Excited cosmic strings with superconducting currents
EExcited cosmic strings with superconducting currents
Betti Hartmann, Florent Michel, and Patrick Peter Instituto de F´ısica de S˜ao Carlos (IFSC), Universidade de S˜ao Paulo (USP), CP 369, 13560-970 , S˜ao Carlos, SP, Brasil Centre for Particle Theory, Durham University, South Road, Durham, DH1 3LE, UK Institut d’Astrophysique de Paris ( G R ε C O ), UMR 7095 CNRS, Sorbonne Universit´es,UPMC Univ. Paris 06, Institut Lagrange de Paris, 98 bis boulevard Arago, 75014 Paris, France. (Dated: December 8, 2017)We present a detailed analysis of excited cosmic string solutions which possess superconductingcurrents. These currents can be excited inside the string core, and – if the condensate is large enough– can lead to the excitations of the Higgs field. Next to the case with global unbroken symmetry, wediscuss also the effects of the gauging of this symmetry and show that excited condensates persistwhen coupled to an electromagnetic field. The space-time of such strings is also constructed bysolving the Einstein equations numerically and we show how the local scalar curvature is modifiedby the excitation. We consider the relevance of our results on the cosmic string network evolutionas well as observations of primordial gravitational waves and cosmic rays. I. INTRODUCTION
Although cosmic strings [1–6], i.e. linear topologicaldefects expected to have formed at phase transitions dur-ing the early stages of the Universe, are no longer ac-cepted as candidates for cosmic microwave background(CMB) primordial fluctuations [7] (See Ref. [8] for anupdate on the cosmic string search in the CMB and themore recent work [9] in which new methods are beingdeveloped), they are still expected to be produced inthe grand unified theory (GUT) framework (see, e.g.,Ref. [10] and references therein), in which case they arevery likely to have bosonic condensates [11] or be current-carrying [12]. The structure of such objects has beenstudied in detail for many models, from the original Wit-ten [13] fermionic [14, 15] or bosonic kind [16–19], leadingto effective equations of state [20, 21] potentially usefulfor large scale network simulations [22, 23]. Until thereason why strings have yet not been observed in theCMB is clarified, it is therefore of utmost importance tounderstand in as many details as possible their internalstructure and the associated plausible cosmological con-sequences.In a previous work [24], by investigating the neutralcurrent-carrying Witten model [17], we identified a newset of excited solutions in which the condensate oscillatesand thus yields a many-valued equation of state, i.e. wefound several (possibly many, depending on the param-eters) different branches in the energy per unit lengthand tension as functions of the state parameter. Wealso argued that those new modes should be unstableand deduced some plausible cosmological consequences.The purpose of this work is to deepen our understand-ing of these modes and to make the argument for theirinstability more rigorous. We also discuss inclusion ofelectromagnetic-like effects [18, 19] if the current is cou-pled to a massless gauge field. Finally, we couple ourmodel to gravity in order to derive the local [25–27] andasymptotic [28–33] geometrical structure.An interesting new outcome of this detailed investiga- tion is that the string-forming Higgs field itself may oscil-late in a restricted regime of parameter space, which leadsto oscillations in the gravitational field around the vor-tex, thus potentially enhancing the gravitational wavesproduced by a network of such strings and leading to theemission of high energy particles.Besides their possible relevance for cosmology, these so-lutions may have close analogues in atomic Bose-Einsteincondensates. Indeed, it is now well known (see for in-stance [34] and references therein) that one-dimensionalvortex lines can arise in rotating condensates. Consider-ing a dilute gas of two types of atoms with different tran-sition frequencies, it should be possible to tune the po-tential to mimic the Higgs field-condensate interactionsin superconducting strings. One would then expect so-lutions with a similar structure and basic properties, al-though the stability analysis would be somewhat differ-ent since non-relativistic condensates obey a first-orderequation in time, so that, in particular, the analoguesof the unstable modes with imaginary frequencies foundin Section III D would be negative-energy modes in thenon-relativistic case. Such analogies between cosmologi-cal phenomena and condensed-matter systems have beenfruitful in the context of black-hole physics [35–39], inparticular clarifying the effects of Lorentz violations onHawking radiation [40] and leading to the discovery ofnew phenomena in condensed-matter systems. It is con-ceivable that a detailed study of such excited vortex linesin condensed matter would also reveal new interestingphysics.The purpose of this paper is to detail and comple-ment the results of the analysis of Ref. [24], in which theelectromagnetic-like U(1) symmetry of the model was infact not gauged, thus corresponding to neutral currentsflowing along the string [17]. This is done in Sec. III.In particular, we present new results related to the back-reaction of the excited condensate on the Higgs field.The effects due to a nonvanishing value of the a r X i v : . [ h e p - t h ] D ec electromagnetic-like coupling are discussed briefly inSec. IV A and the gravitational effects are presented inSec. IV B. In Sec. V we discuss our results and conclude. II. THE MODEL
The underlying toy model describing a current-carrying vortex (superconducting cosmic string) has beenproposed by Witten in 1985 [13]. It consists in twocomplex scalar fields φ and σ , each subject to indepen-dent phase shift invariance, both of which being possiblygauged. The general situation is therefore the so-calledU(1) × U(1) scalar Witten model, which reads L = 12 ( D µ φ )( D µ φ ) ∗ + 12 ( D µ σ )( D µ σ ) ∗ − V ( φ, σ ) − G µν G µν − F µν F µν . (1)Here G µν and F µν denote the field strength tensors of thetwo U(1) gauge fields B µ and A µ respectively, namely G µν = ∂ µ B ν − ∂ ν B µ and F µν = ∂ µ A ν − ∂ ν A µ , (2)and the covariant derivatives read D µ φ = ∂ µ φ − i e φB µ and D µ σ = ∂ µ σ − i e σA µ , (3)where e and e are the coupling constants of the re-spective scalar fields φ and σ to the corresponding gaugefields. Finally, we set the potential to V = λ (cid:0) | φ | − η (cid:1) + λ | σ | (cid:0) | σ | − η (cid:1) + λ | φ | | σ | , (4)which is the most general renormalizable one given thefield content.In what follows, we choose the parameters of the po-tential (4) above in such a way that the U(1) symme-try associated to the fields φ and B µ gets spontaneouslybroken, thereby forming an Abelian-Higgs string, whilethe U(1) symmetry associated to the fields σ and A µ re-mains unbroken. Associated to this unbroken symmetrythe cosmic string will carry a locally conserved Noethercurrent and a globally conserved Noether charge, whichin the gauged case can be interpreted as electromagneticcurrent and charge, respectively. According to the standard model of particle physics however,such a massless U(1) gauge boson corresponds unambiguouslyto the photon and the relevant symmetry to that of actual elec-tromagnetism. We keep referring to an electromagnetic-like cou-pling because the structure we are investigating here might beonly temporary, with the symmetry being only unbroken as anintermediate step in a full GUT symmetry-breaking scheme lead-ing to the standard model.
