Wakes in dilatonic current-carrying cosmic strings
A. L. N. Oliveira*
Instituto de Fı´sica, Universidade de Brası´lia, Brası´lia, Brazil M. E. X. Guimara˜es†
Instituto de Fı´sica Teo´rica/UNESP, Sa˜o Paulo, Brazil
共Received 12 March 2003; published 20 June 2003兲
In this work, we present the gravitational field generated by a cosmic string carrying a timelike current in the scalar-tensor gravities. The mechanism of formation and evolution of wakes is fully investigated in this framework. We show explicitly that the inclusion of electromagnetic properties for the string induces logarith-mic divergences in the accretion problem.
DOI: 10.1103/PhysRevD.67.123514 PACS number共s兲: 98.80.Cq
I. INTRODUCTION
Topological defects are predicted in many gauge models as solitonic solutions resulting from spontaneous breaking of gauge or global symmetries. Among all these solutions, cos-mic strings have attracted attention because they may be the source of large-scale structure in the Universe关1兴. A relevant mechanism to understand structure formation by cosmic strings involves long strings moving with relativistic speed in the normal plane, giving rise to velocity perturbations in their wake 关2兴. The mechanism of forming wakes has been considered by many authors in the general relativity theory
关3,4兴. More recently, Masalskiene and Guimara˜es 关5兴
consid-ered the formation of wakes by long strings in the context of scalar-tensor theories. They showed that the presence of a gravitational scalar field—which from now on we will call generically a ‘‘dilaton’’—induces a very similar structure as in the case of a wiggly cosmic string 关4兴. Further, Bezerra and Ferreira关6兴 showed that, if torsion is present, this effect is amplified.
Our purpose in this paper is twofold. We first generalize the work of Peter and Puy 关7兴 and of Ferreira et al. 关8兴 for the case of a cosmic string carrying a timelike current in the framework of scalar-tensor gravities. From the point of view of purely gravitational physics, it was shown that the inclu-sion of a current hardly affects the metric outside the string
关9兴. However, it is well known that the inclusion of such an
internal structure can change the predictions of cosmic string models in the microwave background anisotropies 关10,11兴. This is our second goal; namely, we analyze the formation and evolution of wakes in this spacetime and we show ex-plicitly how the current affects this mechanism. For this pur-pose, we consider a model in which nonbaryonic cold dark matter propagates around the current-carrying string. The Zel’dovich approximation is carried out in order to treat this motion. We anticipate that our main result is to show that the inclusion of a current brings logarithmic divergences and can actually break down the accretion mechanism by wakes.
This work is outlined as follows. In Sec. II, after setting
the relevant microscopic model which describes a supercon-ducting string carrying a current of timelike type, we present its gravitational field and show the explicit dependence of the deficit angle, the deflection of light, and the geodesics on the timelike current. In Sec. III, we consider the mechanism of formation and evolution of wakes in this framework by means of the Zel’dovich approximation. Finally, in Sec. IV, we end up with some conclusions and remarks.
II. TIMELIKE CURRENT-CARRYING STRINGS In this section we study the gravitational field generated by a string carrying a current of timelike type. We start with the action in the Jordan-Fierz frame
S⫽ 1 16
冕
d 4x冑
⫺g˜冉
⌽˜ R˜⫺共⌽˜ 兲 ⌽˜ ˜g ⌽˜⌽˜冊
⫹Sm关m,g˜兴. 共1兲 g˜ is the physical metric which contains both scalar and
tensor degrees of freedom, R˜ is the curvature scalar associ-ated with it, and Sm is the action for general matter fields which, at this point, is left arbitrary. The metric signature is assumed to be (⫺,⫹,⫹,⫹).
