Dynamics of superconducting cosmic strings

Nuclear Physics B291 (1987) 847-875 North-Holland, Amsterdam

DYNAMICS OF SUPERCONDUCTING COSMIC STRINGS David N. SPERGEL, Tsvi PIRAN* and Jeremy GOODMAN

Institute for Advanced Study, Princeton, NJ 08540, USA Received 5 January 1987 Revised 2 March 1987)

This paper introduces a covariant classical action for a superconducting string and derives the equations of motion for the string and its charge carriers. A formula for the electromagnetic power emitted by the moving string is calculated and is applied in the near-cusp region. The string emits beamed electromagnetic radiation, as well as relativistic particles, from the near-cusp region. A net current along the string can arise from either primordial flux threading the string loop or from a net asymmetry in the quantum number of the charge carriers. The string's motion in an external magnetic field can produce an oscillating current whose amplitude grows secularly. The implications of these results are discussed in the context of current astrophysical models.

1. Introduction

Symmetry breakings in the early universe can produce stable topological defects: monopoles, cosmic strings, and domain walls. In recent years, the gravitational effects of cosmic strings have attracted significant attention (see Vilenkin [1] for a review). Witten [2] has shown, however, that in many theories, cosmic strings can behave like superconductors and carry electric currents. The addition of electromagnetism enriches enormously the interaction of a string with its surroundings. If they exist at all, superconducting strings will be much easier to observe than their nonconducting cousins, and may be of broader astrophysical importance. In the short time since the appearance of Witten's paper, many astrophysical uses for superconducting strings have already been proposed [3-7]. Much of the astrophysical work was done without a thorough investigation of the new physics associated with a superconducting string, and some confusion seems to exist concerning some of the basic macroscopic properties of such strings. Our object in the present work has been to dispel some of this confusion in our own minds and, we hope, in our readers' as well. Hence this is a paper devoted to physics rather than astrophysics. * Also, The Racah Institute for Physics, The Hebrew University, Jerusalem, Israel. 0550-3213/87/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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In sect. 2 of this paper, we introduce the complete, covariant, classical action for a superconducting string in an arbitrary curved background. This action is an effective low-energy theory valid for energies much less than the symmetry breaking scale associated with the formation of the string. From this action we derive equations of motion for the string, for the charge carriers, and for electromagnetic and gravitational fields. As one application of these equations, we derive a formula for the electromagnetic power emitted by a prescribed motion of the string. As another, we find a particular exact solution for the motion of a string containing non-electromagnetic "charge carriers" (i.e. massless longitudinal modes). A free string (i.e., a nonconducting string uncoupled to electromagnetic and gravitational fields) generically attains the velocity of light at isolated points in time and space [1]. Such events are called cusps. These cusps are artifacts of our ignoring the finite width of the string; however in a astrophysical context, these effects only become important when the relativistic Lorentz factor of the string exceeds 10 25. Because of its electromagnetic couplings and the inertia of its charge carriers, a superconducting string will not reach these high Lorentz factors. However, some of the most interesting observable effects of strings occur at near-cusps where the string velocity becomes very large. Therefore, in sect. 3 we offer a coordinate-free definition of a true cusp and derive expressions for the behavior of the string in the neighborhood of a general cusp. These expressions are used to show that the electromagnetic power emitted at a cusp would be infinite if a superconducting string could experience a true cusp. We have not solved the completely self-consistent problem in which the backreaction of the emitted radiation modifies the motion of the string. At cusps, or near-cusps, the current in the string can become very large. Witten [2] has shown that when the current in a string exceeds a certain critical value, Ima~, it becomes energetically favorable for the charge carriers to escape the string (if they are fermions), or for a section of the string to revert to the normal state, thus freeing all of the charge carriers in that section (if they are bosons). By considering the Lorentz-invariant formulation of /max, however, we show in subsects. 2.6 and 3.2 that it is possible for the current to exceed Ima~ in physically important reference frames without the creation of particles. Thus the power radiated at cusps is not necessarily limited by the existence of a critical current. Sect. 4 considers various processes that can generate currents along a superconducting string. As pointed out in refs. [2] and [4], a net current can be created if external magnetic flux passes through a closed loop of string at the time of its formation, and the sources of this external flux are later removed. To create large currents by this mechanism, large initial fluxes are required. We point out, however, that large currents can also be rapidly built up if the string absorbs particles containing net values of some quantum number (e.g. baryon number) and that it is energetically favorable for the string to do so. Finally, we observe that while the conservation of flux limits the net current (in the absence of particle absorption or

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loss), it does not prevent large local currents from building up as the string interacts with external fields. We give an explicit example of such build-up in subsect. 4.3. F o r many astrophysical applications, the net current is less important than the local current, especially at near-cusps. Sect. 5 concludes with a brief discussion of some of the astrophysical implications of this work.

2. Basic equations Witten [2] showed that string superconductivity can arise in grand unified theories with either fermionic or bosonic charge carriers. If there are fermion zero modes that can propagate along the string then these particles can behave like Cooper pairs in a superconducting wire. If a charged Higgs field has a vacuum expectation value in the core of the string, then there are bosonic charge carriers along the string. In this later case, current is carried by excitations of the charged Higgs field. We refer the reader to Witten's seminal paper for a more detailed discussion of the relevant features. Hill and Widrow [8] also provide a useful discussion of fermionic superconductivity. Throughout this paper, we will assume that the fermions are massless modes on the string. This section derives the equations of motion of a superconducting string. The string is idealized as a line of zero thickness. The charges are represented by a massless scalar field, q~, which is confined to the world sheet of the string. This treatment is valid for strings with either fermionic or bosonic charge carriers. The type of charge carriers is important when the discharge of particles is discussed in subsect. 2.6. 2.1. THE STRING'S Mt~TRIC Consider a two-dimensional string world surface Z~(~), (i, j . . . = 0,1;/~, v... = 0 . . . . . 3) embedded in a four-dimensional curved background, described by a four-metric g,~ (with signature + 2). We will often omit the subscript s when it will cause no confusion to do so. The induced two-metric on the string world surface is:

hij= OiZ~OjZ ~ -

oz. oz.

oz. oz

O~i Ol~j - g . ~ O,~i a ~ J

(2.1.1)

Two-dimensional indices i, j . . . . are lowered and raised using h ij and its inverse hiJ. Four-dimensional indices, /~, v .... are raised and lowered using g,,. We define "r = ~0 and o = ~1. These coordinates have the unit of length. We will often consider closed loops of length L; o then varies from 0 to 2~rL along the loop. Dotted quantities denote differentiation with respect to r, while primed quantities denote differentiation with respect to o. A two-vector a i in the world sheet can be converted to a four-vector a t by contracting with OiZ ~' - but the inverse operation

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is possible only if a t is tangent to the world sheet. It is useful to recall that

-det(hij ) - -h=(OoZ~O1Z~)2-(OoZ"OoZ,)(O1Z"OIZ~), h ij =

(-01Z~O1Z~/h 3oZ~ OlZ~/h

3oZ~OlZ~/h ) _ 3oZ~ 3oZ~/h .

