Light from cosmic strings

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<ul><li><p>Light from cosmic strings</p><p>Daniele A. Steer</p><p>APC* 10 rue Alice Domon et Leonie Duquet, 75205 Paris Cedex 13, France</p><p>Tanmay Vachaspati</p><p>Physics Department, Arizona State University, Tempe, Arizona 85287, USA(Received 2 January 2011; published 25 February 2011)</p><p>The time-dependent metric of a cosmic string leads to an effective interaction between the string and</p><p>photonsthe gravitational Aharonov-Bohm effectand causes cosmic strings to emit light. We</p><p>evaluate the radiation of pairs of photons from cosmic strings and find that the emission from cusps,</p><p>kinks and kink-kink collisions occurs with a flat spectrum at all frequencies up to the string scale. Further,</p><p>cusps emit a beam of photons, kinks emit along a curve, and the emission at a kink-kink collision is in all</p><p>directions. The emission of light from cosmic strings could provide an important new observational</p><p>signature of cosmic strings that is within reach of current experiments for a range of string tensions.</p><p>DOI: 10.1103/PhysRevD.83.043528 PACS numbers: 98.80.Cq</p><p>I. INTRODUCTION</p><p>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.</p><p>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 emissionwhich we treat inthe present paperwas 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 G2 where G is Newtons constant</p><p>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</p><p>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 casesfrom 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.</p><p>II. GRAVITATIONAL AHARONOV-BOHM</p><p>The gravitational field of a cosmic string is characterizedby the parameter G which is constrained to be less than107. Hence it is sufficient to consider the case of aweak gravitational field and linearize the metric around aMinkowski background</p><p>g h: (1)Then coupling between the gravitational field and thephoton becomes</p><p>*Universite Paris-Diderot, CNRS/IN2P3, CEA/IRFU andObservatoire de Paris</p><p>PHYSICAL REVIEW D 83, 043528 (2011)</p><p>1550-7998=2011=83(4)=043528(8) 043528-1 2011 American Physical Society</p><p></p></li><li><p>L int 14ffiffiffiffiffiffiffigp FF 12FF Oh2;</p><p>(2)</p><p>where</p><p> h 14h; (3)and the electromagnetic field strength is F @A @A. The coupling is quadratic in A so that to lowest</p><p>order in h photons are created in pairs. The metric</p><p>perturbation, h, due to the cosmic string energy-</p><p>momentum tensor, T, follows from the Einstein equa-</p><p>tions: in Fourier space (denoted by tildes),</p><p>~ ~h 14~h 16G</p><p>k2</p><p>~T 14</p><p>~T</p><p>:</p><p>(4)</p><p>From the Nambu-Goto action, and using the conformalgauge [1]</p><p>Tx Z</p><p>d2 _X _X X0X04x X; (5)</p><p>where X; t is the string world-sheet. A cosmic stringloop trajectory can be written in terms of left- and right-movers</p><p>X; t 12a b; (6)where t, and we will adopt world-sheet coor-dinates such that</p><p>a0 ; b0 ; ja0j 1 jb0j; (7)where primes denote derivatives with respect to the argu-ment. Substituting (6) into (5) yields</p><p>Tx 4Z</p><p>dda0b0 a0b04x X(8)</p><p>which, when Fourier transformed, is</p><p>~T k 4 I;I; I;I;: (9)Here, for the periodic oscillations of a loop of length L</p><p>I X1n1</p><p>k0L</p><p>4 n</p><p>Z L0db0eikb=2; (10)</p><p>I X1n1</p><p>k0L</p><p>4 n</p><p>Z L0da0eika=2: (11)</p><p>It will be important in the following to notice that as aresult of the periodicity of the loop,</p><p>k I 0: (12)We can now calculate the amplitude for the pair creation</p><p>of two outgoing photons of momentum p and p0, andpolarization and 0, respectively, where</p><p>p2 0 p02; p 0 p0 0: (13)This is given by the tree level process shown in Fig. 1, andon using Eqs. (2), (4), and (9) we find</p><p>M p; p0 4Gk2</p><p>IkIkQp; ;p0; 0; (14)</p><p>where momentum conservation imposes that</p><p>k p p0; (15)while</p><p>Q P P 12P; (16)</p><p>P p pp00 p00 : (17)The number of photon pairs produced in a phase space</p><p>volume, the pair production rate, is given by (e.g.Sec. 4.5 of [6])</p><p>dN X;0</p><p>d3p</p><p>231</p><p>2!