PHYSICAL REVIEW D VOLUME 36, NUMBER 12 15 DECEMBER 1987

Coupled field solutions for U(1)-gauge cosmic strings

Pablo Laguna-Castillo and Richard A. Matzner Center for Relativity, The University of Texas at Austin, Austin, Texas 78712

(Received 6 July 1987)

Numerical solutions for the coupled Einstein-scalar-gauge field equations of a static, infinitely long, straight, U(1)-gauge cosmic string are obtained. The boundary conditions at the axis of the string and at radial infinity are chosen such that the solutions represent an isolated string; that is, the metric fields of the string approach Minkowski spacetime minus a wedge at radial infinity. The numerical solutions for the metric, scalar, and gauge fields, and for the components of the energy-momentum tensor, are classified according to the energy scale of symmetry breaking and the mass ratio of the scalar and gauge fields of the strings.

The relatively new cosmic-string scenario has provid- ed a hope of finding a satisfactory model for galaxy for- mation.' Recent calculations2 of correlation functions, density fluctuations, and microwave background pertur- bations indicate that these topological defects, remnants of phase transitions caused by spontaneously broken s y m m e t r y , ~ o l d the promise of great progress along this path. In the cosmic-string scenario, string loops consti- tute one of the main cornerstones because they may play the role of seeds that begin the accretion of matter to form galaxies.4 Although much progress has been made in understanding the evolution of a string network and its comparison with the predicted distribution of galaxies and clusters with observation^,^ the cosmic-string theory of galaxy formation is still young and is based on a con- siderable number of untested assumptions that evidently require more careful attention.

Numerical simulations6 show that, in the early Universe, typically after a phase transition with spon- taneously broken symmetry, a network of approximately 80% infinitely long strings is produced, with the remain- ing 20% consisting of string loops. Even though string loops and infinitely long strings are both capable of ac- creting matter,' string loops are more attractive because, while matter is accreting onto the loops, the loops them- selves oscillate violently leading to the loop decay by gravitational r a d i a t i ~ n , ~ avoiding conflict with the ob- served cosmological density.' It is then of crucial im- portance in order for the cosmic-string scenario to work, that strings are able to exchange partners so string loops can break away from the main network, and so that eventually the Universe arrives at a state in which the dominant species is string loops, with perhaps a few infinitely long strings remaining.

One of the most important and until recently untested assumption in the cosmic-string scenario for galaxy for- mation is the interaction properties of cosmic strings. The process of intercommuting strings is of vital impor- tance because if strings are unable to exchange partners, the number of string loops (which eventually form seeds for galaxy formation) would not be enough to have cosmological relevance. So far, there have been only a

few serious studies that analyze in detail the interaction properties of cosmic strings. The pioneering work in the process of intercommuting cosmic strings was done by ~hellard." His numerical simulations show that when two global strings cross they reconnect the other way under most circumstances. However, these types of string, that arise as a result of a global symmetry break- ing and do not carry gauge magnetic flux, are not suit- able in the classical analysis of string network evolution because the presence of a massless Goldstone responsible for the existence of long-range forces between different parts of the string and for rapid string decay.'' Recent- ly, ~ a t z n e r " has developed a numerical analysis of local cosmic-string interactions, i.e., strings treated as a self- interacting scalar field minimally coupled to a U(1) gauge field. He has found that local cosmic strings also exchange partners under most circumstances, and only for relative velocities close to the speed of light do the strings pass through one another; however, his numeri- cal simulations ignored gravity, which is the dominant long-range interaction of those strings.

