Current-carrying cosmic string loops 3D simulation: towards a reduction of the vorton excess problem

11 Pages·0.4 MB·English

by  Adriana Cordero-Cid

Current-carrying cosmic string loops 3D simulation:
towards a reduction of the vorton excess problem.
Adriana Cordero-Cid∗
Instituto de F´ısica, BUAP, A. P. J-48, 72570 Puebla, Pue. M´exico
Xavier Martin†
Dpto. de F´ısica, CINVESTAV-I.P.N. A.P. 14-74-, 07000 M´exico, D.F., M´exico
Patrick Peter‡
Institut d’Astrophysique de Paris, UPR 341, CNRS, 98bis boulevard Arago, F-75014 Paris, France
(Dated: January 11, 2002)
The dynamical evolution of superconducting cosmic string loops with specific equations of state
2 describing timelike and spacelike currents is studied numerically. This analysis extends previous
0 work in two directions: first it shows results coming from a fully three dimensional simulation
0 (as opposed to the two dimensional case already studied), and it now includes fermionic as well as
2 bosoniccurrents. Weconfirmthatinthecaseofbosoniccurrents,shocksareformedinthemagnetic
n regimeandkinksintheelectricregime. Foraloopendowedwithafermioniccurrentwithzero-mode
a carriers, we show that only kinks form along the string worldsheet, therefore making these loops
J slightly more stable against charge carrier radiation, the likely outcome of either shocks or kinks.
1 Allthesecombinedeffectstendtoreducethenumberdensityofstableloopsandcontributetoease
1 thevorton excess problem. As a bonus, these effects also may providenew ways of producing high
energy cosmic rays.

1
v PACSnumbers: 98.80.Cq,11.27.+d
7
9
0 I. INTRODUCTION yield analytic equations of state. Those can be called
1 respectively transonic, bosonic and zero mode fermionic
0 carrier models.
2 Superconducting cosmic strings are topological de-
The transonic model describes for instance wiggly
0 fects [1] that may be formed at a phase transition in
Goto-Nambu strings [10]. The trapped degrees of free-
/ the early universe [2, 3]. Contrary to their usual coun-
h dom in this case are not particles, but rather transverse
p terpart known as Goto-Nambu strings [4], these possess
string excitations that play the same role. It is self-dual,
- a nontrivial internal structure because of conserved cur-
p meaning that the equation of state algebraic form does
rents that can flow along them [5, 6, 7]. The dynamics
e not depend on whether the current is timelike or space-
of superconducting cosmic strings is, because of the cur-
h like, and it is transonic because the velocities of trans-
: rent structure that has to be taken into account, much
v verseandlongitudinalperturbationshappentobeequal.
more complicated. In order to describe their evolution,
i Amongitsnumerousadvantages,onefindsthatthetran-
X the so-called elastic string formalism, thanks to which
sonicmodelcanbeexactlysolvedinthecaseofflatspace,
the micro-structure can be integrated over to yield the
r and the string loop motion in this model is always sta-
a macroscopicbehavior,wassetupbyCarter[8]. Theonly
ble. It also describes the motion of an ordinary string
requirementis the knowledgeof the equationof state re-
in a five-dimensional space-time, a` la Kaluza-Klein [11].
lating the energy per unit length to the string tension.
In this last case, the state parameter represents the fifth
This is equivalent to giving a single Lagrangianfunction
coordinate of the string location, seen by projection in
(w), depending on a so-called state parameter w, out
L the four dimensional space-time.
of which the dynamical equations can be derived.
Superconducting cosmic string may also involve ac-
VariousmacroscopicLagrangiansfordescribingsuper-
tual particles, the simplest case being that of fermions
conducting cosmic strings have been proposed [9], that
trapped in the defect core in the form of zero modes, as
correspond to different assumptions about the internal
originallyproposedin1985byWitten[2]tointroducesu-
structure. In principle, the currentalong a cosmic string
perconductivity. Under very general circumstances, the
existsbecausesomeextradegreesoffreedoms(e.g. parti-
fermion condensate can be shown [12] to rapidly reach
cles [2]) aretrapped in the defect core. Up to now, there
a zero temperature distribution, so that the integrated
are essentially three models that have been developed to
model does not depend on anything else but the string
characteristic mass scale, m say. The relevant equation
of state in this case is then again self-dual, with the en-
ergyperunitlengthandtensionaddinguptoaconstant
∗Electronicaddress: [email protected]
at all times.
†Electronicaddress: xavier@fis.cinvestav.mx
‡Electronicaddress: [email protected] Bosonic current-carrying cosmic strings are described
2
by an equation of state which is not self-dual. Such these particles to have enough energy to contribute to
models require two different masses [9]: the string scale the highest energy cosmic ray, this paves the way to a
m and a mass characterizing the current, m say, usu- new class of models that may not suffer from the nor-
∗
ally taken to be that of the carrier itself. Such mod- malization constraint [22].