(2.1.2)

(2.1.3)

3oZ~ and 31Z ~ are two tangent vectors to the string world sheet. The string world sheet is timelike with OoZ~OoZ~ < 0, and 31Z~O1Z ~ > 0, except at cusps, which are isolated points where the world sheet of the string is null. At a cusp, the tangent space is spanned by one null and one spacelike vector; in other words, the light cone in the sheet degenerates to a line. At any other point, there are two independent null tangents, which contain the light cone between them. The term "cusp" is unfortunate, since (generically) the world sheet is smooth at such points. 2.2. THE CURRENT AND CURRENT CONSERVATION

J i is the electric 2-current in the sheet: gl

0q,

j i - q ~ Z ~ 3~J'

(2.2.1)

where q is the electric charge and gJ is the antisymmetric symbol with nonzero components e °1 = - e 1° = 1. The current density J ~ in spacetime due to the sheet is

z"s( 4' ) ) = fd2 V

4,x, ( (2.2.2)

The electric current conservation equation J~;~ = 0 becomes:

1

O ( ( ~ J ~)

0,

(2.2.3)

which is satisfied automatically by virtue of eqs. (2.2.1) and (2.2.2). The following relations are useful:

OieP = x/-Lhe~kJk/q,

(2.2.4)

J~= J~g~ O~Z~,

(2.2.5)

J~Jt = JiJi = - qZ Oi~ Oid?,

(2.2.6)

~.ikEjl

qZ cg'ePOJeP= - h JkJ"

(2.2.7)

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In a string with bosonic charge carriers, the current is related to the excitations in the Higgs field by e{ 0j~ = 0~0 + eA~, where ~ is a rotation of the Higgs field in the core of the string and Ai is the electromagnetic field [2]. If the string has fermionic charge carries, then Oo~= n L + n R and 0 1 ~ = n L - - nR, where n L is the number density per unit length of left-movers and n p. is the number density per unit length of right-movers. 2.3. EULER-LAGRANGEEQUATIONS

The action for a superconducting string is:

fd2 '[ s +2, =

+ f d 4 x [ Aaem"{- gravity]

f d2~i¢-~ - 2 T o -

. Oeo Oq, h 0 0~---7,0 ~

[1

+~

+ f¢:7

d4x - - 8rr F~,F

, oz2

]

2J~-i-At,(Zs) ]

,

(2.3.1)

where TO is the (constant) string tension, ~ = q,(~i) is a massless scalar field confined to the sheet, A~ and F~, are the electromagnetic vector potential and Maxwell's tensor and ~ and G are the Ricci scalar and Newton's constant. The Euler Lagrange equations are:

0 8Z# 8 A° 0~' 80,Z" t- ~ = O,

0

8~

(2.3.2)

0,

(2.3.3)

Ox ~' 80,A------,+ -~, = 0 .

(2.3.4)

Or;i 8 0 / 0

0

8Z#

8.~

Recalling that the first two terms in eq. (2.3.1), ~ and .oq°~, look like a two-dimensional cosmological constant and a two-dimensional scalar field we write:

T(,+,)~j -

~1

8Zz°(s+~') 8h'J = -hq[To+~OtepOtep] +Oiepajep,

(2.3.5)

and using

8hq = 8,i OjZ, + 8j 8tZ , , 80iZ~

(2.3.6)

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852 we obtain

0 i (vC._h[hiJ(To_l_lo~l~Oqldp)_oni~Ojd#]g~ajZ~)_l_qeijoojd?oqiZ~F~=O, al~ (2.3.7)

O~i[fZ~ h u Ùjep) - q OoZ~ alZ ~F~ = 0

(2.3.8)

We use eqs. (2.2.4)-(2.2.7) to reexpress (2.3.7)-(2.3.8) in term of the currents:

O~i ~ Z ~ hiy To

2 q2

-h

-~ Jg~ OyZv +J*F"~=O' (2.3.7a)

o~i(vrZhei, Jr) - q2 00Z[, ÙIZ~JF~ = O,

(2.3.8a)

where [ ] stand for antisymmetrization. The equation of motion of a superconducting string includes two terms which do not appear in the nonconducting string equation, the electromagnetic force and the inertia term. The first term is simply the component of the electromagnetic force perpendicular to the string and would arise if we replaced the conducting string by a wire carrying a current. The second term is due to the kinetic energy of the charge carrier field, q~ and gives the field an effective inertia. These charge carriers add an effective mass to the string. These inertia terms do not depend on the charge of the scalar field and they will be present even if the field was not charged. The inertia term will prevent the string from forming a cusp: At no point along its world surface will the string reach the velocity of light. The maximum Lorentz factor, F = [1 - I OoZI 2] 1/2, need not be (2T0/(00~2 + Olt~2))1/2. In spite of what might be expected from a naive inspection of the equation of motion, there is no physical or mathematical reason for this limit. In fact, subsect. 2.7 will show that a circular loop oscillates around this value. It is suggestive, however, to use this value as an order of magnitude estimate. It is interesting to contrast the equation of motion of a superconducting string with that of a charged particle. There are some similarities: a free particle's motion extremizes the length of the world line, while the string extremizes the area of the world sheet. In both cases, the electric force is simply J,F ~. There are, however, important differences. The four-velocity of a charge carrying particle determines uniquely its four-current. A superconducting string, even while carrying a fixed number of charges, has a freedom not available to the point particle: it can

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redistribute its charges across itself. Furthermore, new currents can be generated on the world sheet by external fields without changing the total charge of a string. The electromagnetic field tensor includes both the external fields and the fields generated by the currents along the string. These self-induced fields produce an electromagnetic backreaction which is formally divergent. We defer the discussion of an electromagnetic backreaction to a subsequent paper. The external gravitational and electromagnetic fields, of course, obey the familiar Einstein and Maxwell equations. Variation of the action with respect to A ~ yields the usual equation for the electromagnetic field, r"~;v = 4~rJ" ,

(2.3.9)

together with the gauge condition: A~;. = 0.

(2.3.10)

Variation with respect to g~. yields the Einstein equations:

G

8 G[rs + r ; v + r:m] =

8, Gf d2~l/Ch[[-hiJ(To

+ 10teOOld~)+ Oiq~ogq~]

X~(X ° - Z ° ( ~ k ) ) OiZl~OjZ v] Jr- 8~aTem , (2.3.11)

where we have used the relation: 3h ij 3g.~ = OiZp" OJZ~"

(2.3.12)

2.4. THECONFORMAL GAUGE

So far we have not specified the coordinates on the world sheet. On a two-dimensional surface it is always possible to choose a gauge such that the metric is that of flat space times a conformal factor. We call this a conformal gauge. Recalling (2.1.1), (2.1.2) we obtain the gauge conditions:

vrZ-h = _ OoZ~ OoZ~ = O1ZlzOIZ#, OoZ" OlZ ~ = 0.

(2.4.1) (2.4.2)

This coordinate condition is singular on a cusp since then 3oZ" becomes null while 31Z ~' vanishes by virtue of eq. (2.4.1).