</p><p>d3p0</p><p>231</p><p>2!0jMj2; (18)</p><p>where ! and !0 are the energies of the two outgoingphotons, while the energy emitted in the pairs is</p><p>dE X;0</p><p>d3p</p><p>231</p><p>2!</p><p>d3p0</p><p>231</p><p>2!0!!0jMj2: (19)</p><p>Clearly the crucial relevant quantity is</p><p>jMj2tot X;0</p><p>jMj2 4G</p><p>k2</p><p>2X;0</p><p>II II QQ:</p><p>(20)</p><p>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</p><p>photon polarizations in (20) can be simplified through thereplacement</p><p>X;0</p><p> ! ; (21)</p><p>provided certain conditions hold [6]. More specifically, letus define M via</p><p>p</p><p>p</p><p>FIG. 1 (color online). Feynman diagram showing two photonproduction from a classical string.</p><p>DANIELE A. STEER AND TANMAY VACHASPATI PHYSICAL REVIEW D 83, 043528 (2011)</p><p>043528-2</p></li><li><p>Q 0M; (22)so that</p><p>M N N 12N; (23)</p><p>N p pp0 p0: (24)Then the required condition [6] is that</p><p>pM 0 p0M: (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).</p><p>Then, after quite a bit of algebra and on using (21), wefind that jMj2tot is given by</p><p>jMj2tot 2G</p><p>p p028jp Ij2jp Ij2</p><p> 4p p0fjp Ij2jIj2 jp Ij2jIj2p Ip II I p Ip II Ip Ip II I p Ip II Ig 2p p02fjIj2jIj2 jI Ij2 jI Ij2g;</p><p>(26)</p><p>where jIj2 I I, and in order to simplify the resultwe have made extensive use of p I p0 I whichfollows (12) since k p p0. Finally, we have expressedthe answer in powers of k2 2p p0: this will be impor-tant later when we will see that the k2 ! 0 limit plays acrucial role.</p><p>III. EVALUATION OF jMj2totIn order to calculate the energy radiation in photon pairs,</p><p>we need to evaluate jMj2tot, where the dynamics of cosmicstring loops enters Eq. (26) through the integrals Ik.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.</p><p>Generally, however, it is well known that Ik 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 and I corresponds to a cusp on thestringnamely, j _Xj 1; a saddle point in one of theintegrals and a discontinuity in the other occurs at a kink.</p><p>Finally, when two kinks collide, there is a discontinuity inboth integrands. In all three possibilitiescusp, kink, kink-kink collisionI</p><p> decay as a power law, k0Lq, where</p><p>the index q will be determined below.</p><p>A. Saddle points and cusps</p><p>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 0 p02, let us write</p><p>p !;p !1; p !p;p0 !0;p0 !01; p0 !0p0;</p><p>(27)</p><p>where jpj 1 jp0j. Since k p p0, thenk2 2p p0 2!!01 p p0: (28)</p><p>We also define</p><p>k ; k !!0;p p0: (29)Next consider the integral Ik in Eq. (10) when there is</p><p>a saddle point. This implies that there is a point such that</p><p>k b0 0; (30)where from the gauge conditions Eq. (7) b02 0. Thus, at asaddle point</p><p>k b0; where k k</p><p>; (31)</p><p>so that</p><p>k2 2k2 2b02 0: (32)Thus, k2 is null. Furthermore, from Eq. (28),</p><p>p p0 k b0; (33)where p and p0 were defined in Eq. (27).Apart from a sign, all the above goes through for a</p><p>saddle point in I. Again k must be lightlike, but now k a0 0 1 k a0 (since a0 ). Thus,</p><p>p p0 k a0: (34)Evaluation of the integrals around the saddle points, in</p><p>the L 1 limit can then be carried out in the standardway (see e.g. [9]) and, using the kinematic relations givenabove, leads to</p><p>Isaddle;n ALb0s</p><p>L1=3 iBL2 b</p><p>00s</p><p>L2=3 . . . ; (35)</p><p>Isaddle;n AL a0s</p><p>L1=3 iBL2 a</p><p>00s</p><p>L2=3 . . . : (36)</p><p>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</p><p>refers to evaluation at a saddle point. The delta functions in</p><p>LIGHT FROM COSMIC STRINGS PHYSICAL REVIEW D 83, 043528 (2011)</p><p>043528-3</p></li><li><p>Eqs. (10) and (11) enforce L 4n. 