely, j _Xj ¼ 1; a saddle point in one of theintegrals and a discontinuity in the other occurs at a kink.

Finally, when two kinks collide, there is a discontinuity inboth integrands. In all three possibilities—cusp, kink, kink-kink collision—I

�� decay as a power law, ðk0LÞ�q, where

the index q will be determined below.

A. Saddle points and cusps

As a first step in evaluating I�, we establish certainrelations between the momenta of particles emitted whenthere are saddle points in these integrals. On recalling thatboth p2 ¼ 0 ¼ p02, let us write

p ¼ ð!;pÞ ¼ !ð1; pÞ � !p;

p0 ¼ ð!0;p0Þ ¼ !0ð1; p0Þ � !0p0;(27)

where jpj ¼ 1 ¼ jp0j. Since k ¼ pþ p0, then

k2 ¼ 2p � p0 ¼ 2!!0ð1� p � p0Þ: (28)

We also define

k ¼ ð�; kÞ ¼ ð!þ!0;pþ p0Þ: (29)

Next consider the integral IþðkÞ in Eq. (10) when there isa saddle point. This implies that there is a point such that

k � b0 ¼ 0; (30)

where from the gauge conditions Eq. (7) b02 ¼ 0. Thus, at asaddle point

k ¼ b0; where k � k

�; (31)

so that

k2 ¼ �2k2 ¼ �2b02 ¼ 0: (32)

Thus, k2 is null. Furthermore, from Eq. (28),

p ¼ p0 ¼ k ¼ b0; (33)

where p and p0 were defined in Eq. (27).Apart from a sign, all the above goes through for a

saddle point in I�. Again k� must be lightlike, but now k �a0 ¼ 0 ¼ �ð�1þ k � a0Þ (since a0 ¼ ���). Thus,

p ¼ p0 ¼ k ¼ �a0: (34)

Evaluation of the integrals around the saddle points, inthe �L � 1 limit can then be carried out in the standardway (see e.g. [9]) and, using the kinematic relations givenabove, leads to

Isaddleþ;n ¼ AþLb0s

ð�LÞ1=3 þ iBþL2 b00sð�LÞ2=3 þ . . . ; (35)

Isaddle�;n ¼ A�La0s

ð�LÞ1=3 þ iB�L2 a00sð�LÞ2=3 þ . . . : (36)

where the subscript n on I� refers to the nth term in thesum in Eqs. (10) and (11), and the subscript s on a� and b�

refers to evaluation at a saddle point. The delta functions in

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Eqs. (10) and (11) enforce �L ¼ 4�n. We have droppedan overall phase factor which is irrelevant because it is thesquare of the amplitude that gives a rate. The coefficientscan be evaluated explicitly;

Aþ ¼�

12

L2jb00j2�1=3 2�

3�ð2=3Þ ;

Bþ ¼�

12

L2jb00j2�2=3 1ffiffiffi

3p �ð2=3Þ;

(37)

and A� and B� are given by identical expressions exceptthat b00 is replaced by a00.

A cusp on a string loop occurs when j _Xj ¼ 1 and hence,from (6), when a0 ¼ �b0. However, from Eqs. (33) and(34) this condition requires a saddle point contribution toboth Iþ and I�. In the vicinity of the beam of the cusp,from Eqs. (31) and (33), we get p� b0c ¼ �a0

c, and simi-larly for p0. Therefore from (35) and (36), we estimate

p � Isaddle� �Oðk2Þ; p0 � Isaddle� �Oðk2Þ: (38)

This is an important result: consider jMj2tot given inEq. (26). In the beam of the cusp k2 ¼ 2p � p0 ! 0[Eq. (32)], so one might worry that Eq. (26) diverges dueto the overall factor of 1=ðk2Þ2. However, this divergence isrendered harmless by the scaling in Eq. (38). Indeed, in thek2 ¼ 0 limit, only the last line in Eq. (26) gives thedominant contribution to the emission from the cusp:

ðjMj2totÞcusp ! ð2�G�Þ2 2½jIþj2jI�j2þ jI�þ � I�j2 � jIþ � I�j2� (39)

with I� in Eqs. (35) and (36) [we have dropped the label‘‘saddle’’]. The other terms in Eq. (26) all contain factorssuch as jp � I�j and are higher order in k2.