In what follows, we concentrate our attention on super-conducting vortex configurations which arise from the spon-taneous breaking of the symmetry U(1)⫻Uem(1). There-fore, the action for the matter fields is composed of two pairs of coupled complex scalar and gauge fields (,B) and (,A). Also, for technical purposes, it is preferable to work in the so-called Einstein 共or conformal兲 frame, in which the scalar and tensor degrees of freedom do not mix:
S⫽161G *
冕
d4x冑
⫺g关R⫺2兴⫹冕
d4x冑
⫺g ⫻冉
⫺12⍀2共兲关共D 兲*D⫹共D兲*D兴 ⫺ 1 16共FF ⫹H H兲⫺⍀2共兲V共兩兩,兩兩兲冊
, 共2兲*Email address: andreo@fis.unb.br
where F⫽A⫺A, H⫽B⫺B, and the potential is suitably chosen in order that the pair (,B) breaks one symmetry U(1) in vacuum 共giving rise to the vortex configuration兲 and the second pair (,A) breaks the symmetry Uem(1) in the core of the vortex共giving rise to the superconducting properties兲: V共兩兩,兩兩兲⫽ 8 共兩兩 2⫺2兲2⫹ f共兩兩2⫺2兲兩兩2⫹ 4 兩兩 4 ⫹m 2 2 兩兩 2. 共3兲
We restrict ourselves, then, to the configurations corre-sponding to an isolated, static current-carrying vortex lying on the z axis. In a cylindrical coordinate system (t,r,z,) such that r⭓0 and 0⭐⬍2, we make the following an-satz:
⫽共r兲ei, B ⫽
1
q关Q共r兲⫺1兴. 共4兲
The pair (,A), which is responsible for the superconduct-ing properties of the vortex, is set in the form
⫽共r兲ei(t), A t⫽
1
e关Pt共r兲⫺t兴, 共5兲
where Pt corresponds to the electric field that leads to a timelike current in the vortex. We also require that the func-tions , Q(r), (r), and Pt must be regular everywhere and must satisfy the usual boundary conditions of vortex关12兴 and superconducting configurations关13,14兴.
The action 共2兲 is obtained from Eq. 共1兲 by a conformal transformation
g
˜⫽⍀2共兲g and by the redefinition of the quantity
G
*⍀
2共兲⫽⌽˜⫺1,
which makes evident the feature that any gravitational phe-nomena will be affected by the variation of the gravitation constant G
*in the scalar-tensor gravity, and by introducing a new parameter
␣2⫽
冉
ln⍀共兲
冊
2
⫽关2共⌽˜ 兲⫹3兴⫺1.
Variation of the action 共2兲 with respect to the metric g and to the dilaton fieldgives the modified Einstein’s equa-tions and a wave equation for the dilaton, respectively; namely,
G⫽2⫺gg␣␣⫹8G
*T,
䊐g⫽⫺4G*␣共兲T, 共6兲
where T is the energy-momentum tensor, which is ob-tained by
T⫽ ⫺2
冑
⫺g␦Smat
␦g. 共7兲
We note in passing that, in the conformal frame, this tensor is not conserved, providing us with an additional equation
ⵜT⫽␣共兲Tⵜ.
In what follows, we will write the general static metric with cylindrical symmetry corresponding to the electric case in the form
ds2⫽⫺e2dt2⫹e2(␥⫺)共dr2⫹dz2兲⫹2e⫺2d2, 共8兲 where,␥, are functions of r only.
The nonvanishing components of the energy-momentum tensor using Eq. 共7兲 are
Ttt⫽⫺1 2⍀ 2共兲兵⫺gtt2P t 2⫹grr关
⬘
2⫹⬘
2兴⫹g2Q2其 ⫺81grr再
⫺gttPt⬘
2 e2 ⫹g Q⬘
2 q2冎
⫺⍀ 4共兲V共,兲, Trr⫽⫺1 2⍀ 2共兲兵gtt2P t 2⫺grr关⬘
2⫹⬘
2兴⫹g2Q2其 ⫹ 1 8g rr再
gttPt⬘
2 e2 ⫹g Q⬘
2 q2冎
⫺⍀ 4共兲V共,兲, T⫽⫺1 2⍀ 2共兲兵gtt2P t 2⫹grr关⬘
2⫹⬘
2兴⫺g2Q2其 ⫺81grr再
gttPt⬘
2 e2 ⫺g Q⬘
2 q2冎
⫺⍀ 4共兲V共,兲, Tzz⫽⫺1 2⍀ 2共兲兵gtt2P t 2⫹grr关⬘
2⫹⬘
2兴⫹g2Q2其 ⫺ 1 8g rr再
gttPt⬘
2 e2 ⫹g Q⬘
2 q2冎
⫺⍀ 4共兲V共,兲. 共9兲Therefore, for the electric case, Eqs. 共6兲 are written as 
⬙
⫽8G *e 2(␥⫺)关T 1 1⫹T 3 3兴, 共⬘
兲⬘
⫽4G *e 2(␥⫺)关⫺T 0 0⫹T 1 1⫹T 2 2⫹T 3 3兴, ⬘
␥⬘
⫽共⬘
兲2⫹共⬘
兲2⫹8G *e 2(␥⫺)T 1 1 , 共⬘
兲⬘
⫽⫺4G *e 2(␥⫺)T. 共10兲In order to solve the above equations we will divide the space into two regions: the exterior region r⭓r0, in which only the electric component of the Maxwell tensor contrib-utes to the energy-momentum tensor, and the internal region 0⭐r⬍r0, where all matter fields survive. r0 is the string thickness.