els have been examined with full details [6, 7]. Even Thearticleisarticulatedasfollows. Inthenextsection
though a bosonic condensate can be treated as a sin- we derive the equations of motion in three dimensions,
gle classical field, contrary to it fermionic counterpart, and express them in an appropriate way for efficient nu-
thecorrespondingstringslookrathermoreinvolvedthan merical resolution. In section III, we present the equa-
the fermionic current-carrying ones. Indeed, it turned tions of state for superconducting cosmic strings with
out that two equations of state are required to describe bosonic and fermionic currents, while the results of the
bosonic current-carrying cosmic strings, one for each of simulation are exhibited in section IV. For a loop with a
the availableregime,namelyspacelikeortimelike. What bosonic current, we exemplify cases where the loop de-
previousanalysisrevealedisthateveninthiscase,quan- velops shocks, kinks or folds on itself in much the same
tum effects should be taken into account, due to various wayas in the two dimensional case. The analysisis then
possibleinstabilities thatcouldbe triggeredbythe loops extended to superconducting loops with fermionic zero-
evolution [13]. mode current-carriers. In this case, we find that in some
There are results on the evolution of superconducting instances, the loop shows regions of discontinuous cur-
string loops in one and two dimensions [13, 14] for those vature (kinks) but we do not find shock waves as in the
equations of state corresponding to strings with bosonic bosonic situation. We also observe that in most of the
currents. Theseworkshaveshowntheappearanceofvar- cases we have investigated, the dynamical evolution of
ious singular behaviors as the dynamical evolution can a loop, whether with bosonic or fermionic current, will
drive parts of a given loop outside of the domain of va- drive it out of the elastic regime. This is yet another
lidity of the elastic string description. For instance, the confirmationthatthequantum(microscopic)effectsthat
loopcandevelopshocks[13,15]orfoldonitselfincompli- aretakingplaceinacosmicstringlooparealmostalways
catedshapes. Thefirstquestionthatcouldthenbeasked not negligible, and even more so in the cosmologicalset-
regardingthese effects ishowmuchoftheseis due tothe ting for which those effects might modify drastically the
restriction to one or two dimensions ? In other words, model predictions [23]. We discuss this particular point
are these results real or do they merely arise because of in the concluding section.
someprojectioneffect. Inthe presentpaper,we haveac-
cordingly extended previous studies to the simulation of
II. EQUATIONS OF MOTION
a superconducting cosmic string loop with bosonic cur-
rents in three dimensions. We find that all the singular
behaviorsobservedintwodimensions generalizeto three For a current–carrying cosmic string, the equations of
and cannot therefore be assigned to some projection ar- motioncanbeexpressedastheconservationofthestress
tifact. energy tensor and the equation of state. The stress en-
ergy tensor of a cosmic string canbe expressedin diago-
Furthermore, we also investigated the behavior of a
loop with fermionic currents, using what was recently nal form as [8]:
shown to be the relevant equation of state for such
Tµν =Uuµuν Tvµvν, (1)
strings [12, 16], i.e. the fixed trace model. In this case, −
althoughdynamicalevolutioncanalsodrivetheloopout-
where uµ and vµ are the two orthogonal, respectively
side of the domain of validity of the elastic string de-
timelike andspacelike unit eigenvectors1, which describe
scription and fold it in complicated shapes, shocks never
the string worldsheet, and U and T are the two cor-
develop. All these singular behaviors,both for fermionic
responding eigenvalues identified respectively with the
andbosoniccarriers,areinterestingbecause,theyareex-
energy per unit length and tension of the string. The
pected to lead to charge carrier emission, a mechanism
equations of motion stem from the conservation of the
that has many cosmologicalconsequences.
stressenergytensor,andthey canbe split into twopairs
The dynamics of the string loops under consideration of intrinsic and extrinsic dynamical equations. The in-
in this paper demands to be applied to a network [17] of trinsic dynamical equations are obtained by projection
suchstringsinordertodecideonimportantissuessuchas along the string worldsheet while the extrinsic dynam-
vortonformation[18]andevolution[19]. Anotherrelated ical equations are obtained by projection perpendicular
possibilityconcernstheup-to-nowmysterioushighesten- to the string worldsheet. The intrinsic equations reduce
ergy cosmic ray events [20]. These could be explained to two current conservation laws, one timelike
in terms of topological defects such as superconducting
cosmic strings, and indeed there have been many such ¯ (νuρ)=0, (2)
ρ
∇
proposals[21]. Inthe caseathand,ourconclusionsseem
to imply that much more particles are released during
the normal evolution of a current-carrying cosmic string
loop than during that of an ordinary loop. Assuming 1 uµuµ=−vµvµ=−1,uµvµ=0
3
and the other spacelike ensures that the dynamical model does not break down.
The velocities (11) should also be asked to both be less
¯ρ(µvρ)=0, (3) than unity in order to avoid superluminal propagation,
∇
but this, in practice, never actually happens so that it is
where ¯ = η σ is the covariant derivative on the
∇ρ ρ ∇σ not really a constraint. To summarize, we require that,
string worldsheet, obtained by projecting the usual co-
along the loop trajectories, we have
variant derivative on the worldsheet with the projection
operator ηρσ = uρuσ +vρvσ, and ν and µ are defined 0 c2 1, (12)
through the rela−tions ≤ T,L ≤
constraints that are not always trivially satisfied.
µν = U T, (4)
− We nowhave to fix the gaugeto determine unambigu-
dU
µ = , (5) ously the relevant degrees of freedom. The dynamical
dν equations are solved numerically choosing as unknowns
the string worldsheetcoordinatesxµ(t,ψ) as in [13], and
and can be understood, respectively, as a number den-
as parameters t, the time coordinate, and ψ, a spacelike
sity variable and its associatedeffective mass variable or
curvilinear coordinate chosenas the potential associated
chemical potential.
with the irrotationality equation (6). With this choice,
The intrinsic equations can be equivalently re-
uρ,vρ and ν are expressed as:
expressed as irrotationality equations, respectively as
ǫσρ σ(νvρ)=0, (6) uρ = x˙ρ, (13)
∇ ζ˙
and 1
vρ = (βx˙ρ+ζ˙2x′ρ), (14)
ǫσρ (µu )=0, (7) nζ˙
σ ρ
∇ ζ˙
whereǫσρ istheantisymmetrictangentelementtensorof ν = (15)
n
the string worldsheet, given by
with dots and primes respectively denoting derivatives
ǫσρ =uσvρ vσuρ, (8)
withrespecttotandψandwehaveadoptedthefollowing
−
definitions
in terms of the tangent vectors u and v.
