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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 sec