Volume 202, number 3 PHYSICS LETTERS B 10 March 1988

B O S O N I C S U P E R C O N D U C T I N G C O S M I C S T R I N G S . I. C L A S S I C A L F I E L D T H E O R Y S O L U T I O N S

Ar i f BABUL Princeton University Observatory, Peyton Hall, Princeton, NJ 08544, USA

Tsvi P I R A N ~ and Dav id N. SPERGEL Institute for Advanced Study, Princeton, NJ 08540, USA

Received 14 December 1987

This paper explores a four-dimensionaL field theory in which superconducting bosonic cosmic strings can form. We solve the field equations and explore the parameter space within which the strings are superconducting. We find that the maximum allowed current is often much less than suggested by dimensional analysis. In most of the parameter space, even the maximum current in the strings is not sufficient for many of their proposed astrophysical uses and especially, the current is rarely sufficent to form "'frozen" string loops.

I. Introduction

Symmetry breakings in the early universe can produce stable topological defects: monopoles , cosmic strings, and domain walls. In recent years, the gravi ta t ional effects o f cosmic strings have a t t racted significant a t tent ion (see ref. [ 1 ] for a review). Wit ten [2] showed that in many theories, cosmic strings can behave like supercon- ductors and may carry electric currents. There are two physical mechanisms that can endow cosmic strings with superconduct ing propert ies. I f fermions obta in their mass at the phase t ransi t ion that produces the string, they may be t rapped in the string as massless zero modes, giving rise to fermionic superconduct ing strings. Supercon- duct ivi ty may also arise if a charged Higgs field acquires an expectat ion value in the core of the string. The current in such strings is carr ied by bosonic modes.

In the short t ime since the appearance o f Wi t ten ' s paper, many authors have proposed var ious interest ing astrophysical effects that ought to arise if superconduct ing cosmic strings exist [3 -7 ]. However, all o f these scenarios require the cosmic strings to bear large currents in order to generate observable consequences. Super- conduct ing cosmic strings are of little intererest to astrophysicists i f the m a x i m u m current that they can carry is small. Wit ten [ 2 ] ment ioned that large currents ought to arise natural ly in bosonic strings. This mot ivates our explorat ion of the proper t ies of bosonic strings.

In this paper, we explore a s imple U(1 ) ' × U ( 1 ) field theory. We require the external vacuum to be non- conduct ing and explore the region o f pa ramete r space within which superconduct ing cosmic strings are stable. We calculate the m a x i m u m current that a stable superconduct ing string can carry and f ind that it is often much smaller than previous estimates. We also f ind that the pa ramete r space where " f rozen" string loops can exist is quite small. " F r o z e n " loops occur when the cosmic string carries sufficiently large currents so that the resulting magnet ic field pressure plus the pressure due to the kinetic energy o f the charges can balance the string tension [ 8 -10 ]. The s t ress-energy tensor is calculated for the string solutions and in a companion paper [ 11 ], the tensor

Also at Racah Institute for Physics, The Hebrew University, 91 904 Jerusalem, Israel.

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is used in determining the spacetime curvature around a superconducting cosmic string. Most of our conclusions are valid for any theory that gives rise to superconducting strings.

2. Basic equations

Witten [ 2 ] described a U (1) ' × U (1) theory in which a complex scalar field q~ couples to, a gauge field C and carries U(1 ) ' charge q and a complex scalar field cr couples to the electromagnetic field A and carries U(1 ) charge e, not necessarily the charge of an electron. The lagrangian for this theory is

5°= - ½ [ D " ~ ] D . ~ ] * - ½ [DUa] [ D . a ] * - ( 1 / 1 6 n ) F . . F ~" - ( I / 1 6 n ) H u . H ~'~ - V (G or),

V(O, cr)= ~G( 1012 - 0 2 ) 2 + f l O l 2 Icrl 2 +~2~ [crl 4 - ½ m 2 I~12 , (1)

where DUq~= (0U+iqC")0, D ' a = (0U+ieA ~') a. Our spacetime metric has the signature + 2 and we adopt the units h = c = 1. At low temperatures, O assumes a non-zero expectation value, 0, and breaks the U(1 ) ' symmetry. This phase transit ion can trap cosmic strings as topological defects. In certain cases, it is energetically favorable for the a in the string to assume a non-zero expectation value, breaking the U(1 ) of electromagnetism and allowing the string to carry an electric current.

