Astron. Nachr. /AN 334, No. 9, 1051 – 1054 (2013) / DOI 10.1002/asna.201211993

Synchrotron radiation of superconducting cosmic strings

D.A. Rogozin� and L.V. Zadorozhna

Taras Shevchenko National University of Kyiv, Faculty of Physics, Academician Glushkov Ave. 2, Kyiv, 03680, Ukraine

Received 2012 Nov 15, accepted 2013 May 6Published online 2013 Nov 9

Key words cosmology: theory – early Universe – intergalactic medium – radiation mechanism: non-thermal

Cosmic strings are topological defects which can be formed during the symmetry breaking phase transitions in the earlyUniverse. Their existence finds support in modern superstring theories, both in compactification models and in theorieswith extended additional dimensions. Strings predicted in most grand unified models respond to external electromagneticfields as thin superconducting wires. As they move through the cosmic magnetic fields, strings develop tremendous electriccurrents. Superconducting cosmic strings can serve as powerful sources of nonthermal radiation in a wide energy range,viz. as synchrotron radiation of electrons accelerated by bow shock waves which are created by the magnetosphere ofstrings moving relativistically through the intergalactic medium (IGM). Calculations of the expected radiation fluxes showthat for typical parameters of the strings and the IGM the existing detectors can see loops at average string distances.According to our calculations they can also be observed if they are located in clusters of galaxies.

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1 Introduction

Cosmic strings are topologically stable, one-dimensionaldefects in the vacuum which can appear during appropri-ate phase transitions in an adiabatically expanding earlyUniverse that is cooling down from a very hot initial state(Kampfer 2000; Rocher, Jeannerot & Sakellariadou 2004).During their existence, the strings have formed entanglednetworks which are composed of loops and infinite strings(Dubath, Polchinski & Rocha 2008; Vanchurin 2008). Thenumerical calculations of string evolution indicate that thenetwork of strings evolves in a scale-invariant way which ischaracterized by the presence of several infinite (in the lim-its of the horizon) segments and a collection of loops withvarious lengths in each Hubble volume. The strings oscillateand move with relativistic velocities. The average velocityat the correlation length is 〈V 〉 ∼ 0.15 c, and the root-mean-square velocity of the string in matter dominated epoch isVrms ∼ 0.58 c, where c is the velocity of light (Vilenkin &Shellard 1994).

As the cosmic strings arise at the end of the inflationperiod, they contribute insignificantly to the total spectrumof density fluctuations (Wyman, Pogosian & Wasserman2005). Their contribution is essentially limited by observ-able data. The fluctuations of the temperature of the cos-mic microwave background (CMB) can be a result of somecombinations of quantum fluctuations in the Universe andthe perturbations caused by cosmic strings (Pogosian et al.2009; Fraisse et al. 2008). However, no direct observationsof strings in fluctuations of the microwave emission areavailable till now, which imposes some restrictions for thestring parameters, in particular Gμ/c2 � 10−7, where μ is

� Corresponding author: rogozin d@ukr.net

the tension (or the mass per unit length) of a string and Gthe gravitational constant (Fraisse 2007).

According to realistic particle-physics models, cosmicstrings can possess the property of developing tremendouselectric currents, thus they effectively becoming electricallysuperconducting wires of astrophysical dimensions, insideof which the massless carriers of charge (zero modes) aremoving with no resistance (Ringeval 2001; Ferrer et al.2006). Significant attention is given to the electrodynamicproperties of strings and to their interaction with the cosmicplasma. In the presence of a current along a string, the latterbecomes a sources of electromagnetic emission.

2 Cosmic strings

The energy scale of a phase transition is characterized bythe parameter η that is related to the tension (mass per unitlength) μ of a string by μ ∼ η2/(�c3), where � is the Planckconstant. The dimensionless parameter α characterizes thegravitational action of a cosmic string (i.e. its energy lossrate) and is connected with the tension μ as

α =ΓGμ

c2∼ η2

m2Pl c

4, (1)

where Γ ≈ 50 is dimensionless, and mPl is the Planck mass(Siemens et al. 2006). The typical length of the loops ofcosmic strings is l = αct, where t is the cosmological mo-ment of time (Damour & Vilenkin 2005). Since we considera close region of the Universe, t = t0 = 13.6×109 yr.

