- Email: [email protected]
Cosmic vortons
Nuclear Physics B323 (1989) 209-224 North-Holland, Amsterdam
COSMIC V O R T O N S R.L. DAVIS*
Institute for Theoretical Ph.vsics, Unit,ersi(v of Califi~rnia, Santa Barbara, CA 93106, USA E.P.S. SHELLARD**
Center for Theoretical Pl~vsics, Laboratory for Nuclear Science, and Department of Pl~vsics, MIT, Cambridge, MA 02139, USA Received 21 November 1988
We study the production of vortons in the early universe. Vortons are rings of vortex stabilized by charge, current and angular momentum. They are able to carry all of the quantum numbers of ordinary point particles. We discuss a mechanism by which they may be produced during the damped epoch following a string-producing phase transition. We estimate their density, discuss their stability, and remark on some possible cosmological consequences. We look at two examples where vorton production may be significant: the OTW scenario for explosive formation of large scale structure and electroweak strings in the sun.
1. Vortons In type II superconductors it is common to find vortices, quanta of magnetic field confined into flux tubes running along string-like topological defects. In superfluids similar vortices, but without the magnetic flux tubes, are readily observed experimentally. It would not be surprising if there were other, similar vortex phenomena in nature. One of the most interesting possibilities is that vortices, or cosmic strings, appear in a cosmological phase transition [1]. The resulting network of strings could have influence on the observable features of the universe, and a wide range of such effects are currently being studied and tested. Extensions of the simplest models involve extra degrees of freedom which couple to the vortex [2]. This leads to currents which are bound to the vortex, essentially degrees of freedom living on the two dimensional space-time of the vortex worldsheet. The currents may be neutral, or there may be background gauge fields, such as that of electromagnetism, coupled * On leave from Tufts University, Department of Physics and Astronomy, Medford, MA 02155, USA. Research supported under grants NSF PYI 525135, NSF PHY82-17853 and supplementary funds from NASA. ** Research supported by Department of Energy Grant DE-AC02-76ER03069. 0550-3213/89/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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to them. A possibility which has not yet received much attention is that the vortex may have a spinning particle-like soliton configuration - a vorton state [3, 4]. A vorton is a stationary ring of vortex whose tension gets balanced by the stresses and angular momentum caused by trapped worldsheet charges. Its radius is typically
R v - N/x/~ ,
(1.1)
where tt is the string tension of the underlying vortex, 1 / f ~ - is roughly the thickness of the vortex, and N is an integer specifying the number of particles trapped on the worldsheet. As a very special case one could imagine the vorton with zero angular momentum but with N very l a r g e - so that the vorton may even extend over astronomical distances. These have been called cosmic springs [5-7], and if they can be formed they would give a constraint on cosmological models that combine superconducting cosmic strings with primordial magnetic fields [8]. We will make some comments on springs near the end, but in this paper we are mainly interested in the much larger class of microscopic vortons which spin [3, 4]. These are localized classical solutions to the equations of motion which have a time independent stress-energy tensor - they are stationary and do not radiate classically. At distances large compared to R v they look like point masses with quantized electric charge and angular momentum; in many ways they are like ordinary particles, hence the name vorton. We can summarize a mechanism for the formation of vortons as follows: In the early universe a phase transition will produce a brownian network of vortices as long as the unbroken subgroup is not simply connected. For a period of time after the phase transition the motion of the vortices is strongly damped [1, 9,10]. Vortons may be produced because initially irregularly shaped loops become circular as they relax, their motion damped by the surrounding medium. If there are net quantum numbers in bound states on the ring then vortons are formed when the stresses encountered on squeezing down the trapped charges are large enough to counteract the string tension. Once formed, the vortons would behave like non-relativistic particles, which might later decay. In this report we will examine their role in cosmology. We start by discussing the above mechanism in more detail in sect. 2. After that we look briefly at vorton stability, estimating the vorton lifetime. In sect. 3 we discuss several implications, and sect. 4 are some closing remarks.
2. Vorton formation
Cosmological phase transitions can give rise to a variety of topological defects - monopoles, strings, domain walls and their hybrids. They are an unavoidable consequence of spontaneous symmetry breaking in the standard cosmological
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model: since the present horizon contains many smaller regions which were causally disconnected at early times, a symmetry breaking occurring in the past cannot lead to coherence across the entire horizon at later times [1]. This lack of coherence requires that topological defects exist whenever the vacuum manifold is not simply connected. Vortons can be produced by a similar mechanism, though because they are only partially topological their formation requires further assumptions that we will outline below. To create them we need the following: (1) First, a network of strings must be formed. Such networks have been studied in cosmic string models and we will use those results freely. The main consequence is that a very large number of irregularly shaped brownian string loops are formed at a phase transition. (2) There must also be either bosonic or fermionic bound states on the strings. This may seem an arbitrary requirement, but in the fermionic case it is actually quite a natural consequence of Yukawa couplings between fermions and Higgs fields. If there are b o u n d states then a string loop can carry current and charge quantum numbers. We will explain below that on formation loops necessarily have non-zero values of these charges. Requirements (1) and (2) combined mean that the early universe can be a source of a very large number of string loops with worldsheet quantum numbers. However, these loops will not be circular or stationary to begin with; they are highly excited states of a vorton. Thus the final condition, that this excited vorton can relax into a stationary circular vorton state is crucial: (3) the quantum numbers must not all leave the worldsheet as the loop loses energy and shrinks. As long as some q u a n t u m numbers remain then a loop of initially arbitrary shape will settle into a final vorton state. This last condition is the most difficult to satisfy because as current (in this section we use the term current loosely to mean ordinary electric current, electric charge density, neutral currents or charges, or a variety of more exotic quantities playing similar dynamical roles) builds up in a shrinking loop it becomes easier for the charge carriers to move off the string. If we define the vorton current to be the current needed to counter the string tension, then a vorton can only exist if at the vorton current the likelihood for the bound quantum numbers to leave the string is small. Stability of the stationary vorton will be discussed in the following section; here we will assume that it is sufficiently long-lived to be cosmologically interesting. Apart from vorton stability is the question of whether or not the brownian loops initially formed can settle into a vorton state. For this we must in general require that the charge carriers stay bound to the vortex while it undergoes typically violent motion. For example, a freely collapsing loop can bounce at some finite radius if the energy in the current becomes large enough to stop the collapse. If the initial current was much less than the vorton current then the maximum current at the turn-around point will be much larger than the vorton current, and if the charge carriers can leave the string at this high current then the vorton will not form. Likewise, a loop that is not freely collapsing but initially oscillating will undergo violent, relativistic
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motion as it decays. A current will be driven by these oscillations and can be expected to reach values much larger than the final equilibrium current in a stationary vorton. At cusps, kinks, and other regions of high local curvature there are especially likely to be losses. The range of models that produce a significant density of vortons would therefore be severely restricted (because relativistic motion would require the string to carry currents much higher than the vorton current without losses) except for the fact that the string motion is strongly damped by the surrounding medium for a period of time following the phase transition [1, 9,10]. As long as the string is over-critically damped its current need not build up to levels significantly larger than the vorton current, and the likelihood that the loops relax into a vorton with nearly the same quantum numbers is vastly greater than in the undamped case. For this reason we will focus our attention on a mechanism for vorton formation in the damped epoch.
2.1. THE PHASE TRANSITION The formation of cosmological string networks is discussed by Vachaspati and Vilenkin in ref. [11]. They simulated the network by assigning random values for the phase of the Higgs field on a large cubic lattice with spacing corresponding to the correlation length ~. They then traced the resulting strings by searching for windings of 27r around faces of the cubes, and determined the string distribution. Not surprisingly, the strings were found to be brownian in nature, that is, if R is the distance separating two points on a string then the average length L of string that joins them is given by L = R2/~.