For a superconducting cosmic string, the conserved ζ˙ = x˙ρx˙ ,
ρ
numbers associatedwith the two conservedcurrentscor- −
p
respondtothechargedcurrenttrappedinthestringand ζ′ = x′ρx′, (16)
ρ
the winding number of the string. When a loop is con- q
β = x′ρx˙ ,
sidered, these translate into two integer-valued charges, ρ
N and Z, say, once integration around the loop is per- n = β2+ζ′2ζ˙2.
formed. q
The extrinsic equations of motion can be written as
The equation of motion (6) is automatically satisfied by
µ (Uuν uρ Tvν vρ)=0, (9) the choice of its associated potential ψ as variable, and
ρ ν ν
⊥ ∇ − ∇ the remaining equations of motion (7) and (9) become
where µ , defined by respectively
ν
⊥
⊥µν =gµν −ηµν =gµν −ǫµρǫρν, (10) v·[(n2−β2c2L)x¨−ζ˙4c2Lx′′−2βζ˙2c2Lx˙′]=0, (17)
is the orthogonal projection operator. where v is the spatial part of vρ, and
Thenecessaryquantitiesfordescribingacosmicstring
loop are completed by the knowledge of two other im- w [(n2 β2c2)x¨ ζ˙4c2x′′ 2βζ˙2c2x˙′]=0, (18)
portantparameters,namelythespeedsoftransverseand ⊥1,2· − T − T − T
longitudinal perturbations along the string, written re- with w the spatial parts of two independent
⊥1,2
spectively as c and c , and given by quadrivectorsorthogonaltotheworldsheet,whichweare
T L
choosing as
T dT
c2 = , c2 = . (11)
T U L −dU 1
w = (y′z˙ y˙z′,0, z′,y′), (19)
⊥1 y′z˙ y˙z′ − −
These velocities are useful to explore the stability of a −
loop at equilibrium [26]. Furthermore, they are helpful
and
since they allow us to define what we call elastic regime:
the string dynamics is in the elastic regime if transverse 1
w = (0,y′z˙ y˙z′,x˙z′ z˙x′,y˙x′ x˙y′). (20)
and longitudinal perturbations are stable. This indeed ⊥2 y′z˙ y˙z′ − − −
−
4
1
Then, the equationsofmotioncanbe solvedto find that z¨ = (z′F +K), (23)
x(t,ψ),y(t,ψ) and z(t,ψ) satisfy n2
1
x¨ = (x′F +G), (21)
n2
1
y¨ = (y′F +I), (22) where the functions F, G, I and K are given by:
n2
ζ˙2c2
F = L ζ˙2(c x′′+c y′′+c z′′)+2β(c x˙′+c y˙′+c z˙′) , (24)
n2 β2c2(cid:20) 1 2 3 1 2 3 (cid:21)
− L
ζ˙2c2
G = T ζ˙2 (x′y′′ x′′y′)c +(x′z′′ x′′z′)c +2β (x′y˙′ x˙′y′)c +(x′z˙′ x˙′z′)c , (25)
n2 β2c2(cid:26) (cid:20) − 2 − 3(cid:21) (cid:20) − 2 − 3(cid:21)(cid:27)
− T
ζ˙2c2
I = T ζ˙2 (y′′z′ z′′y′)c +(y′′x′ x′′y′)c +2β (y˙′z′ y′z˙′)c +(y˙′x′ x˙′y′)c , (26)
n2 β2c2(cid:26) (cid:20) − 3 − 1(cid:21) (cid:20) − 3 − 1(cid:21)(cid:27)
− T
ζ˙2c2
K = T ζ˙2 (y′′z′+z′′y′)c +(z′′x′ z′x′′)c +2β (y˙′z′+z˙′y′)c +(z˙′x′ z′x˙′)c , (27)
n2 β2c2(cid:26) (cid:20) 2 − 1(cid:21) (cid:20) 2 − 1(cid:21)(cid:27)
− T
with We shall assume in what follows that the dynamics
obtained by means of the numerical solution is accurate
as long asboth these quantities are conservedatleastat
c1 x′+y˙∆3+z˙∆2, the few % level (worst case during the development of a
≡
c y′ x˙∆ +z˙∆ , (28) shock) all along the trajectory.