A. Field equations
The ansatz for the vector fields in cylindrical coordi-nates ( r, θ, z ) reads B µ d x µ = 1 e [ n − P ( r )] d θ,A µ d x µ = 1 − b ( r ) e ( ω d t − k d z ) , (5)while the scalar fields take the form φ ( r, θ, z ) = η h ( r )e i nθ , σ ( r, θ, z ) = η f ( r )e i( ωt − kz ) (6)We introduce the following dimensionless coordinateand energy ratio x ≡ (cid:112) λ η r, q = η η , (7)and the rescaled coupling constants α i = e i λ , and γ i = λ i λ ( i = 2 , . (8)We also rescale the Lagrangian into the dimensionlessquantity L → ˜ L := L / ( λ η ).With these notations, the equations of motion read (cid:18) P (cid:48) x (cid:19) (cid:48) = α P h x , (9)1 x ( xb (cid:48) ) (cid:48) = α bf , (10)1 x ( xh (cid:48) ) (cid:48) = P hx + h ( h −
1) + γ f h, (11)1 x ( xf (cid:48) ) (cid:48) = ˜ wf b + γ f ( f − q ) + γ f h , (12)where a prime denotes a derivative with respect to x andwe have defined the state parameter w as w := k − ω = λ η ˜ w , thereby defining its rescaled counterpart ˜ w . Thesign of the state parameter w is defined positive for aspacelike current ( w >
0) and negative for a timelikecurrent ( w < w = 0 corresponds to a chiral(lightlike) current.The necessary boundary conditions corresponding to acurrent-carrying vortex then read P (0) = n, b (0) = 1 , h (0) = f (cid:48) (0) = b (cid:48) (0) = 0 , (13)at the origin andlim x →∞ P ( x ) = lim x →∞ √ xf ( x ) = 0 , and lim x →∞ h ( x ) = 1 , (14)at infinity. Although we have produced solutions with n > n = 1 fordefiniteness. B. Integrated quantities
Cosmological consequences of the existence of topolog-ical defects can be studied under the approximation thatthey are infinitely thin in their transverse dimension com-pared with their longitudinal extension. This amounts tointegrating over the transverse dimensions. In our case,the relevant quantities are the energy per unit length U and tension T . Those are calculated as the eigenvaluesof the integrated stress-energy tensor ¯ T µν := (cid:90) (cid:18) − δ L δg µν + g µν L (cid:19) d x ⊥ , (15)where in the present symmetric situation the relevantintegration measure element across the string is givenby (cid:82) πθ =0 d x ⊥ = (cid:82) πθ =0 r d r d θ = 2 π r d r . To figure them,we restrict attention to the worldsheet space coordinates ξ a ∈ { t, z } , i.e., we explicit the matrix ¯ T ab and findthe eigenvalues by solving the characteristic equationdet (cid:0) ¯ T ab − λη ab (cid:1) = 0, with the 2-dimensional Minkowskimetric η ab := diag { , − } . This leads to (cid:18) UT (cid:19) = η (cid:18) ˜ U ˜ T (cid:19) = πη (cid:90) (cid:32) (cid:88) i =1 ε i ± c + u (cid:33) x d x, (16)where ε := h (cid:48) + f (cid:48) , (17) ε := h P x , (18) ε := P (cid:48) α x , (19) c := | ˜ w | (cid:18) b (cid:48) α + f b (cid:19) , (20) u := 12 ( h − + γ f ( f − q ) + γ h f . (21)This form clearly makes all the relevant quantitiesLorentz invariant; in Eq. (16), the meaning of the col-umn vector is that U corresponds to the + sign in frontof the quantity c , while T is calculated with the − sign(this ensures that U ≥ T ). These definitions of U and T are valid even in the electromagnetically coupled case e (cid:54) = 0, even though we mostly concentrate in what fol-lows on the neutral case e = 0.The velocities of longitudinal and transversal pertur-bations which are given by c L = (cid:112) − d T / d U and c T = (cid:112) T /U , respectively, should both be real in order for thestring to be stable [41]. This requires
T /U >
T / d U <
0, conditions which we refer to below as Carterstability conditions. Note that there is a degeneracy in the structureless (currentless)case leading to the usual Nambu-Goto action for which U = T . Another quantity of interest is the current flowingalong the worldsheet. Starting from the U(1) invarianceof σ , one forms the microscopic current J µ := 1 e δ L δA µ = − η f [ ∂ µ ( ωt − kz ) − e A µ ] , (22)where the normalizing factor 1 /e ensures it remains fi-nite in the neutral limit e →
0. Integrating radiallyagain yields the current CC := (cid:90) d x ⊥ (cid:113) | η ab J a J b | . (23)This gives explicitly, in terms of the field functions C = 2 π | v | η (cid:90) f b r d r = 2 π η √ λ ˜ C, (24)where the reduced state parameter is v = sign( w ) (cid:112) | w | = √ λ η ˜ v ; the meaning of this parameter is clear: for aspacelike current, there exists a frame in which ω → w → k , in which case v → k , while for a timelike current,there exists a frame where k →
0, so that v → − ω (thesign is included in order to clearly distinguish betweenspacelike and timelike configurations and for conveniencewhen it comes to plotting). III. SOLUTIONS IN THE NEUTRAL MODEL
In the following, we will concentrate on the case α =0, i.e. the case in which the current along the string isungauged, which implies b ( x ) ≡ A. Linear condensate
To motivate the existence of excited solutions, we workin a regime where the condensate is sufficiently smallto neglect its backreaction on the string-forming Higgsscalar h . To reduce the number of parameters, we de-fine the shifted squared frequency Ω ≡ ˜ w − γ q . Then,Eq. (12) becomes f (cid:48)(cid:48) + 1 x f (cid:48) = (cid:0) Ω + γ h (cid:1) f + γ f . (25)We look for “bound state” solutions which are regularat x = 0, not equal to zero everywhere (i.e., we discardthe trivial solution f = 0), and decay strictly faster than x − / at infinity. One can obtain two bounds on Ω,namely − γ ≤ Ω < ⇒ − m σ ≤ w < M σ , (26) This condition ensures that there is no quadratic conserved fluxat infinity, in accordance with the usual definition of a boundstate. where m σ := (cid:0) λ η − λ η (cid:1) is the rest mass of the cur-rent carrier σ field outside the string where | φ | → η ,and M σ := λ η its mass inside the string where φ → • If Ω ≥
0, since γ > f (cid:48)(cid:48) + x f (cid:48) has everywherethe same sign as f . Assume first f (0) >
0. Since[ xf (cid:48) ( x )] (cid:48) > x > xf (cid:48) ( x ) = 0 at x = 0, this implies that[ xf (cid:48) ( x )] >
0, and therefore that f (cid:48) ( x ) >
0, for suf-ficiently small positive values of x : the function f thus grows. Therefore, in order for f to van-ish asymptotically, it must stop growing at somestage, and hence it must go though a maximum: ∃ x max ; f (cid:48) ( x max ) = 0 & f (cid:48)(cid:48) ( x max ) ≤
0. But we alsohave, by construction, that f ( x max ) >
0, implying f (cid:48)(cid:48) ( x max ) + x f (cid:48) ( x max ) = f (cid:48)(cid:48) ( x max ) >
0, in con-tradiction with the hypothesis. The function f ( x )thus grows indefinitely. For f (0) <