This paper considers a stationary superconducting cosmic string bearing uniform current. The string is as- sumed to be straight, infinitely long, and lying along the x-axis. Cylindrical symmetry simplifies calculations: ~(x) = [~(r) I e inO and a (x ) = Icr(r) I e iv/(z). The only non-zero components of the gauge fields are Az(r) and Co (r) . The action of the string and its envi ronment reduces to

5e= - ~ r d r {½10'(r) 12 + ½1 a ' ( r )12 +Q'(r )2 /8zcq2r 2 + P ' ( r ) 2 / 8 ~ e 2 + 1O(r)[2Q(r)2/2r2 + ½P(r) 2 la(r)12

+~2~[ [O(r)12 - 0 2 ] 2 +J]O(r)I 2 la(r)12 +~)t~ I or(r)14 - ½ m 21 ~r(r)[2}, (2)

where P ( r ) = O~'/Oz+ eAr(r) and Q(r ) = 1 + qCo(r) for a string with n = 1 winding number. Throughout the pa- per, we work in the cooordinate basis. Pr ime denotes partial derivatives with respect to the radial coordinate r. We draw attention to the te rm ½P21 a l 2 in the action, noting that this term represents the kinetic energy density of the charge carriers along the string. We have assumed that the string has no net charge (O~u/0t = 0) and that the current along the string is uniform (0z~,/0z2 = 0).

In order to simplify eq. (2), we scale the variables,

~ ( X ) 2 = IO(y) 12/0 2, # ( X ) 2 = ( , ~ . a / m 2 ) l ~ Y ( X ) [ 2 , P 2 = ( m 2 / j . ~ 2 a 0 4 ) e 2 , (3a)

the coupling constants,

(~2 = q 2 / ~ , ~ 2 . . ~ ( m 2 / , ~ c r y l 2 ) e 2 , (3b)

and the unit of length,

r 2 = x 2 / 2 j 1 2 . (3c)

Hence

50= - -0 2 f X d.x'{ ½~' (x) 2 q-oL, ½~'(x) 2 q -Q ' (x)2 /87~#2x 2 -~-P ' (x )2 /8~o 2 . - { - ~ ( x ) 2 Q ( x ) 2 / 2 x 2 .Jr ½ P ( x ) 2 ~ ( x ) 2

+ ~ [~(x) 2 - 1] 2 + ce2~(x )26(x ) 2 + a3[ ~6(x) 4 - ½a(x):] }. (4)

Eq. (4) allows for an easy recognition of the free parameters in the theory:

Of I = m2/~a02 , oL2=fm2/) l .e~2aO 2, 0(. 3 - ' :m4/~o~.~rO 4 . (5a)

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Volume 202, number 3 PHYSICS LETTERS B 10 March 1988

In order to understand the physical significance of the a-parameters, we note that

a, ~ [ a ( 0 ) / O ( ~ ) ] 2, o~21al ~ f12+, o~31al ~ ( M~IM,) 2 . (5b)

where the f ~_are the two quartic scalar couplings in the theory, M~ ~ m is the mass parameter of the a field, and M0~ ~/x/2o is the mass of the massive component of the 0 field.

We are interested in the region of parameter space in which the vacuum is non-conducting and the cosmic string is superconducting. A cosmic string will exist in any region of space where the phase of the ~ field exhibits a non-zero winding number upon integration along a closed path. The necessary boundary conditions for a cosmic string,

q~(0)=0, q ~ ( ~ ) = l , Q ' ( 0 ) = 0 , Q ( ~ ) = 0 . (6)

are sufficient to constrain the ~ and Q fields. Our vacuum is non-conducting. This restricts the interesting parameter space: the potential energy of the

vacuum state V( I a l = 0, 101 = q) = 0 must be less than that of the superconducting state V( I a l @ 0, I O l = 0):

o:3<½ , (7)

and the non-conducting state must be a global minimum of the potential. The condition for the vacuum state to be a minimum is 82V(la[ =0, I~1 = q ) / S a z < 0 . This necessitates

a3 < 2 a 2 . ( 8 )

In principle, the potential may have another minimum at ¢ ~ 0, a ~ 0 but the above constraints guarantee that the non-conducting vacuum state is indeed a global minimum of the potential. These conditions are essential to obtain a string with non-conducting exterior.