The loop density depends on time as given by Damour& Vilenkin (2005):

n =1

α(ct)3. (2)

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1052 D.A. Rogozin & L.V. Zadorozhna: Synchrotron radiation of superconducting cosmic strings

Table 1 Characteristics of cosmic strings with various tensions.

α String Parameters

l (pc) ds (pc) θ (′′)

5×10−6 2.1×104 7.1×107 60.310−6 4.2×103 4.2×107 20.610−8 41.7 9.0×106 110−11 0.042 9.0×105 0.01

For given α, the average distance of the loop positions (ordistance from loop to observer) is ds = n−1/3 = α1/3ct.The mean angular size of a loop as observed on Earth isθ ∼ l/ds = α2/3 and depends only on the tension of thestring (see Table 1).

3 Superconducting cosmic strings in theintergalactic plasma

Let strings move with a Lorentz factor γs through the inter-galactic medium with the follwing typical parameters: pro-ton and electron densities ne ∼ np = n1 =10

−7n−7 cm−3,B1 = BIGM = 10−7B−7 G is the magnetic field (here-inafter B−7 = B/10−7, n−7 = n/10−7 etc.).

During the oscillations of a loop in the intergalacticmagnetic field, an electric current is generated therein witha mean amplitude of

i = ki q2eBIGM l/� , (3)

where qe is the electron charge and ki � 1 a constant. Thecurrent generates a proper magnetic field around the stringof (Witten 1985)

Bmag(r) = 2i/(c r), (4)

where r is the distance to the string.The ionized cosmic plasma cannot penetrate into the re-

gion with a strong magnetic field near the string; thus theshock wave is formed at some distance rs. Behind the front,the flow of the plasma, in the reference system of the string,is decelerated and flows around the “magnetosphere” of thestring, i.e the region where the high pressure of the mag-netic field is balancing the dynamical pressure of the plasmabehind the shock. Therefore, the radius of the shock wavecan be determined. With the accepted notations we get afterZadorozhna & Hnatyk (2009a):

rs =kiq

2eBIGM l

2�c2γsh√

πn1mp

= 3.1×1015ki γ−1sh B−7 α−8 n

−1/2−7 cm, (5)

where mp is the proton mass, γsh = γs is the Lorentz-factorof the shock wave.

The characteristics of a shock wave – its Lorentz factorγsh and the Lorentz factor of the plasma behind the wave

front γ2 – (both are given in the laboratory reference system,Blandford & McKee 1976) is

γ2 �√(γ2

sh + 1)/2 . (6)

Particle density n2 and energy density e2 behind the shockfront are (Zadorozhna & Hnatyk 2009a; Zadorozhna &Hnatyk 2009b)

n2 ≈ 4γ2 n1, (7)

e2 = etot � ep = γ2 n2 mp c2 ≈ 4γ22 n1mp c2. (8)

Relativistic protons give the main contribution to the energydensity behind a relativistic shock wave, ep � e2. Magneto-hydrodynamic processes behind the front of a shock wavelead to the transfer of some part of the thermal energy ofprotons to electrons, ee = εee2, εe < 1, and to the genera-tion of a turbulent magnetic field, eB = εBe2, εB < 1. Itsvalue is (Zadorozhna & Hnatyk 2009a)

B2 = 2γ2

√8πc2εB mp n1 = 3.9×10−5γ2n

1/2−7 ε

1/2

B,−1 G .

