Radiation of Nambu-Goldstone bosons from infinitely long cosmic strings

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  • PHYSICAL REVIEW D VOLUME 44, NUMBER 12 15 DECEMBER 1991

    Radiation of Nambu-Goldstone bosons from infinitely long cosmic strings

    Maria Sakellariadou Research Group in General Relativity, Universit; Libre de Bruxelles, Campus Plaine, CP231, 10.70 Brussels, Belgium

    (Received 4 April 1991

    Nambu-Goldstone-boson radiation from an infinitely long string with a helicoidal standing wave is studied in the weak-field approximation. The radiation power and the spectrum are calculated. In the case of a single-small-amplitude helicoidal standing wave, the radiated power goes as el0, indicating that the Nambu-Goldstone-boson radiation mechanism is efficient only for large-amplitude waves. A simple analysis of the radiation due to a wide spectrum of small-amplitude waves indicate that in general the emitted power goes as c4.

    I. INTRODUCTION

    According to the current understanding of high-energy physics, the underlying field theory describing nature has a high degree of symmetry, most of which is broken at low energy. Consequently, it is theoretically expected [I] that during the evolution of the Universe, as it cooled through its expansion, there were phase transitions asso- ciated with the spontaneous breakdown of gauge and glo- bal symmetries, leaving behind relics of the old sym- metric phase. Among the macroscopic topological de- fects formed in this way, both the two-dimensional (domain walls) and pointlike (magnetic monopoles) ones are inherently cosmologically dangerous [2]. This is not so for the linelike ones, known as cosmic strings. These vortex lines are the most interesting topological defects, not only because their existence is compatible with the observed Universe, but also because they can generate density fluctuations sufficient to explain galaxy formation [3] and can lead to a number of profound and distinctive observational effects [4].

    Particle-physics models containing a symmetry group G, which is spontaneously broken down to a subgroup H such that the first homotopy group n,( G / H ) is nontrivi- al, predict gauge or global string formation, depending on whether G is a gauge or a global symmetry group. The condensed-matter analogues of gauge and global strings are quantized tubes of magnetic flux in superconductors and vortex lines in liquid helium, respectively.

    At formation, the main part of the energy density of a string network is in very long (longer than the horizon) Brownian strings and only a small part is in a scale- invariant distribution of closed-loop trajectories. In the course of the universal expansion, strings intercommute and self-intersect, chopping off closed loops. The main energy-loss mechanism for oscillating closed-loop trajec- tories of gauge strings is the emission of gravitational ra- diation, while for global strings it is the emission of Nambu-Goldstone bosons. The rate of this radiation determines the loop's lifetime and has been calculated for both gauge [5] and global strings [6 ] . However, a straight gauge or global string does not radiate energy and, there- fore, does not dissipate with time, even when modulated by traveling waves [8].

    High-resolution numerical simulations [9] of gauge

    string networks verify the validity of Kibble's one-scale- model description [lo] only for the large-scale behavior of the network and not for the revealed small-scale struc- ture on the long strings. As string segments intercom- mute forming closed loops, discontinuities in their veloci- ty and direction (kinks) are developed, which result in the presence of wiggles on the long strings. These wiggles, in the form of closely spaced kinks, are excitations with wavelengths much smaller than the network correlation length and contribute nearly half to the total energy of the string network. Moreover, the small-scale structure determines the size of the produced stable loops, resulting in a reduction of the expected gravity-wave amplitude and suggesting that wakes formed behind rapidly moving segments of long strings will be the ones to play the dom- inant role in structure formation via string-producing gravitational perturbations.

    One might expect that the kink density would decrease with time, by either universal expansion or the tendency of the more wiggly pieces of string to chop off the net- work. Both those mechanisms have been checked per- forming numerical simulations of gauge string networks [ l l ] and have been found to be ineffective in smoothing out kinks. Gravitational and Nambu-Goldstone-boson radiation back reaction is the third possible smoothing- out mechanism for local gauge and global cosmic strings, respectively. The analytical calculation, in the weak- field approximation, of the gravitational radiation emit- ted from a helicoidal infinitely long gauge string has been presented in Ref. [12] and extended in Ref. [13] for arbi- trary (but small-amplitude) waves.

    In this paper the analytical calculation of the emitted power of Nambu-Goldstone bosons by an infinitely long global string with a single-amplitude helicoidal standing wave and its generalization in the case of a wide spectrum of small-amplitude waves will be studied in the weak-field approximation. Throughout the paper the system of units in which f i = c = l is used, and ( - , $ . , + , + I is chosen as the signature for the metric.