Even if these conditions are satisfied, it is not always favorable to have a superconducting phase inside the string. Superconductivity requires that the overall contribution to the total energy of the system due to the introduction of ] a(0) I ~ 0 be less than the total energy of the non-conducting cosmic string. Except when I ~ ~/, the presence of the a field does not appreciably affect the ~ field configuration; therefore, the above energy condition may be expressed as

Uo < 0 , (9a)

where

(1 f xdxP(X)20(x) 2 A. P ' (x)2X~ , U° =P(0)2 + 8~e2 f X ~ x d x ½0'(x) 2 , 2 d P ( 0 ) 2 u ' ~ ] "t- a 1

"[-Og 2 f xdx#(x)2O(X)2--OI3 ; xdx[lO(x)2-10(x)4]. ( 9 b )

In the above equation, the integrals are of order unity or less. The P(0) 2 term, the first term in eq. (9b), is non- zero only when there is a current in the string. Zhang [ 14] and Hawes, Hindmarsh and Turok [ 15 ] have shown that an additional constraint must be imposed in order to suppress the decay of current within a Hubble time due to quantum effects. We do not consider quantum effects in our classical analysis.

Inequalities ( 7 ) - (9a ) suggest that it will be difficult to find superconducting string configurations. In partic- ular, inequality (9a) very strongly limits the parameter space for superconductivity compared to the parameter space envisioned by Witten [2]. If a3 is large enough to satisfy eq. (9), it will often violate (7) or (8): I f it is energetically favorable for the string to be superconducting, then it is also likely that it will be energetically favor- able for the vacuum to be superconducting.

The amplitude of the a and P fields determine the current density

J(x) =e[a(x) ]2p(x) =O(x)2 ff(x)'e~',~o~'/2 . (10)

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¢,1

- .01

- . 02

- . 03 0

a

, h ~ i I i L , i I

.05 .1 [ I I [ ] I I I

.15 I

.2

' I

I .25

.12

.1

D8

.06

.04

.02

o , I I ~ I I I ~ I I .05 .1

\ i i i i I i i I ~ ' ] ' ~

.15 .2 ~o

I

25 I

.3

Fig. 1. (a) The contribution to the energy density per unit length, U~ as a function of Po in units of q2. This figure was calculated with c+ [ = 0.32, c+2 = 0.1, c+3 = 0.19 and 4g ~2 = 4n~ 2 = 0.1. (b) The current as a function of Po. Same parameters as (a).

The total current carried by the string is

I = 2 n f r d r J ( r ) = K 0 q , K-2nfxdx6(x)2 f i (x ) . (11)

For a given set of oL parameters, Uo increases as Po=P(O) increases (see fig. la ) . Eventually, Uo does not satisfy inequali ty (9a) and superconductivi ty is no longer energetically favorable. 6(0) is maximal when Po = 0 and it decreases as Po increases. ~ (0) = 0 when U~ = 0. Since J ( x ) ~ ~ (x) 2/3(x), I = 0 when Po = 0 or U~ = 0. Fig.

lb displays the total current as a function of Po. The maximal current/max occurs at some intermediate value of

Po. In a current bearing string (Po+~0), when the proper length of a section of the string near a cusp shrinks,

Po = O~/Oz will increase [ 9 ]. Eventually, the current in this section of the string will reach its maximum. If the decreasing branch in fig. lb is unstable [ 13 ], we expect that this port ion of the string will undergo a first-order phase t ransi t ion to the non-conduct ing state.

The stress-energy tensor for the superconducting cosmic string can be calculated from the lagrangian:

2 ) +rr+OO( r) 1 ( O 0 O~ _Jf_gOOQ2~/)2 V(~),G)-- 8ne~ k, Or] 8 g q 2 ' r + = - g++ + g = : e + + 2 _ +

( 2 + 1 ) - - 2 gr+g°° ( ~ ) 2 grrg=Z (OP~ -l- T~r = ½grr OrO(~ -~rO~7 2 -½(g°°QZ~2Fgzzp2~72)-V(~'cr)-F 8ge 2 \ O r J ~ '

ZI ( -~rOO 2"~-grr ~ 2 ) grrgzz (OP']2-Fgrrg°° ( or 8--~q 2 TO=½gOOQ202 grr +g~p2a2 --V((p,a)-- ~--T-\~-.] , (12)

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Volume 202, number 3

I 00

S0

- S O

PHYSICS LETTERS B lOMarch 1988

i00

, [ ~ 7 I i , , , I ' ' ~ '

J

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

L ~ ' z / o 2

i k I I I I I I I I I I I I I I I

0 i 2 3

Fig. 2. The integrated components of the stress-energy tensor as a function of current. The string becomes a "spring", when M~< 0. In this figure, the a parameters have been fixed near the values that allow the largest currents; a~ = 1.0, oQ = 0.25, and o! 3 = 0.499.