A. The exterior metric
Due to the specific properties of the Maxwell tensor
T⫽0 and T␣T␣⫽1
4共T ␣T
␣兲␦ 共11兲 the Einstein equations 共10兲 may be transformed into some algebraic relations called Rainich algebra关15,16兴 which, for the electric case, have the form1
R⫽Rzz⫹Rrr⫹R⫹Rtt⫽2e2(⫺␥)
⬘
2, 共Rt t兲2⫽共R r r⫺2grr⬘
2兲2⫽共R 兲2⫽共R z z兲2, Rtt⫽⫺R, Rtt⫽Rrr⫺2grr⬘
2, R⫽Rzz. 共12兲 Then we have ⬙
⫹1 r⬘
⫺⬘
2⫺␥⬙
⫽D 2 r2 , 共13兲 ␥⬙
⫹1 r␥⬘
⫽0. 共14兲 The solution is ␥⫽m2ln冉
r r0冊
, ⫽n ln冉
rr 0冊
⫺ln冉
r/r0⫹k 1⫹k冊
. 共15兲Therefore, the exterior metric for a timelike current-carrying string is dsE2⫽
冉
r r0冊
2m2⫺2n W2共r兲共dr2⫹dz2兲 ⫹冉
rr 0冊
⫺2n W2共r兲B2r2d2⫺冉
r r0冊
2n 1 W2共r兲dt 2, 共16兲 where W共r兲⬅共r/r0兲 2n⫹k 1⫹k . 共17兲The constants m,n,k,D will be determined after the inclu-sion of the matter fields.
B. The internal metric and the matching
In order to solve the Einstein equations共6兲 in the internal region 共where all matter fields contribute to the
energy-momentum tensor兲 we use the weak-field approximation. Therefore, we can expand the metric and the dilaton fields to first order in G0 such that
g⫽⫹h, ⫽0⫹1.
In this way, Eqs.共6兲 reduce to
ⵜ2h
⫽⫺16G0
冉
T(0)⫺ 1 2T(0)
冊
共18兲 in a harmonic coordinate system such that (h⫺12␦h),
⫽0, and the linearized equation for the dilaton field is ⵜ2
(1)⫽⫺4G0␣共0兲T
(0). 共19兲
In the above equations 共18兲 and 共19兲, T(0) is the energy-momentum tensor evaluated to zeroth order in G0, and T(0) its trace. In Cartesian共harmonic兲 coordinates the nonvanish-ing components of T(0) are
T(0)tt ⫽⫺1 2
再
2P t 2⫹⬘
2⫹⬘
2⫹ 1 r2 2Q2冎
⫺81再
⫺Pt⬘
2 e2 ⫹ 1 r2 Q⬘
2 q2 ⫹冎
⫺V共,兲, T(0)xx ⫽冉
cos2共兲⫺1 2冊
再
⬘
2⫹⬘
2⫺ 2Q2 r2 ⫺ Pt⬘
2 4e2冎
⫺12再
⫺2P t 2⫺ 1 4 Q⬘
r2q2⫹2V共,兲冎
, T(0)yy ⫽冉
sin2共兲⫺1 2冊
再
⬘
2⫹⬘
2⫺ 2Q2 r2 ⫺ Pt⬘
2 e42冎
⫺12再
⫺2P t 2⫺ 1 4 Q⬘
2 r2q2⫹2V共,兲冎
, T(0)zz ⫽⫺1 2再
⫺ 2P t 2⫹ ⬘
2⫹⬘
2⫹ 1 r2 2Q2冎
⫺81再
⫺Pt⬘
2 e2 ⫹ 1 r2 Q⬘
2 q2冎
⫺V共,兲. 共20兲In this paper, in order to solve Eqs.共18兲 and 共19兲 with source given by Eqs.共20兲, we use a method that was first applied by Linet in Ref. 关17兴 and later by Peter and Puy 关7兴 and by Ferreira et al. 关8兴. The trick consists in writing down the
1In scalar-tensor theories, these relations are modified by a term