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Cosmic strings

The equation of motion of a free string becomes a simple wave equation in this gauge. The equations of motion are still greatly simplified in the superconducting case as well. Eqs. (2.3.7) and (2.3.8) become

8 8~ °

[(

To _

( 00(~)2-~-( 01(~)2) 2 8oZ~OoZ~

gx,

8o z~ _

Oqot~ O01t~ 81z~ ]

81Z~SiZ gx~

0 [( (~0~)2+(01t~) 2) Or;1 T°-

2 OIZ" OIZ.

00t~01@

]

gx~ OaZ" OoZ~'aoZ~,g~ O°Z~

+qe i j Oiq~OiZvF ~ , __ -O,

(2.4.3)

a2~

a2~

(8~o)2

(8~1)2

qSoZ~81Z~,=O,

(2.4.4)

It is straightforward but tedious to verify that if Z ~ satisfies initially eqs. (2.4.1) and (2.4.2) and if Z ~ and ~ satisfy eqs. (2.4.3) and (2.4.4) then eqs. (2.4.1)-(2.4.2) hold. For the free string, it is often useful to use an additional gauge freedom and to define z o o -) = x ° = z, which is a solution of the free string equation. We point out that Z ° = ~" is not a solution for a string carrying charge carriers, or even for a nonconducting string moving in a curved background.

2.5. ELECTROMAGNETICRADIATIONFROMA STRING Jackson's analysis of the electromagnetic radiation from a particle moving along its world-line [9] can be generalized to yield the electromagnetic radiation emitted from a string moving along its world sheet. In this section, we make the gauge choice that t = • and 8oZ~O1Z ~ = O. We begin with the solution to the inhomogeneous Maxwell equation:

A~(x~') = 2 f d4x',,C~'(x'V)O(x°- x ' ° ) 3 [ ( x V - x'V)2] .

(2.5.1)

Substituting in (2.5.1), the string's current (2.2.2) yields,

.4.(x,)

= 2 f do

f

J"(o, )O(x o- zo(o,

[(x

z (o, (2.5.2)

For convenience, we define c ~ =- eij 8 iepOjZ ~ = ~

J~.

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D.N. Spergel et al. / Cosmic strings

I n o r d e r to evaluate,

a"A.(x~)= 2f do f d.~c.e(x°- Z°) a'~[(x~- Z~)]~

(2.5.3)

w e e x p r e s s the derivative of the ~ function at c o n s t a n t o as, x" - Z ~

d

)

OoZ (x -z ) a--;(

(2.5.4)

a n d i n t e g r a t e b y parts. T h e electromagnetic field tensor is then a function of the c u r r e n t a l o n g the intersection of the observer's b a c k w a r d light-cone a n d the string w o r l d sheet:

X ~)

=

f do f O-°c~(x"- Z~) -- O°---c"!x--~--Z")+__cZ OoZ~((~ -

c a OoZ"

z j ( aoZ~ ) ) ~

((~- zO aoZ~)~

×[(~- z.)aoOoZ.- OoZ.aoZ.]l ,

(2.5.5)

! ret

w h e r e ret m e a n s evaluated at x ° - Z ° = t - ~"= Ix - Z I = R. It is useful to define a u n i t null vector, n ~, such that x" - Z ~ = R(o, "r)nL T h e e l e c t r o m a g n e t i c field t e n s o r c a n n o w b e split into a velocity term which is p r o p o r t i o n a l to 1/R 2 a n d an a c c e l e r a t i o n o r r a d i a t i o n term which is p r o p o r t i o n a l to 1/R:

l oc n ocn cn c n

F."(~) = fdo ~ +fdo

(..<)~

R~ [

(n~12rt) 3

(nn2~n)2

+

l)

/'/uZvJ.ret

(n~2~n)3

(2.5.6)

~ ]Jret

T h e F o u r i e r t r a n s f o r m of the r a d i a t i o n t e r m in F ~",

F/ad(0~, ~ ) =

fFAd(t , ~)ei'~tdt

(2.5.7)

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C o s m i c strings

can be simplified by changing the variable of integration to T = t - R ( t , ~):

rr~a~(,~,a)= f

e,,~,,+",d~-f~

~oCgl,l v - -

OoCr/,/g

cVn u _

,,%

C,ttnr

]

(?./rtZrl)2F/vZr"]

(2.5.8)

Since the observer is far from the string, h is constant with time and the distance to any point on a string can be approximated in terms of r, the distance to a fixed point on the string: R ( z, o ) = r - ~ . Z ( z, o ) = r - "r - n"Z~. Note that d [ c~'n" ] = OoCt'n ~ - OoeÈn ~'

e"n~' - c~'n~n'TZ,

(2.5.9)

allows simplification of eq. (2.5.8) by integration by parts:

~fdofd'r(c~n"-c"n")exp[-i~o(nnZ,)].

i~oe i~r

F~'~(~,~)

(2.5.10)

The energy radiated per unit frequency per solid angle can by calculated from the Poynting vector, T op = F°XF~/4er: d 2Erad

d~0 dg2

(2.5.11)

r 2 n ~ T °~.

After a little algebra, eqs. (2.5.10) and (2.5.11) can be combined to yield: d2Erad d~0 dI2

~d2

= -4~r - f do f d,r(~×c)exp[-i,~(n~Z~)]

2. (2.5.12)

This expression is evaluated in subsect. 3.6 for a general cusp solution. 2.6. PARTICLE DISCHARGE Up to this point, we have viewed the cosmic string as purely one-dimensional. In order to study particle creation at and near the string, we must consider the finite width of the string. Particle creation occurs differently for bosonic and fermionic string so we will also need to consider the cases separately. Witten [2] described how particle creation can occur in fermionic strings when the Fermi energy of the charge carriers exceeds their vacuum rest mass. This condition defines the maximum current, Imax -- q m where rn is the mass of the charge carriers in the vacuum. Since the fermion gets its mass at the phase transition that formed the string, m can not exceed g~. ~ is the scale of the symmetry breaking that

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formed the string and g is the coupling constant associated with the symmetry breaking. For simplicity, we set g = q. The Fermi momentum of a charge carrier depends on its density per proper length of string. This is clearly not an invariant quantity, therefore we consider J"J~ which measures the square of the current density in the frame in which there is no net charge for spacelike J" and the square of the charge density in the frame in which there is not net current for timelike J~. Note that we can rewrite J~J~ as 2J+J_//x]-~, where J+ = qO+q~is the right-moving current, J _ = -qO_ep is the left-moving current. We define,

0_+= (01+ a0), o + = ~ - 2 ( o _ + r ).