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;</p><p>A </p><p>12</p><p>L2jb00j21=3 2</p><p>32=3 ;</p><p>B </p><p>12</p><p>L2jb00j22=3 1ffiffiffi</p><p>3p 2=3;</p><p>(37)</p><p>and A and B are given by identical expressions exceptthat b00 is replaced by a00.</p><p>A cusp on a string loop occurs when j _Xj 1 and hence,from (6), when a0 b0. However, from Eqs. (33) and(34) this condition requires a saddle point contribution toboth I and I. In the vicinity of the beam of the cusp,from Eqs. (31) and (33), we get p b0c a0c, and simi-larly for p0. Therefore from (35) and (36), we estimate</p><p>p Isaddle Ok2; p0 Isaddle Ok2: (38)This is an important result: consider jMj2tot given in</p><p>Eq. (26). In the beam of the cusp k2 2p p0 ! 0[Eq. (32)], so one might worry that Eq. (26) diverges dueto the overall factor of 1=k22. However, this divergence isrendered harmless by the scaling in Eq. (38). Indeed, in thek2 0 limit, only the last line in Eq. (26) gives thedominant contribution to the emission from the cusp:</p><p>jMj2totcusp ! 2G2 2jIj2jIj2 jI Ij2 jI Ij2 (39)</p><p>with I in Eqs. (35) and (36) [we have dropped the labelsaddle]. The other terms in Eq. (26) all contain factorssuch as jp Ij and are higher order in k2.</p><p>The above analysis assumes that k is in the direction ofthe cusp. If k is at some small angle, , to b0, we can writek b0 2=2 and repeat the above analysis as in [7]. Theestimate is valid for</p><p> m; 4Ljb00j2ffiffiffi</p><p>3p</p><p>1=3</p><p>: (40)</p><p>Similarly in the case of I</p><p> m; 4Lja00j2ffiffiffi</p><p>3p</p><p>1=3</p><p>: (41)</p><p>This estimate assumes k2 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</p><p> L1=3.</p><p>B. Discontinuities and kinks</p><p>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 uk, one canextract the dominant contribution;</p><p>Idisc Z uk</p><p>da0eikaka0uk...</p><p>Zuk</p><p>da0eikaka0uk...</p><p> 2i</p><p>a0</p><p>k a0 a</p><p>0k a0</p><p>eikak ; (42)</p><p>where a0 refers to the value of a0 on either side of thediscontinuity, and ak a uk. Similarly we can findI, and the result is</p><p>Idisc 2</p><p>i</p><p>b0</p><p>k b0 b</p><p>0k b0</p><p>eikbk : (43)</p><p>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</p><p>k I 0. Hence if k2 ! 0 then p p0 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. Thecase when k b0 0 or k a0 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 and I corresponds to a</p><p>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,</p><p>jp Idisc j2 Ok2; jp0 Idisc j2 Ok2; (44)which should be compared to Eq. (38) for a cusp. This canbe most clearly seen by writing, for example,</p><p>p Idisc p </p><p>a0k a0</p><p> a0</p><p>k a0</p><p>!!0</p><p>2p a0a0 p a0a0</p><p>k a0k a0</p><p> p0 p:</p><p>(45)</p><p>Therefore, p Idisc Ojp0 pj from which (44) followssince !!0p0 p2 k2 [see Eq. (28)].</p><p>C. No saddle point or discontinuities</p><p>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 / en where n L=4 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</p><p>DANIELE A. STEER AND TANMAY VACHASPATI PHYSICAL REVIEW D 83, 043528 (2011)</p><p>043528-4</p></li><li><p>cusp or a kink diverges and needs to be cut off due to thethickness of the string.</p><p>IV. POWER EMITTED</p><p>We now evaluat...</p></li></ul>