    11. PRELIMINARIES

    The Goldstone model of a self-interacting complex sca- lar field 4 provides the simplest theory giving rise to glo- bal strings. The Lagrangian

    3767 01991 The American Physical Society

  • 3768 MARIA SAKELLARIADOU 44

    has a global U(1) symmetry, which is spontaneously bro- ken by the vacuum expectation value 141 =v. The magni- tude of 4 is substantially different from 7 only within the string core defined by its radius 6:

    Outside the string core, 4 is effectively given in terms of a single degree of freedom 0 ( x ) as

    $-veie '~) , (3)

    where 0 ( x ) is the massless Nambu-Goldstone field associ- ated with the spontaneous breaking of the global U(1) symmetry. In contrast with the case of gauge strings, where the energy per unit length of a straight string is all localized within the string core, for global strings p is logarithmically divergent:

    where R is a large-distance cutoff. For cosmic global strings, R is of the order of the distance between strings, which is typically of the order of the horizon.

    In the low-energy limit, when the string curvature ra- dius is much greater than its core radius, the interaction of global strings with Nambu-Goldstone bosons can be described [6,7] by a model of strings interacting with an antisymmetric tensor field A,, with a particular choice of the coupling constant g, namely, g =2~77. A con- venient choice of gauge condition is [14]

    where f P ( a , 7 ) stands for the string world sheet; overdots

    and primes denote derivatives with respect to T and a , re- spectively. Consequently, the string trajectory can be de- scribed by the vector function f ( a , t ) , and the field equa- tions and string equations of motion read [6]

    where po is the bare mass density of the string and FpVp is the tensor field defined by

    To calculate the radiation from oscillating global strings, it will be assumed [6] that in Eq. (7) noninteracting loop trajectories may be used. I t should be mentioned that doubts have been raised on the consistency of such an as- sumption, on the basis of computer simulations [15]. However, arguments in favor of its validity have also been given, based both on numerical simulations [16] and on analytical studies [17] of global strings.

    The spacetime trajectory of an infinitely long string having the form of a helix along the z axis can be written as

    where f is a modulation orthogonal to the z axis. Follow- ing the same analysis as in Ref. [ I 11, the full string trajec- tory reads

    where R is the breathing frequency of the helix and the parameter E (0 I E < 1) determines the winding number per unit length.

    Because of the symmetry of the problem, it is convenient to work in cylindrical coordinates. Thus vectors are broken into a component orthogonal to the z axis and a component along z, i.e., x = ( p cos0, p sine, z ) ~ p p ^ + z ' i and

    For this particular helicoidal string trajectory, j,,(x,t) turns out to be periodic along the z axis, with period 27r( 1 - E ~ ) ' / ~ / R , and periodic in time, with period 27r/R.

    I t is not difficult to show [ l l ] that for large p the field A,, is a superposition of plane waves, namely, and q is defined by

    ~ ( ~ 2 - ~ 2 ) 1 / 2 (14)

    Clearly, K is the component of Q along the z axis. Note that when the source has periodicity in z, as in the case examined here, the Fourier component of j,, appearing in Eq. ( 13) is

    where the "wave vector" Q and the "polarization tensor" j,,=-J 1 dxj , , (x ,w)exp(- iQ-x) , h o < z S h (15)

    e,, are explicitly given by

  • ?!? RADIATION OF NAMBU-GOLDSTONE BOSONS FROM . . . 3769

    where A is the period along z. The present symmetry suggests to consider the power radiated through a cylin- drical surface centered on the source and having a radius much larger than the source's orthogonal size. The power radiated through such a surface in the angle be- tween 8 and 8 +do, and z and z + dz, from a single- frequency source, is

    Hence the power of Goldstone-boson radiation emitted per unit time from an infinitely long string through a sur- face of unit length, centered on the string and having ra- dius p much larger than the source's orthogonal size, reads

    Using Eqs. (12)-(141, the above equation simplifies to

    111. GOLDSTONE-BOSON RADIATION FROM AN INFINITELY LONG GLOBAL STRING

    WITH A HELICOIDAL STANDING WAVE

    In this section the Nambu-Goldstone radiation for the string trajectory specified by Eq. (10) will be calculated in the weak-field approximation. First, the components of j , , ( x , t ) will be calculated from Eqs. (7) and (10) .