T - = ½ g z z p 2 a 2 - ~ grr -t-gOOQ2(~ 2 -- - - • (12 cont 'd) Or V(~,0.)+ 8ge2 \ 8 r J 8zcq 2

Without current, a string, superconducting or ordinary, is invariant under transformation in the z - t plane; thus, T~ = T',. However; with a # 0, the string acquires a mass [ 11 ]. The presence of current breaks this symmetry: there is now a preferred Lorentz frame in which there is no net charge. In this frame, T~ need no longer equal T',.

An examination of the equation for T§ reveals that the cosmic string's intrinsic tension is opposed by the pressure arising from the mutual repulsion o f the closed magnetic field lines (the P ' 2 term) and by the pressure generated by the inertia of the charge carriers (the pz] 0.12 term). The charge carriers affect the tensile properties of the superconducting cosmic string in much the same way as water flowing through a hose affects the hose- /water system - as the flow increases, the system becomes stiffer and more rigid. If a string can support suffi- ciently large currents, TZ, which is usually negative, can become zero or positive. When T~> 0, the string behaves like a "spring" [8 -10] and string loops stop oscillating. String solutions that can bear the most amount o f current (IMAx= 3.7q), Occur in the vicinity o f a,--- 1, c~2=0.25, and a3=0.499 . In this region, we were able to find string solutions where T§ changed sign and become positive with increasing current. Fig. 2 illustrates the variation in M't/~l 2 and M§/~/2 with increasing current, where MU~ - - 2 7 ~ f T ~ r dr, for a string with c~, = 1.0, C~z= 0.25, a3 = 0.499. M§ < 0 implies T§ > 0. As I > q, we note that the approximate version of the third con- straint (eq. (9)) is no longer valid and we must compare the total energy of the superconducting string (includ- ing current) with the total energy of a non-conducting string in order to determine whether superconducting strings with large currents are energetically favourable.

In spacetime with spherical symmetry, the gravitational field in the empty space external to a source is inde- pendent o f the distribution of mat ter-energy interior to the source. This is Birkhoff 's theorem. However, for cylindrically symmetric spacetime, such a theorem does not exist. The spacetime geometry at infinity depends on how the mass-energy is distributed in the interior o f a linear source. Furthermore, in the case o f a cosmic string bearing an electric current, the situation is further complicated because the space external to the string is no longer empty but is permeated by a magnetic field. In a companion paper [ 11 ], we use the above stress- energy tensor to determine the exterior metric around a superconducting cosmic string.

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3. Numerical method

PHYSICS LETTERS B lOMarch 1988

In order to find the lowest energy solution of the field equation that satisfies the boundary conditions, we searched numerically for the lowest energy solution that extremizes the action.

The calculation is simplified by a change of variables, r= tan ¢ so that the integration range becomes finite: 0~<~<7r/2 and

~/2f {COS2~ [1 (d_.X 2 jl_ l~l ( d ~ 2 l (dO~2 jr. 1 ( ~ ) ] d P 2 t l 2 - - J d~tan - - - \ d ~ J d~J + 8~z~Ztan2~ \ d ~ } 8~f i 0

+ ¢O~$2~ \2 t - ~ T n 2 ~ l ( ~ 2 Q 2 +-~/~2~2 +~(~2 1 )2.÷OL2~2~2 avOL3(I~4 1~2))} • (13)

We define the variables: ~, #~, Qi and /~ on an equally spaced grid: ~i= ( i - 1)A~ ( i= 1 ..... N+ 1, where A~= 7 r / 2 N ) . The action, expressed in discrete variables,

= --*'~1= tan ¢,+,,2 cos2~i+,/2 (~+'2A~-q~)2 + - , (Q,+,_Q,)2 + (p,+,_:p,)2)

2A~ + 87r~ 2 tanZ~i+lnA~ 8zr~- J

X~, tan ¢i A~( ~2Q~ ,-2 2 ) - ,=~ ~ - ~ \ 2 t--~7-n2~i +~Pi ai +~ (q~2 _ 1)2 +0~2q~2~2 +0~3(~64 _½~2) , (14)

is varied with respect to q~i, r/i, Qi and/~ to yield a set of 4Ncoupled non-linear algebraic equations for the fields. These equations are solved by the SOR method. (See ref. [ 12 ] for a review of SOR techniques in field theory.)