2 1 3
≡ −
c z′ x˙∆ y˙∆ ,
3 2 3
≡ − −
and III. EQUATION OF STATE
∆1 y′x˙ y˙x′, Theequationofstatedescribesthesubstructureofthe
≡ −
∆ z′x˙ z˙x′, (29) current–carryingcosmic string considered by relating its
2
≡ −
∆ z′y˙ z˙y′. internal parameters, for instance its energy density and
3
≡ − tension. Therearethreeinterestingcurrent–carryingcos-
As a test of the deviations of the numerical approxima- mic string models: the transonic model and the bosonic
tionfromtheexactsolutionitwillbeconvenienttocalcu- and fermionic superconducting models. The transonic
latethe(conserved)totalenergyandangularmomentum modelis usedto describe the macroscopicdynamicalbe-
of the system given, with our choice of unknowns, by (l haviorofawiggly“Goto–Nambu”cosmicstringandcar-
being the string arc length) [13] riesacurrentassociatedwiththewiggles[10]: itpermits
to neglect the dynamics of the strings on small scales by
E dlT00, (30) integratingoverthesescales,effectivelyraisingthestress-
T ≡I energy tensor degeneracy, and thereby to introduce an
effective current. This model also possesses the nice fea-
which reads, explicitly
tureofbeing algebraicallysolvableinflatspace[24],and
thus can provide useful analytical examples with which
U[ν(ψ)]
E = dψ (n2 β2c2), (31) one cancompare the accuracyof a simulationcode. The
T I nζ˙2 − T
othermodels we shalldealwithareused to describecos-
micstringsinwhichacurrentofparticles(eitherbosonic
and the angular momentum [8]
or fermionic) has condensed.
Uβn For superconducting strings with a bosonic current,
J =ZN =Ψ dψ (1 c2), (32)
T I ζ˙2 − T the model requires two mass scales, namely the cosmic
string mass scale m, which is the string forming symme-
where Z is the particle quantumnumber, N is the wind- trybreakingenergyscale,andm whichisthemassscale
∗
ingnumberandΨistheparametriclengthofthe loopin of the current carrier. In this case, one can write down
the ψ coordinate (which is conserved,actually being one twodifferentequationsofstatedependingonwhetherthe
of N or Z). conserved current along the string is timelike or space-
5
like[6]. Thesetworegimesofthemodelarecalledrespec- which gives
tively electric and magnetic in accordance with the situ-
ν2 ν2
ationwhere the currentis coupledto an electromagnetic U =m2+ , T =m2 , (41)
field [7]. Indeed, in this case, a timelike current would 2 − 2
induce an essentially electric field (in the sense that, as where m is, as before, the string scale. Here, the pertur-
E2 >B2,the magneticcomponentcanbeframedaway), bation velocities are simply
while a spacelike current leads to a magnetic-like field.
2m2 2m2 ν2
Usingatoymodelwhichproducessuchstrings[2,6],the c2 = 1= − , (42)
equationofstateforUasafunctionofTinthemagnetic T U − 2m2+ν2
regime is [9] c2 = 1. (43)
L
In this case, the string can leave the domain of elasticity
m2 m m2
U =m2+ ∗ ∗ ∗ m2+T, (33) only if the velocity of transverse perturbations c2 can
8 − 2 r16 − T
become negative, i.e. if ν √2m. We shall see that
| | ≥
and the transversaland longitudinal speeds are given by this indeed happens.
1 2m2(1+ν2)2 ν2(1 ν2)
c2 = ∗ − − ∗ , (34)
T 1+ν2 2m2(1+ν2)+ν2 IV. EVOLUTION
∗ ∗
1 3ν2
c2 = − ∗, (35) We studied the evolution of loops in three dimensions
L 1+ν2
∗ after having reproduced, as a check, exactly the two-
where ν = ν/m . In this case, c2 can become negative dimensional results of Ref. [13]. Then, the evolution of
∗ ∗ L loops in three dimensions was investigated with the var-
so that the string may become unstable with respect to
ious superconducting equations of state discussed above.
longitudinal perturbations and thus leave the domain of
The class of initial configurations with which we have
elasticity. In the electric regime,the toy model yields an
been concerned was obtained by perturbing the circular
equation of state of the form
equilibrium solution in the following way:
U =T +m2∗ e2(m2−T)/m2∗ −1 , (36) x(t=0,ψ) = Rcos ψ (44)
(cid:2) (cid:3)
(cid:18)Rγν (cid:19)
and the two perturbations velocities are 0
c ψ
x˙(t=0,ψ) = T sin (45)
c2 = 2(m2/m2∗)+ln(1−X∗) (1−X∗) , (37) −1+ǫ (cid:18)Rγν0(cid:19)
T 2(m(cid:2)2/m2∗)+ln(1−X∗) (1(cid:3)−X∗)+2X∗ y(t=0,ψ) = Resin ψ (46)
(cid:2) (cid:3) (cid:18)Rγν (cid:19)
0
where
c ψ
y˙(t=0,ψ) = T cos (47)
2ν2+1 4ν2+1 1+ǫ (cid:18)Rγν (cid:19)
X = ∗ − ∗ , (38) 0
∗ 2νp2 ℓψ
∗
z(t=0,ψ) = Acos (48)
(cid:18)Rγν (cid:19)
and 0
ℓψ
1 z˙(t=0,ψ) = Bsin , (49)
c2 = . (39) − (cid:18)Rγν0(cid:19)
L 1+4ν2
∗
where ν and γ (1 x˙2 y˙2 z˙2)1/2 are respectively
p 0
≡ − − −
In this regime,it is clear thatonly c2 maybecome nega- the current and the Lorentz factor of the circular equi-
T
tive,atwhichpointthestringmaybecomeunstablewith librium solution, e is the ellipticity of the configuration,
respect to transverse perturbations. ψ is the spacelike coordinate, chosen to vary from zero
The final model of current-carrying cosmic strings to 2πRγν0 using the gauge freedom in the worldsheet,
which we shall consider here concerns the case in which ǫ is a dimensionless parameter which measures the de-
fermions may be bound to cosmic strings in the form of viation in velocities from the circular equilibrium state
two lightlike currents of zero modes propagating in op- obtainedwhene=1andǫ=0,andℓisanintegerphase
posite directions. The resulting total fermionic current parameter for the transverse z motion. Note that the
can then be timelike, spacelike or lightlike. Because the amplitude A of the perturbation in the transverse (with
fermion condensate tends to a zero temperature distri- respecttothestringloopplane)direction,andthatofthe
bution, the equation of state is self-dual, that is of the associatedvelocityB areindependentsincethosepertur-
same form whether the fermionic current is timelike or bations have been shown [26] to be absolutely stable.