0, the sameargument applies in the opposite direction, show-ing that f ( x ) decreases for all values of x , whilethe case f (0) = 0 would lead to the trivial solution f ( x ) = 0 for all x . As a result, we deduce that ∀ x > , | f ( x ) | ≥ | f (0) | . This is clearly in contra-diction with the assumption that f goes to zero atinfinity, so we must set Ω < • Let us now show that Ω ≥ − γ . To this end, it isconvenient to define the function s ( x ) := √ xf ( x ).Eq. (25) may be rewritten as s (cid:48)(cid:48) + s x = (cid:0) Ω + γ h (cid:1) s + γ x s . (27)To simplify the notations, let us also define the twoquantities K := − (Ω + γ ) andΘ( x ) := − x + γ (cid:2) h ( x ) − (cid:3) + γ x s ( x ) , (28)in terms of which Eq. (27) becomes s (cid:48)(cid:48) ( x ) = − K s ( x ) + Θ( x ) s ( x ) , (29)which gives, upon multiplication by 2 s (cid:48) ( x ) on bothsides, dd x (cid:0) s (cid:48) + K s (cid:1) = 2Θ s (cid:48) s. (30)Eq. (30) is our main tool to prove the desired result.Indeed, as we now show, if K >
0, i.e., Ω < − γ ,then the “energy” s (cid:48) + K s does not go to zeroat infinity, in contradiction with the definition of alocalized state.For clarity, let us list explicitly the properties ofthe functions s and h we will use. First, we assume that s is not identically zero, i.e., that a condensateis present inside the string. Second, we use that h and f , and thus s , converge to zero exponentiallyat infinity, as shown in [17]. This implies that1. h − x ∈ [0 , + ∞ [;2. s /x is integrable on the interval x ∈ [1 , + ∞ [;3. s (cid:48) ( x ) goes to zero as x → ∞ .The function Θ is thus absolutely integrable at in-finity. If K (cid:54) = 0, there thus exists x > (cid:90) ∞ x | Θ( x ) | d x < (cid:112) | K | . (31)This is the crucial point, which allows us to boundthe variation of the “energy” s (cid:48) + K s .We now have all the elements to prove the desiredresult. As in the first point, we proceed by con-tradiction. Let us assume that K > M s ≡ sup x>x | s (cid:48) s | . Since s is not a con-stant function, s s (cid:48) takes nonvanishing values, so M s >
0. Moreover, since we demand that s ( x ) and s (cid:48) ( x ) must vanish asymptotically, | s ( x ) s (cid:48) ( x ) | goesto zero in this limit, so M s is reached at some point x ≥ x . Using that (cid:104) s (cid:48) ( x ) ± √ K s ( x ) (cid:105) ≥
0, oneobtains s (cid:48) ( x ) + K s ( x ) ≥ √ K | s (cid:48) ( x ) s ( x ) |≥ √ K M s . (32)On the other hand, from Eq. (30), (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ∞ x dd x (cid:0) s (cid:48) + Ks (cid:1) d x (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ∞ x s (cid:48) s d x (cid:12)(cid:12)(cid:12)(cid:12) ≤ M s (cid:90) ∞ x | Θ | d x < √ K M s , (33)where Eq. (31) was used in the last step. We thushave: (cid:2) ( s (cid:48) ) + K s (cid:3) ∞ x < √ K M s . (34)Combining Eqs. (32) and (34), we deduce thatlim x →∞ (cid:2) s (cid:48) ( x ) + K s ( x ) (cid:3) > √ KM s , in contradiction with the assumption that s and s (cid:48) both go to zero in this limit. We conclude thatsolutions can exist only if K ≤
0, i.e., if Ω ≥ − γ . There is, of course, an infinite number of possible choices: anysufficiently large value of x will satisfy this property. In order to motivate the existence of our excited modes,we further assume that the nonlinear term in (12) is neg-ligible, and we work with the following simple continuousbut non differentiable ansatz for the function h : h ( x ) = (cid:26) κ x for 0 ≤ x < /κ, x > /κ. (35)This simple form provides a strong motivation for theexistence of excited solutions and allows to determinesome of their expected properties.For x ≥ /κ , f satisfies a modified Bessel equation [42].The only solutions going to zero sufficiently fast at infin-ity are f ( x ) = C K (cid:16)(cid:112) Ω + γ x (cid:17) , C ∈ R . (36)To solve the equation in the interior region x < /κ ,it is useful to define the variable Y ≡ √ γ κx and thefunction F by f ( x ) = exp[ − Y ( x ) / F [ Y ( x )]. Doing this,we obtain Y F (cid:48)(cid:48) + (1 − Y ) F (cid:48) + A F = 0 , A ≡ − (cid:18) Ω4 √ γ κ + 12 (cid:19) . (37)This is the confluent hypergeometric equation [42]. Theonly regular solutions are F ( Y ) ∝ L A ( Y ), where L A de-notes the Laguerre function with parameter A . So, for x > /κ , f ( x ) = C e −√ γ κx / L A ( √ γ κx ) , C ∈ R . (38)Since (25) has no singularity at x = 1 /κ , f and f (cid:48) must becontinuous at that point, and this provides two matchingconditions. A straightforward calculation shows they canbe simultaneously satisfied if and only if2 L (cid:48)A L A = 1 + K (cid:48) (cid:0) √ µ + 1 ξ (cid:1) √ µ + 1 K (cid:0) √ µ + 1 ξ (cid:1) , (39)where µ ≡ Ω /γ and ξ ≡ √ γ /κ . Then, C and C arerelated through C C = L A (cid:0) √ γ /κ (cid:1) K (cid:0) √ Ω + γ /κ (cid:1) e −√ γ κ/ . (40)To our knowledge, (39) can not be solved analyticallyin general. However, it greatly simplifies in the limit ξ (cid:29)
1, i.e., for very small κ . Then f is negligible for x > /κ and is approximately given by a globally regularsolution of the Laguerre equation going to zero at infin-ity. The latter are the Laguerre polynomials [42], whichexist if and only if A ∈ N . Moreover, the n th Laguerrepolynomial has n − f thus has m = n − m (cid:28) ξ/
4. We thus expect that solutions with γ /
43 1 / x f / f ( ) FIG. 1. Fundamental (continuous) and first three excited(dashed for m = 1, dotted for m = 2, and dot-dashed for m = 3) solutions in the limit ξ (cid:29) m nodes exist up to a maximum value m max close to ξ/ x ≈ (cid:0) γ κ (cid:1) − / , (41)while for the second excited solution, the two roots areat x ≈ (cid:104) (2 ± √ / (cid:112) γ κ (cid:105) − / .We solved Eq. (39) numerically for various values of ξ and found few deviations from the above picture. Inparticular, solutions with m nodes exist for m between 0and a maximum value m max , approximately equal to ξ/ ξ (cid:29)
1. We also solved Eq. (25) numerically using ashooting method to see the effects of the nonlinear termas well as that of a more realistic profile for h . Concerningthe former, we found its main effect is to decrease thevalue of µ of each solution, by a term quadratic in f (0).For each value of m , there is a critical value of | f (0) | above which the solution disappears as the correspondingvalue of µ drops below −
1, as shown in Fig. 2 for thefundamental solution with ξ = 4. The nonlinear termalso has the tendency to widen the condensate, althoughthis becomes significant only close to the critical value.Similarly, we found that replacing the above profile of h with a hyperbolic tangent does not change the qualitativebehavior of the solutions. Its main effect is to increase m max , which seems to come from the slower convergenceof h towards 1. B. Numerical construction
We have solved numerically the coupled set of differ-ential equations (9), (11) and (12), subject to the appro-priate boundary conditions (13) and (14). x f / f ( ) ξ = f (0) =0.1 f (0) =0.3 f (0) =1.0 f (0) =2.0 f (0) =3.0 f (0) =3.5 f (0) ν ξ = µ FIG. 2. Top panel: Fundamental solutions for ξ = 4 anddifferent values of f (0), expressed in units of κ/ √ γ . Bottompanel: Values of the parameter µ for these solutions.