(2.6.1)

1 2 Thus, if j+j_/f-L~> ~[rnax ' it is energetic favorable for a left-mover to scatter off of a right-mover and for both charge carriers to leave the string. This process occurs rapidly, since the mean free path for this process, -[(q2/m)~2J+J_/x/'Zh-m2/q2] -1, is much smaller than the length of the string. We can view this process as being nearly instantaneous. The particles created will barely have enough energy to escape from the string. Although they will be at rest in the frame in which there is no net charge (J0 = 0), to a distant observer, these particles will be emitted at ultrarelativistic velocities. In the vacuum, these charge carriers decay. The interaction of these carriers with their environment produces interesting astrophysical effects [5, 7]. In a bosonic string, dissipation of current through intense electromagnetic radiation will occur in the portion of the string where J"J~ exceeds I~a~. Where this occurs, it becomes energetically favorable for the string to undergo a phase transition from the superconducting to the normal state [2]. If J~J~< 0, then the current is a time-like vector and it is possible to transform to a frame in which there is no net current and the electromagnetic field is purely E-like. Strong E fields destabilize the vacuum and produce electron-positron pairs. A portion of the string with a net positive charge behaves like Z >> 137 nucleus: It would eject positrons and electrons would screen the current at that point on the string. Thus in equilibrium, the left-handed and fight-handed movers must have opposite charges and the current can nowhere change sign.

2.7. AN EXACT SOLUTION FOR AN INERTIAL STRING

As a simple application of the equations of motion developed in subsects. 2.1-2.4, we consider a string in a flat spacetime whose current-carrying field, q~, has a negligible charge (q = 0). This "inertial" string has no electromagnetic interactions, but its dynamics are influenced by the kinetic energy of the ff field. In particular, the inertial string can form a static and stable closed loop.

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858

Eq. (2.4.4) for q~ becomes the free two-dimensional wave equation,

02*

02*

0o 2

0~.2 = 0,

(2.7.1)

which has the general solution ~b(o, ~') = ~ + ( o + ~-) + ~ _ ( o - ~-).

(2.7.2)

Here ~+ and q~ correspond to left- and right-moving modes. For simplicity, we shall assume that q~_ = 0 (left-movers only), whence 1[( 00q~)2 q_ ( 01~)2] = 00q~01~ = n2 '

(2.7.3)

where n is a constant - it is proportional to the number of charge carriers per unit coordinate length. We seek a solution for the motion of the string in the form

'('1 ) .

l r(,c)q(o)

(2.7.4)

The constraint equation (2.4.2),

(i'r)(q. q) = 0,

(2.7.5)

suggests that the string is circular and q is a vector of constant length, which we normalize to unity, Iql = 1. The second constraint equation (eq. (2.4.1)) yields i 2 = ~2 + r2(q,)2.

(2.7.6)

Since q' depends on o only, it follows that (q,)2 must be constant, and if o ranges from 0 to 2~rL, this constant should be L. Substitution of (2.7.3)-(2.7.6) into the spatial part of the string equation of motion eq. (2.4.3) yields

q --~

To+--r- 7 -

~

n2L2) = 0 . -do---- 7 To r - - -

If we contract this with q and note that (q)2 = 1 implies q. q' = - ( q ' ) 2 find that d2s d-----5 + L-2s = 0,

(2.7.7)

~_

__

L-2, we

(2.7.8)

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859

where n 2L2 )

s('r)=

Tor-

-

r

.

(2.7.9)

Hence s oscillates around zero, and r oscillates around the value r0 = n L / ~ o . Of course, r cannot vanish: the loop is prevented from shrinking to zero by the angular momentum of the q~ modes. In particular, the static solution r = r 0 in which the centrifugal force on the q~ modes just balances the tension in the string is possible.

3. The cusp

In four dimensions, a solution to the free string equation of motion, Z~ = Z ~'', generically has at least two cusps every oscillation. The cusp occurs when the two null vectors, (0 0 _+ 0x)Z~ become parallel. At this moment, the velocity of a bit of the string reaches the speed of fight. Thus, it is not surprising that exciting physics occurs near the cusp. This section will explore some of the phenomena that occur near this point on the string world sheet. The cusp is an artifact of the low-energy effective theory. On length scales comparable to the width of the string, ~, physics associated with the fundamental field theory that produced the strings become important and modifies the cusp. As we will show below, the relativistic Lorentz factor at a distance 7/from the cusp is of order v/L/~I. Since in astrophysical applications, the string length is typically greater than 10 5o times the string width, we expect that our idealized analysis around the cusp is valid for "t < 10 25. In the first part of this section, a general cusp solution is developed for a free string. The second subsection considers particle creation near the cusp. The third subsection explores how the inertia of the charge carriers can prevent the formation of a cusp. In the fourth subsection, the electromagnetic power radiated from a cusp is calculated. 3.1. GENERAL CUSP SOLUTION We have defined the 0 coordinate on the string to be everywhere timelike. On a cusp there is no timelike direction and the cusp condition is: OoZ~ OoZ ~ = 0,

(3.1 .la)

i.e. aoZ~ becomes null. In the conformal gauge this yields O1ZtxO1Zl~ = O,

(3.1.1b)

which means that there is a coordinate singularity and that 01Z" = 0. At a cusp, the

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860

string velocity reaches the speed of light, 12~'1 = 0 and the physical length of the string per o, I Z ' l drops to O. W e define this point on the string world sheet as ~"= O, o = 0 and expand 2~ and Z ' a r o u n d this point: = a o + r a I + o a 2 + 'r2a3 + 2 0 " r a 4 + 0 2 a 5 ,

Z ' = 'rb 1 + o b 2 + r 2 b 3 + 2 o ' r b 4 + 0 2 b 5 .

T h e differentiability of Z: d 2 Z / d o : d ~ = d 2 Z / d r d o a2=b

1,

a4=b

3,

(3.1.2)

yields

a 5 = b,.

(3.1.3)

T h e free string equation of motion, Z = Z " produces further constraints: al=b

2,

a3 =b4,

a 4 = bs-

(3.1.4)

W e c o m b i n e these constraints with the requirement that 2 ~ - Z ' = 0 up to second order in ~" and o to obtain additional relations: a0.al=0, a0"a2=0, a 0"a 3=

_½(a

l.a l+a

2.a2),

a 0 • a 4 = -a 1 • a 2 ,

a 0 •a 5 = a 0 •a 3 .

(3.1.5)

T h u s near a cusp, the string acceleration vector ( a l ) and its length vector ( a 2 ) are spacelike vectors normal to the string's direction of motion. These conditions (3.1.3)-(3.1.5) permit us to represent the vectors a, and b~ in an orthonormal basis ( @ 82, 83): ao= 80 , a 1 = b 2 = ae I ,

a 2 = b 1=/~81 + T82, a3 = b4 = a5 = _ ½(a2 +/~2 + T2)80 + ~81 + E82, a 4=

b 5 = b 3 = -- ( a ] ~ ) 80 q- ~81 ~- ~82 .