(2.1)
The simulations also demonstrated that about 80% of string length is in open strings and the remainder in a scale-invariant distribution of closed loops. The number density of loops of average radius R between R and R + dR was given by n ( R ) = vR
4dR,
(2.2)
where v = 6 + 2 . The next step is to discuss our expectations for the current and charge that become trapped on the strings. To simplify the discussion we assume that the charged particles become non-relativistic at the same temperature that the strings form, and that the associated coherence length ~ is the same. We will also assume a second order phase transition, so our mechanism is only strictly applicable to that case. A string segment of length L will pass through approximately - L / ~ uncorrelated regions, so the net particle number would be N ~ ( L / ~ ) 1/2. For particles with charge e the net charge is thus Q ~ e ( L / ~ ) 1/2. Further, the velocities fluctuate in the same manner so the net current is, independently, J ~ e ( L / ~ ) 1/2. For a
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brownian loop of average radius R and length L we get typically N ~ V/~
- R/~.
(2.3)
The charge carriers may be either fermions or bosons. The precise relationship between the particle number and winding number for bosonic charge carriers will be summarized in the next section. The key feature of eq. (2.3) is that because we expect ~ - T¢ 1, the thermal wavelength at the phase transition, and because the thickness of the string 1 / ~ - is also typically -T~ -l, this equation together with eq. (1.1) imply R - Rv ,
(2.4)
or in words, the size R of a brownian loop when it is formed is typically the radius of a circular vorton with the same particle number N. This has important consequences as will be seen below. It should be noted that the net charge on a single loop of size R is the same order of magnitude as the total charge due to random fluctuations inside the volume of radius R. This is because the net charge inside the volume R 3 is determined by the surface fluctuations, also - R / ~ . That there are many many other loops inside the volume, each with a net charge, is not paradoxical because charges between different loops will be correlated. 2.2. THE DAMPED EPOCH Given that loops acquire a net charge and current we can now consider their subsequent evolution. The phase transition is followed by a period when string motion is strongly damped by the surrounding medium. Assuming negligible entropy production, this damping period lasts for several orders of magnitude in time until [10]
t, - (G
)-ltc.
(2.5)
During the damped phase strings will straighten out with the initial small scale fluctuations disappearing before larger ones, the coherence length growing from ~c << tc to ~. - t.. String motion can be overcritically (or just critically) damped and not oscillating or subcritically damped and oscillating. Following ref. [10], for overcritical damping the velocity of a section of string with local radius of curvature R is determined by setting the force due to tension - I ~ / R equal to the frictional force - T3v, giving v R - t~/T3R.
(2.6)
For R < i t / T 3 this relation breaks down, meaning the tension is larger than the frictional force and the string will undergo damped oscillations on that scale. Since
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initially R >_ Tc t the entire network will be overcritically damped to begin with, and the time it takes for a fluctuation of size R to move a distance R,
is the time it takes for that scale to straighten out. Without worldsheet currents and charges an initially brownian loop of size R will have collapsed in time ~-. Since small scale fluctuations get smoothed out first the loop becomes more and more circular as it collapses, proceeding from a highly damped period of slow collapse and rapid smoothing to a final stage where the radius is small enough that eq. (2.6) breaks down and the loop relaxes subcritically damped. As it gets smaller and the temperature goes down the frictional damping turns off and the loop oscillates, damped only by radiation. With worldsheet currents and charges the picture is different. Assuming that charge and current losses are negligible, and recalling eq. (2.4), the smoothing period results in a nearly circular loop already at or close to its equilibrium radius. An originally brownian loop can thus reach its vorton state without ever collapsing significantly or undergoing relativistic motion. This overcritical damping mechanism for vorton production is correct, however, only if at the end of the smoothing time the loop, originally of size - R v, has undergone negligible cosmological stretching. If there is stretching then when it becomes smooth the loop is at some radius larger then R v. Because string fluctuations at that scale are just passing over to subcritical damping, it will therefore have to relax by undergoing damped oscillations around R v. If the stretching is significant then the loop will not have time to relax before damping begins to turn off at that scale, so the final stages of relaxation must involve violent oscillations damped only weakly by radiation. Stretching becomes important when the smoothing time is comparable to the Hubble time, or, equivalently, when the string velocity on the scale R in eq. (2.6) is comparable to the Hubble velocity - R / 2 t . This means that the worldsheet quantum numbers on loops with R >>
2(Gl~)l/4tc
(2.7)
will have to survive some relativistic motion if a vorton is to form. For these loops the initial smoothing process and frictional damping continue to act until ~ R - t,. We can therefore identify three regimes in which vorton formation should be considered. The first, and most likely to form vortons, is the overcritically damped period in which cosmological stretching is small. Brownian loops of size R get smoothed out into vortons of roughly the same size. Some oscillations may occur as long as the string does not reach relativistic speeds. Even delicately balanced vortons unstable to relativistic perturbations should form. In the second period the loops are initially overcritically damped as they get smoothed out, but because
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stretching has given them a much larger radius than a vorton with the same quantum numbers they must relax by undergoing significant oscillations that are subcritically damped. Since points along the string would reach relativistic speeds, a vorton would have to be somewhat robust to come out of this period. In the third period irregular loops come into the horizon later than t,. They must undergo violent, irregular, relativistic motion damped only by radiation. These are the least likely to form vortons. There will be a smooth cross-over from one regime to another, so it is difficult to estimate what the cross-over points are. Using eqs. (2.7) and (1.1) we can probably conclude that loops with N c r - 2(G/.t) 1/4
(2.8)
will relax overcritically damped. We have assumed entropy is not produced in a significant amount, but if it were then this early epoch is just that in which most would appear, and this would tend to increase N~r because damping would decrease more slowly. As stated, loops with N > Nc~ will have to be somewhat robust in order to survive some relativistic motion. However, as long as they relax in the second regime these loops are strongly damped at first and are more likely to stabilize than those that break free from the network after t,. Taking into account stretching, the maximum value of N for loops which feel significant damping is
Nm~_ t~,t~_(Gt~ ) a
(2.9)
Now let us suppose that vortons are produced relatively efficiently in the damped phase. It will be shown in the next section that there is usually a value of N below which the vortons are quantum mechanically unstable. Assuming Nmin < NmaX and using eqs. (2.2) and (1.1), their energy density can be estimated. The mass of a vorton of size R is - / , R so the energy density in vortons at t c would be
fR.,~ dR E - - ~P]Rmm "-~
1~2v N2in ,
and since they redshift like matter
[r
(2.1o)
N~in I Tc We have said nothing about loops breaking up or joining together. If a loop were to break then the pieces would still satisfy eq. (2.4) and would relax according to the preceding discussion. It is the same if two were to come together. The physics of these processes is complex, though in the end we expect a scale invariant spectrum
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of vorton radii. As a result, the value of ~ in eq. (2.10) may be different from that in eq. (2.2). The mechanism described here makes it plausible that vortons could form even when unstable to highly relativistic perturbations. The uncertainties involved prevent us from making a precise prediction, but even if an exceedingly tiny number of vortons were produced the cosmological implications would be serious. To place this in perspective consider the related monopole problem. Monopoles, topologically stable knots arising in G U T models, proved to be a major obstacle in reconciling such models with the standard cosmology [12]. Monopoles would appear in the early universe because the correlation length ~ of the Higgs field is bounded above by the causal horizon /H(t) -- 2t. A density in monopoles of one per horizon at the G U T time, redshifted like matter, would be
3 3 OM-- mouT~,mGUT/mp) T
.
Since m o n o p o l e - a n t i m o n o p o l e annihilation is ineffectual at this density, naive extrapolation to the present day shows monopoles and baryons would be nearly of equal abundance, n M - n B [13]. Using m(~vv - 1016 GeV gives
P M - IO16pB"
(2.11)
This illustrates the enormity of the monopole problem. One can readily observe that the production of even one vorton per 1016 horizon volumes at this early epoch (with comparable or greater mass) could also be a major cosmological disaster. To summarize this section, vortons can form in the damped phase for N < Nm~x, while those with N < Ncr are especially easily formed because they undergo overcritically damped relaxation. Assuming the vortons are stable for N > Nmin, then if Nmi. < Nm~, eq. (2.10) gives their contribution to the energy density. As indicated in the preceding paragraph this could lead to an enormous overdensity.