1. A case study
In what follows, we concentrate on the solutions for q = 0 . α = 0 .
01 and γ = 10 and study the effects ofthe variation of γ . This case is complementary to thestudy done in [24], where the couplings γ and γ hadbeen chosen one to two orders of magnitude larger. Firstlet us recall that restrictions on the couplings exist in thismodel, in particular we have γ ≤ γ , such that in thefollowing we will study solutions for γ ∈ ]0 : 100[. Notethat the second requirement q γ ≤ γ is automaticallyfulfilled within this interval of the parameter γ .We have constructed solutions with up to 3 nodes inthe condensate field function. We observe that for allvalues of m solutions exist in a limited interval of thecentral value of the condensate field, f (0) ∈ [0 : f (0) max ]such that for f (0) → f ( x ) ≡
0. Thiscorresponds to a value of the state parameter ˜ w whichwe will denote ˜ w ,m in the following. Our results for m = 0 , , , m σ = γ − q γ . We observe that˜ w ,m is a linear function of γ and is parallel to − ˜ m σ for all values of m . The difference ∆ m := ˜ w ,m − ( − ˜ m σ )decreases with increasing node number m . The values aregiven in Table III B 1. For the given parameter values, FIG. 3. The value of ˜ w cr and ˜ w in dependence on γ for q = 0 . γ = 10, α = 0 .
01 and m = 0 , , , − m σ = q γ − γ . m ∆ m γ (eq ,m )2 .
70 71 2 .
20 342 0 .
41 803 0 .
05 94TABLE I. Some characteristic values of the ˜ w ,m and ˜ w cr ,m curves shown in Fig. 3. we hence find that the formula˜ w ,m = ∆ m − γ + q γ holds.Our numerical results indicate that f (0) can be in-creased up to a maximal value f (0) max which dependson the values of the couplings in the model. We willdenote the corresponding value of the state parameter˜ w cr ,m in the following. The value of ˜ w cr ,m is a decreasingfunction of γ . The qualitative behaviour is similar forall values of m : ˜ w cr ,m = 0 for γ → w cr ,m = − γ → − m σ .Let us denote the value of γ at which ˜ w cr ,m = ˜ w ,m by γ (eq ,m )2 , the numerical values of which are given in TableIII B 1 for m = 0 , , , γ = γ (eq ,m )2 , the qualitative dependence of ˜ w onthe central condensate value f (0) changes. For γ >γ (eq ,m )2 the state parameter ˜ w decreases for increasing f (0) such that for ˜ w → ˜ w cr ,m the value of the condensate f (0) becomes very large and, in fact, as our numericalresults indicate, tends to infinity, f (0) max → ∞ . Thiscase has been studied in detail in [24]. Here we presentour results for the energy per unit length ˜ U , the tension˜ T and the current ˜ C as functions of the state parameter˜ v for α = 0 . q = 0 . γ = 99 and γ = 10 in Fig.4.For increasing m the range in ˜ v , for which supercon-ducting string solutions exist decreases. At ˜ v the energyper unit length and tension are equal and the current ˜ C becomes zero. At ˜ v cr the current diverges. We find thatindependent of the value of m , ˜ v cr = − f (0) max corresponding to this criticalvalue is (nearly) independent of the node number. Giventhe interpretation put forward in [43], namely consider-ing the current ˜ C and ˜ v as a conjugate pair, in which | ˜ v | is the particle number density and ˜ C the chemical poten-tial or effective mass per particle, we find that solutionsexist only above a certain particle number density, whichincreases with increasing node number m . Furthermore,for a given particle number density | ˜ v | , the effective massper particle is largest for the m = 0 solution and de-creases with increasing node number. At the maximalpossible particle number | ˜ v cr | the current diverges.For γ < γ (eq ,m )2 , on the other hand, we find that f (0)can be increased up to a maximal value f (0) max , whichis finite, and that ˜ w is an increasing function of f (0).From this maximal value of f (0) a second branch of so- FIG. 4. Top : The energy per unit length ˜ U (dashed) andthe tension ˜ T (solid) as function of the state parameter ˜ v for α = 0 . q = 0 . γ = 99, γ = 10 and m = 0 , , , C . The phase frequencythreshold of Eq. (26) is clearly visible as the divergence pointat ˜ w th = − γ + q γ (cid:39) − lutions exists for decreasing f (0), while the state param-eter ˜ w further increases. We will discuss the origin ofthe existence of this branch and the physical phenomenaassociated to it in subsection III B 3.
2. Higgs field oscillations
During the study of smaller values of the couplings γ and γ , we observed a new phenomenon that is notpresent for the cases presented in [24]. The reason forthis is that the central value of the condensate function, f (0) can have larger values for smaller values of γ and γ , respectively. For sufficiently large values of f (0) wefind that the oscillations of the condensate function cantrigger an oscillatory behavior in the Higgs field function.This is shown for γ = γ = 10, q = 0 . α = 0 .
01 and m = 2 in Fig. 6, in which we also show the condensatefield function f ( x ) together with the Higgs field function h ( x ) for increasing values of f (0) up to the maximal pos-sible value of f (0) ≈ . f (0),here f (0) = 0 .
01 and f (0) = 0 .
1. But as soon as f (0) islarge enough, we see that the Higgs field starts to showan oscillating behaviour, see the profiles for f (0) = 0 . f (0) = 0 . γ and γ even smaller – and consequently f (0) much larger – we observe oscillations of the Higgsfield with large relative amplitudes on a finite interval ofthe radial coordinate x , on which the Higgs field functionpossesses nodes. As a first approximation this can beunderstood by considering (11) and assuming that theterms in P and h can be neglected with respect to the f term. Assuming further that the oscillations appearaway from the origin x = 0, we can also neglect (to firstapproximation) the h (cid:48) term such that the equation reads h (cid:48)(cid:48) ≈ h (cid:2) γ f ( x ) − (cid:3) , which has oscillating solutions for γ f ( x ) − <
0. Our numerics confirms this and we findthat the Higgs field oscillations occur in an interval of x which is bounded by those two values of x for which γ f ( x ) − m = 2, α = 0 . γ = 10 − , γ = 0 . q = 0 . w = − . f (0) ≈ . x , where γ f ( x ) − < FIG. 5. The Higgs field function h ( x ) together with (cid:2) γ f ( x ) − (cid:3) / (cid:2) γ f (0) − (cid:3) for m = 2, α = 0 . γ =10 − , γ = 0 . q = 0 . w = − . We have also investigated cases with different values of m > γ and confirm that for the parameter rangestudied here, the back-reaction of the condensate func-tion induces oscillations in the Higgs field function when f (0) becomes large, i.e. when non-linear back-reactioneffects can no longer be neglected.Note that we do not observe oscillations in the limit f (0) → m = 0, hence this phenomenon isrestricted to a regime of the parameter space which al-lows for large values of f (0). We believe this phenomeonto be very rich and to have important implications. Adetailed numerical analysis, which is outside the scope of FIG. 6. The string-forming Higgs field profile h ( x ) and thecondensate f ( x ) as functions of the rescaled core radius x for α = 0 . q = 0 . γ = γ = 10, m = 2 and various valuesof f (0) ∈ { . , . , . } (from top to bottom). this paper, is hence left as future work.