The above analysis assumes that k is in the direction ofthe cusp. If k is at some small angle, �þ, to b0, we can writek � b0 ¼ �2þ=2 and repeat the above analysis as in [7]. Theestimate is valid for

�þ �m;þ ��4Ljb00j2ffiffiffi

3p

�

�1=3

: (40)

Similarly in the case of I�

�� �m;� ��4Lja00j2ffiffiffi

3p

�

�1=3

: (41)

This estimate assumes k2 ¼ 0 but it holds even if k isperturbed so that it is not precisely null. To summarize,the estimate (39) holds in a cone of opening angle

�þ � �� � ð�LÞ�1=3.

B. Discontinuities and kinks

Next we find the contribution of a discontinuity to theintegrals I�. On expanding the integrands on both sides ofthe discontinuity, which is say in a0 at �� ¼ uk, one canextract the dominant contribution;

Idisc� �Z uk

d��a0�e�ik�ðakþa0�ð���ukÞþ...Þ

þZuk

d��a0þe�ik�ðakþa0þð���ukÞþ...Þ

¼ � 2

i�

�a0þ

k � a0þ� a0�

k � a0�

�e�ik�ak ; (42)

where a0� refers to the value of a0 on either side of thediscontinuity, and ak ¼ að�� ¼ ukÞ. Similarly we can findI�þ, and the result is

Idiscþ �� 2

i�

�b0þ

k � b0þ� b0�

k � b0�

�e�ik�bk : (43)

It is important to observe that Idisc� ���1 decays fasterwith frequency than Isaddle� [Eqs. (35) and (36)].The estimates in Eqs. (42) and (43) preserve the relation

k � I� ¼ 0. Hence if k2 ! 0 then p� ¼ p0� ¼ k� andagain p � Idisc� ! 0 and p0 � Idisc� ! 0. These relations areimportant to see that the expression for the invariant matrixelement in Eq. (26) is not singular in the k2 ! 0 limit. The

case when k � b0� ¼ 0 or k � a0� ¼ 0 is very special becausenow there is a saddle point in addition to a discontinuityand we shall not consider its consequences.While a saddle point in both I�þ and I�� corresponds to a

cusp on the string, a saddle point in one of the integrals anda discontinuity in the other occurs at a kink. Thus thedominant contribution to photon production from a kinktakes place exactly when k2 ! 0, namely, in the forwarddirection, when p and p0 are collinear. In this limit,

jp � Idisc� j2 �Oðk2Þ; jp0 � Idisc� j2 �Oðk2Þ; (44)

which should be compared to Eq. (38) for a cusp. This canbe most clearly seen by writing, for example,

p � Idisc� � p ��

a0þk � a0þ

� a0�k � a0�

�

¼�!!0

�2

½ðp � a0þÞa0� � ðp � a0�Þa0þ�k � a0þk � a0�

�� ðp0 � pÞ:

(45)

Therefore, p � Idisc� ¼ Oðjp0 � pjÞ from which (44) followssince !!0ðp0 � pÞ2 ¼ k2 [see Eq. (28)].

C. No saddle point or discontinuities

If a loop has neither cusps or kinks, then I� decayexponentially with �L. In that case the energy radiatedby the loop per unit time, which is proportional to jMj2tot,also decays exponentially as _En / e��n where n ¼�L=ð4�Þ is the harmonic number and � is a coefficient.In other words, a loop with no kinks or cusps will radiate afinite amount of energy. We have checked explicitly thatthis is the case by considering the radiation from a chiralcosmic string loop for which there are no cusps and kinks.On the other hand, as we now show, the radiation from a

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cusp or a kink diverges and needs to be cut off due to thethickness of the string.

IV. POWER EMITTED

We now evaluate the power emitted from cusps, kinksand kink-kink collisions. We divide (19) by L to get theaverage power radiated in the nth harmonic

_En ¼ 2�

L2nZ d3p

ð2�Þ31

2!