3. Vorton stability In the simplest bosonic U(1) × U(1) model quantum numbers exist on the vortex because a complex scalar field 6 has bound states in a potential well degenerate along the vortex. For a vorton this degeneracy is in the shape of a circle. We only consider the bosonic case, though with fermionic bound states the results are similar. A bound state means there is a classical solution to the equations of motion in which the value of 161 is non-zero only along the circle. This we call the o-condensate, and it is zero off the vortex but rises to a non-zero value on the vortex core [2]. We assume that the radius of the vorton is large compared to both the thickness of the vortex and the thickness of the condensate of 6. It is useful then to
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217
define the integral of a2 over the vortex cross section
z = So2
(3.1)
This number is highly model dependent (it was argued in ref. [16] that for quantum mechanical reasons it must be >__20 for the strings to sustain currents approaching the string scale). Also, for now we will neglect the effect of any electric and magnetic fields, treating the vorton as neutral. Comments on electromagnetic vorti are made later. Stationary solutions to the equations of motion, if they exist, are of the form
8(x) = o(x)exp{ i[N + ( t - l) + N_(t + l ) ] / R v },
(3.2)
where l is the arc length around the vorton and N+ and N are conserved worldsheet charges, o(x) is appreciably non-zero only inside a narrow torus with big radius R v. The underlying vortex has energy proportional to its length so there is a centripetal force, exactly opposed by the repulsive stresses encountered on squeezing down the current and charge. In refs. [3, 4, 14] we studied the conditions for a vorton to exist. A range of quantum numbers N+ and N are possible, but of special interest are chiral vortons, where one or the other is zero. These are the most stable, and in the following we take N = 0 and N + = N. The stationary solutions are then 8(x)exp[i(,-
l)N/Rv].
(3.3)
In general 2J depends on the current and charge density, but for a chiral vorton it is constant, stabilized classically by angular momentum. Assuming circular symmetry the total energy of a neutral vorton is E v = 2rrRv/x + 4~r~N2/Rv,
(3.4)
w h e r e / , is the vortex string tension and the second term comes from eq. (3.3). This energy has a minimum at the vorton radius, where
N / R v = VI-~/2~ , but stability under general non-circular perturbations is difficult to show because there are so many degrees of freedom. An alternative is to simulate a vorton numerically, making an existence proof for classical stability. The problem of quantum stability requires more thought. At the quantum level the solution eq. (3.3) is a coherent state of quanta, all with momentum N / R v. Being
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chiral their energy is also N / R v- For large N the number of quanta is # q u a n t a - 27rN2:.
(3.5)
Without getting into the details of a model we can study the quantum stability of a vorton by estimating the probability that the ~ current will emit a particle which then escapes to infinity. Recall that a particle is bound because it is massless on the vorton but massive in the background spacetime. We denote the mass of a free particle mo. The energy of a sigma particle on the vorton is equal to its momentum N / R v , while if it were located at some other fixed radius it would have energy E = ~m2o + ( N / R )2 =-rho( R ). If there is no radius R, greater than Rv, at which rho(R ) = N / R v , then the vorton is an absolute minimum of the potential for o and the vorton appears to be quantum mechanically stable. The condition for this is that m o > N/Rv. If there is a radius at which N o ( R ) = N / R v then the potential barrier has a finite thickness and the vorton may decay by tunnelling. Thus tunnelling is possible as long as rn o < N / R v = ffi-/2.~ . The height of the barrier depends on R according to A E ( R ) = r~o( R ) - N / R v = ~m2~ + ( N / R ) 2 - N / R v , and the barrier width is obtained by setting rh°(R v + AR) = N / R v ,
yielding
A R -~ Nm2/(4qrt~ ) 3/2, to leading order in too~ V~. We can estimate the decay rate easily because it is essentially a semi-classical one-dimensional quantum tunnelling event: a particle is first trapped in a circularly degenerate potential well located on the vortex, then it tunnels radially outward through a barrier of thickness z~R, beyond which it dissipates classically. It is simple to compute the bounce. A similar calculation for neutral fermions was done in ref. [4]. The vorton has a lifetime F 1
1
e x p [ N ( m 2,. o/,47r/~)~3/=Ij ,
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219
and in order for a vorton to survive until the present time t o we must have Smi .
-
In(to 4 f f ~ f i ) ti m 2o /" 4
Tr/t)a-3/2 .
(3.6)
When the condensate U(1) is electromagnetism, the vorton carries electric charge and current. There are strong electromagnetic fields in the vicinity of the vorton, and this m a y offer a new decay mode. A pure electric field will pair-create o particles at a rate that goes as - exp( - 2 m / e E ) . The largest electric fields are expected to be on the vorton, just outside the core of the vortex. Moving inside, the field gets small again. We can estimate the magnitude of IEI at the vorton to be Emax ~ m o This kind of vacuum polarization would permit the charge on the vorton to disappear by annihilation with an antiparticle produced in the surrounding vacuum. F o r the chiral vortons we are considering this is not a problem because at the vortex, where the vorton looks cylindrical, there is always a magnetic field as well as an electric field, and IEI = I B I .
Having a perpendicular magnetic field suppresses the pair creation of the pure electric field. In the case of null fields E 2 - B e = 0 and pair creation stops. Thus at very short distances where the curvature of the vortex can be neglected pair creation will not occur. Only at distances on the scale of the vorton radius will the electromagnetic field depart significantly from a null field, while far away we will have 3~(~" e) - ,~ E = 2~rRvl R 2 ,
B = 7rR2I
R3
,
where for the chiral vorton I is both the current and the charge per unit length, and b is the unit vector in the direction of the vorton magnetic moment. The exponent in the amplitude for pair creation will thus be suppressed by the ratio of the vortex thickness to the vorton radius. For non-chiral vortons stability is more difficult to analyze. This is because either quenching or antiquenching will occur, depending on whether there is more or less current than charge [3,4]. For example, in eq. (3.2) the net charge would be N++ N while the net current would be N + - N . In the case that the current dominates over the charge quenching occurs and X goes to zero as the current gets larger. This tends to destabilize the vorton. In the second case the charge dominates and X antiquenches, getting larger as the charge density increases. This tends to stabilize the vorton unless it gets so large that the charged particles are no longer
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well localized on the vortex and can escape. In this paper we assume that an appreciable fraction of the vortons are or become approximately chiral with either N+>> N_ or N+<< N_.
4. Cosmological implications The phase transition that would produce vortons could occur at any temperature above the weak scale. Because efficient vorton formation would produce enormous quantities, even at the lower temperatures there are interesting effects. We will only discuss two possibilities, the G U T scale and the electroweak scale. 4.1. G U T V O R T O N S :
T~ - 1016 GeV
Here, from eqs. (2.8) and (2.9), the loops which feel significant damping have Nm~ -
10 6
and
Nor- 102.
F r o m eqs. (2.10) and (2.11) and the discussion vortons must have Nmin > 108 in order to avoid since undamped motion tends to suppress vorton long as Nmin > Nm~.,. Putting Nmin - 106 into eq.
at the end of sect. 2, these G U T a monopole type of problem, but formation, we are probably safe as (3.6) tells us that we must have
mo < 0.25/*
in order to avoid a vorton overdensity. Since the critical current on a superconducting string is essentially m o, this implies that models needing superconducting strings with critical currents approaching the string scale itself could have a vorton problem. The OTW scenario for explosive formation of structure is such a model [6]. That the density of vortons is potentially immense may lead to a constraint, even ruling out the model, though given the multiple-parameter dependence of the models, the large uncertainties in estimating vorton production and the sensitivity of the vorton lifetime to these factors we find it difficult to say anything precisely. In particular, we cannot at present argue that a vorton problem is in any sense generic to OTW type models. It is important to note, however, that we have neglected to consider the additional damping due to primordial magnetic fields [15]. Perhaps further study will be more fruitful constraining the model. A previous attempt to rule out the OTW picture has centered around the notion of a cosmic spring [5-7,16]. A spring is a vorton in the limit of no angular momentum and large winding number N, where N / 2 = N + = N_ in eq. (3.2), and the field configuration is completely static: d( x ) = o( x )exp( iNl/Rv ) .