(9)

4 Synchrotron radiation fromsuperconducting cosmic strings

We assume that the electrons develop a power law energydistribution in the post-shock region with an exponentialcutoff at high energies:

N(γe) = K ′γ−pe e

−γe

γe,max ,

N(Ee) = KE−pe e

−Ee

Ee,max , (10)

where K and K ′ are the proportionality coefficients, γe,max

and Ee,max are maximal Lorentz factor and maximal energyof the electrons, respectively, and p = 2.2 (p ≈ 2.25 fromthe analysis of data from gamma-ray bursts). For this case,the spectral emissivity of the electrons is as follows (Lon-gair 2011):

j(ν) =

∫Ee

J(ν, Ee)N(Ee) dEe, (11)

where J(ν, Ee) is the spectral emissivity of one electron. Inour paper we use an approximation for this function whichwas proposed by Aharonian (2004):

J(ν, Ee) =

√2e3B

me c2x

∞∫x

K5/3(η) dη

=

√2e3B

me c2Cx1/3 e−x. (12)

Here x =ν

νc

=2ν

3γ2eνg

=4πme c ν

3eBγ2e

= bν

γ2e

, where

b =4πmec

3eB, C ≈ 1.85, me is the electron mass and B the

magnetic field.

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Astron. Nachr. /AN 334, No. 9 (2013) 1053

The density of the electrons, ne,2, and the density oftheir thermal energy, ee,2, can be calculated as

ne,2 =K

p − 1

(1

Ep−1min

− 1

Ep−1max

), (13)

ee,2 = εeγ2 n mp c2 =K

p − 2

(1

Ep−2min

− 1

Ep−2max

). (14)

Now, with the assumption of a very high maximal electronenergy (in comparison with their minimal energy), K canbe found from Eq. (14) (where p = 2.2, γsh = 2):

K � 4(p − 2)εe n1γ22 mpc2(mec

2γe,min)p−2

= 1.5×10−12γ2.22 n−7 ε1.2

e,−1 erg1.2 cm−3. (15)

While K ′ equals

K ′ =K

(mec2)p−1= 3×10−5γ2.2

2 n−7 ε1.2e,−1 cm−3. (16)

The minimal Lorentz-factor of electrons can be ex-pressed as

γe,min =mp

me

p − 2

p − 1γ2 εe = 31γ2 εe,−1. (17)

The maximal Lorentz-factor of the electrons is es-timated from the equality of the acceleration time taccand time of synchrotron cooling tsyn (Sari & Esin 2001;Zadorozhna & Hnatyk 2009a):

tacc =cRL

v2A

, (18)

tsyn =γemec

2

Psyn

, (19)

where RL = γemec2/(qeB2) is the Larmor radius of an

electron, vA is the Alfven velocity (v2A/c2 � 2εB), Psyn =

4/3σTc eBγ2e is the emission power of one electron in the

local system of coordinates (the post-shock plasma), and σT

is the Thompson cross-section. As a result, γe,max is as fol-lows:

γe,max =

√12πεBqe

B2σT

= 8.1×109γ−1/22 n

−1/4−7 ε

1/4

B,−1. (20)

Another restriction for γe,max can be found considering thesize of the acceleration region (∼rs): Ee,max = qersB2.The previous expression can be rewritten as (Zadorozhna &Hnatyk 2009a)

γe,max =qersB2

mec2≈ 7.1×107γ2γ

−1sh kiB−7α−8ε

1/2

B,−1. (21)

Hence, if a plasma is passing through the front of ashock-wave, the power spectrum of relativistic electrons be-hind the front can be described by Eqs. (10) and (16), mini-mal energy (17) and maximal energy (20) or (21). This radi-ation is one of the main manifestations of superconductingcosmic strings in the IGM.

Fig. 1 Expected spectral energy flux of synchrotron radiationfrom a loop of a cosmic string in the intergalactic medium at theaverage distance with different tensions (for n1 = 10−7 cm−3,B = 10−7 G, γsh = 2, εe = 0.1, εB = 0.1).