Z d3p0

ð2�Þ31

2!0

jMj2n;tot�ðn� ð!þ!0ÞL=4�Þ; (46)

while the total power is the sum over all harmonics. Notethat � ¼ !þ!0 ¼ 4�n=L.

A. Emission from a cusp

In order to get the power radiated from a cusp, we firstinsert the expressions for Isaddle� given in Eqs. (35) and (36)into Eq. (39), remembering to include the sum and deltafunctions in Eqs. (10) and (11). Not all the terms in Isaddle�contribute since the string constraint Eqs. (7) imply

ja0j2 ¼ 0 ¼ jb0j2; a0 � a00 ¼ 0 ¼ b0 � b00 (47)

and, in addition, since a0 ¼ �b0 at the cusp,

b0c � a00c ¼ 0 ¼ a0c � b00c (48)

(where the subscript c denotes the cusp). Then to leadingorder we find jI�þ � I�j ¼ jIþ � I�j and the nonvanishingcontributions in (39) come from the jIþj2jI�j2 term whichis proportional to ja00c j2jb00c j2, so that

ðjMj2n;totÞcusp ¼ 2ð2�G�Þ2jIþj2jI�j2

� ðG�Þ2L4 1

n8=3ja00

cLj�2=3jb00cLj�2=3; (49)

where we ignored an overall numerical factor of order 1.In order to calculate (46) and estimate _En, we next

rescale the momenta p and p0 by 4�n=L. Then

Z d3p

ð2�Þ31

2!¼ 1

2ð2�Þ3�4�n

L

�2 Z

d �! �!Z

d2p (50)

where �! ¼ Ljpj=4�n. The integration over �! gives anorder 1 numerical factor. The integration over the directionp is estimated as

Zd2pd2p���2m;þ��2m;��ja00

cLj4=3jb00cLj4=3n�4=3; (51)

where we have used �m;� given in Eqs. (40) and (41).

Now putting together the results in Eqs. (46), (49), and(51), we finally obtain

_E njcusp ��G�

L

�2ja00

cLj2=3jb00cLj2=3: (52)

The most important feature of this estimate [10] is that itdoes not depend on the harmonic number n: all factors of n

have cancelled! This result was noted earlier in [2] in thecontext of scalar radiation and in [3] in the context offermion radiation, though for specific loop trajectories.Our analysis here is more general. Additional features ofinterest are the 1=L2 dependence that will result in largerradiation from smaller loops, and the dependence on the(dimensionless) cusp acceleration vectors, La00

c and Lb00c .The n-independent spectrum of gravitational AB radia-

tion implies that photons of arbitrarily high frequencieswill be emitted from cusps. Naively this would imply adivergence in the emitted power. However, the Nambu-Goto action ignores the thickness of a field theory string,and this suggests a cutoff for the highest harmonic that canbe emitted—the highest harmonic emitted from a cusp isthe string width in the rest frame of the string. To work outthe Lorentz factor we start out by noting that the saddlepoint analysis gives contributions from the region aroundthe cusp of size jk � a0j< 1 (in the case of I�) where thephase in the integrand is not oscillating rapidly. (We followthe discussion in [11].) This gives a time and lengthinterval on the string world sheet

j�tj; j��j< L

ð�LÞ1=3 : (53)

In this region

1� _x2 � ð��Þ2L2

� ð�LÞ�2=3: (54)

Requiring that � be less than the Lorentz boost factortimes the inverse string width, and ignoring numericalfactors we get

�<MffiffiffiffiffiffiffiffiML

p; (55)

where M is the mass scale associated with the string. Interms of the harmonic number, this gives a cutoff

nc � ðMLÞ3=2: (56)

Hence, Eq. (52) applies for n nc.The result in Eq. (52) gives the power emitted in the nth

harmonic. Observationally, it is the energy flux at theobserver’s location that is relevant. To get this flux wedivide _En by the cross-sectional area of the beam emittedfrom the cusp at a distance r from the cusp

_En

��2mr2� 1

r2

�G�

L

�2n2=3; (57)

where we have estimated the dimensionless cusp accelera-tion to be order unity, and the width of the beam as in (41).The flux grows with n suggesting that it is advantageous

to observe at high frequencies. However, the beam athigher n is narrower and hence event rates are lower athigh frequencies. These are observational issues that wewill return to in future work.