Springs might appear if there are coherent primordial magnetic fields and superconducting cosmic strings: string loops are formed with some magnetic flux linking
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221
them, so as they decay and shrink huge winding numbers, much larger than those considered in this paper, yield currents stretching over macroscopic distances. There will not, however, be a corresponding net charge on a loop. If it is possible for the current to get high enough the current stresses could stabilize the loop, making a gigantic spinless vorton state, or cosmic spring. The size of the spring suggested in ref. [16] is roughly the size of the solar system, with N - e ~°°. For the OTW [8], the primordial fields and resulting large oscillating currents are needed to power the formation of voids in the large scale structure of the universe, and if cosmic springs did form like this they would rule out or constrain the model. There are several difficulties with trying to rule out OTW in this way, One is that because there is negligible angular momentum current quenching occurs. Without angular momentum X is a decreasing function of N / R in eq. (3.4), and it usually goes to zero before a stabilizing current is reached. This would prevent currents from ever getting large enough to stabilize a vorton, though examples where quenching does not happen for a straight static string are discussed in refs. [6, 16]. Another problem is that these are big loops, coming into the horizon long after the damped phase is over. The loops are thus originally oscillating relativistically, and when the current gets up to mo it is possible for it to peel off the vortex. The evidence shown for springs in ref. [16] is that for an idealized straight static string it is possible for the net string tension to go to zero before quenching, but this is far from sufficient proof if the string loops are originally undergoing violent oscillations. In fact, the authors require a value m~ << ~/~* which makes current losses due to oscillation virtually certain. In the undamped epoch any extrapolation of results obtained from looking at the profiles of straight strings must be done very carefully. A final drawback of the spring constraint is that even if it were established that macroscopic springs could form in large enough quantities to rule out OTW, then certainly microscopic vortons, intrinsically more stable because they have angular momentum [3,4], would be produced at the phase transition, and, because small vortons dominate over large ones this would be a much more serious problem. Our study leaves open the question of whether meaningful constraints can be put on OTW or not, though we have clarified many aspects of the problem. The extra damping due to primordial magnetic fields, which we have not considered here, may prove to be decisive. We should point out that vortons may be a boon, rather than a difficulty, for the OTW scenario. This is because with no added features the models may provide dark matter at an acceptable density. As dark matter, they have unique properties. As seen in sect. 3, vortons may be decaying, thus producing showers of * This can be seen clearly from fig. 2 of ref. [16]. The v e ~ slow fall-off of the condensate compared to the exponential rise of the Higgs field is due to the small mass of a free condensate particle in our notation m,, - needed by the authors. The condensate is not well localized on the vortex, which means that the charge and current is weakly bound to the string and can easily pecl-off when the string is set to oscillate. When bound states are poorly localized the straight static string results should not be trusted to hold true for oscillating strings.
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high energy cosmic rays. For N - 103 they are Planck mass particles. OTW vortons will be highly charged magnetic dipoles. Any charge >137e gets screened by vacuum polarization, and the rest will be neutralized by ions in the plasma. They are, in fact, C H U M P s (charged ultra massive particles) and will be trapped inside planets and stars, their thermal flux possibly making them detectable [17]. 4.2. ELECTROWEAK VORTONS: T~ -
i0 2
GeV
As our second example we consider the electroweak superconducting strings looked at by Chudnovsky and Vilenkin [18]. In this case Nm~ - 10 36
and
Ncr- 10 9,
so the likelihood of producing long-lived vortons at late times is greater than at the G U T scale. The model assumes high critical current, which means m o ~ ~ . Now let us only consider those vortons that would form by overcritically damped relaxation. Setting Nmin = Nor in eq. (3.6) requires mo < 0.02V/~ in order to avoid production of long-lived vortons. But since the authors want m~ - V@- we can expect Nmin < Nor and large numbers of vortons to be produced in the model. Using eq. (2.10) and supposing these vortons give ~2 = 1 today, we obtain Nmi n ~
2 x 10 5 .
Using eq. (3.6), this gives rn, ~ 0.32~/~-,
(4.1)
which is consistent with the high critical currents assumed in the model. The lightest vortons would be - 107 GeV. Since these dominate over the more massive ones, the stringent laboratory constraints on charged dark matter particles with mass 108 GeV and below would seem to apply here (see the discussion in ref. [17]), and critical currents as large as that in eq. (4.1) appear to be ruled out. 5. Conclusion
We have sketched the main features of vortons in standard cosmology and pointed out some of the consequences. The discussion has been necessarily qualitative because of the large uncertainties and model dependencies; our focus has been more to elucidate the underlying physics rather than make definitive statements and predictions. Certainly progress can be made in this direction. For clarity, let us review our various assumptions and caveats. That a string producing phase transition may occur in which worldsheet charges appear on the strings follows from standard notions of cosmology and phase transitions. The
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223
mechanism for vorton production proposed in sect. 2 assumed the charges get trapped on the string in a second order phase transition at roughly the same temperature as that at which the strings form. If the charge-trapping temperature were appreciably less then it would decrease the particle number on a loop originally of size R, and proportionately decrease both Nor and Nm~. This is therefore the best way to avoid a vorton overdensity. Since this temperature is also the typical maximum current for a freely oscillating string, one may decrease the vorton density by decreasing the critical current. We also assume the universe is radiation dominated during the damped epoch, with negligible entropy production. This may not be the case, although more entropy would mean stronger damping and vorton production would be enhanced. To get the estimate of eq. (2.10) we assumed that the scale invariant distribution of the original brownian loops is maintained in the vorton distribution after the loops have relaxed, even though loop breaking and reconnection will occur, and that an appreciable fraction of the loops fall into approximately chiral vorton states. Finally, in the case of electrically charged vortons the role of electromagnetic drag was not considered. This must be remedied, and for OTW the effect of primordial magnetic fields on the damped epoch is especially important to study. Again, these effects result in additional damping which will enhance vorton production. In summary, vortons are produced in the early universe much like monopoles, but with several notable differences. Most importantly, while we have made a case here for how they could be formed in immense quantities, they are far from necessary, and thus do not pose any serious theoretical problem like monopoles. They must be considered, however, in assessing the virtues of any model that uses superconducting strings with high critical currents. A final remark is that while we have concentrated on vorton production, numerical work suggests that our mechanism may be applied to get non-topological solitons as the final state of an unstable vorton. There may be other consequences of the damped period following the formation of strings or other topological defects. We believe that this period has been insufficiently studied. We hope to return to these and other questions in a future publication. We are indebted to Alan Guth, D. Spergel and M. Hindmarsh and J. Freiman for useful discussions. We are especially grateful to Alex Vilenkin for comments on an early version of this paper. E.P.S.S. wishes to thank the Aspen Center for hospitality while part of this work was being done.