string and its component fields by means of␦ functions.2In doing this, we are allowed to write the string’s energy-momentum tensor as
T⫽diag共U,⫺Z,⫺Z,⫺T兲, 共21兲
where U, Z, and T are macroscopic quantities which are defined as the energy per unit length, the transverse compo-nents per unit length, and the tension per unit length, respec-tively: U⬅Mtt⫽⫺2
冕
0 r0 Tttrdr, X⬅Mrr⫽⫺2冕
0 r0 Trrrdr, Y⬅M⫽⫺2冕
0 r0 Trdr, T⬅Mzz⫽⫺2冕
0 r0 Tzzrdr. 共22兲From the transverse quantities X and Y we define one single Cartesian component Z:
Z⫽⫺
冕
rdrdTxx⫽⫺冕
rdrdTyy. 共23兲 On the other hand, the energy-momentum in the exterior region 共and the corresponding definition of the electric cur-rent兲 can be rewritten in Cartesian coordinates asTemtt ⫽Temzz⫽ I 2 2r, Temk j⫽⫺ I 2 2r4共2x kxj⫺r2␦ k j兲. 共24兲 Therefore, we have T(0)tt ⫽U␦共x兲␦共y兲⫹ I 2 4ⵜ 2
冋
ln冉
r r0冊册
2 , T(0) zz⫽⫺T␦共x兲␦共y兲⫹ I 2 4ⵜ 2冋
ln冉
r r0冊册
2 , T(0)k j⫽⫺I2冋
␦k j␦共x兲␦共y兲⫺kjln共r/r0兲 2册
. 共25兲Solving the linearized Einstein equations, we obtain
ⵜ2h tt⫽⫺4G0
再
I 2ⵜ2冋
ln冉
r r0冊册
2 ⫹2共U⫺T⫺I2兲␦共x兲␦共y兲冎
⇒htt⫽⫺4G0再
I2冋
ln冉
r r0冊册
2 ⫹共U⫺T⫺I2兲ln冉
r r0冊冎
, 共26兲 ⵜ2h zz⫽⫺4G0再
I2ⵜ2冋
ln冉
r r0冊册
2 ⫹2共U⫺T⫹I2兲␦共x兲␦共y兲冎
⇒hzz⫽⫺4G0再
I2冋
ln冉
r r0冊册
2 ⫹共U⫺T⫹I2兲ln冉
r r0冊冎
, 共27兲 ⵜ2h k j⫽⫺4G0冋
⫹2I2kjln冉
r r0冊
⫹2␦k j共U⫺T⫺I2兲␦共x兲␦共y兲册
⇒hk j⫽2G0I2r2kjln冉
r r0冊
⫺4G0␦k j共U⫹T⫺I2兲ln冉
r r0冊
. 共28兲In order to match the internal and external solutions and obtain the parameters m, B, n, and D, we need to return to the cylindrical coordinate system:
gtt⫽⫺
冉
1⫹4G0再
I2冋
ln冉
r r0冊册
2 ⫹共U⫺T⫺I2兲ln冉
r r0冊冎冊
, 共29兲 gzz⫽1⫺4G0再
I2冋
ln冉
r r0冊册
2 ⫹共U⫺T⫹I2兲ln冉
r r0冊冎
, 共30兲 grr⫽1⫺2G0I2⫺4G0共U⫹T⫺I2兲ln冉
r r0冊
, 共31兲 g⫽r2冋
1⫹2G0I2⫺4G0共U⫹T⫺I2兲ln冉
r r0冊册
. 共32兲Before presenting the final result we note that the metric共29兲 does not preserve the feature gRR⫽gzz. We need again to make a change of coordinates, to first order in G0,
R⫽r
再
1⫹a1⫺a2ln冉
r r0冊
⫺a3冋
ln冉
r r0冊册
2冎
,where a1⫽G0(4T⫺I2), a2⫽4G0T, and a3⫽⫺2G0I2. Without loss of generality, let us make R⫽r and expand the parameters in powers of ln(r/r0). In doing that, we finally get
2The calculations are straightforward but lengthy. We will skip the
details here and provide the results directly. For the details of these calculations, we refer the reader to Refs.关7,8兴.