spacelike, given by [12, 16] For a given equation of state, the stability of the per-
turbed circular equilibrium solution can easily be de-
U +T =2m2, (40) rived [26]. The primary purpose of this work is to find
6
outthefateoftheunstableconfigurations. Beforestudy- Another important effect that may take place along a
ing such configurations within the framework of bosonic superconducting loop is the appearance of kinks. These
andfermionicsuperconductingequationsofstate,wefirst are regions where the curvature of the string becomes
checked the program against stable ones with the tran- 0.25 0.25
sonic equation of state. In this case, everything runs 0.2 t=1 0.2 t=60
0.15 0.15
smoothly and there is nothing particular to note about 2ν* 2ν*
0.1 0.1
theloopsdynamics. Thus,noevolutionoftransonicloops
0.05 0.05
is shown (see however Ref. [13] for a discussion of these 0 0
0 10 20 30 40 50 0 10 20 30 40 50
configurations;thethreedimensionalsimulationnowpre- ψ ψ
0.25 0.25
sented merely confirms this reference results as we were 0.2 t=90 0.2 t=120
able to check that they were not artifacts of the two di- 0.15 0.15
2ν* 2ν*
mensional projection). 0.1 0.1
More interesting is the case of the bosonic current 0.05 0.05
0 0
where singular dynamics where observed in two dimen- 0 10 20 ψ 30 40 50 0 10 20 ψ 30 40 50
sions [13]. The analysis in this case is presented next. 0.25 94.86
t=150
It is then followed by an investigation of the fermionic 0.2
0.15
current case which is completely new. 2ν* ET94.85
0.1
0.05
0 94.84
0 10 20 30 40 50 0 50 100 150
A. Bosonic Currents ψ t
FIG. 1: Time evolution of the state parameter ν2 along the
∗
For superconducting strings with bosonic currents, we
string worldsheet for an elliptic configuration of a loop with
confirm that all the singular behaviors observed in two a bosonic current in the magnetic regime, having large ellip-
dimensions, i.e., shocks in the magnetic regime, kinks in ticitye=0.6andvelocityparameter ǫ=1,andwithaphase
theelectricregime,foldingloopsandexitfromtheelastic parameter ℓ = 3 (and A = 3, B = 0.2). In the last panel
−
regime in general, persist in three dimensions ; they are the variation of the energy shows two important things. On
not projection artifacts and represent the real physical the one hand the small peaks in the graph are an evidence
time evolution of superconducting cosmic string loops. that the loop size decreases because the initial velocity has
been reduced. On the other hand, nearly at the end of the
In the following, we show examples that illustrate each
simulation, the rare variation of the energy gives a signal of
case in turn.
theshocksproducedalongtheloopstring. Therelativevaria-
Ashockwavemaybeformedatpointsalongthestring
tionofthetotalenergy,shownonthelastpanel,isstillrather
worldsheet,and according to previous results [13, 15] we
small even as the simulation reaches its veryend. The situa-
further confirm that a shock can only take place in the tionissimilarfortheangularmomentum,andthesimulation
magneticregime. Anexampleofsuchashockinthreedi- eventually breaksdown.
mensions, using equation of state (33) is shownin Fig. 1
on which is also represented the space variations of the
state parameter ν = ν/m for different instants in time
∗ discontinuous. We found that giving an extra dimension
for a transversely perturbed loop with ℓ = 3 ; note that
for the loop to evolve into did not suppress the appear-
such a loop, from the transverse perturbation point of
ance of kinks. Fig. 2 shows the evolution of a loop in
view, should be strictly stable. In this case, the geomet-
the electric regime, and therefore using the equation of
ric evolution does not reveal any visible change in the
stategivenbyEq.(36). Inthiscaseagain,thesimulation
loop,butthe time evolutionofthe stateparameterν2 as
∗ showsthat2ℓkinks arebeing formedatpoints alongthe
a function of the ψ parameter shows the development of
loop. Toillustratethispointfurther,weplottedinFig.3
shockfrontsalongthe stringworldsheet. Atthesepoints
the curvature of the loop as a function of the spacelike
the current in the string becomes discontinuous and the
string internal coordinate ψ. This is, in general, given
code cannot handle the evolution anymore,as shown for
by [8]
instance by the fact that the total energy is no longer
conserved. Although such transverse modes in principle
Kµ =ηρνKµ , (50)
wouldhavebeenexpectedtobestable,itturnsout,with ρν
the values of the parameters chosen, that nonlinear ef-
i.e., in the case at hand,
fects rapidly become dominant, effectively coupling the
transversemodetothelongitudinalone. Thatthisreally
occursismadeclearbythefactthatthenumberofshock K2 =(w⊥ρ1Kρ)2+(w⊥ρ2Kρ)2, (51)
frontsgeneratedduringthis evolutionis 2ℓ,thus directly
related to the phase parameter ℓ. where
7
1
wρ K = n2 β2 (y¨z′ z¨y′)+2βζ˙2 y′z˙′ z′y˙′ +ζ˙4(y′z′′ z′y′′) , (52)
⊥1 ρ ∆3n2ζ˙2 h(cid:0) − (cid:1) − (cid:0) − (cid:1) − i
1
wρ K = (n2 β2)(∆ x¨ ∆ y¨+∆ z¨) 2βζ˙2(∆ z˙′ ∆ y˙′+∆ z˙′) ζ˙4(∆ x′′ ∆ y′′+∆ z′′) .(53)
⊥2 ρ ∆3n2ζ˙2 h − 3 − 2 1 − 3 − 2 1 − 3 − 2 1 i
t=1 t=23 200 200
t=1 t=23
5 5 150 150
K100 K100
0 0
20 20 50 50
−150 5 0 −5 −10 −20 0 −150 5 0 −5 −10 −20 0 00 2 ψ4 6 8 00 2 ψ4 6 8
t=43 t=63 200 200
t=43 t=63
5 5 150 150
K100 K100
0 0
20 20 50 50
−150 5 0 −5 −10 −20 0 −150 5 0 −5 −10 −20 0 00 2 ψ4 6 8 00 2 ψ4 6 8
t=68 90.5 200
t=68
5 150
0 ET89.5 K100
−5 20 50
10 5 0 −5 −10 −20 0 88.50 10 20 30 40 50 60 70 00 2 4 6 8
t ψ
FIG. 2: Configuration evolution in the electric case with FIG.3: InvariantmagnitudeK =√K2ofthecurvature,from
bosoniccurrentswiththesameellipticityasinFig.1,ǫ=0.1 Eq.(51), for the configuration shown in Fig. 2.