References [1] T.W.B. Kibble, Phys. Rep. 67 (1980) 183 [2] E. Witten, Nucl. Phys. B 249 (1985) 557 [3] R.L. Davis and E.P.S. Shellard, Phys. Lett. B 209 (1988) 485
224 [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
R.L. Dat,is, E.P.S. Shellard / Cosmic t,ortons
R.L. Davis, Phys. Rev. D38 (1988) 3722 E. Copeland, M. Hindmarsh and N. Turok, Phys. Rev. Lett. 58 (1987) 1910 C. Hill, H. Hodges and M. Turner, Phys. Rev. D37 (1988) 263 D.N. Spergel, T. Piran and J. Goodman, Nucl. Phys. B 291 (1987) 847 J. Ostriker, C. Thompson and E. Witten, Phys. Lett. B 180 (1986) 231 A.E. Everett, Phys. Rev. D24 (1981) 177 A. Vilenkin, Phys. Rep. 121 (1985) 263 T. Vachaspati and A. Vilenkin, Phys. Rev. D30 (1984) 2036 J. Preskill, Phys. Rev. Lett. 43 (1979) 1365 A. Guth and S.-H. Tye, Phys. Rev. Lett. 44 (1980) 631,963(E) R.L. Davis and E.P.S. Shellard, Phys. kett. B 207 (1988) 404 A. Vilenkin, private communication D. Haws, M. Hindmarsh and N. Turok, Phys. Lett. B 209 (1988) 255 M. Fukugita, P. Hut and D.N. Spergel, Dark matter and luminous planets, IAS, Princeton preprint E. Chudnovsky and A. Vilenkin, String in the sun?, preprint TUTP-88-1
COSMIC V O R T O N S R.L. DAVIS*
Institute for Theoretical Ph.vsics, Unit,ersi(v of Califi~rnia, Santa Barbara, CA 93106, USA E.P.S. SHELLARD**
Center for Theoretical Pl~vsics, Laboratory for Nuclear Science, and Department of Pl~vsics, MIT, Cambridge, MA 02139, USA Received 21 November 1988
We study the production of vortons in the early universe. Vortons are rings of vortex stabilized by charge, current and angular momentum. They are able to carry all of the quantum numbers of ordinary point particles. We discuss a mechanism by which they may be produced during the damped epoch following a string-producing phase transition. We estimate their density, discuss their stability, and remark on some possible cosmological consequences. We look at two examples where vorton production may be significant: the OTW scenario for explosive formation of large scale structure and electroweak strings in the sun.
1. Vortons In type II superconductors it is common to find vortices, quanta of magnetic field confined into flux tubes running along string-like topological defects. In superfluids similar vortices, but without the magnetic flux tubes, are readily observed experimentally. It would not be surprising if there were other, similar vortex phenomena in nature. One of the most interesting possibilities is that vortices, or cosmic strings, appear in a cosmological phase transition [1]. The resulting network of strings could have influence on the observable features of the universe, and a wide range of such effects are currently being studied and tested. Extensions of the simplest models involve extra degrees of freedom which couple to the vortex [2]. This leads to currents which are bound to the vortex, essentially degrees of freedom living on the two dimensional space-time of the vortex worldsheet. The currents may be neutral, or there may be background gauge fields, such as that of electromagnetism, coupled * On leave from Tufts University, Department of Physics and Astronomy, Medford, MA 02155, USA. Research supported under grants NSF PYI 525135, NSF PHY82-17853 and supplementary funds from NASA. ** Research supported by Department of Energy Grant DE-AC02-76ER03069. 0550-3213/89/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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to them. A possibility which has not yet received much attention is that the vortex may have a spinning particle-like soliton configuration - a vorton state [3, 4]. A vorton is a stationary ring of vortex whose tension gets balanced by the stresses and angular momentum caused by trapped worldsheet charges. Its radius is typically
R v - N/x/~ ,
(1.1)
where tt is the string tension of the underlying vortex, 1 / f ~ - is roughly the thickness of the vortex, and N is an integer specifying the number of particles trapped on the worldsheet. As a very special case one could imagine the vorton with zero angular momentum but with N very l a r g e - so that the vorton may even extend over astronomical distances. These have been called cosmic springs [5-7], and if they can be formed they would give a constraint on cosmological models that combine superconducting cosmic strings with primordial magnetic fields [8]. We will make some comments on springs near the end, but in this paper we are mainly interested in the much larger class of microscopic vortons which spin [3, 4]. These are localized classical solutions to the equations of motion which have a time independent stress-energy tensor - they are stationary and do not radiate classically. At distances large compared to R v they look like point masses with quantized electric charge and angular momentum; in many ways they are like ordinary particles, hence the name vorton. We can summarize a mechanism for the formation of vortons as follows: In the early universe a phase transition will produce a brownian network of vortices as long as the unbroken subgroup is not simply connected. For a period of time after the phase transition the motion of the vortices is strongly damped [1, 9,10]. Vortons may be produced because initially irregularly shaped loops become circular as they relax, their motion damped by the surrounding medium. If there are net quantum numbers in bound states on the ring then vortons are formed when the stresses encountered on squeezing down the trapped charges are large enough to counteract the string tension. Once formed, the vortons would behave like non-relativistic particles, which might later decay. In this report we will examine their role in cosmology. We start by discussing the above mechanism in more detail in sect. 2. After that we look briefly at vorton stability, estimating the vorton lifetime. In sect. 3 we discuss several implications, and sect. 4 are some closing remarks.
2. Vorton formation
Cosmological phase transitions can give rise to a variety of topological defects - monopoles, strings, domain walls and their hybrids. They are an unavoidable consequence of spontaneous symmetry breaking in the standard cosmological
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211
model: since the present horizon contains many smaller regions which were causally disconnected at early times, a symmetry breaking occurring in the past cannot lead to coherence across the entire horizon at later times [1]. This lack of coherence requires that topological defects exist whenever the vacuum manifold is not simply connected. Vortons can be produced by a similar mechanism, though because they are only partially topological their formation requires further assumptions that we will outline below. To create them we need the following: (1) First, a network of strings must be formed. Such networks have been studied in cosmic string models and we will use those results freely. The main consequence is that a very large number of irregularly shaped brownian string loops are formed at a phase transition. (2) There must also be either bosonic or fermionic bound states on the strings. This may seem an arbitrary requirement, but in the fermionic case it is actually quite a natural consequence of Yukawa couplings between fermions and Higgs fields. If there are b o u n d states then a string loop can carry current and charge quantum numbers. We will explain below that on formation loops necessarily have non-zero values of these charges. Requirements (1) and (2) combined mean that the early universe can be a source of a very large number of string loops with worldsheet quantum numbers. However, these loops will not be circular or stationary to begin with; they are highly excited states of a vorton. Thus the final condition, that this excited vorton can relax into a stationary circular vorton state is crucial: (3) the quantum numbers must not all leave the worldsheet as the loop loses energy and shrinks. As long as some q u a n t u m numbers remain then a loop of initially arbitrary shape will settle into a final vorton state. This last condition is the most difficult to satisfy because as current (in this section we use the term current loosely to mean ordinary electric current, electric charge density, neutral currents or charges, or a variety of more exotic quantities playing similar dynamical roles) builds up in a shrinking loop it becomes easier for the charge carriers to move off the string. If we define the vorton current to be the current needed to counter the string tension, then a vorton can only exist if at the vorton current the likelihood for the bound quantum numbers to leave the string is small. Stability of the stationary vorton will be discussed in the following section; here we will assume that it is sufficiently long-lived to be cosmologically interesting. Apart from vorton stability is the question of whether or not the brownian loops initially formed can settle into a vorton state. For this we must in general require that the charge carriers stay bound to the vortex while it undergoes typically violent motion. For example, a freely collapsing loop can bounce at some finite radius if the energy in the current becomes large enough to stop the collapse. If the initial current was much less than the vorton current then the maximum current at the turn-around point will be much larger than the vorton current, and if the charge carriers can leave the string at this high current then the vorton will not form. Likewise, a loop that is not freely collapsing but initially oscillating will undergo violent, relativistic
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motion as it decays. A current will be driven by these oscillations and can be expected to reach values much larger than the final equilibrium current in a stationary vorton. At cusps, kinks, and other regions of high local curvature there are especially likely to be losses. The range of models that produce a significant density of vortons would therefore be severely restricted (because relativistic motion would require the string to carry currents much higher than the vorton current without losses) except for the fact that the string motion is strongly damped by the surrounding medium for a period of time following the phase transition [1, 9,10]. As long as the string is over-critically damped its current need not build up to levels significantly larger than the vorton current, and the likelihood that the loops relax into a vorton with nearly the same quantum numbers is vastly greater than in the undamped case. For this reason we will focus our attention on a mechanism for vorton formation in the damped epoch.