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B. Emission from kink

Now we estimate the power emitted from a kink[Eq. (46)], say when there is a discontinuity in a0 thatcontributes to I�, and a saddle point that contributes toIþ. Then I� ¼ Idisc� is given by (42) and Iþ ¼ Isaddleþ by(35). Note that, for a given k�, there will be a saddle pointcontribution to Iþ if there exists a �þ such thatk � b0ð�þÞ ¼ 0. Since b0ð�þÞ describes a curve parame-trized by �þ, there is a saddle point contribution for everyk in the direction of a point on the b0 curve. Hence thesaddle point contribution applies to emission along a curveof directions.

Since there is a saddle point in Iþ, the kinematic rela-tions discussed in Sec. IVA [Eqs. (27)–(33)] apply. Notingthe estimate in Eq. (44), we find that the dominant termsin (26) in the �L � 1 limit are given by

ðjMj2n;totÞkink ! 2ð2�G�Þ2p �p0 jIþj2½2jp � I�j2 þp �p0jI�j2�;

(58)

where we have also used jI�þ � I�j ¼ jIþ � I�j. Now we cansubstitute I� from Eq. (42) and Iþ from (35). Ignoringnumerical factors, the result can be written as

ðjMj2n;totÞkink � ðG�Þ2L4 1

n10=3ðshape factorsÞ; (59)

where the ‘‘shape factors’’ include various scalar productsthat depend on a0�—the shape of the loop—and the direc-tions of the momenta. The shape factors can be writtendown using the expression in (58) but the result is notilluminating. Equation (59) should be compared with theanalogous one, Eq. (49), for a cusp.

We now turn to the phase space integrals in (46). Thephase space volume is given in (50) except that, unlike inthe case of the cusp, there is a whole curve of saddle pointsthat are relevant, and the emission is along a curve ofdirections. Hence, the integration over momentum direc-tion is estimated as

Zd2p� 2��m;�; (60)

where �m;� are given in (40) and (41).

Putting together all the pieces in Eq. (46) and using (50),we obtain

_E njkink ��G�

L

�2ðshape factorsÞ: (61)

As in the cusp case, an important feature of this result isthat it is independent of the frequency of emission.

We now denote the cutoff in harmonic number by nk,and estimate it by the thickness of the string. What isdifferent from the cusp is that the velocity of the string atthe kink is not ultrarelativistic and there is no correspond-ing large Lorentz boost factor. Therefore, the estimate in(61) applies for

n < nk �ML: (62)

The energy flux from a kink at distance r is now given by

_En

�mr2� 1

r2

�G�

L

�2n1=3: (63)

C. Emission from kink-kink collision

This is the situation in which both I� and Iþ get con-tributions from discontinuities in a0 and b0, respectively.Therefore, in the general formula (26) we have to insert(42) and (43). This gives

jMj2tot ! 2

�2�G�

p � p0

�2½4jp � Iþj2jp � I�j2

þ 2ðp � p0Þ1fjp � Iþj2jI�j2 þ jp � I�j2jIþj2gþ ðp � p0Þ2jIþj2jI�j2�: (64)

The estimate in Eq. (44) shows that all the terms are non-singular if we take k2 ! 0.To evaluate the power radiated from a kink-kink colli-

sion, we see from Eqs. (43) and (42) that I� are Oð1=nÞ.Hence jMj2 � n�4. Also, since there is no beaming in thekink-kink collision

Zd2p� �: (65)

