Astroparticle Physics l(1993) 239-243 North-Holland
Cosmic gamma rays from collapsing cosmic strings
Xinyu Chi, Charles Dahanayake *, Jerzy Wdowczyk 2 and Arnold W. Wolfendale Department of Physics, Univerdy of Durham, Durham, DHl 3LE, UK
Received 6 July 1992; in final form 19 November 1992
In a recent paper [Astropart. Phys. 1 (1992) 1291 we showed that collapsing strings could not generate the highest energy cosmic ray particles if they were protons because of unacceptably large fluxes of cosmic gamma rays produced along with the protons. Here we show that it is conceivable that the particles detected above IO l9 eV are, in fact gamma rays - in which case the collapsing string model can be retrieved. The parameters characterizing the spectrum of gamma rays at injection are defined, including the requirement that the gamma ray cascading in the vicinity of the strings must be small.
The origin of the cosmic rays of the highest energies is still one of the great problems of astrophysics. A variety of factors, not at least the apparent ankle in the energy spectrum at about 1019 eV, have led to the idea that the particles of higher energy probably come from extragalactic sources . In fact, we have recently made the case for a significant fraction of some of these particles even being of Galactic origin  but, nevertheless, there still appears to be a significant extragalactic ~mponent. It is usually presumed that the majority of the particles are protons.
Even invoking extragalactic sources there are, of course, still problems associated with the na- ture of the sources. What are they? Collapsing cosmic strings might appear to offer the distinct ~ss~bili~ of providing an explanation in that the cohapsed strings are supposed to release X-par- ticles of mass 1O24 eV which decay to yield very
Correspondence to: X. Chi, Department of Physics, University of Durham, Durham, DHl 3LB, UK. On leave from Un~ersi~ of Kelaniya, Kelaniya, Sri Lanka. 2 On leave from the Institute of N&ear Studies, W-950
energetic particles 14,5]. Bhattach~jee and Rana  in particular have developed such a model (see the very recent work by Bhattacharjee et al,  for a continuation of the earlier work). Unfor- tunately, as we have shown 111, the gamma rays produced along with the protons carry so much energy that their cascade through the universe gives rise to an excessive ffux of low energy gamma rays. Specifically, we have shown [ll that the predicted flux at 100 MeV is some twenty times observation when the proton flux, after modulation is normalized to observation at 102 eV.
With this recent analysis has come the realiza- tion that if the particles detected by the EAS arrays above 1019 eV were in fact gamma rays, rather than protons, then the origin problems might be soluble. In the present work we start by examining whether or not the hypothesis of pri- mary gamma rays is tenable. Deciding that it is, we evaluate the conditions necessary for the y-ray spectrum produced by the collapsing strings and see whether the parameters derived are reason- able.
Finally, an anaIysis is given of experimental checks that might be made.
0927-6505/93/$06.00 0 1993 - Elsevier Science Publishers B.V. All rights reserved
240 X. Chi et al. / Cosmic gamma rays from collapsing cosmic strings
2. Can cosmic rays above 1019 eV be mainly gamma rays?
Although it is usually assumed that the extra- galactic particles are protons this can hardly be regarded as proven and in this section we exam- ine the possibility that they are, instead, mainly - or entirely - gamma rays.
It is well known that gamma-ray-initiated showers are deficient in muons [7,8], although it must be remarked that the apparent normality of the muon content of showers from the direction of Cygnus X-3 191 immediately causes some doubt. Nevertheless, we continue along the conventional path.