ds2⫽
再
1⫺4G0冋
共U⫺T⫹I2兲ln冉
r r0冊
⫹I2ln2冉
r r0冊册冎
共dr 2⫹dz2兲⫹r2冋
1⫺8G 0冉
T⫺ I2 2冊
⫺4G0共U⫺T⫺I 2兲ln冉
r r0冊
⫺4G0I2ln2冉
r r0冊册
d2⫺再
1⫹4G0冋
I2ln2冉
r r0冊
⫹共U⫺T⫺I 2兲ln冉
r r0冊册冎
dt2. 共33兲In the next section, we use this metric to compute the deficit angle and the deflection of light rays by a current-carrying string.
C. Deficit angle and the deflection of light rays by a current-carrying string
Now that we have calculated the metric共33兲 of a timelike current-carrying string, we can compute some quantities as-sociated with it, such as the deficit angle and the deflection of light rays. The deficit angle is calculated with the help of the expression ␦⫽2
冉
1⫺ 1冑
g d d冑
g冊
.For the metric 共33兲 it is, to first order in G0,
␦⫽4G0共U⫹T⫺2I2兲⫹O共G0
2兲. 共34兲 It is clear that, if I is zero共the neutral string case兲, the deficit angle共34兲 reduces to␦⫽8G0U which is of order⬃10⫺6 for grand unified theory 共GUT兲 strings.
The deflection of the light ray is given by the expression
⌬⫽2
冕
min ⬁ d冉
⫺g 2 p⫺2 ggtt ⫺ g g冊
⫺1/2 ⫺. 共35兲As in关7兴 we obtain, knowing that p is the impact parameter,
⌬⫽4G0
再
U⫺I2冋
3 2⫺ln冉
p r0冊册冎
⫹4ln共2兲I2G0⫹O共G02兲. 共36兲From expression共36兲, we can observe some remarkable fea-tures of the deflection of light due to a timelike current-carrying string in the scalar-tensor gravity. First of all, there is no dependence on the tension in the deflection of light rays. This feature is reminiscent of the neutral string. The second feature is the dependence on the impact parameter p due to the presence of the current. This is due to the fact that the inclusion of an electromagnetic field actually increases the total energy with the distance to the string core and, as a consequence, the deflection of light is more intense. Finally, we notice that the dilaton field amplifies this effect since its inclusion also contributes to increasing the total energy of the system.
III. FORMATION AND EVOLUTION OF THE WAKES AND THE ZEL’DOVICH APPROXIMATION A relevant mechanism to understand structure formation by cosmic strings involves long strings moving with relativ-istic speed in the normal plane, giving rise to velocity per-turbations in their wake 关2兴. Matter through which a long string moves acquires a boost in the direction of the surface swept out by the string. Matter moves toward this surface by gravitational attraction and, as a consequence, a wake is formed behind the string. In what follows, we will study the implications of a timelike current in the formation and evo-lution of a wake behind the string that generates the metric
共33兲. For this purpose, we will mimic this situation with a
simple model consisting of cold dark matter composed of nonrelativistic collisionless particles moving past a long string. In order to make a quantitative description of accre-tion onto wakes, we use the Zel’dovich approximaaccre-tion, which consists in considering the Newtonian accretion prob-lem in an expanding universe using the method of linear perturbations.
To start with, we first compute the velocity perturbation of massive particles moving past the string. If we consider that the string is moving with normal velocityvsthrough matter, the velocity perturbation can be calculated with the help of the gravitational force due to the metric 共33兲:
u⫽8G0Uvs␥⫹ G0 vs␥
冋
2␣共0兲2共U⫹T⫹I2兲⫺共U⫺T⫺I2兲 ⫺2 ln冉
rr 0冊
I2册
共37兲 with␥⫽(1⫺vs 2)⫺1/2. The first term in Eq.共37兲 is equivalent to the relative velocity of particles flowing past a string in general relativity. The other terms come as a consequence of the scalar-tensor coupling of the gravitational interaction and the superconducting properties of the string.