Another aspect of the cosmic-string scenario which constitutes the main focus of attention in this paper and, despite being intensively investigated, still requires a more complete analysis is that of the gravitational field of a cosmic string. From dimensional arguments, the thickness of a cosmic string is 6=r]-' , where r] defines the energy scale of symmetry breaking. Typically, for grand unification scales r ] = 1016 GeV, leading to an ap- proximate thickness for the strings of 6 = cm. In most of the processes involving cosmic strings, it is thus "natural" to neglect their thickness; however, the analysis of intercommuting strings is one of the examples in which the diameter of the strings cannot be neglected because of its local nature. The interaction of cosmic strings is then one of the processes that demonstrates the importance of investigating the properties of cosmic strings as objects with "finite" thickness.

Unfortunately, the majority of the studies of the gravi- tational effects of straight cosmic strings, where their thickness is not neglected, lack consistency. This was first pointed out by ~ a r f i n k l e ' ~ ; he recalls that a local

3663 @ 1987 The American Physical Society

3664 PABLO LAGUNA-CASTILLO A N D RICHARD A. MATZNER 36

cosmic string is, after all, a configuration of scalar and gauge fields, and thus it is not enough to assume an energy-momentum tensor, e.g., energy density equals tension along the axis, and solve for the fields. Con- sistency can only be achieved if the coupled Einstein- scalar-gauge field equations are solved. Recently, an at- tempt along this direction was made when we intro- duced a model of a cosmic string which provides analyti- cal solutions for the gravitational, scalar, and gauge fields consistent with the coupled field equations.I4 However, the applicability of our model is strongly con- strained by the assumptions introduced in order to allow the existence of analytical solutions.

The purpose of this paper is then to provide solutions of the coupled Einstein-scalar-gauge field equations for a static, infinitely long, straight, U(1)-gauge cosmic string with no assumptions other than that the solutions exhib- it boundary conditions which represent an isolated string. We begin reviewing the full coupled Einstein- scalar-gauge field equations for a static, cylindrically symmetric, U(1)-gauge cosmic string. We then discuss the set of boundary conditions for the gauge, scalar, and gravitational string fields as well as the numerical method that was used to solve the coupled system of equations. It is found that the class of solutions which represent an isolated string, that is, solutions in which the metric fields approach at radial infinity Minkowski spacetime minus a wedge, can be classified according to the value of 7 (the energy scale of symmetry breaking) and the mass ratio of the gauge and scalar fields. We show that for energy scales 7 < 10" GeV, in agreement with what has been already predicted, the metric fields exhibit no significant differences from Minkowski space- time minus a wedge not only at radial infinity but also near the core of the string. Since we are interested in analyzing cases in which the gravitational field inside and near the string is not negligible, our numerical study of the scalar, gauge, and gravitational fields includes en- ergy scales 77 > 10" GeV. We also present a detailed description for the behavior of the energy-momentum tensor c o m ~ o n e n t s in terms of the only free ~ a r a m e t e r s in the theory, the symmetry-breaking scale, and the ratio of the scalar to gauge field masses. Furthermore, we ob- tain that the quantity ,u/.rrr12, with p the linear energy density of the string, depends only on the ratio of the scalar and gauge masses. Finally, we show that the ap- proximation he= 8?rp for the deficit angle of the string at radial infinity is correct within 0.1% for energy scales 7 < 10'' GeV.

The model of a cosmic string considered in this paper is that of a single complex scalar field 4 minimally cou- pled to a U(1)-gauge field A,, with the complex scalar field interacting with itself through the standard "Mexi- can sombrero" potential. In terms of these fields the to- tal Lagrangian reads

where D, EV, +ieA,, Fa, =V, A, -Vb A,, V( d )

= h( d 1 2-72)2/8 , and R is the Ricci scalar. (In our

units f i=c = G = 1.) Varying the fields d , A,, and gab independently, one obtains

DaDad-2aV/a4* = O , (2a)

-ie [d (D ,d )* - d * ( D a d ) ] / 2 = 0 , (2b)

and

Gab = grTab , (2c)

where the energy-momentum tensor is given as

Tab= [ ( D a d ) * D b 4 + ( D b d ) * D a d ] / 2

+ F a c F C b / 2 + L ~ g a b

with LM the matter Lagrangian

L , = - (V,d+ieA,d)(Vad+ieAad)*/2

-h ( d 2 - r 1 2 ) 2 / 8 - ~ a b ~ a b / 4 . (3b)