Calculations at energies near 1015 eV have shown that the muon content in the gamma-ray- initiated showers should be only about 3% of that in particle-initiated showers [7,8]. However, what was not realized in the early work was that the muon fraction will increase considerably for the higher energy showers. Using results on photo- produced muons within normal showers [lo] it can be shown that the number of muons in gamma showers increases more rapidly than the primary energy, Nk a E; with (Y = 1.0-1.1. In proton-ini- tiated showers, on the other hand, the relation- ship is N, a EF and therefore (N,),/(N,), a Es with p = 0.2-0.3. At 102 eV, in consequence, we expect the muon fraction to have increased to 30%-95% of the norm. Thus, the total number of muons in gamma-ray- and proton-initiated show- ers should be almost comparable and one cannot rule out the possibility of the primary particles above 1015 eV being gamma rays on the basis of inadequate muon numbers.
There is another reason why primary gamma rays are in with a chance and this concerns the width of the showers. Figure 1 shows a compari- son of the lateral distribution of muons in proton- and gamma-ray-induced showers, using the calcu- lations of Wdowczyk  for gamma ray and Greisen [ll] for protons. The calculations relate to a primary energy of 1015 eV. It will be noted that at 1000 m from the shower axis the ratio of densities is about 0.3 and clearly this ratio will increase considerably at the higher primary ener- gies.
I 100 rlml
Fig. 1. Comparison of the later distribution of muons for primary gamma rays and protons; the primary energy is 10
eV and the results relate to near vertical showers.
Bearing in mind the oft-quoted increase in width of lateral distributions with primary energy, e.g. ref , it will be evident that the gamma ray fraction above 1019 eV could indeed be quite significant. It should be added that there is an extended literature on this topic, which bears out the early conclusions [12,131. There is thus some justification for the present examination of the cosmic string hypothesis.
3. The model
It is assumed that the result of the decay of an X-particle (of mass 10% eV> is the generation of a gamma ray spectrum of simple power law form down to an energy E,, the maximum energy being 10% eV. The multiplicity of gamma rays generated No, is allowed to be a variable. It is true that models of X-particle decay, involving jet production, have been formulated but they have to be extrapolated in energy considerably and, furthermore, the effect of cascading, etc., in the environment of the collapsing string - with its
X. Chi et al. / Cosmic gamma rays from collapsing cosmic strings 241
consequent effect on the gamma spectrum - is unknown. For illustrative purposes, therefore, we write
/ EmAE-YE dE = 1O24 eV,
/ EmAE-y dE = N 0'
These two equations define the relationship between E, and y for fixed No.
The gamma ray spectrum generated by an injection spectrum of the form AEpY will in first approximation be AEpY X A(E)/R where h(E) is the interaction length for y-yCMB collisions and R is the linear dimension of the universe. We appreciate that there will also be contributions to the gamma ray flux through cascading from higher energies but these are ignored in the present calculations - approximations in the input do not warrant an accurate treatment. A(E)/R is shown in fig. 2 for both -y-~c~n collisions and for p- yCMB collisions, the latter being given because of the contribution to the ambient flux of cosmic rays from the protons generated by the cosmic strings in the manner discussed in our previous work.
It is important to note that, for the same injection spectrum of protons and gamma rays,
18 I 1o17
lo l9 *O 21 22 23 EleV)
Fig. 2. The ratio A(E)/R, of the interaction length h(E) to the linear dimension of the universe R, for y - yCMB collisions and for p--rCMB collisions. The usual processes are included in the calculations, viz. e +e- and pion production for p- -rCMB; under y-yCMB we also include y-starlight and y-radio as well as trident production in y- ycMn (see ref. 
and references therein).
1020 21 22 23
Fig. 3. The gamma ray spectrum at earth for various values of the spectral index y. The flux at earth has been normalized to
the observed (particle) flux at 10 eV.
protons would dominate in the modulated spec- trum received at the earth below about 5 x lo* eV and one would need a ratio of y/p of about 50 at 102 eV at injection in order to achieve unity at the earth. However, in the present case, as we shall show, the exponent for the injection spectrum of gamma rays ( N 2.4) differs from that adopted for protons (1.5) and yields a ratio of y/p at injection of N 60 if the energy content in protons and gamma rays is the same, i.e. some- what bigger than the 50 referred to above. This means that we can ignore the proton component from now on.