(3.1.6)

D.N. Spergel et aL / Cosmic strings

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T o zero orderth, the cusp is described by a null vector. Two additional vectors orthogonal to the direction of motion are needed to characterize the cusp up to first order. Up to rotations, this amounts to three free parameters: a, r , and T. Four more parameters (8, e, 4, ~) appear in the second order expansion but these additional parameters are generally unimportant and do not appear, for example, in the calculations of the electromagnetic radiation from the cusp in lowest order. The general expansion around a cusp can be expressed as: ,g

(3.1.7)

Z=

Alternatively, this solution can be written in terms of o e = f~-~(o +_ ,r): 0+--0"

28{03 2

3

2

s

2

(3.1.8) The Lorentz factor, F, of the string near the cusp diverges only at a point: (3.1.9)

/~ = 121-a = la~¢ + aeo1-1

If either lall = 0 or [a21 = 0, the solution is special and the cusp occurs at a line on the world sheet rather than at a point. However, for a generic solution, only a small proper length of the string, A L = L / F 2, has a Lorentz factor in excess of F. These ultrarelativistic velocities are achieved for only brief moment of proper time, At = L / F 2. The area of the string world sheet that exceeds a given Lorentz factor,

A(F)=

[L/rfL/r¢J0

Z 2

dod,=

(3.1.10)

"0

is only a.small fraction of the total area. This small area, however, is responsible for most of the string's energy loss.

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3.2. CURRENT LOSSAT THE CUSP Consider a test current on a string with a cusp, i.e. we take the limit that the amplitude of the field q~--, 0 and there are no external fields. In this case Z ~ satisfies the free field equation and the constraint equations, eqs. (2.4.1), (2.4.2). The physically interesting question is whether the current diverges at the cusp. In the conformal gauge the square of the electric current in the string is:

j~j=q2(

( O°d'b)2

(Oldp)2) I 31Z~12 •

I 00Z~12

(3.2.1)

In general both terms diverge and since J"J~ is a scalar it will diverge in any other gauge as well. J~J, will be finite if 00q~= 01t~ = 0 at the cusp or if aoq~_ 91q~= 0 there. Eq. (3.2.1) can be rewritten in terms of the left- and right-moving currents:

J+J_ J~'J~,= 2 ~

- 2

q20+epO_dp ~/- h

(3.2.2)

If J + --- J _ = I, then discharge will occur when the string Lorentz factor reaches/"sat:

Inlax /-'sat = T

(3.2.3)

When the string achieves this velocity, it becomes energetically favorable for left-handed and right-handed movers to scatter and for their end products to leave the string. The heavy fermions quickly decay in the vacuum and their decay products interact with the surrounding magnetic field. The astrophysical implication of these discharges have been explored in refs. [5] and [7]. Since Z ~ and ~ have the same periodicity the same field ~ will appear when the cusp forms a second and third time. If the string discharges in such a case it is likely that the current will rearrange itself so that the particle emission will be less severe in future appearances of the cusp. Discharge will deplete equal numbers of left- and right-movers. Since their charges were produced in different regions of the string, there is likely to be an asymmetry in the ratio of left- to- right-handed movers. Therefore, annihilation will deplete the cusp region of one type of charge carrier and all the charge carriers near the cusp will move in one direction. When either J+ or J is zero, then their scalar product does not diverge; however, the four current that enters in the inhomogeneous Maxwell equation still

D.N. Spergelet al. / Cosmicstrings

863

diverges:

1 J+O+Z~-J_O_Z"

jr = -

2

(~-h

(3.2.4)

The charge carrier inertia term in the stress-energy tensor also diverges at the cusp. The next two sections explore these effects.

3.3. D O T H E C H A R G E C A R R I E R S P R E V E N T T H E CUSP?

The argument proceeds by contradiction. First we assume that the effects of the current on the motion are negligible, so that Z ~ obeys the free string equation

h ij a i adz ~ =

(3.3.1)

0 ;

then we show that this assumption leads (except in a special circumstance described below) to a divergent expression for the total energy of the string. According to eq. (2.3.11), the contribution of the q~ field to the total energy of the string is

= f d ~ 1 OZOl d-~ Ot ( Oid~oJ~ --

l hiJ

OmOOmd~) aiZ° OjZ O,

(3.3.2)

where we have assumed that the external spacetime is fiat and have used Minkowski coordinates x ~ = (t, x). The integrand in the second line above is to be evaluated at the value of 4 ° that makes Z ° (fo, 41)= t. The contribution of Ts°° (the tension term) to the energy is finite, since we know that cusps do occur in the absence of currents, and that of Tern °° = (E2 + B2)/8~r is positive. In the special conformal gauge for which 4 ° = Z ° = t, and writing o for fl, we have

G = fdo

(00,) 2 + (al~) 2 Gh

(3.3.3)

As discussed in subsect. 3.1 ff~h- = C - 2 oc ( o -

a~) 2 '

(3.3.4)

if the cusp occurs at (~0, ~1) = (to, oc), i.e. if we have chosen to evaluate E , at the instant t = tc in which the cusp occurs. Here F = [1 - (a0Z)2] -z/2 is the Lorentz factor of the string.

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D.N. Spergel et al. / Cosmic strings

Substituting (3.3.4) into (3.3.3), we see that the total energy will diverge unless

Oo~( tc, °o) = Ol~( tc, oc) = 0.

(3.3.5)

Now q~ obeys the equation of motion (2.15). If q = 0 (non-electromagnetic "charge" carriers), then it is clear that the cusp is not a special point for this equation, and there is no natural reason for (3.3.5) to hold. Of course, special initial conditions could be found for which (3.3.5) does hold, but this requires a "conspiracy". In this case, therefore, we can place an upper limit on Fma~ by requiring that E , not exceed the total energy of the string, which is -- 2~rLTo. Using (3.3.3) and (3.3.4), we obtain

/'max < 27rTo lim

-77- •

q--*0 Jtyp

(3.3.6a)

The expression in parentheses is the reciprocal of the mass per unit length contributed by the charge carriers. We can obtain a stricter, but less rigorous, estimate of the maximum F by requiring that E , is less than the tension in the cusp region LTo/Fm~. This implies, /'m2~ < 2~rTo lira

q-o /

.

(3.3.6b)

If q 4= 0, then the term involving F~, in (2.15) may be very large at the cusp because of the fields created by the particles themselves, so we must consider the possibility that the electromagnetic backreaction could quench the current. As we have been unable to treat the back reaction problem in general, we cannot say how much of the energy radiated comes out of the current itself (as opposed to the motion of the string). However, the global conservation of current described in subsect. 4.1 suggests that the energy radiated does come from the string's motion. 3.4. ELECTROMAGNETIC RADIATION FROM THE CUSP

Since the string velocity is largest near the cusp, this region will be the dominant source of electromagnetic radiation. We can use the general cusp expansion obtained in subsect. 3.1 in the formalism developed in subsect. 2.6 to calculate the power radiated. If there was no backreaction, a point of the string would have reached the speed of light; the inertia of the charge carrying field and the electromagnetic backreaction, however, limit the maximum string velocity. We include this effect in an ad hoe fashion by adding a t e r m Fm2el to the string's velocity near the cusp. This adds I'm2r to the first column in (3.1.7) and limits the string's maximum Lorentz factor to irma~. Subsect. 3.2 showed that the current near the cusp flows primarily in one direction in a fermionic string, thus we will set O0q~= 01~ = NL/L, where NIJL is the excess current moving in the left direction. The current near the cusp can now

D.N.Spergelet aL / Cosmicstrings

865

be expressed up to first order in o+: c ~ = qe q 0 / 0 0 j Z ~' = ~l~ qNL 0 Z ~' L qNL = ---L-- (1, - 1 , ( a

- fl)o_, -7o_).