3. The second branch
When increasing the central value f (0) of the conden-sate function we observe that the structure associated tothe oscillation of the condensate function remains closeto the string axis. This changes on the second branchof solutions mentioned above. Decreasing f (0) from itsmaximal value, the value of ˜ w increases further on thesecond branch. We observe that, although the value of f (0) decreases, it does so slowly. However, with increas-ing ˜ w the structure associated to the condensate fieldoscillations moves to larger values of x , i.e. we obtain so-lutions with h ( x ) ≈ f ( x ) ≈ constant (cid:46) f (0) max onan interval x ∈ [0 : δ ], where δ increases with increasing˜ w . This is shown for m = 2, α = 1 . γ = 1 . γ = 10, q = 0 . w in Fig. 7. For w = − . f (0) = 0 .
1. Increasing ˜ w up to ˜ w c r = − . f (0) and with that the condensate and cur-rent increase close to the string axis. The maximal pos-sible value of f (0) in this case is f (0) = f (0) m ax ≈ . w further leads now to the decrease of f (0)and the increase of δ . For ˜ w = − . w = − . f (0) = 1 . f (0) = 1 . FIG. 7. The condensate field function f ( x ) for m = 2, α =1 . γ = 1 . γ = 10 . q = 0 . w .Note that ˜ w = − .
47 corresponds to ˜ w cr . The fact that the structure moves out to infinity canalso be clearly seen when investigating the location ofthe zeros of the condensate function. This is shown for asolution with m = 2 nodes, α = 1 . γ = 1 . γ = 10 . q = 0 . x and x ,respectively, of the two nodes in dependence of ˜ w .Decreasing ˜ w on the second branch of solutions turnsout to be numerically very difficult, but we believe itto be very reasonable that this second branch can beextended backwards all the way to f (0) = 0, in thelimit of which the structures moves to infinity and theenergy per unit length U and the tension T tend to in-finity. We can understand this dependence by consid-ering the condensate field equation (12) on the interval x ∈ ]0 : δ ], where h ( x ) ≡ f ( x ) = constant . = f (0)Excluding the possibility f ( x ) ≡
0, this implies that˜ w + γ (cid:2) f (0) − q (cid:3) = 0. We demonstrate that our numer-ical data joins this curve for three different sets of param-eters, see Fig. 9. Hence, though the numerics becomesvery hard at the end points of the respective numericaldata curves, the analytically given curves are (very likely)the proper continuation. We do not see any indicationsin the numerics that the curves should stop.Finally, let us explain qualitatively why two branchesof solutions in f (0) exist in our model. This is easily FIG. 8. The values x and x of the first and second node ofthe condensate function f ( x ) in dependence on ˜ w for m = 2, α = 1 . γ = 1 . γ = 10 . q = 0 . understood when remembering that we have rescaled theradial coordinate r → x = r/ Λ as well as the state param-eter w → ˜ w = Λ w , where Λ = ( √ λ η ) − is the lengthscale associated to the Higgs field. When increasing ˜ w on the first branch of solutions, we increase the conden-sate close to the string axis until we reach the maximalpossible value of the condensate related to a value of ˜ w ,which stays fixed on the second branch of solutions andis negative in the case studied above. Now to increasethe value of ˜ w further, i.e. make it tend to zero frombelow, we need to decrease Λ. But this in turn impliesthat the rescaled radial coordinate x increases. This isexactly what we observe in our numerics – the non-trivialstructure in the fields moves out to larger values of x .
4. Strings with n > We have also constructed superconducting string so-lutions with n >
1, motivated by a recent study donein a very similar model [44, 45]. As our stability analy-sis below shows, the qualitative behaviour of our resultsis independent of n . To demonstrate this, we have con-structed numerically solutions with n = 2 and n = 3 andcompared these to the n = 1 case.Our results for a solution with m = 2, α = 1, γ = 1, γ = 10, q = 0 .
1, ˜ w = − f ( x ) is practically unchanged, althoughwe observe a small decrease in the central value f (0)with increasing n (see Table II for the numerical values).Moreover, the oscillations in the Higgs field function h ( x )that we observed for n = 1 persist for n = 2 ,
3, althoughslightly modified. As far as the integrated quantities areconcerned, we observe that the energy per unit length ˜ U per winding n , i.e. ˜ U /n slightly decreases indicating thatfor our choice of couplings a superconducting string withhigher n can be interpreted as a bound state of n super-conducting strings with winding n = 1. The numerical0 FIG. 9. The value of ˜ w as function of f (0) for three differentsets of parameter choices with γ = 10. We give the numericaldata (solid blue) as well as the analytic curve γ ( q − f (0) )(dashed green). values of ˜ U /n as well as ˜
T /n are given in Table II. Thisrelates to the observations made in [44, 45]. Finally, letus mention that we also find that the value of the current˜ C decreases with increasing n . A more detailed analysisof this fact is out of the scope of this paper and is left asfuture work. FIG. 10. The profiles of the condensate, Higgs and gauge fieldfunctions (from top to bottom) for m = 2, α = 1, γ = 1, γ = 10, q = 0 .
1, ˜ w = − n = 1 , , n f (0) ˜ T πn ˜ U πn ˜ C π | ˜ v | .
245 0 .
314 7 .
679 1 . .
227 0 .
234 3 .
672 1 . .
190 0 .
234 2 .