Let us suppose now that the wake was formed at ti
⬎teq. The physical trajectory of a dark particle can be writ-ten as
h共xជ,t兲⫽a共t兲关xជ⫹共xជ,t兲兴 共38兲
where xជ is the unperturbed comoving position of the particle and (xជ,t) is the comoving displacement developed as a consequence of the gravitational attraction induced by the wake on the particle. Suppose, for simplification, that the wake is perpendicular to the x axis关assuming that dz⫽0 in the metric共33兲 and r⫽
冑
x2⫹y2] in such a way that the onlynonvanishing component of isx. Therefore, the equation of motion for a dark particle in the Newtonian limit is
h¨⫽⫺ⵜh⌽, 共39兲
where the Newtonian potential⌽ satisfies the Poisson equa-tion
ⵜh
2⌽⫽4G
0, 共40兲
where(t) is the dark matter density in a cold dark matter universe. For a flat universe in the matter-dominated era,
a(t)⬃t2/3. Therefore, the linearized equation forx is ¨⫹ 4
3t˙⫺ 2
3t2⫽0 共41兲
with appropriate initial conditions: (ti)⫽0 and ˙ (ti)⫽
⫺ui. Equation共41兲 is the Euler equation whose solution is easily found: 共x,t兲⫽35
冋
uiti 2 t ⫺uiti冉
t ti冊
2/3册
.Calculating the comoving coordinate x(t) using the fact that
h˙⫽0 in the ‘‘turnaround,’’3 we get
x共t兲⫽⫺6 5
冋
uiti2 t ⫺uiti冉
t ti冊
2/3册
. 共42兲 With the help of Eq.共42兲 we can compute both the thicknessd(t) and the surface density(t) of the wake关1兴. We have, then, respectively共to first order in G0),
d共t兲⬇12 5
冉
t ti冊
1/3再
8G0Uvs␥⫹ G0 vs␥冋
2␣共0兲2共U⫹T⫹I2兲 ⫺共U⫺T⫺I2兲⫺2 ln冉
r r0冊
I2册冎
共t兲⬇25 v1 s␥t冉
t ti冊
1/3冋
8U共vs␥兲2⫹2␣共0兲2共U⫹T⫹I2兲 ⫺共U⫺T⫺I2兲⫺2 ln冉
r r0冊
I2册
, 共43兲where we have used the fact that (t)⫽1/6G0t2 for a flat universe in the matter-dominated era and that the wake was formed at ti⬃teq. Clearly, from Eq. 共43兲 we see that the presence of the current causes the accretion mechanism by wakes to diverge.
IV. CONCLUSION
Inclusion of a current in the internal structure of a cosmic string could drastically change the predictions of such mod-els in the microwave background anisotropies. In particular, a current of a timelike type could bring some divergences, leading to some unbounded gravitational effects 关9兴. In this work we studied the effects of a timelike current string in the mechanism of wake formation. For this purpose, we first studied the gravitational properties of the spacetime gener-ated by this string in the framework of scalar-tensor gravi-ties. We analyzed the dependence of the deficit angle and the deflection of light on the current in this spacetime. Then we carried out an investigation of the mechanism of formation and evolution of wakes in this framework, showing the ex-plicit contribution of the current to this effect.
Wakes produced by moving strings can provide an expla-nation for filamentary and sheetlike structures observed in the universe. A wake produced by the string in one Hubble time has the shape of a strip of width ⬃vsti. With the help of the surface density共43兲 we can compute the wake’s linear mass density, say˜ ,
共t兲⬇25 ␥1t
冉
tt i冊
1/3冋
8U共vs␥兲2⫹2␣共0兲2共U⫹T⫹I2兲 ⫺共U⫺T⫺I2兲⫺2 ln冉
r r0冊
I2册
. 共44兲If the string moves more slowly or if we extrapolate our results to earlier epochs, we see that the logarithmic term will cause divergences and the mechanism of forming wakes will break down.
ACKNOWLEDGMENTS
The authors would like to thank CAPES for partial finan-cial support in the context of the PROCAD program. A.L.N.O. would like to thank CAPES for a Ph.D. grant. M.E.X.G. is on leave from Depto. de Matema´tica, Univer-sidade de Brası´lia.
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