(3.4.1)

The ultrarelativistic motion of the string will beam its electromagnetic radiation in the ~1 direction. Therefore, we can expand h, the unit vector along the line between the string and a distant observer: e=(l_

~ 2, 0cosq,, 0 sin,#) ~0

(3.4.2)

Substituting eqs. (2.5.12), (3.1.8), (3.4.1), and (3.4.2) in the expression for the incident flux yields,

d2Erad qZw2 NL~2 ~ d~oda2 = 16~r -L-) / d o + f

exp{liw[(O2+F~x)O+

do_

-2

- ( ( a + fl)cos • + T sin,)0o I + ~((a + fl) 2 Jr- ~2)O3 - (0 2 + Fm~) o_ + ( ( , - fl)cos * - V sin ~) Oo2-

- ½ ( ( a - fl)2+ T : ) o 3 ] } ( [ _ ( a - fl)sinq~- Tcosq~)]Oo , 2.

-Osineo+Y°-,(a-fl)°-+Oc°seo)

(3.4.3)

This integral can be expressed in terms of Bessel functions:

2 2

2

[

q~6~ N2-~-K?/3(~+)c°s21 2°~Q3+

d2Erad doo d~2

x

k 3R2- ] 0sin,~- ~__T

+

2~oQ3_ 2~oQ3_ ) [ xsin 2 ~ +K2/3(¢_)X~cos 2 ~

--fl) X- 2/3(~-) sing ~ ] j ,

2x -4 K 2

Ocoseo-(a-fl)-~_ ]2

(3.4.4)

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D.N. Spergel et al. / Cosmic strings

where P + = O 2 + l~m~,

Q_ = - o [(a + B)cos 4, + "~sin,C,],

R_+=(a+B):+C X± =

R+ (t~±P++_- Q 2 ) 3/2

~ + = co

(3.4.5)

3R 2

It is useful to define cutoff frequencies, co~ such that ~+(co+, 0 = 0) = 1: 3 3rm'ax

C

(3.4.6)

a n d cut-off angles such that ~±(w << wc, 0c) = 1:

coL

1/3

(3.4.7)

Recall the asymptotic behavior of the Bessel functions:

for x << 1

K.(x)~]

2 tx] _ _ e-X

(3.4.8) for x >> 1.

T h u s the power falls off exponentailly for co >> coc and 0 >> 0c. W e can n o w evaluate the peak intensity,

q2NZLI.(~]2F(2)2RS3R_2

d2Erad -

568~r

" "

(3.4.9)

D.N. Spergel et al. / Cosmic strings

867

and the total power emitted: Erad = f0'°cdw ~-d--~Odd2Erad 2

N o t e that if the cusp was not truncated, the total power would have diverged. In a bosonic string, the current may be nearly uniform up to Fsat, the Lorentz factor at which the string goes normal. In this case, c ~ = qa0~ a l Z " .

(3.4.11)

Again, the radiated flux can be expressed as Bessel functions. The power emitted has the same dependence on F in both cases.

3.5. COMPARISON OF POWER RADIATED IN ELECTROMAGNETICAND GRAVITATIONAL WAVES The result (3.4.10) shows that the electromagnetic energy radiated in the vicinity of a near-cusp scales as the maximum Lorentz factor, Fmax, that is achieved. Hence the energy radiated by a true cusp (Fm~x ---' oo) would be infinite. On the other hand, Vachaspati and Vilenkin [10] have shown that the total power radiated in gravitational waves by a nonsuperconducting string forming a cusp is finite. Yet we know that the formal similarities between electromagnetic and weak-field gravitational radiation are very strong. In this subsection, we shall explain the origin of this important difference between these two forms of emission at cusps. Consider a near-cusp that reaches maximum Lorentz factor Fma~ over coordinate distances and times /Io~ - L / F m ~ , ,

AT e - L / F m ~

(3.5.1)

Atc - L/r m x

(3.5.2)

and proper distances and times Arc -

L/r m x,

(see subsect. 3.1 for these scalings). If we assume that the cusp dominates the electromagnetic emission, then the total energy radiated as viewed from the approximate rest frame of string from the near-cusp can be estimated using Larmor's formula (cf. ref. [9]): A/7,em --(s)_ EAtc

Q2i~2At c

(for c = 1).

(3.5.3)

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D.N. Spergel et aL / Cosmic strings

Here we should use for Q~ the total charge in the cusp region, which is

qUL Q = q - ~ Ao~

['max ,

(3.5.4)

where we have assumed, as in subsect. 3.4, that left-moving charge carriers dominate. Furthermore, t3 is the proper acceleration, and since F changes by order unity during the cusp event, we can estimate this as

(3.5.5)

e - c/ tc- r mJL. Combining (3.5.3), (3.5.4), and (3.5.5), we find that

The energy radiated as measured in the "lab" is Fm~x A E(S~, in agreement with (3.4.10). In the gravitational, nonsuperconducting case, (3.5.3) must be replaced by AE(S) grav

-

GMc2v2i~2Atc

(3.5.7)

This can be "derived" either from the quadrupole formula / ~ - G l d 3 I / d t 3 1 2 by putting d S l / d t 3 - d 3 ( M x x ) / d ~ ' 3 - M v b , or by reasoning that the conserved "charge" in this case is the momentum M v and hence replacing Qc with G1/2Mv in (3.5.3). For M, we use the total mass in the near-cusp,

Mc = 7"oat - 7"oL/CLx.

(3.5.8)

Note that in comparison with (3.5.4), we have here an extra power of Fm~, which is squared in (3.5.7), and so we obtain the estimate LIE grav ~2~ - GT2L/F2max - e M c

(~ =- GTo).

(3.5.9)

This would lead to zlE 0ab) ~x/-,~1. However, (3.5.9) is only the energy radiated by the immediate vicinity (3.5.1) or (3.5.2) of the cusp; since Vachaspati and Vilenkin [10] showed that the total radiated energy is finite and nonzero when Fm~, ~ ~ , we conclude that the total emission is not dominated by the immediate vicinity of the cusp. The final form of (3.5.9) shows, however, that the portion of the emission attributable to the near-cusp itself is of order e times the mass-energy available in the cusp. Had the local emission been larger than eM c by a positive power of Fm~x, then we would have concluded that gravitational backreaction prevents the cusp.

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D.N. Spergel et aL / Cosmicstrings

Our actual result suggests that the gravitational radiation backreaction will not prevent cusps.