When Po vanishes (there is no current in the string),/~ vanishes everywhere and we can use ¢= ~r/2 as the upper boundary for the integration. When Po~ 0,/~(oo) ¢ 0 and the total action diverges (the energy density of the magnetic field around an infinite straight line is infinite). In astrophysical environments, we are interested in either string segments that cross the horzon or in closed loops. Either the horizon size or the loop's radius provides a natural cut-off length: log ( cut-off length/string width) ~ 100.

Fig. 3 shows a solution obtained with a l= 0.32, oL 2 = 0.1 and a3 = 0.19. the cosmic string carries a current, I= 0.1 q. We have assumed that 41r~ 2 = 4zt~72 = 0.1. Doubling and quadrupling of the number of grid points pro- vided a check on the stability and accuracy of the solution.

1

.4 "'-.. "'. -

i 0 2 4 6 8 I0

x

Fig. 3. The solutions to the field equations for P, Q, ~ and a for a stable superconducting cosmic string configuration associated with a current, I= 0.1 q. This figure was calculated for the same parameters as fig. la.

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Volume 202, number 3 PHYSICS LETTERS B 10 March 1988

4. Results

The three-dimensional a parameter space was explored numerically with 4 g ( 2 and 47gq 2 fixed at 0.1. Fig. 4 shows that in most of the parameter space, superconductivity is energetically favorable only for small values of c~ t. While only contours corresponding to the maximum value of a~ are illustrated, the full parameter space was explored. Constraints (7) and (8) clearly delineate the allowed region in the c~2-c% plane.

Most of the astrophysical applications of superconducting cosmic strings require large currents. The Ostriker, Thompson and Witten [4] scenario for galaxy formation requires the string's luminosity in low frequency elec- tromagnetic radiation to be at least comparable to the string's luminosity in gravitational radiation. A cosmic string e m i t s Egrav ~ Gq 4- If the average current in the string is I, the string also emits E r a d ~ 12. Since I~</max,

/ ~ r a d / / ~ g r a v , ~ 2 4 I ~ a × / G ? I = ( K ~ ) 2 m ~ x / G r l 2 . ( 1 5 )

The factor K was not included in previous estimates of the maximum current. The ratio of the magnetic field energy density to the string's total energy density is

Em~g/Etot ~ (K~) 2 ln(R/ro) ~ 100(K~) 2 , (16)

where we have assumed that the a field and the current do not appreciably affect the 0 field configuration and therefore, E~o~ .~ lw 2.

For 4g~2=0.1 , fig. 5 shows that in a large portion of parameter space the maximum current that a stable superconducting cosmic can carry is less than the current required to generate many effects of interest to astro- physicists. Fig. 6 shows that when ~2 is tuned to a higher value, the current threshold for fixed a2 and a3 is increased. The figures display contours corresponding to the maximum value of (K~) 2, the ratio/~raJ/?grav in units of (Gr/2)- ~ in the a2-a3 plane. In most of parameter space, I~ax/~/2 = (K~) 2 is less than e2~ 10 -2, the value assumed in the Ostriker et al. [4] galaxy formation scenario.

The above difficulty is not a property of the toy U( 1 ) × U(1 )' theory, but rather indicative of a general di- lemma: Superconducting bosonic cosmic strings can only exist when it is energetically favorable for the vacuum to be non-conducting while the string is superconducting. This paper shows that these two conditions are nearly contradictory and that superconductivity can only occur in a small region of parameter space. Fermionic strings

1.2 L t 1.2

i #

a,=O.O1~ /

.8 ,,'

a,=o l / / 1 VACUUM IS

c7.6 ,' / SUPERCONDUCTING

/ /

"2 i O "'" a,=l .O~- ' " i i [ i ~ i '

.2 .4 .6 8

a a

Fig. 4. A contour diagram showing the maximum value o f a l in the a2-0~3 plane. Superconducting solutions exist only for o~ less than the maximum value. For larger values ofoq, it is not ener- getically favorable for the string to be superconducting. This fig- ure was calculated with 4 g ( 2 = 4n~2 = 0.1.