and A=3 (B =0). One clearly sees the development of six
kinks corresponding to twice the phase parameter ℓ = 3. In
thiscase,thepeaksseenintheenergyareformedatthesame magnetic regime, or c2 in the electric regime, becoming
T
time as the kinksare present. negative. In the latter case, the tension reaches nega-
tive values and the string actually behaves like a spring.
In Fig. 5 is shown the evolution of the longitudinal per-
It is clear in the figure that where kinks appear, large turbation velocity as a function of the spacelike parame-
variations of the curvature are developed, showing that ter along the string loop at different instants of time in
the kinks are not gauge artifacts but really occur. As a typical example for which the dynamical evolution of
expected, the location of the kinks corresponds to the the loop drives it out of the elastic regime when c2 be-
L
points with large curvature. In general the effects found comeslessthanzero. Whateverhappenspastthistimeis
in the two dimensional planar case are still present in presumably not accounted for correctly by the program
three dimensions. They are however usually affected (or which, being an effective macroscopic description, does
even initiated) by the transverse perturbation in z. not recognize the instability at the microscopic level.
In some cases, the string loop shows points where in- All these results are reminiscent of what had previ-
tercommutation, which is not taken into account in our ously been obtained in the two dimensional case [13].
code, should be occurring. A systematic analysis shows There now is no more ambiguity about them being pro-
that a loop in three dimensions tends to fold on itself in jection artifacts. The conclusions derived in the 2D case
any regime, magnetic or electric. This is similar to what therefore still apply.
happens to Goto-Nambu loops for which it was even ar-
guedthatthisintercommutingprocesscouldbethelead-
ing process for a string network to loose energy [23, 25]. B. Fermionic Current
The conclusions that can be drawn from this fact for a
superconducting string network are therefore the same, In this section, we consider current–carrying cosmic
and will be discussed later. In Fig. 4 is shown an exam- strings with fermionic currents, obeying the equation of
ple of such a missed intercommutation for a loop in the state (40). The behavior in this case follows somewhat
electric case. that observed in the bosonic case: the loop can fold on
Another possibility thatis openfora loopto leavethe itself and leave the elastic domain. We shall exemplify
region where the elastic description is valid is to locally both cases in turn, but first let us remark the important
have one of its squared perturbation velocity, c2 in the following point. Since in the fermionic case the equation
L
8
t=1 t=9.5 0.6 0.6
t=1 t=15
2 2 0.4 0.4
0 0 −5 2cL0.2 2cL0.2
−2 0 −5 −2 0
10 5 0 −5 −10 5 10 5 0 −5 −10 5 0 0
t=14.5 t=24.5 0 20 40 60 80 0 20 40 60 80
ψ ψ
2 2
0.6 0.6
0 0 t=30 t=45
5 −5 0.4 0.4
−2 0 −2 0
t=34.5 −10 −5 0 5 10 −5 t=41 10 5 0 −5 −10 5 2cL0.2 2cL0.2
2 2 0 0
0 −5 0 5 0 20 4ψ0 60 80 0 20 4ψ0 60 80
−2 0 −2 0 0.6 137.32
10 5 0 −5 −10 5 −10 −5 0 5 10 −5 t=60
t=50 53.5 0.4
02 ET53 2cL0.2 ET137.316
5 0 137.312
−2 −10 −5 0 5 10 −5 0 52.50 10 20 30 40 50 0 80 0 20 40 60
t ψ t
FIG. 4: A typical string loop configuration showing points FIG. 5: Time evolution of the squared perturbation veloc-
where intercommutation should take place, with ℓ = 2, el- ity,c2,inthemagneticregimeforabosonicsuperconducting
lipticity e = 0.3, A = 1, B = 0.1, and ν∗20 = 0.1. At these loop.LThedynamicsdrivestheloop out oftheelastic regime,
points, two loops should be formed and the subsequent evo- as seen on the last evolution panel, in which one finds a re-
lution given by thecode is nolonger physically relevant. gion having c2 0. The last panel shows the total energy
L ≤
integratedalongtheloop asafunctionoftime: itsconstancy
uptothepointwherec2 becomesnegativerevealsthatnoth-
L
ing particular took place and the physicaldescription can be
of state is self-dual, there cannot be any qualitative dif-
trusted. Past this point, the energy is very badly conserved
ference betweenthe magnetic andthe electric regimesat
and the program, as expected, has difficulties handling the
the level of the equations of motion. However, the two problem. This figureis for A=2 and B =0.