4. The origin of the current in the string This section explores the origin of the current on the string. The first subsection will show that if there is no discharge from the string, then the string will behave like a superconductor: the induced current depends only on the flux initially threading the loop and the imposed external field. The second subsection shows the violation of the assumption of no particle absorption or emission modifies this conclusion. The string can, in fact, exploit the chemical potential due to the baryon asymmetry and produce a current. 4.1. STRING IN AN EXTERNALMAGNETIC FIELD The interaction of a string with the electromagnetic fields can be understood in terms of the dynamics of a circuit. Integrating eq. (2.3.8) around the loop yields, ×B).O/

J.dl

_

q2

d~

(4.1.1)

dt '

where dl = ~-L--h-(oZ/~o)do, o_L is the string velocity perpendicular to its length, and ~ is the magnetic flux through the string loop. Chudnofsky et al. [3] split ~ into two pieces: an external flux, ~ext, and ~selr = M J . M is the properly weighted loop inductance and J is the average current: J = ( 1 / 2 ~ r L ) ~ J . dl. The circuit equation can now be reexpressed as:

dI,

t ] dOex ~-

-d-TtJr\2~r L +1 Y-dl = - q

.

(4.1.2)

The equation implies that the string behaves like a laboratory superconductor: If external flux is added and then removed from the loop, no net current is induced. On the other hand, if as Ostriker, Thompson, and Witten [4] suggested, the string was formed threaded by primordial flux, then a permanent current was established on the string. Field and Press (private communication) have shown that eq. (4.1.2) implies that the radiation reaction does not reduce the string's current. If the string is truly periodic, then after every oscillation, it returns to the same shape and M and d~ext/dt are unchanged, thus ~J- dl will be conserved.

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D.N. Spergelet aL/ Cosmicstrings

4.2. BARYONNUMBER CONSERVATIONAND THE GENERATION OF CURRENT Witten [2] discusses a superconducting string produced by the breaking of O(10)---, S U ( 5 ) × U(1). In this model, this symmetry breaking gives mass to the quarks. The u and fi quarks travel in one direction on the string, while the d, d, e +, and e - travel in the other direction on the string. If a hydrogen atom is near the string, then it is energetically favorable for its composite fermions to "jump" onto the string where they are massless modes. The string can thus convert external baryon number into current. Particle discharge is the reverse process which converts current into baryon number. The string's baryon number, ~stnng, depends on the current on the string: ~stn.s = ( 1 / q ' ) ~ J . d l . Noting that emission or absorption of particles does not change ~sui,g + ~ x t , where ~ x t is the baryon number of the universe exterior to the string, we add an additional term to the circuit equation:

~[~(q2M+l)J'dl]

q 2--~ d [~ext+-~'xt] '

(4.2.1)

where q' is the net charge per baryon number. For Witten's O(10) model, q ' / q - 8 Since there is a net cosmic baryon asymmetry, the string can exploit this "chemical potential" to generate a current. The rate of growth of the number of charges, 2q, depends upon the number density of the baryon asymmetry, n~-10-6(1 + z) 3 cm -3, and the string cross section for particle capture, Scapt IV = n ~ S captC .

(4.2.2)

We will estimate Scapt ~ L)t, where ?t is the Compton wavelength of the charge carriers of mass m. Note that this process is the inverse of particle emission. When the current on the string is less than/max, it is energetically favorable for particles to move onto the string. When the current exceeds/max, particle emission is energetically favored. The string will continue to accumulate charges until the total number along string exceeds Nma x ---L/?t. Once this critical current is reached, it is no longer energetically favorable to add charges to the string. Using the standard hot big-bang values for the time-temperature relation, we can estimate how rapidly saturation current is achieved: d ( N d(logT) ~

)

"'~l° - 2 T , --- 10 mGev ~V,

(4.2.3)

where m oev is the charge carrier's mass in GeV and TGev is the black body temperature of the universe in GeV. If the charge carriers are light quarks, the string reaches saturation current in a fraction of an expansion time. This mechanism can produce a current in a string with more massive charge carriers if there is a net

D.N. Spergel et aL / Cosmic strings

871

baryon number associated with the massive fermions. Note that this mechanism would be ineffective for inducing currents of the strength required by Ostriker, Thompson, and Witten [4], whose charge carriers must have mass, m = 1015 GeV. A variant of this current generation mechanism requires an asymmetry in the string's cross section for particle capture. A slight difference between the particle and anti-particle capture cross sections would be sufficient to induce the maximum current in the most massive strings. The particle emission mechanism may also provide a channel for inducing net current from an external field. If an external field is applied to the string, and some of the induced current is emitted as particles at the cusp and these particles escape from the string environment, then when the field is removed, a current may remain on the string. Thus for a superconducting cosmic string, the generation of new flux is possible only when dissipative effects are included. Astrophysicists are familiar with similar constraints for the generation of new flux in M H D plasmas. 4.3. GENERATION OF AC CURRENTS BY EXTERNAL FIELDS

The flux conservation theorem derived in section 4.1 is a global theorem on net current. It does not prevent the generation of variable ("AC") currents along the string. Consider a string in flat space with external magnetic and electric fields. We assume that the inertia of the charge carriers, as well as the electromagnetic forces acting on the string can be neglected, the string behaves then like a free string. Following ref. [11], we describe the string, in its center of mass rest frame as: t ) ½[a(o_) + b ( o + ) ] '

Z"=

(4.3.1)

where e_+ = V~-2(° + ~') and la'l = Ib'l = 1. Substitution of eq. (4.3.1) to the equation of motion of the charge carriers' field ep, eq. (2.4.4), yield the current generation equation:

0%

02¢

012

002 = ½q(b' X a ' ) " B c m - ½q(a'+b')"Ecru,

(4.3.2)

where Ecru and Bcm are the electric and magnetic field in the string center-of-mass rest frame. a and b are periodic functions, with a period 2~r. We expand the derivatives, a' and b' as:

a'= E a , e i"a-= E a , e i"(°-'), n

b'=

Eb.e

ino+

?1

(4.3.3)

I1

- Eb.e i"(°+O. __

n

(4.3.4)

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D.N. Spergel et al. / Cosmic strings

We consider, now, homogeneous and time independent external fields B and E. We expand q~ as: q~(~-, o) = E q , n(~')e in°

(4.3.5)

n

to obtain

~n + n2dpn= - ½q[an" Ee-in~ + bn.Ee in" E(bm>(an_m)Bei(2m-n)r].

(4.3.6)

m

We observe that there the first two terms, on the 1.h.s. of eq. (4.3.6), are resonant and therefore generate an AC current whose amplitude grows linearly with time. The current built up moving with velocity v for time At is of order: q2BcvAt (or order 3 x 1014 A, for B = 10 -9 gauss and q2 = a, the fine structure constant). The oscillating current induced may be up to vAt/L larger than the net current due to the initial flux threading the loop. In a homogeneous background field, the third term gives rise to a DC current whose magnitude oscillates in time. In an inhomogeneous background field, the third term can produce an AC term whose amplitude may be as large as cAt/L. The energy sources for these currents are the translational motion of the string and its kinetic energy in oscillatory motion.