I I

,(Ke...)2 1O 4 / /

.8 / / / / /

.4 I / "'" "'"'" .2 "" "'" "' ""''''iK~)'~=t 0 a

0 I L

0 .2 .4 cl a

I

4ff~Z=O 1

VACUUM IS

SUPERCONDUCTING

i I i l l

.6 8

Fig. 5. A contour plot showing the maximum possible value of Erad/L'gr, v, (K() 2 in units of ( GO 2) - t, in the a2-0t3 plane. For each a2 and a3, the value o fa~ was tuned to maximize the maximal

• -2 ~7 current. This figure was calculated using 4he = 4 n q - = O. I.

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Volume 202, number 3 PHYSICS LETTERS B 10 March 1988

1.2

#.6

.4

I

' ' ; ' I / ' ' '

, , (K~ '~=10- '

/ ]

/ / / /

,' / /

[ / "

i' /(Ke-,)~ =5xi0-3 /

/

/; / , " _.,(K~2 ~1o

.2 .4 o. a

t

4 ~ = I00

VACUUM IS

SUPERCONDUCTING

I i k

.6

1 Fig. 6. Same as fig. 5 with 4n02= 100.

face a similar dilemma. The massive fermions that carry charges along the string must be unstable in the vacuum yet stable along the string. These condit ions may be difficult to satisfy and suggest that superconducting strings with very large currents may not be generic.

While we were completing this ~¢ork, we learned of two related studies: a variational analysis of the field equations by Hill, Hodges and Turner [ 10 ] and a numerical study of the field equations by Amsterdamski and Laguna-Castillo [ 16 ]. Both groups obtained equivalent constraints to inequalities ( 7 ) - (9a) which delineate the

allowed region for superconductivity. Our results are consistent with those of the above two studies. We all agree that the max imum current in the bosonic strings will, in general, be significantly less than Wit ten 's estimate [ 2 ].

Acknowledgement

We would like to acknowledge C. Thompson, M. Hindmarsh, N. Turok and B. Paczyfiski for helpful discus- sions. We would like to thank M. Turner for helpful comments on the manuscript . T.P. thanks the Aspen Center for st imulating his interest in cosmic strings. T.P. and D.N.S. are supported by NSF grant PHY-862066. A.B. is

supported by an NSERC (Canada) post-graduate fellowship.

References

[ 1 ] A. Vilenkin, Phys. Rep. 121 (1985) 263. [2] E. Witten, Nucl. Phys. B 249 (1985) 557. [ 3 ] E.M. Chudnovsky, G.B. Field, D.N. Spergel and A. Vilenkin, Phys. Rev. D 34 (l 986) 944. [4] J.P. Ostriker, C. Thompson and E. Witten, Phys. Lett. B 180 (1986) 231. [ 5 ] C.T. Hill, D.N. Schramm and T.P. Walker, Ultra-high energy cosmic rays from superconducting cosmic strings, FERMILAB pre-

print (1986). [6] A. Vilenkin and G.B. Field, Nature 326 (1987) 772. [7] A. Babul, B. Paczyfiski and D.N. Spergel, Astrophys. J. 316 (1987) L49. [8] E. Copeland, M. Hindmarsh and N. Turok, Phys. Rev. Lett. 58 (1987) 1910. [9] D.N. Spergel, T. Piran and J. Goodman, Nucl. Phys. B 291 (1987) 847.

[ 10] C.T. Hill, H.M. Hodges and M.S. Turner, Phys. Rev. D 36 (1987) 1; Phys. Rev. Lett. 59 (1987) 2493. [ 11 ] A. Babul, T. Piran and D.N. Spergel, IAS preprint. [ 12] S.L. Adler and T. Piran, Rev. Mod. Phys. 56 (1984) l. [ 13] C. Thompson, J.P. Ostriker and E. Witten, private communication. [ 14] Y. Zhang, Phys. Rev. Lett. 59 (1987) 2111. [ 15] C. Hawes, M. Hindmarsh and N. Turok, in preparation (1987). [ 16 ] P. Amsterdamski and P. Laguna-Castillo, Phys. Rev. D (1987), to be published.

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