One sees that d has a low-energy true vacuum expec- tation value 1 ( d ),, / =v. Defining d l and d2 by q5=&l+id2, one can make use of the gauge freedom to arrange that d 2 = 0 , and thus one can introduce a new field 4; =4 , -7 , where I ( d l )o 1 = 7 . Substitution of these expressions into the Lagrangian (3b) leads to the physical interpretation of an ordinary massive vector meson A , , and a massive neutral scalar field d', wlth masses given by rn *'= hV2 and rn A = ( e j2, respective-

ly. Equations (2a)-(2c) represent the coupled Einstein-

scalar-gauge field equations; however, in order to extract from those equations solutions which describe a U(1)- gauge cosmic string, further assumptions needs to be im- posed on the scalar, gauge, and metric fields. Since we are interested in dealing with an infinitely long, static, and straight cosmic string, it will be assumed that in the usual cylindrical coordinates ( t , z , p , e ) the scalar, gauge, and gravitational fields take the form

and

respectively, where the field functions ( A , B, C, Q, and P) depend only on the radial coordinate p . Equation (4a) is the assumption that the winding number for the string here considered is unity.

For the assumed symmetry, the spacetime in the pres- ence of a string has three commuting Killing fields: a timelike Killing field ( a /&) ' normalized to - 1, a space- like Killing field (a /az)" normalized to + 1, and another spacelike Killing field (a/aeIa with closed orbits normal- ized such that, along a closed integral curve, the param- eter 8 varies from 0 to 2 7 ~ with 8 = 0 and 8 = 2 ~ identified. This choice for the normalization of the Kil- ling fields provides boundary conditions for the metric fields: A ( 0 ) = B ( 0 = 0 along the axis, and exp( C) -p2

36 - COUPLED FIELD SOLUTIONS FC )R U(1)-GAUGE COSMIC STRINGS 3665

as p-0. Furthermore, it can be shownI3 that for the given symmetry and the above boundary conditions the metric fields A and B are equal everywhere; i.e., the string possesses explicit Lorentz invariance along its axis.

By construction, the string has a core of false vacuum and a tube of magnetic field due to the form of the effective potential and the presence of the gauge field, re- spectively. Their radii are determined by the Compton wavelength of the scalar and gauge fields. The radius of the false-vacuum core is Sd= md- ' = ( hT2 ) - 'I2, and the radius of the magnetic field tube is 6, =m, - I = (erl)- ' .

For convenience, let us introduce the definitions

and

That is, with the chosen normalization of the radial coordinates given by Eq. (5c), the radii of the core false vacuum and magnetic field tube are at r d = 1 and r, respectively.

We shall later need the components of the energy- momentum tensor since they will play a significant role in the interpretation of the physical properties of the string. The energy-momentum tensor (3a) can be rewrit- ten as

where

and p^"=(a/ap)" are a set of orthonormal vector fields; the components of the energy-momentum tensor in terms of the definitions (5a)-(5d) read

and

where ( ) ' = d /dr and a = -P,. That is, the energy den- sity u of the string equals the magnitude of the negative pressure P, (tension) along its axis. This is an expression of the Lorentz invariance of the string along its axis; however, contrary to what is usually assumed in the study of the gravitational properties of a cosmic string,

strictly speaking a cosmic string presents also radial P p and angular Po components in the energy-momentum tensor.