Returning to the gamma rays alone, fig. 3 shows the gamma ray spectrum at earth for vari- ous choices of y, normalization to observation having been made at 10 2o eV as usual. Figure 4 shows the corresponding values of E, from eqs. (1) and (2). Figure 4 also shows the energy den- sity in the gamma radiation coming from the losses by y-yCMB collisions in extagalactic space.
In an earlier work  of Wdowczyk and Wolfendale, based on previous studies , it is shown that the energy density, needed to give a
242 X. Chi et al. / Cosmic gamma rays from collapsing cosmic strings
Energy Dens.1 ty
-16 IeV cm?
1 1 I 4 1 2 3 4
Y Fig. 4. The lower limit E, to the energy of particles generated by the decay of an X-particle, the upper limit being 10z4 eV, satisfying eqs. (1) and (2). Also shown (right-hand scale) is the energy density in the gamma radiation resulting from v-yCMB collisions. We require E, greater than lo* eV and an energy
density less than 3 x 10d6 eV cmw3.
gamma ray flux at 100 MeV equal to that meas- ured for the apparently isotropic extragalactic flux, was N 3 x lO-j eV cmm3 so that this value sets a limit on y and N,, specifically, 3000 2.4.
The extent to which these values are reason- able is discussed in the next section.
4. The derived parameters
There is some guidance on the value of N,, and to a lesser extent on y, from observations of jets in hadron-hadron collisions. At first sight one might put No to be several hundred; by extrapolation of overall multiplicity from lower energy pp collisions and y = 1.5, as derived by Bhattacharjee and Rana . However, this would be too simplistic insofar as the extrapolation will be far from linear, in view of the cross section for jet production rising much more rapidly than the total; thus N,, will certainly be bigger than several
hundreds. Secondly, the exponent for y-rays will surely be greater than that for protons at the very high jet multiplicities involved. Of overriding im- portance is the likelihood of cascading in the environment of the collapsing string. The strong magnetic fields and photon densities involved will surely cause this, but in the present model it is apparent that not much can be allowed - less than a factor of about 2 in intensity at 102 eV.
The last mentioned point needs further discus- sion. At first sight it might be thought that near- to-source cascading would eliminate the model but in fact the cascading problem affects each and every origin model. Thus, if any discrete-ob- ject source is proposed (e.g. pulsar) losses in the local environment, e.g. intense magnetic field, should be crucial. Moderately extended sources such as active galactic nuclei fare no better, with their intensive photon fields. Very extended sys- tems, such as colliding galaxies, might fare better but not obviously so because of the extended time period over which the acceleration occurs. Cos- mic strings might in fact be less affected than the other sources because we are dealing with pri- mary gamma rays which pursue straight line paths and can, in principle, escape rapidly from the high loss region.
The conclusion to be drawn about the accept- ability of the parameters is that they are not unreasonable. It is unlikely that, in the foresee- able future, a stronger statement than that will be possible.
5. Conclusions and suggestiijns for farther work
Although at first sight the cosmic string hy- pothesis appears a little far fetched it is not obvious that it can be dismissed. Recent studies of ripples in the CMB have strengthened the Big Bang model of the origin of the universe and with it the likelihood of cosmic string production so that this role in cosmic ray production war- rants further examination.
A more detailed study of the characteristics, muon/electron number, radial and zenith angle distribution of the particles in extensive air show- ers at the highest energies will surely pay divi-
X. Chi et al. / Cosmic gamma rays from collapstng cosmic strings 243
dends. An interesting phenomenon, to be sought, is the anisotropy in the extragalactic component. Seemingly discrete sources of low mass parti- cIes (probably gamma rays, in fact) at high Galactic Iatitudes would be indicative of the cos- mic string hypothesis.
The authors are indebted to the editor, T.K. Gaisser, for focusing our attention on the prob- lem of jet-cascading at source.
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