311 1 . C. Carter stability
The macroscopic stability criterion of superconduct-ing strings [41] relates the velocities of longitudinal andtransversal perturbations to the energy per unit length U and the tension T . In the neutral limit e →
0, thedefinitions above imply that all the integrated quantities U , T and C are positive definite. One also finds, fromthe definitions, the useful relationship U − T = | v | C ⇐⇒ ˜ U − ˜ T = | ˜ v | ˜ C, (42)from which one can prove [17] that there exists a finiteneighborhood around v = 0 for which the string is macro-scopically stable, i.e. both the transverse ( c T ) and thelongitudinal ( c L ) velocities, defined above, are real. In-deed, let us first consider the spacelike case for which v ≥
0. In that case, the energy per unit length happensto equal the Lagrangian from which one deduces the fieldequations (9) to (12), so that differentiating U with re-spect to v reduces merely to differentiating the explicitappearance of v . Looking at Eq. (16), one sees that thisamounts to v ≥ U d v = C = ⇒ (42) d T d v = − v d C d v . (43)Similarly, for v ≤
0, the Lagrangian yielding the fieldequations now being T , one obtains, v ≤ T d v = C = ⇒ (42) d U d v = − v d C d v . (44)We noted earlier that C ≥
0, and given its definition (24),it is clear that lim v → C = 0: this implies that for v ≥ v = 0 such thatd C/ d v ≥
0. In this region, the first equality in Eq. (43) ensures that d
T / d v ≤
0, which, combined with the sec-ond one stating that d U/ d v ≥
0, implies that c L ≥ v ≤ v . These arguments depending only on the def-inition of the integrated quantities and on the equationsof motion that are satisfied by the fields together withthe boundary conditions, show that there must exist a fi-nite region of state parameter in which the ground stateand the excited configurations are Carter stable for bothelectric (timelike) and magnetic (spacelike) currents.In the region of parameter space studied in Sec. III Bhowever, the condensate exists only for strictly negativevalues of the state parameter, and therefore the argu-ment cannot apply, although it does apply in many otherregimes, such as that discussed in Ref. [24]. Here, onemust resort to the numerical solution, such as that shownin Fig. 4. We see that Carter criterion for stability isindeed fulfilled, so it would appear our modes are macro-scopically stable. We must therefore now move on to alocal analysis to show the microscopic instability leadingto the cosmological consequences drawn in Ref. [24] andfurther elaborated in our concluding section V. D. Linear stability analysis and decay rate
To determine the possible physical effects of the excitedsolutions, a crucial piece of information is whether theyare stable – and, if not, what is the typical time scale oftheir decay. While a full stability analysis is beyond thescope of the present paper, useful information can be ob-tained from the study of linear perturbations, on whichwe now concentrate. As we wish to determine the evo-lution in time of the solutions after small perturbations,we need the field equations for t - and z -dependent fields.For simplicity, we consider only those solutions where σ and B µ are independent on θ . The field equations are ∂ t σ − ∂ z σ − ∂ r σ − r ∂ r σ + 2 ∂ σ ∗ V = 0 , (45)( ∂ t − i eB t ) φ − ( ∂ z − i eB z ) φ − ( ∂ r − i eB r ) φ − r ( ∂ θ − i eB θ ) φ − r ( ∂ r − i eB r ) φ + 2 ∂ φ ∗ V = 0 , (46) ∂ ν ∂ ν B t − r ∂ r B t − ∂ t (cid:18) ∂ ν B ν − r B r (cid:19) + i e φ ∗ ( ∂ t − i eB t ) φ − φ ( ∂ t + i eB ) φ ∗ ] = 0 , (47) ∂ ν ∂ ν B z − r ∂ r B z − ∂ z (cid:18) ∂ ν B ν − r B r (cid:19) + i e φ ∗ ( ∂ z − i eB z ) φ − φ ( ∂ z + i eB z ) φ ∗ ] = 0 , (48)and ∂ ν ∂ ν B r + 2 r ∂ θ B θ − ∂ r ∂ ν B ν + i e φ ∗ ( ∂ r − i eB r ) φ − φ ( ∂ r + i eB r ) φ ∗ ] = 0 . (49)In the following, in order to keep the equations as simple as possible, we assume φ ∗ ∂ µ φ ∈ R for µ (cid:54) = θ .2Let us assume we have a solution φ = φ (0) , σ = σ (0) , B µ = B (0) µ of the form given in Eqs. (5) and (6). We look forperturbed solutions of the form φ = φ (0) + δφ , σ = σ (0) + δσ , B µ = B (0) µ + δB µ , where δφ ( t, r, θ, z ) = p ( r ) exp (cid:104) i (cid:16) nθ + (cid:112) λ η ν t − (cid:112) λ η κz (cid:17)(cid:105) ,δσ ( t, r, θ, z ) = s ( r ) exp (cid:110) i (cid:104) ( ω + (cid:112) λ η ν ) t − ( k + (cid:112) λ η κ ) z (cid:105)(cid:111) ,δB θ ( t, r, θ, z ) = a ( r ) exp (cid:104) i (cid:16)(cid:112) λ η ν t − (cid:112) λ η κz (cid:17)(cid:105) , (50)( ν, κ ) ∈ (i R ) , and a , p , s are three real-valued functions. We work in the gauge ∂ µ δB µ = 0 and assume B r = B t = B z = 0. One can easily show that the resulting system of equations is self-consistent provided the algebraic relation ω ν = k κ is satisfied. When allowing ν and/or κ to be more general complex numbers, B r , B t and B z are sourcedby the imaginary part of φ ∗ ∂ µ φ and can thus not be set to zero, which is why we restrict attention to perturbationsatisfying (cid:61) m ( φ ∗ ∂ µ φ ) = 0.The system to be solved is then x ∂ x ( x∂ x s ) = (cid:0) ˜ w + κ − ν + 3 γ f − γ q + γ h (cid:1) s + 2 γ f hp, x ∂ x ( x∂ x p ) = (cid:18) − ν + κ + P r + 3 h − γ f (cid:19) p + 2 γ hf s − hPx a,x∂ x (cid:18) x ∂ x a (cid:19) = (cid:0) − ν + κ + α h (cid:1) a − α P h ( p + p v ) , (51)with the boundary conditions p (0) = s (cid:48) (0) = a (0) = 0and p ( ∞ ) = s ( ∞ ) = a ( ∞ ) = 0. If there exists ν ∈ i R − such that this system has a solution, then the backgroundsolution is linearly unstable in the sense that it sup-ports perturbations growing exponentially in time. Find-ing numerical solutions to this system is challenging, asits exponentially-growing solutions make it difficult toreach a satisfactory numerical precision for the boundedones we are interested in. However, as explained in Ap-pendix A, one can already obtain information about thelinear stability of the solution by viewing the Higgs andgauge fields as nondynamical in the linear analysis, i.e.,setting p = a = 0. The system (51) then reduces to1 x ∂ x ( x∂ x s ) = (cid:0) ˜ w + κ − ν + 3 γ f − γ q + γ h (cid:1) s. (52)In the present work, since our main aim is to study thenonlinear solutions rather than linear perturbations weshall work only with Eq. (52). A more general stabilityanalysis may be interesting, but is outside of the scope ofthe present study; besides, as we also argue below, sincethe system exhibits instabilities already for this limitedrange of perturbation shapes, it can only be shown to beeven more unstable than what we obtain here.An instability corresponds to a spatially boundedmode growing exponentially in time (in a given refer-ence frame), i.e., to a bounded solution of Eq. (52) with ν − κ <
0. Since the above derivation requires ω ν = k κ ,such solutions make sense only for the magnetic case w >
0. We shall motivate below that the unstable char-acter of the solutions persists in the case w <
0. Fig. 11shows the eigenvalues ν − κ of Eq. (52) for γ = 10, γ = 200, and q = 4, for the condensates with one, two,and three nodes computed in a fixed Higgs field back-ground h ( x ) = tanh( x ). Although only the solutionswith ν − κ < • For ν − κ > ˜ w + γ − γ q , the solutions oscillatein the large x region, with an amplitude decayingas x − / . Bounded solutions thus always exist, pro-viding the continuous spectrum of Eq. (52). • For ν − κ < ˜ w + γ − γ q , the solutions areexponentially increasing or decreasing at infinity.When imposing the boundary condition s (cid:48) (0) = 0,they are thus spatially bounded only for a discreteset of values of ν − κ , and represent the discretespectrum of Eq. (52). • Among these discrete eigenvalues, one, two, andthree are negative for the solutions with one, two,and three nodes, respectively.The third point is the most important one: it meansthat the solution with m nodes (for these parameters,and m ranging from 1 to 3) has m unstable modes. Thisproperty happens to be satisfied for all the sets of param-eters we tried numerically. We also verified it holds whenworking with the actual profile of the Higgs field [solvingEqs. (9) – (12)] instead of the hyperbolic tangent ansatz.We found no instability for the solutions with m = 0.3 FIG. 11. Eigenvalues of Eq. (52) for the solutions with m =1 (top), m = 2 (middle), and m = 3 (top) nodes in themagnetic case w >