Then, putting together factors in Eq. (46) and using (50),we obtain

_E njk-k ��G�

L

�2ðshape factorsÞ (66)

exactly as in the estimate for the cusp and the kink, thoughthe shape factors are different in all three cases and in thiskink-kink case they may vary with direction of the mo-menta. Once again, the result is independent of the har-monic number n and holds up to nk ¼ ML, as in the kinkcase.The energy flux from a kink-kink collision at

distance r is

_En

r2� 1

r2

�G�

L

�2: (67)

The estimate (66) is due to contribution of the disconti-nuities at fixed values of both �þ and ��. Thus, theemission is coming from a single point on the string atone instant of time. This corresponds to the point where aleft-moving kink and a right-moving kink collide. Hence,the temporal duration of the burst is set by the stringthickness. On the other hand, emission at a frequency !cannot be temporally resolved in a time interval less than�!�1. For this reason, the observed burst duration atfrequency ! is set by !�1.The total energy emitted in harmonic n during a kink-

kink collision can be estimated from (66), which is the

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emitted power averaged over a time period L. Hence thetotal energy emitted in the nth harmonic in one kink-kinkcollision is

Enjk-k � ðG�Þ2L

ðshape factorsÞ: (68)

The continuum version may now be written as

dE

d!

��������k-k�ðG�Þ2c ac bðother shape factorsÞ; (69)

where the ‘‘sharpness’’ c a is defined by [12]

c a ¼ 12ð1� a0þ � a0�Þ ¼ 1

4ða0þ � a0�Þ2 (70)

and similarly for c b. In Eq. (69) we have pulled out factorsof the sharpness since the result must vanish if the sharp-ness vanishes. The ‘‘other shape factors’’ will in generalalso depend on the direction of emission.

V. CONCLUSIONS

In this paper we have calculated the flux of photons fromcosmic strings. In general this falls off exponentially withharmonic number n. However, as we have shown, thepower emitted from cusps, kinks and kink-kink collisionsdoes not fall off with n—rather, it is n independent. Thusthe emission from these features on the string dominatesover the emission from the rest of the string, at least at highfrequencies. If we denote the differential energy flux atfrequency !0 by F, i.e.,

F � d3E

dtd!0d�s

; (71)

where �s denotes a solid angle, then our results can besummarized as follows:

Fcusp � ðG�Þ2L

ð!0LÞ2=3; �s < ð!0LÞ�2=3; (72)

Fkink � ðG�Þ2L

ð!0LÞ1=3; � < ð!0LÞ�1=3; (73)

Fk-k � ðG�Þ2L

: (74)

The cusp emits a beam within a solid angle, the kink emitsalong a curve, while the kink-kink emission is in alldirections. The duration of the cusp and kink beams isgiven by Eq. (53), while the duration of the kink-kinkradiation is given by the wavelength at which the emissionis observed. Also, the cusp radiates at frequencies

!0 <MffiffiffiffiffiffiffiffiML

pwhereas the kink and kink-kink collisions

radiate for !0 <M, where M is the string scale.Our results so far provide the emission characteristics

from certain features on strings. Now we briefly discuss thecumulative effect of having many such features on a givenloop of string. The cusp and kink emissions are beamed

and this makes the analysis more involved. However, theemission due to kink-kink collisions is not beamed and iseasier to estimate.Equation (69) gives the energy emitted from a single

kink-kink collision. To get the energy emitted from a stringsegment, we need to sum over all the kink-kink collisionsoccurring on that string segment of length �l in an intervalof time �t

dE

d!� ðG�Þ2

Zdc adc bc ac b

dnadc a

dnbdc b

�l�t; (75)

where naðc a; tÞ is the number of kinks of sharpness c a attime t per unit length of string, and similarly for nb.We will consider emission from a loop of string that

formed at time tf by breaking off a long string. The loop

inherits a large number of (shallow) kinks from the longstring and from Ref. [12] we can write

Zdc ac a

dnadc a

� 1

tf

�tft�

��: (76)

The exponent � is�0:7 in the radiation-dominated epoch.In the matter-dominated epoch, strings contain all the kinksaccumulated until the epoch of matter-radiation equality,teq, and from then on the scaling in (76) has �� 0:4 [13].