    Defining two new parameters,

    allows for writing the final formulas for the components of j,, in concise form. After some algebra the obtained expressions for the components of j , ,(x, t ) read

    --'A1 j 1 3 = - j 3 1 = -- cosy sin( a t ) , 2P

    j z3 = - j32 =- -''A' sin^ sin( a t ) , 2P

    Next, the Fourier transform of j , , ( x , t ) will be calculated in two steps. First, j , , (x ,w) will be computed and then j , , ( q , ~ , w ) . Because of the periodicity of j,, in both time and z variables, one can denote the discrete values of w and K by wn and K, , with w , = n R and K , = m a / ( 1 - e2 Clearly,

    with

    Again, it is convenient to define a couple of intermediate parameters:

    to present the outcome of the integral in Eq. (23) . With these definitions the components j , , ( x , o , ) read

    The spatial Fourier transform j , , ( q , ~ , , a , ) is calculated from

  • 3770 MARIA SAKELLARIADOU 44

    Using Eq. (26), with and p being the polar angles of the vectors p and q, respectively. After some lengthy manip- ulations, relatively simple expressions are obtained in terms of Bessel functions. These are best presented intro- ducing a few definitions:

    Hence the components of j,,( q, K, , W, ) read

    I

    jI2= -jZ1 = ( ~ E ~ E / ~ ) ( J ~ - ~ J ~ + ~ - J ~ + ~ J ~ - ~ ) ,

    ~ 1 3 = - ~ 3 1

    = - [ E ( ~ - E ~ ) ~ / ~ E / ~ ] [ ~ ~ ~ ( J ~ - ~ J ~ - J ~ J ~ - ~ )

    +e-'p(Jl+lJ, -JIJ,+,)] ,

    j z2=0 ,

    ~ 2 3 = - ~ 3 2

    = [ ~ E ( ~ - E ~ ) ~ / ~ E / ~ ] [ ~ ' ~ ( J ~ - ~ J , - J ~ J ~ - ~ ) -

    -e 'p(Jl+lJ,-JIJ,+,)l,

    j 3 3 = 0 ,

    where the argument of the Bessel functions is

    p - ( ~ / 2 ) [ n ~ - r n ~ / ( l - ~ ~ ) ] ~ / ~ . (30)

    Using the Fourier components of j,, given in the above equations and the relation J- , , (z)=( - 1 )"J,(z), the emit- ted Nambu-Goldstone radiation can easily be obtained from Eq. (18). In particular, the power emitted within a cylinder of unit length, whose axis coincides with the string axis, is

    where the prime in the sum over rn further restricts rn to those values for which rn +n is even.

    I t is easy to show analytically that for m = O the radiat- ed power given by the above equation vanishes identically for any value of the parameter E. Thus, in the summation over m, the lowest value of rn is 1, while in the summa- tion over n, the lowest value of n is 3.

    IV. RESULTS AND DISCUSSION

    A careful inspection of Eq. (31) reveals that the power radiated by the considered infinitely long global string with a helicoidal standing wave diverges as E--t 1. In fact, the divergence arises from the factor 1 --c2 appearing in the denominator of the expression for the total power. The reason for this is simply understood by realizing that in such a limit the amount (length) of string per unit dis- tance along z goes also to infinity. This can be immedi- ately seen by writing the energy per unit length ( E ) along the z axis. From Eq. (10) one gets Az =( 1 - E ~ ) ' / ~ A E /p , where AE =PACT. Hence

    Clearly, the winding number of the helix R/[27r( 1 - E ~ ) ' / ~ ] defined as the number of revolutions per unit length along its axis, is also diverging in the limit of E-1. On the other hand, for E = O the emitted radia- tion vanishes, since in such a case the trajectory reduces to that of a static infinitely long straight string.

    One can numerically evaluate the radiated power as a function of E . One finds that P (E ) has a strong depen- dence on E, both for E around 1, where it diverges, and for small E, where vanishes very quickly. This is best seen from Fig. 1, where P ( E ) / E " is plotted against E. To ex- hibit even more clearly the behavior of the radiated power, one finds convenient to introduce a reduced power through

    Clearly, as may be appreciated from Fig. 2, is a smooth function of E, which vanishes as E = O and remains other- wise of order 1.

    One can also analyze for given E the frequency spec- trum of the Nambu-Goldstone-boson radiation. In Fig. 3 such a spectrum is reported for (a) ~ = 0 . 9 9 and (b)

  • RADIATION O F NAMBU-GOLDSTONE BOSONS F R O M . . . 3771

    FIG. 1. Power P radiated by an infinitely long global string with a helicoidal standing wave, in the weak-field approxima- tion, vs E. E is the parameter determining the winding number of the helix, and P i s in units Q ? 7 2 ~ ' 0 .