perturbation velocities are not equal and the string may
become unstable only with respect to transversal per-
turbations. As a result, the dynamical evolution of the
shown as a function of the spacelike string parameter ψ.
stringcanleadittodeveloponlykinks2,althoughinthis
Thesameremarksasforbosonicstringsapply,inpartic-
casewecannotsaythattheeffectisexclusiveoftheelec-
ular concerning the non conservation of the total energy
tric or magnetic regime. In Figs. 6 and 7, we show the
whenevertheelasticregimeisleft. Noteheretheappear-
time evolution and the curvature (snapshots at different
anceofa numericalinstability thatpropagatesalongthe
times)foraloopwithfermioniccurrent,againwithphase
string and that is seen in the form of (unphysical) oscil-
parameter ℓ=3 and initial value of the state parameter
lations.
set as ν2 = 0.1. The appearance of kinks is seen as the
∗0
curvatureofthestringbecomesdiscontinuous. Whenthe
kinkisformedthetotalenergyisnolongerconservedand
V. DISCUSSIONS AND CONCLUSIONS
the simulation ends.
Similarly to the case of cosmic string loops endowed
This work is the direct follow-up of previously dis-
with bosonic currents, loops with fermionic currents
cussed two dimensional string loop simulations [13]. It
can fold on themselves, exhibiting points where inter-
extends this old work in two directions. First it now
commutation should take place, and with very much
provides results about the dynamics of such string loops
the same rate of occurrence in the evolution. Self-
in three dimensions, thereby answering the question as
intercommutation for loops loosing their energy again
to whether the observedeffects were actually physicalor
seems to be in this case a favored mode.
mere projection artifacts. This new simulation clearly
Thestringloopcanalsoleavethedomainofvalidityof
shows that indeed the effects presented in Ref. [13] are
the elastic formalism when c2 becomes negative. Fig. 8
T physical. AnotherextensionofRef.[13]thatisdonehere
depicts the time evolution of the transverse velocity c2
T concerns the equation of state, which can now describe
up to the point where it vanishes. The velocity here is
fermionic [12] currents as well.
In the case of strings endowed with spacelike bosonic
currents, it was shown that most of the loops develop
shock. These shocks are seen as discontinuities in the
2 Notethatsinceshockscanonlyforminthemagneticregimeand
current and occur because the squared longitudinal per-
asinthisself-dualsituation,theelectricandmagneticregimeare
turbationvelocityc2 canbecomenegative. Intheelectric
undistinguishable, shocks cannot form in the fermionic current L
case. regime of timelike currents, it is the squared transverse
9
t=1 t=20 0.4 0.4
0.3 t=1 0.3 t=15
5 5 0.2 0.2
2cT 2cT
0 0 0.1 0.1
20 20 0 0
−150 5 0 −5 −10 −20 0 −150 5 0 −5 −10 −20 0 −0.10 20 40ψ 60 80 −0.10 20 40ψ 60 80
t=45 t=70 0.4 0.4
0.3 t=30 0.3 t=45
5 5
0.2 0.2
0 0 2cT0.1 2cT0.1
20 20 0 0
−150 5 0 −5 −10 −20 0 −150 5 0 −5 −10 −20 0 −0.10 20 40ψ 60 80 −0.10 20 40ψ 60 80
t=110 t=115 0.4 126.47
0.3 t=69
5 5 0.2 126.466
2cT ET
0 0 0.1
20 20 0 126.462
−150 5 0 −5 −10 −20 0 −150 5 0 −5 −10 −20 0 −0.10 20 40ψ 60 80 0 10 20 30t 40 50 60
FIG. 6: Configuration evolution for fermionic equation of
FIG. 8: Time evolution of thespeed of transversal perturba-
state. In this case, the shape of the loop changes until kinks
tions for a cosmic string loop with fermionic currents (with
are formed along the string, at which point the elastic string
A = 2 and B = 0.2). The dynamical evolution of the loop
description ceases to be sufficient. Quasi kinks are formed drivesit out or t−heelastic regime, where c2 0.
first,thataresubsequentlyroundedoff,untiltheactualkinks T ≤
reallyform. Here,andinthefollowing figure,theparameters
A=3 and B=0.2 havebeen used.
self-dual.
In both cases, fermionic and bosonic currents, and for
40 40
30 t=1 30 t=20 whateverkindofcurrent,theloopstendtofoldonthem-
K20 K20 selves, generating contact points where intercommuta-
10 10 tion should occur. The almost systematic occurrence of
0 0
0 5 10 15 0 5 10 15 such configuration leads us to conclude that one may
ψ ψ
40 40
30 t=45 30 t=70 divide superconducting string loops into essentially two
K20 K20 species: those loops which tend to divide into smaller
10 10 loops, and the other ones that are identified with the
0 0
0 5 10 15 0 5 10 15 proto-vortons of other authors [19]. Our results tend to
ψ ψ
40 40
30 t=110 30 t=115 indicate that the latter category is much less populated
K20 K20 than the former, as was already postulated in earlier in-
10 10 vestigations [19]. It also implies that these intercommu-
0 0
0 5 10 15 0 5 10 15 tations will be, in the case of superconducting cosmic
ψ ψ
91.5
strings,much more frequent than in the case of ordinary
ET91.45 strings.