2.1. THE PHASE TRANSITION The formation of cosmological string networks is discussed by Vachaspati and Vilenkin in ref. [11]. They simulated the network by assigning random values for the phase of the Higgs field on a large cubic lattice with spacing corresponding to the correlation length ~. They then traced the resulting strings by searching for windings of 27r around faces of the cubes, and determined the string distribution. Not surprisingly, the strings were found to be brownian in nature, that is, if R is the distance separating two points on a string then the average length L of string that joins them is given by L = R2/~.
(2.1)
The simulations also demonstrated that about 80% of string length is in open strings and the remainder in a scale-invariant distribution of closed loops. The number density of loops of average radius R between R and R + dR was given by n ( R ) = vR
4dR,
(2.2)
where v = 6 + 2 . The next step is to discuss our expectations for the current and charge that become trapped on the strings. To simplify the discussion we assume that the charged particles become non-relativistic at the same temperature that the strings form, and that the associated coherence length ~ is the same. We will also assume a second order phase transition, so our mechanism is only strictly applicable to that case. A string segment of length L will pass through approximately - L / ~ uncorrelated regions, so the net particle number would be N ~ ( L / ~ ) 1/2. For particles with charge e the net charge is thus Q ~ e ( L / ~ ) 1/2. Further, the velocities fluctuate in the same manner so the net current is, independently, J ~ e ( L / ~ ) 1/2. For a
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213
brownian loop of average radius R and length L we get typically N ~ V/~
- R/~.
(2.3)
The charge carriers may be either fermions or bosons. The precise relationship between the particle number and winding number for bosonic charge carriers will be summarized in the next section. The key feature of eq. (2.3) is that because we expect ~ - T¢ 1, the thermal wavelength at the phase transition, and because the thickness of the string 1 / ~ - is also typically -T~ -l, this equation together with eq. (1.1) imply R - Rv ,
(2.4)
or in words, the size R of a brownian loop when it is formed is typically the radius of a circular vorton with the same particle number N. This has important consequences as will be seen below. It should be noted that the net charge on a single loop of size R is the same order of magnitude as the total charge due to random fluctuations inside the volume of radius R. This is because the net charge inside the volume R 3 is determined by the surface fluctuations, also - R / ~ . That there are many many other loops inside the volume, each with a net charge, is not paradoxical because charges between different loops will be correlated. 2.2. THE DAMPED EPOCH Given that loops acquire a net charge and current we can now consider their subsequent evolution. The phase transition is followed by a period when string motion is strongly damped by the surrounding medium. Assuming negligible entropy production, this damping period lasts for several orders of magnitude in time until [10]
t, - (G
)-ltc.
(2.5)
During the damped phase strings will straighten out with the initial small scale fluctuations disappearing before larger ones, the coherence length growing from ~c << tc to ~. - t.. String motion can be overcritically (or just critically) damped and not oscillating or subcritically damped and oscillating. Following ref. [10], for overcritical damping the velocity of a section of string with local radius of curvature R is determined by setting the force due to tension - I ~ / R equal to the frictional force - T3v, giving v R - t~/T3R.
(2.6)
For R < i t / T 3 this relation breaks down, meaning the tension is larger than the frictional force and the string will undergo damped oscillations on that scale. Since
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initially R >_ Tc t the entire network will be overcritically damped to begin with, and the time it takes for a fluctuation of size R to move a distance R,
is the time it takes for that scale to straighten out. Without worldsheet currents and charges an initially brownian loop of size R will have collapsed in time ~-. Since small scale fluctuations get smoothed out first the loop becomes more and more circular as it collapses, proceeding from a highly damped period of slow collapse and rapid smoothing to a final stage where the radius is small enough that eq. (2.6) breaks down and the loop relaxes subcritically damped. As it gets smaller and the temperature goes down the frictional damping turns off and the loop oscillates, damped only by radiation. With worldsheet currents and charges the picture is different. Assuming that charge and current losses are negligible, and recalling eq. (2.4), the smoothing period results in a nearly circular loop already at or close to its equilibrium radius. An originally brownian loop can thus reach its vorton state without ever collapsing significantly or undergoing relativistic motion. This overcritical damping mechanism for vorton production is correct, however, only if at the end of the smoothing time the loop, originally of size - R v, has undergone negligible cosmological stretching. If there is stretching then when it becomes smooth the loop is at some radius larger then R v. Because string fluctuations at that scale are just passing over to subcritical damping, it will therefore have to relax by undergoing damped oscillations around R v. If the stretching is significant then the loop will not have time to relax before damping begins to turn off at that scale, so the final stages of relaxation must involve violent oscillations damped only weakly by radiation. Stretching becomes important when the smoothing time is comparable to the Hubble time, or, equivalently, when the string velocity on the scale R in eq. (2.6) is comparable to the Hubble velocity - R / 2 t . This means that the worldsheet quantum numbers on loops with R >>
2(Gl~)l/4tc
(2.7)
will have to survive some relativistic motion if a vorton is to form. For these loops the initial smoothing process and frictional damping continue to act until ~ R - t,. We can therefore identify three regimes in which vorton formation should be considered. The first, and most likely to form vortons, is the overcritically damped period in which cosmological stretching is small. Brownian loops of size R get smoothed out into vortons of roughly the same size. Some oscillations may occur as long as the string does not reach relativistic speeds. Even delicately balanced vortons unstable to relativistic perturbations should form. In the second period the loops are initially overcritically damped as they get smoothed out, but because
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215
stretching has given them a much larger radius than a vorton with the same quantum numbers they must relax by undergoing significant oscillations that are subcritically damped. Since points along the string would reach relativistic speeds, a vorton would have to be somewhat robust to come out of this period. In the third period irregular loops come into the horizon later than t,. They must undergo violent, irregular, relativistic motion damped only by radiation. These are the least likely to form vortons. There will be a smooth cross-over from one regime to another, so it is difficult to estimate what the cross-over points are. Using eqs. (2.7) and (1.1) we can probably conclude that loops with N c r - 2(G/.t) 1/4
(2.8)
will relax overcritically damped. We have assumed entropy is not produced in a significant amount, but if it were then this early epoch is just that in which most would appear, and this would tend to increase N~r because damping would decrease more slowly. As stated, loops with N > Nc~ will have to be somewhat robust in order to survive some relativistic motion. However, as long as they relax in the second regime these loops are strongly damped at first and are more likely to stabilize than those that break free from the network after t,. Taking into account stretching, the maximum value of N for loops which feel significant damping is
Nm~_ t~,t~_(Gt~ ) a
(2.9)
Now let us suppose that vortons are produced relatively efficiently in the damped phase. It will be shown in the next section that there is usually a value of N below which the vortons are quantum mechanically unstable. Assuming Nmin < NmaX and using eqs. (2.2) and (1.1), their energy density can be estimated. The mass of a vorton of size R is - / , R so the energy density in vortons at t c would be
fR.,~ dR E - - ~P]Rmm "-~
1~2v N2in ,
and since they redshift like matter
[r
(2.1o)
N~in I Tc We have said nothing about loops breaking up or joining together. If a loop were to break then the pieces would still satisfy eq. (2.4) and would relax according to the preceding discussion. It is the same if two were to come together. The physics of these processes is complex, though in the end we expect a scale invariant spectrum
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of vorton radii. As a result, the value of ~ in eq. (2.10) may be different from that in eq. (2.2). The mechanism described here makes it plausible that vortons could form even when unstable to highly relativistic perturbations. The uncertainties involved prevent us from making a precise prediction, but even if an exceedingly tiny number of vortons were produced the cosmological implications would be serious. To place this in perspective consider the related monopole problem. Monopoles, topologically stable knots arising in G U T models, proved to be a major obstacle in reconciling such models with the standard cosmology [12]. Monopoles would appear in the early universe because the correlation length ~ of the Higgs field is bounded above by the causal horizon /H(t) -- 2t. A density in monopoles of one per horizon at the G U T time, redshifted like matter, would be
3 3 OM-- mouT~,mGUT/mp) T
.