0. The parameters are γ = 10, γ = 200,and q = 4. The background condensate is computed for aHiggs field h ( x ) = tanh( x ). The shaded area shows the region ν − κ > w + γ − γ q , in which the modes oscillate at infinityinstead of decreasing exponentially. As explained in the text,only negative values of ν − κ correspond to instabilities. As mentioned above, the electric case w < .However, since the solutions we found are smooth in thelimit w → + , we conjecture that the aforementioned in- The reason is that terms in B t ∂ t ϕ and B z ∂ z ϕ will then appearin the perturbed Lagrangian, which can thus not be written inthe form (A24). stabilities will still be present, at least for small values of − w . To further motivate this, we show in Fig 12 eigen-values obtained for the condensate with one node, for thesame parameters as in Fig. 11. To obtain them, we canno longer make the assumption ω ν = k κ and Eq. (52)becomes (cid:20) ˜ w + γ (cid:0) f − q (cid:1) + γ h − ν + κ − (cid:16) ˜ ω ν − ˜ k κ (cid:17) − x · ∂ x · x · ∂ x (cid:21) s + γ f s ∗ = 0 . (53)We work in a frame where ˜ k = 0 and look for solutionswith κ = 0. Notice that the spectrum is invariant un-der complex conjugation because Eq. (53) is unchangedunder s → s ∗ , ν → ν ∗ . It is also invariant as well as un-der the symmetry transformation (˜ ω, ν ) → ( − ˜ ω, − ν ). Asshown in Fig. 12, at least one eigenvalue with a negativeimaginary part is present in most of the domain of w forwhich the solution with one node exists. Although theargument of Appendix A does not apply to this case, thissuggests that these solutions are also unstable. This com-pletes the argument that excited current-carrying cosmicstrings are unstable.Although we are mostly interested in the case wherethe winding number n is equal to 1, one may wonderif and how choosing a larger value would affect theseresults. At the level of the Higgs field, the main differencelies in the behaviour close to the string axis where h ( x )is proportional to x n . To get a first idea of the structureof the set of solutions for n >
1, we thus solved Eq. (25)numerically in a background field given by h ( x ) = tanh( κx ) n , (54)for n from 2 and 3, for the same parameters as in Fig. 12and with w = 1. We obtained similar results: first, onesolution with m nodes exists for m between 0 and a maxi-mum value (equal to 4 for n = 2 and 5 for n = 3); second,the solution with m nodes has m unstable modes. Wethus conjecture that the results obtained in this work,concerning both the structure of excited solutions andstability, remain qualitatively valid for n >
1, as isalso confirmed by our numerical construction, shown inIII B 4. A systematic analysis of this case is left for afuture work.
IV. ELECTROMAGNETIC ANDGRAVITATIONAL EFFECTSA. Solutions in the U(1) gauge × U(1) gauge model
In this subsection, we discuss the effects of the cou-pling of the current to an electromagnetic field. Figure 13shows the field profiles for various values of α in the case m = 1. Similar results were obtained for various values4 FIG. 12. Imaginary (left) and real (right) parts of some modes of the solution with one node in the electric case w <
0. Theparameters are γ = 10, γ = 200, and q = 4. The background condensate is computed for a Higgs field h ( x ) = tanh( x ). of m , showing that, as for the background mode [18], theinternal structure of the current-carrying cosmic string isessentially not modified by inclusion of electromagneticeffects, the latter being, if anything, only capable of longrange interactions on the macroscopic behavior of thestrings [19, 46]. The figure also shows clearly the ex-pected behavior of the gauge potential sourced by an in-finitely long current-carrying string, i.e. b ( r ) ∼ ln ( r/r σ ),where r σ (cid:39) m − σ , the Compton wavelength of the currentcarrier σ , provides an order of magnitude estimate of theelectromagnetic radius of the vortex. B. Gravitational effects
The space-time of a superconducting string possessesa deficit angle ∆ ∼ U + T , similar to that of a Nambu-Goto string [29], while locally there exists an attractiveforce towards the string [27, 28, 30], potentially leadingto observable effects [47].The existence of a deficit angle is responsible for a num-ber of physical effects (for a recent review see [6]). Whenthe string moves, it creates wakes that could e.g. beobservable in the 21cm radiation from hydrogen, whilethe so-called Kaiser-Stebbins-Gott effect [48] leads to dis-continuities in the CMB. Furthermore, the deficit anglewould lead to gravitational lensing that is quite distinctfrom that caused by other spatially extended objects. Fi-nally, so-called kinks and cusps on strings as well as theoscillations of string loops are believed to emit gravita-tional waves.In order to discuss gravitational effects, we couple themodel (1) minimally to gravity and choose the followingparametrization of the metric tensord s = N ( x )d t − d x − L ( x )d θ − K ( x )d z . (55)This model has already been studied in [33] and we re-fer the reader for more details to this paper. Let us just remark here that there is an extra dimensionless cou-pling in the model, which corresponds to the ratio be-tween the symmetry breaking scale η and the Planckmass M Planck = G − / : β = 8 πG N η , (56)with G N the Newton constant.Given the numerical solutions to the coupled matterand gravity equations, we can read of the deficit angle ofthe space-time from the behaviour of the metric function L ( x ):∆ = 2 π (1 − c ) , where L ( x → ∞ ) → c x + c (57)where c and c are constants that have to be determinednumerically.Solving the coupled matter and Einstein equations nu-merically we determined the deficit angle for the oscillat-ing string solutions, which is given by the sum of the en-ergy density U and the tension T . This is nothing new incomparison to the fundamental string solutions, however,we now have a dicrete set of values of the deficit anglefor one fixed set of coupling constants . Hence, measur-ing the deficit angle, e.g. by gravitational lensing, doesnot uniquely determine the values of the couplings in themodel.We also observe a new effect that is related to theoscillations of the Higgs field appearing for sufficientlylarge values f (0). We find that these trigger an oscillationin the local scalar curvature. This is demonstrated inFig. 14 for a m = 2 solution and various values of thecondensate. V. CONCLUSIONS
In this paper, we have studied excited cosmic string so-lutions with superconducting currents. These solutions5