In the spatial components of the stress-energy tensor (P,, P,, and Po ), the term (x2- 1 12/8 in Eqs. (7a)-(7c) arises from the effective potential and, as usual, contrib- utes with a negative pressure (tension). The term e 2 A ( ~ ' ) 2 / 2 a ~ 2 is purely due to the presence of the gauge field, which by assumption has only 0 component and p dependence. Thus, there is a "magnetic" field of the string which runs along its axis, and in analogy with electromagnetism, the gauge term's negative sign (ten- sion) in Eq. (7a) for the P, = -a component is just a consequence of the work that one would need to provide in order to stretch the lines of magnetic field. On the other hand, in Eq. (7b) for the Pp component, the same gauge term, e 2 A ( ~ ' ) 2 / 2 a ~ 2 , exhibits now a positive sign (pressure), meaning that work is required to squeeze the lines of the magnetic field toward the axis in the radial direction. Positive sign in the terms on Eq. (7c) for ihe Po component indicates pressure in the direction of 8 a,

and negative sign means tension in the same direction. The remaining terms in the spatial components of the

energy-momentum tensor can be interpreted similarly to the pure-gauge term. The coupling term ( e , x P / K ) ~ / ~ enters as tension in the axial and radial directions but as pressure in the angular direction. In contrast, the pure scalar term ( ~ ' ) ~ / 2 contributes as tension in the axial and angular directions but as pressure in the radial direction.

In terms of the definitions (5a)-(5d), substitution of the expressions (4a)-(4c) for the fields into the coupled field equations (2a)-(2c) leads" to

and

Equations (8a) and (8b) correspond to the scalar and gauge field equations (2a) and (2b), respectively. Equa- tions (8c) and (8d) arise from the two independent com- ponents of Einstein equations for the symmetry imposed on the string. Equations (8a)-(8d) represent thus the full system of coupled metric-scalar-gauge field equations for a static, infinitely long, straight U(1)-gauge cosmic string and form the basis of our numerical analysis.

As discussed before, the normalization of the Killing fields provides the boundary conditions for the metric fields at the axis:

and

3666 PABLO LAGUNA-CASTILLO A N D RICHARD A. MATZNER - 36

The last two conditions guarantee that near the axis of the string the spacetime metric is smooth and the nor- malization of the Killing field (a/aO)u is preserved.

We still require a fourth boundary condition for the gravitational second-order differential equations (8c) and (8d). As we shall later show, near the axis of the string the scalar field X and the derivative of the gauge field P are linear in r. Thus Eq. (8c) can be rewritten near the axis as

where we have used that K -+re *. Since in this equation the term on the right-hand side inside the square brack- ets is constant, Eq. (10) is a Bessel or modified Bessel equation for the field e *; in order to have regular solu- tions at the axis, A ' must vanish. The boundary condi- tions to be imposed upon the gravitational fields are then conditions (9) plus A1=O. They force the spacetime near the axis of the string to be nearly flat ( e A = l , ec12=r) . On the other hand, ~ a r f i n k l e ' ~ has shown that there exists a class of solutions which, at radial infinity, approaches Minkowski spacetime minus a wedge; that is, as r --t m , e ", and e '12/r must reach constant values.

Knowing that the spacetime is almost flat near the axis and Minkowski minus a wedge at radial infinity, it is then perfectly safe to assume that the boundary condi- tions for the scalar and gauge fields can be obtained from the corresponding asymptotic solutions for those fields in a flat background. In a flat background, i.e., K = r and e "= 1, one has from Eqs. (8a) and (8b) that, as r --to,

X = a r , (1 la )

and, as r -+ w ,

with a,b,c,d positive constants. Thus, the boundary conditions for the scalar and gauge fields are given at the axis of the string by X(O)=O, P(O)= 1 and at radial infinity by X ( m ) = l , P ( C O )=O.

The numerical technique employed to solve the system of nonlinear ordinary differential equations (8a)-(8d) used a two-points boundary-relaxation routine for the scalar and gauge field equations (8a) and (8b), and a Runge-Kutta routine for the gravitational field equations (8c) and (8d), with those routines exchanging data after each iteration.