The time t� denotes the epoch at which frictional effectson strings became unimportant. Hence Eq. (75) can bewritten as

dP!

dld!� ðG�Þ2

t2f

�tft�

�2�; (77)

where P! denotes the power emitted at frequency !.We now obtain some numerical estimates, leaving a

detailed analysis for future work. The photons emittedfrom loops deep into the radiation epoch will get thermal-ized. Hence the emission from loops in the post-recombination era is most relevant for direct observation.Loops at the epoch of recombination could have beenproduced in the radiation epoch and for simplicity weconsider a loop that was formed at the epoch ofradiation-matter equality, tf¼ teq�1011 s. With G��10�8, t�� tP=ðG�Þ2�10�27 s [1], where tP�10�43 s isthe Planck time, and � ¼ 0:7, we get

dP!

dld!¼ ðG�Þ2þ4�

t2�2�f t2�P

� 10�22 ergs

cm: (78)

Detectors on Earth observe a flux of photons and it is morerelevant to calculate the number of photons arriving at thedetector. This follows from E ¼ N!!, where N! is thenumber of photons of frequency ! emitted by the string,

dN!

dtdl� 10�22 ergs

cm

d!

!: (79)

If the loop of length teq � 1021 cm is at a distance compa-

rable to the present horizon, r� 1027 cm, then the flux of

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photons at the detector is obtained by multiplying (79)by teq=r

2,

dN !

dtdA

��������1 loop� 10�10

km2 � yr

d!

!

�G�

10�8

�2ð1þ2�Þ

; (80)

where N ! denotes the number of photons arriving at thedetector with collecting area dA. If at teq there was one

loop of length teq per horizon, the number of loops that can

contribute to the flux at the detector is given by the numberof horizons at teq that fit within a comoving volume

equal to our present horizon volume: t30=ðteqzeqÞ3, wheret0 � 1017 s and zeq � ðt0=teqÞ2=3. Therefore the number of

contributing loops is �t0=teq � 106 and the photon flux

due to all of these loops is

dN !

dtdA

��������loops at rec� 10�4

km2 � yr

d!

!

�G�

10�8

�2ð1þ2�Þ

: (81)

This estimate suggests that a detector with collecting areað100 kmÞ2—comparable to the Auger observatory—willdetect �1 photon=year in every logarithmic frequency in-terval emitted by string loops from the recombination era.The estimate (81) holds for frequencies all the way up to thestring scale �1015 GeV but it does not take into accountany propagation effects. Neither does it take into accountthe network of long strings and the spatial and lengthdistribution of loops. Note that the emission falls steeplywith decreasing string tension. The effect can only be usefulfor small G� if the amount of string in a horizon volume isinversely proportional to some high power of G�.

The pattern of photon emission from a string is linealand this may be helpful to distinguish it from conventional

sources. It is also possible that the emission from thestring will be polarized (though our analysis so far hassummed over polarizations and hence erases the polariza-tion information). The beamed emission from cusps andkinks may provide distinctive events that can signal thepresence of strings. We plan to explore these signatures infuture work.We would like to close with a cautionary note. The

emission rate from kink-kink collisions is greatly enhancedby the factor ðtf=t�Þ� in Eq. (76). This factor is due to the

accumulation of kinks on strings from the time, t�, whentheir dynamics became undamped. The exponent, �, de-pends on dynamical factors, such as Hubble expansion,that tend to straighten out the kinks, but the estimate doesnot take radiation backreaction into account and this willhave a tendency to reduce the emission rate. However, it ispossible that emission at frequencies much lower than thestring scale remain relatively unaffected by the backreac-tion. We can be more confident of our estimates of lightfrom cosmic strings only once this issue is satisfactorilyresolved.

ACKNOWLEDGMENTS

We are grateful to Y.-Z. Chu, E. Copeland, and T. Kibblefor comments and discussions. T. V. was supported by theU.S. Department of Energy at Case Western ReserveUniversity and also by Grant No. DE-FG02-90ER40542at the Institute for Advanced Study, where this work waspartially done. D.A. S. also thanks the Institute forAdvanced Study, where much of this work was done, forsupport.

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