Finally, the spectral emissivity of electrons with a powerlaw energy distribution can be written as

j(ν) =

∫Ee

J(ν, Ee)N(Ee) dEe

= A

γe,max∫γe,min

(ν

γ2e

)1/3

e−b ν

γe2 γ−p

e e−

γe

γe,max dγe,

(22)

where A = CK ′√2 e3B b1/3/(mec

2)2.Let us now consider which spectral flux densities (flux

per unit frequency) can be observed from superconductingcosmic strings located in the intergalactic medium. They areeasily expressed by the spectral emissivity of the electronsj(ν), the emission region Vem and by the average string dis-tance ds:

Fν =Vemj(ν)

4πd2s

. (23)

The emission region can be expressed by its radius andlength:

Vem =3

2πr2

s l ≈ 2×10−4γ−2sh k2

i B2−7 α3

−8 n−1−7 pc3. (24)

5 Fluxes from strings which are located inthe intergalactic medium

In our paper string fluxes were calculated for dif-ferent tensions α, and the results are represented inFig. 1. Also the sensitivity limits of the VLA (VeryLarge Array) and the HST (Hubble Space Telescope)which work in the radio and optical regions, respec-tively, are given. The NRAO VLA Sky Servey con-sists of objects which have energy fluxes νFν in theradio region above 3×10−3 Jy = 4×10−17 erg/(cm2 c)(http://www.vla.nrao.edu). The Hubble Space Telescope

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1054 D.A. Rogozin & L.V. Zadorozhna: Synchrotron radiation of superconducting cosmic strings

Fig. 2 Expected spectral energy flux of synchrotron radiationfrom a loop of a cosmic string in clusters of galaxies which arelocated at different distances from the observer (for α = 10−6,n1 = 10−3(1 + z)3 cm−3, B = 10−6(1 + z)2 G, γsh = 2,εe = 0.1, εB = 0.1).

can observe objects with spectral energy fluxes νFν in Vabove 2.5×10−18 erg/(cm2 c) (http://www.nasa.gov). It isobvious from the figure that cosmic strings with α ≥ 10−6

can be detected with existing telescopes.

6 Fluxes from strings which are located inclusters of galaxies

In a next step let us assume that a superconducting cosmicstring with α = 10−6 is located in a cluster of galaxies witha typical magnetic field of B = 10−6(1 + z)2 G and densi-ties of electrons and protons of n1 = 10−3(1 + z)3 cm−3,where z is the redshift. We computed spectral fluxes forclusters with the following distances from the observer: 20,50, 150, 500, and 4000 Mpc. The distance is connected toredshift z via d = 3t0c(1 + z)1/2

[(1 + z)1/2 − 1

]. The

others parameters were the same: γsh = 2, εe = 0.1 andεB = 0.1.

Our results are shown in the Fig. 2 together with thesensitivities of VLA and HST. One can see that also in thiscase superconducting cosmic strings in the nearest clusterscan be observed by current telescopes.

7 Conclusions

In our work we considered the motion of a superconduct-ing cosmic string with an electric current through the inter-galactic medium and the process of interaction of its mag-netosphere with an ambient intergalactic plasma. Also theformation of a relativistic shock wave around the string andthe acceleration mechanism of electrons by the shock wereexamined.

One new result is the computation of synchrotron radia-tion fluxes from the loops of superconducting cosmic stringsusing an appropriate approximation for the spectral emissiv-ity of one electron and the assumption of an exponential cut-off of the energy distribution of electrons at high energies.Strings with different tensions which are located in the inter-galactic medium were considered. Also, synchrotron radia-tion fluxes from strings located in clusters of galaxies wereobtained.

The possibility to detect fluxes from superconductingcosmic strings by existing facilities of terrestrial and orbitaltelescopes was estimated, too. It was shown that strings withthe largest tensions and which are located in the intergalac-tic medium are well suited for observations with currenttelescopes. On the other hand, strings which are located inclusters of galaxies can be detected with the Hubble SpaceTelescope or the VLA only for the nearest clusters.

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