Now when a string loop self-intercommutes, the topo-
91.4
0 50 100 logical stability that confines the Higgs and other parti-
t
clesinsidethedefectisraisedandsomeoftheseparticles
FIG.7: Timeevolutionofcurvatureagainstthepointlabeling areexpelledaway. Althoughstablewhilecontainedinthe
function ψ for the configuration of the Fig. 6. The curvature defects, these particles are often unstable. In particular,
inthisexamplevariesalmostbytwoordersofmagnitude. As stringsareexpectedtoformattheGrandUnified(GUT)
in thebosonic case, thevariations of theenergy givea signal scale, and the relevant Higgs and gauge fields to decay
of the kinksformation.
therefore almost instantaneously by cosmological stan-
dards. These particles will then initiate showersof lower
massonesdividingtheoriginalenergy. Thisisthe mech-
perturbation velocity c2 that can vanish and reverse its anism that is at work in most Top-Down scenarios [27]
T
sign. As it turns out, this implies that kinks, namely ofUltra High EnergyCosmic Rays(UHECR) whoseori-
regions of discontinuous curvature, develop. gin is still mysterious [20]. Ordinary cosmic strings do
Fermionic strings, being described by an equation of not produce much of these particles, because intercom-
state for which only the squaredtransverseperturbation mutations are not that frequent in Goto-Nambu strings
velocity can become negative, are subject only to the sincemostoftheenergyofthenetworkinthatcaseisex-
possibility of building kinks, and this in a way that is pected to be lost through gravitational radiation [3]. In
independent of the spacelike or timelike nature of the fact [22], the resulting flux is found a disappointing ten
underlying current since the relevantequation of state is ordersof magnitude below the observedone. In the case
10
ofcurrent-carryingstrings,loops havea higher probabil- alsothat the current-carrieris indeed likely to get outof
ityofself-intercommuting,sothemechanismisenhanced thestringsince,asthestringcurvatureincreases,thecar-
accordingly. riermomentumacquiresanorthogonal-to-the-worldsheet
The proto-vortoncase now takesus to the rareconfig- component thanks to which the wavefunction is spread
urations that do not intercommute frequently. Note at over larger distances, thereby effectively increasing the
this point that the scarcity of these states is not enough probability for the particle to tunnel out. This mech-
toensurethevortonexcessproblemtobealtogetheralle- anism can ease the vorton excess problem while in the
viated. Indeed,forGUTscalevortons,theratiobetween sametime providingmoreefficiencyforUHECRproduc-
proto-vortonsanddoomedloopsshouldalreadybe much tion through Top-Down scenarios.
less than 10−20 in order for the present-day vortons not Shocks develop only for bosonic spacelike currents. In
todominatethe Universe[19]iftheresultingvortonsare this case, the state parameter ν, which would end up
stable over cosmologically relevant timescales. being discontinuous were the shock actually occurring,
Gatheringalltheresultsobtainedovertherecentyears, is well represented by the collective momentum of the
we now find the following: for a realistic underlying mi- trappedparticles[6,7],thismomentumbeingalwaysless
croscopic model, the equation of state is such that at than the rest mass of the carrier particles themselves in
least one of the perturbation propagation speed can be- order for the confinement to take place. As we argued
come imaginary, leading to possible instabilities. The before [13], it is rather dubious that actual shock could
main result of the present paper is to show that these form in cosmic strings, as the shocks are always accom-
instabilities will systematically develop in every loop as panied,eitherfollowedorpreceded,byatransientregion
it evolves. As a result,the elastictreatmentturns out to inwhichc2 0. However,inthis region,the momentum
L ≤
be valid for limited periods of time and one must resort ofthecarrierishigheroratleastofthe sameorderasits
to the full microscopic description in between. rest mass. Therefore, it is energetically favored to some
The elastic description breaks down in two possible of the trapped particles to move out of the string.
ways, namely through the formation of kinks or shocks. In all the situations which we have studied, we found
In both cases, the meaning is that the microstructure that the expected dominant effect was particle radia-
shouldbetakenintoaccount. Inparticular,closetothese tion. We are therefore led to the conclusion that an
points, it is no longer appropriate to consider the string elastic treatment of cosmic strings is not completely ap-
as effectively two dimensional and we must consider the propriate. It should be supplemented with some semi-
effects that are due to the finite thickness. classical, non conservative, forces. The dissipative ef-
First of all, kinks cannot exist if the string is a tube fect then introduced can serve to ease the vorton excess
instead of a Dirac line as it would be impossible for in- problem,whileprovidingnewinterestingwaystoproduce
stance to match the Higgs field and its derivatives, as is UHECR.
however required by the underlying field theory, at the
points where the curvature is discontinuous. Therefore,
thestringmustsomehowfindawayofsmoothing,which
Acknowledgments
meansinthatcontextloosingenergy. Theonlynonkine-
matical (in the sense of necessarily breaking the elas-
tic description) way of realizing that is through particle We shouldlike to thank B.CarterandC. Ringevalfor
emission,andweareledtoconcludethat,asinthecaseof many fruitful and enlightening discussions as well as A.
intercommutation, but for completely different reasons, Zepeda forsupportduring the preparationofthis paper.
someparticles(currentcarriers,butalsoHiggsandgauge L.A.C.acknowledgesthesupportreceivedfromCONA-
fields) will be radiated away as the kinks develop. Note CyT.
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