One sees from the scaled coupled equations (8a)-(8d) that the solutions for the fields can be classified accord- ing to the only two free parameters of the theory, the en- ergy scale of the symmetry breaking and the gauge to scalar mass ratio ( m , / m n ) 2 = e 2 / h = a . Thus, in the sets of numerical solutions that were obtained, in one case the value of the symmetry-breaking parameter was kept fixed, and in the other case the ratio of the field masses remained constant.

Figures l (a ) - ( le ) show the solutions for the scalar and

gauge fields, as well as the components of the energy- momentum tensor, when the scale of symmetry breaking is v= in units where the Planck mass is the unit mass. For this value of 7 , five solutions were obtained corresponding to mass ratios ( m A /m , 1 2 = f, t, 1, 2, and 4. Figures ](a) and l (b) reproduce the typical behavior for the scalar and gauge fields as functions of the physi- cal radius exp( C / 2 ) in the coordinates ( t , z , r , 6 ). These figures also show that the scalar and gauge fields more rapidly approach their respective asymptotic values as the gauge field becomes more massive than the scalar field. That is, as ( m , /md )2 grows, the gradients of the scalar and gauge fields increase.

It was found that at this energy scale ( v = l o p 2 ) , the departures of the metric fields from Minkowski space- time are not very significant, at the most 0.03% in the exp( A ) metric field. Furthermore, within our range of tolerance, the metric field e x p ( C / 2 ) can be approximat- ed at radial infinity as exp(C/2)- -sp , with s a positive constant that not only depends on the energy scale 77 but also on the ratio im , /m,) [see discussion after Eq. (1411.

In Fig. l (c) one sees that at the core of the string, the magnitude of the energy density v , and therefore also the magnitude of the spatial component P,, increases with the ratio ( m , /m,l2. One can easily understand that the magnitude of these quantities increases as the gauge field becomes more massive if one recalls, from Figs. l ( a ) and l(b) , that for this case the gradients of the scalar and gauge fields, which enter in the components of the energy-momentum tensor, are larger for larger ( m , /m, 12. Notice also that the magnitudes of the energy-momentum components v and P, are in all cases more than seven times larger than the corresponding values for the remaining spatial components P p and Po . One finds here the justification for neglecting such spa- tial components and thus treating the string stress- energy tensor with only energy density equal tension along its axis; as we mentioned before, this approxima- tion cannot hold in processes such as intercommuting strings since the components Pp and P o play also an im- portant role in the interaction of the strings.

From Fig. lid) one notices that P p < 0 (tension) in the case in which rn, < m, since in this situation the dom- inant contributions to P, come from the coupling term t ( e ,XP / K ) ~ and the effective potential (x2 - 1 )2/8, and both enter with a negative sign in Eq. (7b) for P,. We have mentioned before that the gradients of the scalar and gauge fields increase considerably with m, /m [see Figs. l (a ) and l(b)]. Thus, in the opposite case when m, > m n , the terms in P,, that involve those gradients are now dominant; leading in this situation to Po >O (pressure).

Finally, one sees in Fig. l(e) that the most interesting behavior arises in the angular component P , of the stress-energy tensor. As in the case of the P , com- ponent, for m, < m, the most important c o n t r i b h o n at the core of the string arises from the false vacuum in the effective potential i x 2 - 1 12/8. Since the effective poten- tial enters into the spatial components of the energy-

COUPLED FIELD SOLUTIONS FOR U(1)-GAUGE COSMIC STRINGS

P H Y S I C A L R R O I U S

PHYSIC91. RRCIbS P H Y S I C R L R R D I U S

0 0.2 0.4 0.6 0.8 1.0 1 .2 1 .4 1 . 6 1 .8 2 . 0 2 .2 2 .4 2 . 6 2.8 P H Y S I C A L RROIUS

FIG. 1. Solutions for the (a) scalar and (b) gauge fields of a U(1) cosmic string and its (c) a , (dl P,, and (e) P o stress-energy- tensor components, as functions of the physical radius exp(C/2), for a symmetry-breaking scale of v= lo-'. The dash patterns (-),(---),(----- ), ), and ( ---- ---- -- ) correspond to the fields mass ratio ( m , /md )2 = a, f, 1, 2, and 4, respectively.