5. Astrophysical implications This paper has developed a formalism for studying the dynamics of superconducting cosmic strings and has begun the exploration of the physics of these objects. This section explores some of the astrophysical implications of our investigation. There are several mechanisms that can generate current along the string. If an initial flux exists inside the loop, then when this flux is removed, a current is induced such that the flux within the loop is 1/(1 + a I n ( L / , / ) = 0.5) of the initial flux. The string can also convert quantum number into flux. In fact, it can exploit the net baryon asymmetry to generate a current of up to 1014 A! Whether the string can carry this large current is determined by the mass of its charge carriers. If the loop moves through a region of magnetic flux, a DC current is induced in the string. However, when the string leaves this region, this DC current is removed. There are, however, resonant AC terms that can grow secularly when the string moves through a region of uniform flux. These AC modes are exactly the oscillation modes of the string. Discharges, either at the cusp or when the current is spacelike at some point along the string, will convert some of this AC current into DC current. The amplitude of the AC terms generated by a loop of length L moving with velocity v for time At in an external magnetic field is likely to be vAt/L larger than the D C current due to the initial flux threading the loop. The radiation due to these AC terms is likely to be more isotropic than that predicted from a DC term and will be important in the Ostriker, Thompson, and Witten model [4]. The AC

D.N. Spergel et al. / Cosmic strings

873

current will also produce the energetic discharges evoked in [5-7]. As the string shrinks, both the AC and DC modes grow as 1/L. The current in the string will continue to grow until one of two possible equilibrium states are reached. In a bosonic string, if/t', the mass per unit length of the string in the normal state exceeds 2/t, the mass per unit length of the string in the superconducting state, then the loop will eventually freeze: The kinetic energy of the charge carriers and the magnetic field pressure of the flux in the loop will balance the tension of the string (a similar conclusion was reached by Copeland, Hindemarsh and Turok [12]). On the other hand, if the string is fermionic or if it is 1 p bosonic and /t > ~/~, then the pressure terms are too feeble to balance the tension terms and the inevitable fate to the string is evaporation into gravitational and electromagnetic radiation. In this later case, the current in the string will eventually reach a value at which its growth due to the shrinkage of the string is balanced by the loss of current at the cusp. If the cusp precesses rapidly, so that every oscillation, a new piece of current is lost, then the current will equilibrate at ALIm~/L, where A L is the length of string lost at the cusp. If on the other hand, the cusp precesses slowly, then the same charges appear at the cusp every oscillation and very little current is lost. This allows the average current to reach a much higher value. Several important processes occur near the cusp. As the string's velocity increase, its length shrinks as the kinetic energy of its charge carrying field increases. There is a critical Lorentz factor, F s a t = lmax/I , above which the current saturates. If the string is bosonic, then it may become energetically favorable for the string to flip to the normal state. If the string is fermionic, then the charge carriers begin to escape from the string. These charge carriers are at rest in the frame of zero current. This frame; however, is moving rapidly relative to a distant observer. The charge carriers are emitted from the string with energy F s a t m from a portion of string of length %at" The average number of charge carriers per unit length is N/2~rL, thus the string will radiate (N/2~rL)%atFsat m in particles. Recalling that a = L/F, we can reexpress this emission rate as Nm/2~r. This conclusion will modify the estimates of Hill, Schramm and Walker, who ignored the beaming of radiation emitted at the cusp and the large relativistic boost given to the emitted particle. Since the string will lose equal numbers of left- and right-moving charge carriers near the cusp, it is likely that all of the remaining charge carriers will be moving in one direction. There are two processes that prevent any portion of the string from reaching the speed of light. As the string's velocity increases, so does the kinetic energy of the charge carrying field. The backreaction of this field is included in the equations of motion described in sect. 2 and will limit the velocity of the string. This paper did not consider an additional backreaction due to the emitted electromagnetic radiation. Since the energy associated with the kinetic energy of the charge carriers exceeds the electromagnetic energy and since both scale in the same way, we suggest that the inertial backreaction is the dominant force limiting the string's maximal velocity. We define Fbr as the maximal F achieved by the string before it is stopped

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by a backreaction. If Fbr exceeds Fsat, then the string will emit particles before the backreaction becomes important and particle emission will be the dominant mechanism for energy loss. If Fsat exceeds Fbr, then the backreaction will stop the string from reaching saturation current and the loop will evolve towards the stationary state described earlier. In this later case, energy will be emitted only in the electromagnetic radiation. Most of the string electromagnetic radiation is emitted in a powerful burst of low frequency radiation. (A conclusion reached independently by Vilenkin and Vachaspati [13].) This radiation is emitted in a beam of width 1/Fc, where /'c----min(Fsat, /'br)" The duration of the burst is brief, At ~ - L / ( c F 3) and it is radiated in the direction of the string's motion. This beam radiation is of little use for creating primordial galaxies [4]. Vilenkin and Field [6] have suggested that these collimated beams could be the jets of quasars. This model requires some (as yet unknown) mechanism for aligning the cusps. Some of this energy may be visible in the form of T ray bursts [7]. The formalism developed in this paper can be used to numerically calculate the power radiated in the isotropic component required by the Ostriker, Thompson and Witten model. There are several important questions about the dynamics of superconducting strings that are still unresolved. There is a need to properly treat the electromagnetic backreaction. It is important to understand how this backreaction (together with the backreaction due to the inertia of the charges) will effect the near-cusp region. Numerical integrations of the equations of motions may resolve under what conditions Fbr exceeds Fsat and the rate of precession of the cusps. It is also important to determine if the new terms in the equations of motion are likely to produce frequent self-intersections. While this is not likely far away from the cusp where the current is small, it is certainly plausible when these terms are important in the near-cusp region. Superconducting strings, if they do exist, could be powerful sources of electromagnetic and particular radiation. This paper has developed a framework for the investigation of the dynamics of superconducting strings that will hopefully be of use to workers in the field. We have benefitted from discussion with A. Babul, J. Ostriker, B. Paczynski, W. Press, C. Thompson, and E. Witten. T.P. thanks the Aspen Center for stimulating his interest in cosmic strings. This work was supported by a W.M. Keck Foundation Fellowship (held by J.G.) and by NSF grant PHY8217352.

References

[1] A. Vilenkin,Phys. Reports 121 (1985) 263 [2] E. Witten,Nucl. Phys. B249(1985) 557 [3] E.M. Chudnovsky,G.B. Field,D.N. Spergel and A. Vilenkin,Phys.Rev. D34 (1986) 944

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[4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

J.P. Ostriker, C. Thompson and E. Witten, Phys. Lett. 181 (1986) 243 C.T. Hill, D.N. Schramm and T.P. Walker, Fermi.lab preprint (1986) A. Vilenkin and G.B. Field, CFA preprint, submitted to Nature (1986) A. Babul, B. Paczynski and D.N. Spergel, Princeton Obs. preprint, Ap. J. Lett. 316 (1987) L49 C.T. Hill and L. Widrow, Fermilab preprint (1986) J.D. Jackson, Classical electrodynamics (Wiley, New York, 1975) T. Vachaspati and A. Vilenkin, Phys. Rev. D31 (1985) 3052 N. Turok, Nucl. Phys. B242 (1984) 520 E. Copeland, J. Hindemarsh and N. Turok, Phys. Rev. Lett. 58 (1987) 1910 A. Vilenkin and T. Vachaspati, Phys. Rev. Lett. 58 (1987) 1041

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