Since m o n o p o l e - a n t i m o n o p o l e annihilation is ineffectual at this density, naive extrapolation to the present day shows monopoles and baryons would be nearly of equal abundance, n M - n B [13]. Using m(~vv - 1016 GeV gives
P M - IO16pB"
(2.11)
This illustrates the enormity of the monopole problem. One can readily observe that the production of even one vorton per 1016 horizon volumes at this early epoch (with comparable or greater mass) could also be a major cosmological disaster. To summarize this section, vortons can form in the damped phase for N < Nm~x, while those with N < Ncr are especially easily formed because they undergo overcritically damped relaxation. Assuming the vortons are stable for N > Nmin, then if Nmi. < Nm~, eq. (2.10) gives their contribution to the energy density. As indicated in the preceding paragraph this could lead to an enormous overdensity.
3. Vorton stability In the simplest bosonic U(1) × U(1) model quantum numbers exist on the vortex because a complex scalar field 6 has bound states in a potential well degenerate along the vortex. For a vorton this degeneracy is in the shape of a circle. We only consider the bosonic case, though with fermionic bound states the results are similar. A bound state means there is a classical solution to the equations of motion in which the value of 161 is non-zero only along the circle. This we call the o-condensate, and it is zero off the vortex but rises to a non-zero value on the vortex core [2]. We assume that the radius of the vorton is large compared to both the thickness of the vortex and the thickness of the condensate of 6. It is useful then to
R.L. Davis, E.P.S. Shellard / Cosmic vortons
217
define the integral of a2 over the vortex cross section
z = So2
(3.1)
This number is highly model dependent (it was argued in ref. [16] that for quantum mechanical reasons it must be >__20 for the strings to sustain currents approaching the string scale). Also, for now we will neglect the effect of any electric and magnetic fields, treating the vorton as neutral. Comments on electromagnetic vorti are made later. Stationary solutions to the equations of motion, if they exist, are of the form
8(x) = o(x)exp{ i[N + ( t - l) + N_(t + l ) ] / R v },
(3.2)
where l is the arc length around the vorton and N+ and N are conserved worldsheet charges, o(x) is appreciably non-zero only inside a narrow torus with big radius R v. The underlying vortex has energy proportional to its length so there is a centripetal force, exactly opposed by the repulsive stresses encountered on squeezing down the current and charge. In refs. [3, 4, 14] we studied the conditions for a vorton to exist. A range of quantum numbers N+ and N are possible, but of special interest are chiral vortons, where one or the other is zero. These are the most stable, and in the following we take N = 0 and N + = N. The stationary solutions are then 8(x)exp[i(,-
l)N/Rv].
(3.3)
In general 2J depends on the current and charge density, but for a chiral vorton it is constant, stabilized classically by angular momentum. Assuming circular symmetry the total energy of a neutral vorton is E v = 2rrRv/x + 4~r~N2/Rv,
(3.4)
w h e r e / , is the vortex string tension and the second term comes from eq. (3.3). This energy has a minimum at the vorton radius, where
N / R v = VI-~/2~ , but stability under general non-circular perturbations is difficult to show because there are so many degrees of freedom. An alternative is to simulate a vorton numerically, making an existence proof for classical stability. The problem of quantum stability requires more thought. At the quantum level the solution eq. (3.3) is a coherent state of quanta, all with momentum N / R v. Being
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chiral their energy is also N / R v- For large N the number of quanta is # q u a n t a - 27rN2:.
(3.5)
Without getting into the details of a model we can study the quantum stability of a vorton by estimating the probability that the ~ current will emit a particle which then escapes to infinity. Recall that a particle is bound because it is massless on the vorton but massive in the background spacetime. We denote the mass of a free particle mo. The energy of a sigma particle on the vorton is equal to its momentum N / R v , while if it were located at some other fixed radius it would have energy E = ~m2o + ( N / R )2 =-rho( R ). If there is no radius R, greater than Rv, at which rho(R ) = N / R v , then the vorton is an absolute minimum of the potential for o and the vorton appears to be quantum mechanically stable. The condition for this is that m o > N/Rv. If there is a radius at which N o ( R ) = N / R v then the potential barrier has a finite thickness and the vorton may decay by tunnelling. Thus tunnelling is possible as long as rn o < N / R v = ffi-/2.~ . The height of the barrier depends on R according to A E ( R ) = r~o( R ) - N / R v = ~m2~ + ( N / R ) 2 - N / R v , and the barrier width is obtained by setting rh°(R v + AR) = N / R v ,
yielding
A R -~ Nm2/(4qrt~ ) 3/2, to leading order in too~ V~. We can estimate the decay rate easily because it is essentially a semi-classical one-dimensional quantum tunnelling event: a particle is first trapped in a circularly degenerate potential well located on the vortex, then it tunnels radially outward through a barrier of thickness z~R, beyond which it dissipates classically. It is simple to compute the bounce. A similar calculation for neutral fermions was done in ref. [4]. The vorton has a lifetime F 1
1
e x p [ N ( m 2,. o/,47r/~)~3/=Ij ,
R.L. Davis, E.P.S. Shellard / Cosmic vortons
219
and in order for a vorton to survive until the present time t o we must have Smi .
-
In(to 4 f f ~ f i ) ti m 2o /" 4
Tr/t)a-3/2 .
(3.6)
When the condensate U(1) is electromagnetism, the vorton carries electric charge and current. There are strong electromagnetic fields in the vicinity of the vorton, and this m a y offer a new decay mode. A pure electric field will pair-create o particles at a rate that goes as - exp( - 2 m / e E ) . The largest electric fields are expected to be on the vorton, just outside the core of the vortex. Moving inside, the field gets small again. We can estimate the magnitude of IEI at the vorton to be Emax ~ m o This kind of vacuum polarization would permit the charge on the vorton to disappear by annihilation with an antiparticle produced in the surrounding vacuum. F o r the chiral vortons we are considering this is not a problem because at the vortex, where the vorton looks cylindrical, there is always a magnetic field as well as an electric field, and IEI = I B I .
Having a perpendicular magnetic field suppresses the pair creation of the pure electric field. In the case of null fields E 2 - B e = 0 and pair creation stops. Thus at very short distances where the curvature of the vortex can be neglected pair creation will not occur. Only at distances on the scale of the vorton radius will the electromagnetic field depart significantly from a null field, while far away we will have 3~(~" e) - ,~ E = 2~rRvl R 2 ,
B = 7rR2I
R3
,
where for the chiral vorton I is both the current and the charge per unit length, and b is the unit vector in the direction of the vorton magnetic moment. The exponent in the amplitude for pair creation will thus be suppressed by the ratio of the vortex thickness to the vorton radius. For non-chiral vortons stability is more difficult to analyze. This is because either quenching or antiquenching will occur, depending on whether there is more or less current than charge [3,4]. For example, in eq. (3.2) the net charge would be N++ N while the net current would be N + - N . In the case that the current dominates over the charge quenching occurs and X goes to zero as the current gets larger. This tends to destabilize the vorton. In the second case the charge dominates and X antiquenches, getting larger as the charge density increases. This tends to stabilize the vorton unless it gets so large that the charged particles are no longer
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well localized on the vortex and can escape. In this paper we assume that an appreciable fraction of the vortons are or become approximately chiral with either N+>> N_ or N+<< N_.
4. Cosmological implications The phase transition that would produce vortons could occur at any temperature above the weak scale. Because efficient vorton formation would produce enormous quantities, even at the lower temperatures there are interesting effects. We will only discuss two possibilities, the G U T scale and the electroweak scale. 4.1. G U T V O R T O N S :
T~ - 1016 GeV
Here, from eqs. (2.8) and (2.9), the loops which feel significant damping have Nm~ -
10 6
and
Nor- 102.