3668 PABLO LAGUNA-CASTILLO AND RICHARD A. MATZNER - 36

momentum tensor with a negative sign, Po near the axis ( r < r d = 1 ) i%negative meaning that there exists a ten- sion in the 0 " direction. Near a region close to the string's surface ( r , the dominant term in the energy-momentum tensor components is the scalar- gauge coupling term + ( e A ~ ~ / ~ ) 2 because that region is where neither the scalar nor the gauge fields vanish. Since this coupling term enters into Eq. (7cLfor Po with a positive sign, it induces a pressure in the 0 " direction. In contrast, for the2ituation m, > m d , there is a posi- tive pressure in the 0 " direction at the core of the string ( r S r , ~ a - " ~ ) where the gradients of the gauge field dominate. At the surface of the string ( r d = 1 ), the op- posite situation, i.e., P, < 0 , occurs since for this case the gradients of the scalar field are now the ones dominant.

For the transition case m, = m d ( a= l ) , both P, and P, vanish15. T o understand that P, =Po =0, one substi- tutes ( a= 1 ) and exp( A ) = 1 into Eqs. (8a) and (8b) for the gauge and scalar fields. One then notices that

and

X 'K =XP

satisfy the field equations (8a) and (6%) automatically and also make expressions (7b) and (7c) for P, and Po to identically vanish. It is obviously for this situation where the usual approximation of the string stress- energy tensor as energy density equal tension along its axis is most justifiable.

The behavior of the components of the energy- momentum tensor near the axis of the string can be better understood if one substitutes the flat values for the metric, gauge, and scalar field ( 1 1) around the string axis into expressions (7a ) - (7~) . In this case, P,, and Po read

and

respectively. In each of these expressions the first term comes from the gradients of the scalar field, ( x ' ) ~ = a '; the second is due to the coupling term, ( e * x P / K ) ' = ~ '( 1 -2br2); the third is the effective po- tential (x' - l l2 = ( l - 2a *r ); and the last term arises from gradients of the gauge fields, e 2 A ( ~ ' ) 2 / ~ '=4b '. In the limit when r+O.

and

It is clear then from Eq. (12a) that, independent of the values for a = (m ;l /mm 12, a is positive and P, negative

near the axis of the string. On the other hand, from Eq. (12b), P, and Po could be negative or positive depending on the value for a = ( m , /md l 2 and b, where b arises from the gradients of the gauge field and therefore in- creases with m, /md . For example, in the case a = ( m , /md )2 < 1, one obtains from the numerical solu- tions that 2 b 2 < a / 8 , and consequently Pp and P, are negative. The opposite situation appears when a = ( m , /md 12> 1, in this case 2b2 > a / 8 , which yields P, and Po positive.

It was found that as one increases the symmetry- breaking energy scale, the behavior for the scalar and gauge fields, and the components of the energy-

F-''l"""""n' 0.025 1 ( a )

i

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 CIXROINRTE RROIUS

CBBRDINRTE RROIUS

FIG. 2. Solutions for the metric fields (a) 1-exp( Aj and (b) exp(C/2) , as functions of the coordinate radius r, in the case 4= lo- ' . The dash pattern definitions are the same as in Fig. 1.