    ~ = 0 . 9 . Of course, at large frequency the radiated power decreases with increasing frequency. I t is also apparent that the contribution from even frequencies becomes less and less important as E approaches 1. In fact, for the considered global string with a helicoidal standing wave, the lowest radiating even frequency is bigger than 2 R / ( 1 - E ~ ) " ~ , which diverges in the limit E - 1. On the other hand, the lowest odd frequency which appears in the spectrum of the emitted power is 3R. Figure 4 shows the ratio between the power emitted at 3 0 and the total power as a function of E . Clearly, for small E , essentially all the power is emitted at the l_owest allowed frequency. In such a case ( E

  • 3772 MARIA SAKELLARIADOU - 44

    [Eqs. (33) and (3411 with the oscillating energy

    The characteristic damping time of the oscillations can be estimated from

    In the small+ limit, this gives

    For cosmic global strings, R is of the order of the dis- tance between strings, which is typically of order of the horizon size t. For example, for t -- 10" yr and T-- 1014 GeV, ln(R /S) -- In( t ~ ) = 130; for axion strings at QCD time, t = 1 0 - ~ sec and ~ - - 1 0 ' ~ GeV, leading to ln(tr])=70. Consequently, ln(R /6 )= lo2, and Eq. (37) leads to

    For E = 1, Eq. (38) should also give the right order of magnitude. One concludes from Eq. (38) that Nambu- Goldstone-boson radiation is rather efficient for large- amplitude waves ( - I ) , but becomes less and less efficient as E decreases.

    V. RADIATION FROM A STRING WITH A WIDE SPECTRUM OF SMALL-AMPLITUDE WAVES

    In this section the radiation of Nambu-Goldstone bo- sons from infinitely long global strings with a superposi- tion of wiggles scanning a wide range of small-amplitude

    helicoidal waves will be studied. The string trajectories are given by

    f ( u , t ) = ( l - ~ ~ ) ' / ~ u $

    where AN and BN satisfy the conditions

    and

    Note that in what follows the Cartesian coordinate sys- tem will be used instead of the cylindrical one chosen above. To be specific, one can choose Q = ~ % + K , $ , where K, = m a/( 1 - E' )I/'. One can easily calculate the components of the Fourier transform jp,(q,~,,w, ) in this case. In particular,

    where on = n f l (n fO) and

    with 6 [6--27~( 1 - E ~ ) ~ ' ~ / o ] and T ( 7 = 2 v / f l ) denoting the period along the z axis and time, respectively. Analyzing Eqs. (18), (39), and (42) and keeping in mind that the helicoidal waves have been taken to have small amplitudes ( 0 5 E < 1 ), one can argue that only few terms of the infinitely many ones appearing in Eq. (41) will need to be retained. So, after some algebra, one can show that the nonzero components of j,,(q,O,~, , w, are

  • 44 RADIATION O F NAMBU-GOLDSTONE BOSONS FROM . . . 3773

    Consequently, from Eqs. (18), (42), and (43), the Nambu-Goldstone-boson radiation emitted within a cylinder of unit length, whose axis coincides with the axis of a string with a wide spectrum of small-amplitude waves, reads

    where the prime in the sum over m further restricts m to those values for which m + n is even.

    Note that making the substitutions

    and

    A - N , , =O, A - N , , =O, BN,, =O, BN,, = O

    for N f 1 ,

    one recovers the trajectory for the single-amplitude hel- icoidal infinitely long string [Eq. (lo)], and as can be easi- ly checked, in this case Eq. (44) correctly gives zero power to order e4.

    Equations (34) and (44) lead to the conclusion that in the s m a l l - ~ limit, the radiated power from a global string

    with single-amplitude helicoidal waves is smaller by c6 than the power radiated from a string having a generic wide spectrum of small-amplitude wiggles. In this respect one may note that the form of the power radiated by a long string strongly depends on the specific com- bination of waves on the string. In fact, a single traveling wave does not produce any radiation at all (see, e.g., Ref. [8]), whereas a helicoidal wave does give a nontrivial spectrum. Thus the fact that s m a l l - ~ behavior of the ra- diated power depends strongly on the choice of waves on the string should not come as a surprise.

    ACKNOWLEDGMENTS

    It is a pleasure to thank both the R G G R Group and Gaetano Senatore for support and inspiring discussions throughout this study.

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