F r o m eqs. (2.10) and (2.11) and the discussion vortons must have Nmin > 108 in order to avoid since undamped motion tends to suppress vorton long as Nmin > Nm~.,. Putting Nmin - 106 into eq.
at the end of sect. 2, these G U T a monopole type of problem, but formation, we are probably safe as (3.6) tells us that we must have
mo < 0.25/*
in order to avoid a vorton overdensity. Since the critical current on a superconducting string is essentially m o, this implies that models needing superconducting strings with critical currents approaching the string scale itself could have a vorton problem. The OTW scenario for explosive formation of structure is such a model [6]. That the density of vortons is potentially immense may lead to a constraint, even ruling out the model, though given the multiple-parameter dependence of the models, the large uncertainties in estimating vorton production and the sensitivity of the vorton lifetime to these factors we find it difficult to say anything precisely. In particular, we cannot at present argue that a vorton problem is in any sense generic to OTW type models. It is important to note, however, that we have neglected to consider the additional damping due to primordial magnetic fields [15]. Perhaps further study will be more fruitful constraining the model. A previous attempt to rule out the OTW picture has centered around the notion of a cosmic spring [5-7,16]. A spring is a vorton in the limit of no angular momentum and large winding number N, where N / 2 = N + = N_ in eq. (3.2), and the field configuration is completely static: d( x ) = o( x )exp( iNl/Rv ) .
Springs might appear if there are coherent primordial magnetic fields and superconducting cosmic strings: string loops are formed with some magnetic flux linking
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221
them, so as they decay and shrink huge winding numbers, much larger than those considered in this paper, yield currents stretching over macroscopic distances. There will not, however, be a corresponding net charge on a loop. If it is possible for the current to get high enough the current stresses could stabilize the loop, making a gigantic spinless vorton state, or cosmic spring. The size of the spring suggested in ref. [16] is roughly the size of the solar system, with N - e ~°°. For the OTW [8], the primordial fields and resulting large oscillating currents are needed to power the formation of voids in the large scale structure of the universe, and if cosmic springs did form like this they would rule out or constrain the model. There are several difficulties with trying to rule out OTW in this way, One is that because there is negligible angular momentum current quenching occurs. Without angular momentum X is a decreasing function of N / R in eq. (3.4), and it usually goes to zero before a stabilizing current is reached. This would prevent currents from ever getting large enough to stabilize a vorton, though examples where quenching does not happen for a straight static string are discussed in refs. [6, 16]. Another problem is that these are big loops, coming into the horizon long after the damped phase is over. The loops are thus originally oscillating relativistically, and when the current gets up to mo it is possible for it to peel off the vortex. The evidence shown for springs in ref. [16] is that for an idealized straight static string it is possible for the net string tension to go to zero before quenching, but this is far from sufficient proof if the string loops are originally undergoing violent oscillations. In fact, the authors require a value m~ << ~/~* which makes current losses due to oscillation virtually certain. In the undamped epoch any extrapolation of results obtained from looking at the profiles of straight strings must be done very carefully. A final drawback of the spring constraint is that even if it were established that macroscopic springs could form in large enough quantities to rule out OTW, then certainly microscopic vortons, intrinsically more stable because they have angular momentum [3,4], would be produced at the phase transition, and, because small vortons dominate over large ones this would be a much more serious problem. Our study leaves open the question of whether meaningful constraints can be put on OTW or not, though we have clarified many aspects of the problem. The extra damping due to primordial magnetic fields, which we have not considered here, may prove to be decisive. We should point out that vortons may be a boon, rather than a difficulty, for the OTW scenario. This is because with no added features the models may provide dark matter at an acceptable density. As dark matter, they have unique properties. As seen in sect. 3, vortons may be decaying, thus producing showers of * This can be seen clearly from fig. 2 of ref. [16]. The v e ~ slow fall-off of the condensate compared to the exponential rise of the Higgs field is due to the small mass of a free condensate particle in our notation m,, - needed by the authors. The condensate is not well localized on the vortex, which means that the charge and current is weakly bound to the string and can easily pecl-off when the string is set to oscillate. When bound states are poorly localized the straight static string results should not be trusted to hold true for oscillating strings.
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high energy cosmic rays. For N - 103 they are Planck mass particles. OTW vortons will be highly charged magnetic dipoles. Any charge >137e gets screened by vacuum polarization, and the rest will be neutralized by ions in the plasma. They are, in fact, C H U M P s (charged ultra massive particles) and will be trapped inside planets and stars, their thermal flux possibly making them detectable [17]. 4.2. ELECTROWEAK VORTONS: T~ -
i0 2
GeV
As our second example we consider the electroweak superconducting strings looked at by Chudnovsky and Vilenkin [18]. In this case Nm~ - 10 36
and
Ncr- 10 9,
so the likelihood of producing long-lived vortons at late times is greater than at the G U T scale. The model assumes high critical current, which means m o ~ ~ . Now let us only consider those vortons that would form by overcritically damped relaxation. Setting Nmin = Nor in eq. (3.6) requires mo < 0.02V/~ in order to avoid production of long-lived vortons. But since the authors want m~ - V@- we can expect Nmin < Nor and large numbers of vortons to be produced in the model. Using eq. (2.10) and supposing these vortons give ~2 = 1 today, we obtain Nmi n ~
2 x 10 5 .
Using eq. (3.6), this gives rn, ~ 0.32~/~-,
(4.1)
which is consistent with the high critical currents assumed in the model. The lightest vortons would be - 107 GeV. Since these dominate over the more massive ones, the stringent laboratory constraints on charged dark matter particles with mass 108 GeV and below would seem to apply here (see the discussion in ref. [17]), and critical currents as large as that in eq. (4.1) appear to be ruled out. 5. Conclusion
We have sketched the main features of vortons in standard cosmology and pointed out some of the consequences. The discussion has been necessarily qualitative because of the large uncertainties and model dependencies; our focus has been more to elucidate the underlying physics rather than make definitive statements and predictions. Certainly progress can be made in this direction. For clarity, let us review our various assumptions and caveats. That a string producing phase transition may occur in which worldsheet charges appear on the strings follows from standard notions of cosmology and phase transitions. The
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mechanism for vorton production proposed in sect. 2 assumed the charges get trapped on the string in a second order phase transition at roughly the same temperature as that at which the strings form. If the charge-trapping temperature were appreciably less then it would decrease the particle number on a loop originally of size R, and proportionately decrease both Nor and Nm~. This is therefore the best way to avoid a vorton overdensity. Since this temperature is also the typical maximum current for a freely oscillating string, one may decrease the vorton density by decreasing the critical current. We also assume the universe is radiation dominated during the damped epoch, with negligible entropy production. This may not be the case, although more entropy would mean stronger damping and vorton production would be enhanced. To get the estimate of eq. (2.10) we assumed that the scale invariant distribution of the original brownian loops is maintained in the vorton distribution after the loops have relaxed, even though loop breaking and reconnection will occur, and that an appreciable fraction of the loops fall into approximately chiral vorton states. Finally, in the case of electrically charged vortons the role of electromagnetic drag was not considered. This must be remedied, and for OTW the effect of primordial magnetic fields on the damped epoch is especially important to study. Again, these effects result in additional damping which will enhance vorton production. In summary, vortons are produced in the early universe much like monopoles, but with several notable differences. Most importantly, while we have made a case here for how they could be formed in immense quantities, they are far from necessary, and thus do not pose any serious theoretical problem like monopoles. They must be considered, however, in assessing the virtues of any model that uses superconducting strings with high critical currents. A final remark is that while we have concentrated on vorton production, numerical work suggests that our mechanism may be applied to get non-topological solitons as the final state of an unstable vorton. There may be other consequences of the damped period following the formation of strings or other topological defects. We believe that this period has been insufficiently studied. We hope to return to these and other questions in a future publication. We are indebted to Alan Guth, D. Spergel and M. Hindmarsh and J. Freiman for useful discussions. We are especially grateful to Alex Vilenkin for comments on an early version of this paper. E.P.S.S. wishes to thank the Aspen Center for hospitality while part of this work was being done.
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