36 - COUPLED FIELD SOLUTIONS FOR U(1)-GAUGE COSMIC STRINGS 3669

momentum tensor are analogous to the case v = discussed before. The main difference in this situation is that now the metric fields exp( A ) and exp(C/2) present considerable departures from Minkowski spacetime. The numerical solutions of the metric variables 1 - exp( A ) and exp( C / 2 ) as functions of the coordinate radius r for symmetry-breaking energy scale of q = lo- ' are represented in Figs. 2(a) and 2(b). Figure 2(a) shows that, for m, < m,, the metric coefficient exp( A ) is smaller than the flat value exp( A ) = 1, which means that the proper time of a test particle becomes smaller than the coordinate time. The behavior reverses for m, > m , where one obtains exp( A ) > 1. Since exp( A ) is also the coefficient of the z component of the metric, a similar analysis to that done for the t component can be per- formed.

These numerical results about the metric field exp( A ) can be better understood if we recall that near the axis of the string the equation for exp( A ) can be rewritten from (10) as

where P p =. PH = hv4[- $+ ( ~ ' ) ~ / 2 a y * ] --const. We have seen before that for m , < m , both P, and Po are negative. In this case Eq. (13) is a Bessel equation and the only regular solution is exp( A 1% J , (ap 1, thus exp( A ) < 1. In the opposite case, that is m , > m m , P,, and PH are positive. Equation (13) becomes now a modified Bessel equation, and exp( A ) = I , ( ap ) , which is now larger than one.

Figure 2(b) shows the metric field exp(C/2) for different values of m, /m,. Notice that although in each case at radial infinity this function grows linearly with the radial coordinate, the slope is different for each case. In order to physically interpret this behavior, we see that as p + CC, A + AO, and exp(C/2)+sp. Thus at radial infinity the metric (4c) becomes

which can be brought to Minkowski form changing vari- ables to t l=exp( AU/2) t , z l=exp( Ao/2)z, and h = s e . For this spacetime the parameter h of the Killing field ( a / ah Ia with closed orbits takes the values 0 5 h52n-s. As expected, the spacetime of a string approaches, away from its axis in the radial direction, Minkowski space- time minus a wedge with a deficit angle A8=27r( 1 -s) . Garfinkle13 has given an elegant argument to show that the deficit angle of the string is given by

where the linear energy density of the string is obtained from

It has been recently pointed out1' that the linear ener- gy density p, of a string when the masses of the gauge and scalar fields are equal is given by yo=n-772. Table I

FIG. 3 . Plot of the ratio p/po as a function of m , /mm This curve is well approximated by p /po= ( m , /m , O '.

shows the deficit angle A@, as well as the ratio y / p o , for different values of m , /md and 7 , both obtained from numerical integration of Eqs. (15a) and (15b), respective- ly. First, one notices that obviously the deficit angle in- creases with the energy scale of symmetry breaking; second, when the masses of the gauge and scalar fields are equal, one reproduces the result y,=.rr?72; finally, the ratio p/pO seems to be independent of the energy scale q , and consequently to be only determined by m A /m,. Figure 3 shows that p/p, is approximately given as

This result agrees with the ones by Jacobs and ~ e b b i , ' ~ and recently by Hill, Hodges, and ~ u r n e r , " both of them obtained by a variational approach without includ- ing gravitational effects.

Finally, Figs. 4(a)-4(g) represent the numerical solu- tions for the scalar, gauge, and metric fields, and the stress-energy-tensor components for the case in which the ratio m , /md is now kept fixed, equal to i. These graphs correspond to values v = loP2 , 5X and lo-' . They show that although no considerable change is obtained in the behavior of the gauge and scalar fields [see Figs. 4(a) and 4(b)], the metric fields present significant differences from their corresponding flat values as one increases the energy scale 7 . In particular Figs. 4(d) clearly manifests how the deficit angle at radial infinity depends on the symmetry-breaking scale.

Based on the study done by GarfinkleI3 analyzing the gravitational properties of a local U(1) cosmic string, we have solved the coupled Einstein-scalar-gauge field equa- tions for a static, infinitely long, straight cosmic string constructed of a self-interacting scalar field coupled to a U(1)-gauge field. We have obtained numerical solutions for the metric, scalar, and gauge fields with boundary

PABLO LAGUNA-CASTILLO AND RICHARD A. MATZNER - 36

TABLE I. Values of the deficit angle A 0 of the string and the ratio p/pO as a function of the symmetry-breaking parameter and the mass ratio ( m A / m d 12. --

77 ( r n ~ / r n * ) ~ A 0 (degrees) P I T~~

lo-' 64 0.2126X lop2 0.4700

lo--' 3 2 0.2371 X lop2 0.5240

lo- ' 16 0.2662 X 0.5883

lo- ' 8 0.3009X lop2 0.6652

10-i 4 0.3426X lo- ' 0.7573

lo - ' 2 0 . 3 9 2 6 ~ lop2 0.8679

lo- -3 1 0 . 4 5 2 4 ~ 0.9999

l o - 3 - 7 I 0.5233 X 1.1157

l o p 3 - I 4 0.6066X 1.3407

lo- ' - I 0.7052 x 1.5528 8

lop3 - I 0.81 12 X 1.7930 16

lo--' - I 0.9328 X 2.062 1 32

lo- ' - I 1.0756X 2.3773 64

conditions chosen such that these solutions represent an wedge. In the cases analyzed, the angular component P, isolated string; the solutions were classified according to of the stress-energy tensor presented a very interesting the values of the energy scale for the symmetry-breaking behavior. This component became positive or negative parameter and to the mass ratio of the gauge and scalar depending on whether in the region the gauge or scalar fields. It was shown that for values of 17 < the contributions were dominant. spacetime is practically Minkowski spacetime minus a Although at the moment for most grand unified

36 - COUPLED FIELD SOLUTIONS FOR U(1)-GAUGE COSMIC STRINGS 3671

strings of cosmological interest 7 < loW2, the theory is still young and perhaps in the future that limit could be relaxed. We thus extended our analysis to obtain solu- tions for values of 7 > lop2, where the gravitational

0 0.2 0 .4 0.6 0.8 1 .0 1.2 1.4 1 . 6 1.8 2 .0 PHYSICRL RRDIUS

0 0 .5 1.0 1 . 5 2 . 0 2.5 3 . 0 3 . 5 4 . 0 4 . 5 5 . 0 5 . 5 CBBRDINRTE RRDIUS

effects become relevant. We showed that even though the behavior of the gauge and scalar fields did not change significantly, the metric fields presented impor- tant departures from Minkowski spacetime inside and

0 0.2 0.4 0.6 0.8 1 . 0 1.2 1.4 1 .6 1 .8 2 . 0 PHYSICAL RRDlUS

CBBRDINRTE RROIUS

FIG. 4. Solutions for the (a) scalar and (b) gauge fields, (c) exp( A ) and (d) exp(C/2) metric components, and (e) o, (D P,, and (g) P, stress-energy-tensor components for a mass ratio (m, /mdj2= a . The dash patterns i----), i--------), ) , cor- respond to the symmetry-breaking parameter T = lo-', 5 X and l o ' , respectively.

3672 PABLO LAGUNA-CASTILLO AND RICHARD A. MATZNER - 36

PHYSICRL RROlUS

-0 .055 0 0.5 1 . 0 1.5 2 . 0 2.5 3.0 3 .5 4.0 4.5 5.0 5.5

PHYSICRL RROIUS

PHYSICRL RROIUS

FIG. 4. (Continued).

near the string. Solutions were found which show that ly under development for the case of superconducting the deficit angle and linear energy of the string depend strings.18 not only on the energy scale of the symmetry breaking, bu t also on the ratio of the masses of the gauge and sca- lar fields. Furthermore. the time and axial coordinates W e thank David Garfinkle and Andreas Albrecht for may become larger o r smaller than their corresponding helpful discussions, and in particular P. Amsterdamski proper values also depending on that mass ratio of the for his numerical advice. This work was supported in fields. part by a National Science Foundation G r a n t No.

A similar analysis to the one here presented is current- pHY840493 1.

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