FIG. 13. Profiles for the string-forming Higgs field h ( x ) (toppanel), the gauge condensate f ( x ) (middle) and the asymp-totically logarithmic behaving gauge potential b ( r ) − α . possess a number of nodes in the condensate field func-tion and can trigger – for sufficiently large condensates –oscillations in the Higgs field function as well as in the lo-cal scalar curvature in the space-time around the string.Though some of these solutions are macroscopically, i.e.Carter stable, we show that they are microscopically un- FIG. 14. Ricci scalar as a function of string core distancefor an m = 2 oscillatory mode with various values of thecondensate interior value f (0). stable and would decay rapidly after formation.In the macroscopic description of cosmic strings, whichcharacterizes them solely in terms of their energy perunit length U and tension T , our results are interest-ing because they imply that for a given set of physicalparameters, a discrete number of cosmic strings with dif-ferent values of U and T exist. Assuming that at theformation of cosmic string networks in the primordialuniverse these excited solutions can be formed, the evo-lution of the network would involve (from a macroscopicpoint of view) a number of different types of strings, ofwhich some are unstable. Certainly, this will modifiy thedynamics and evolution of string networks and the ques-tion arises immediately whether and how these networksreach a scaling solution.Moreover, the gravitational effects of cosmic stringsare determined by the deficit angle in their space-time(which in turn is determined solely by U and T ), leadingto a number of observable effects such as gravitationallensing as well as wakes and the Kaiser-Stebbins-Gotteffect. Now since strings with different values of U and T exist, these will lead to different effects e.g. in theCosmic Microwave background (CMB) spectra.From a microscopic point of view, the instability ofexcited solutions leads to emission of high energy par-ticle radiation that could e.g. be observed in the formof cosmic rays. Moreover, since the local curvature ofthe space-time around the string is modified by the con-densate, it is conceivable that when decaying an addi-tional emission of primordial gravitational waves can beexpected.Finally, let us state that our analysis clearly shows thatthe underlying field theoretical structure plays a very im-portant role – even when considering only the macro-physics and, hence, integrated quantities. For Nambu–Goto simulations of cosmic string network evolution (seee.g. [49–51]) the existence of a network of strings withdifferent tensions and the emission of gravitational waves6from excited strings could be of relevance, while forAbelian–Higgs string simulations (see e.g. [52–54] as wellas [55] and references therein) the excitations of the Higgsfield as well as the existence of high energy particle radi-ation could be interesting to take into account. ACKNOWLEDGMENTS
We thank ´Arp´ad Luk´acs for his careful reading ofthe manuscript and his suggestions. BH would like tothank FAPESP for financial support under grant num-ber and CNPq for financial support under
Bolsa de Produtividade Grant 304100/2015-3 . PP wouldlike to thank the Labex Institut Lagrange de Paris (refer-ence ANR-10-LABX-63) part of the Idex SUPER, withinwhich this work has been partly done.
Appendix A: A note on instabilities
In this appendix we show that, under some conditions,it is possible to study the stability of a field configura-tion without solving the full set of linearized field equa-tions. More precisely, assuming the field theory has aHamiltonian structure and the boundary conditions aresuch that the relevant operator can be diagonalized, weshow that finding one unstable mode when viewing allthe fields except one as nondynamical implies that thefull theory, with all fields dynamical, also has an insta-bility. Moreover, the growth rate of perturbations in the“restricted” problem with only one dynamical field givesa lower bound on the growth rate of the most unstablemode in the “full” problem. We first focus on the sim-pler case of a classical particle in two dimensions, whichprovides some intuition as to why adding one degree offreedom generally does not make a system more stable.We then generalize the results to an arbitrary finite num-ber of dimensions and to field theory. Finally, we explainwhy they apply to the model dealt with in the main text.
1. A toy-model: Classical point particle in a 2Dpotential
Let us consider a classical particle with mass m > V . Tomake things simple, let us assume V is quadratic: V ( x, y ) = A x + B y + C xy, (A1) This result can be easily extended to an arbitrary number ofdynamical fields following the same steps. For conciseness, werestrict here to the case of one single dynamical field, used in themain text. where A , B , and C are three real numbers. The equationof motion is m∂ t (cid:18) xy (cid:19) = − (cid:18) A CC B (cid:19) (cid:18) xy (cid:19) . (A2)To determine the stability of the equilibrium position x = y = 0, one can look for solutions whith ( x, y ) ∝ e i ν t with ν ∈ C : the equilibrium is stable if all possible values of ν are real, and unstable otherwise. Plugging this ansatzinto Eq. (A2), one finds nontrivial solutions exist if andonly if (cid:12)(cid:12)(cid:12)(cid:12) A − mν CC B − mν (cid:12)(cid:12)(cid:12)(cid:12) = 0 . (A3)The eigenvalue equation is thus:( A − mν )( B − mν ) − C = 0 . (A4)A straightforward calculation gives the possible eigenval-ues as ν = ( A + B ) ± (cid:112) ( A − B ) + 4 C m . (A5)Although it is easy from this expression to determinedirectly the stability condition, we will here follow a dif-ferent route which will be easier to generalize to a largernumber of degrees of freedom.This two-dimensional case is very particular in that alleigenvalues can be computed explicitly. However, this isgenerally not the case in the presence of a large numberof degrees of freedom. One possible way to simplify thecalculations is to assume some of them are not dynamical.In the present case, for instance, one could set by hand y = 0 and consider only the stability in the x direction.Then, the equilibrium position will be stable if A >
A <
0. Moreover, in the latter case thegrowth rate is: Im( ν ) = (cid:112) − A/m .Let us now return to Eq. (A5) and see what the con-dition
A < y set to 0 by hand can tell us about the stabilityof the “full” problem where x and y are both dynamical.Taking the − sign in this equation, one gets ν = ( A + B ) − (cid:112) ( A − B ) + 4 C m ≤ ( A + B ) − ( A − B )2 mν ≤ Am . (A6)So, the “full” problem also shows an instability, with agrowth rate larger than or equal to (cid:112) − A/m . This illus-trates a general fact: instabilities obtained when freezingsome degrees of freedom give a lower bound on the growthrate of the strongest instability in the “full” problem.
2. Generalization to a finite number of degrees offreedom
Let us generalize this to any finite number N of realdegrees of freedom. Let Φ be the vector of perturbations7with respect to some equilibrium point. We assume theLagrangian has the form L = L (0) + 12 ( ∂ t Φ) T ( ∂ t Φ) −
12 Φ T · K · Φ + O (Φ ) , (A7)where L (0) is evaluated at the equilibrium point (and thusindependent on Φ), a superscript T denotes vector trans-position, and K is a real matrix. The term O (Φ ) denoteshigher-order terms. Without loss of generality, one canassume K is symmetric, since if K is not symmetric, onecan replace it with its symmetric part ( K + K T ) /
2, whichdoes not change the value of L . Neglecting higher-orderterms in Φ, the evolution equation for perturbations is: ∂ t Φ = − K · Φ . (A8)Solutions with Φ ∝ e i ν t exist if and only if ν is aneigenvalue of K .The above analysis can be straightforwardly general-ized to complex degrees of freedom by separating theirreal and imaginary parts, provided the perturbed La-grangian can be written in the form (A7) with vectortransposition replaced by hermitian conjugation. Theoperator K can then be chosen to be hermitian withoutloss of generality.Let us define the “restricted” problem by assumingthat only the M < N first degrees of freedom are dy-namical. We denote wih a superscript ( R ) quantitiespertaining to the “restricted” problem. So, Φ ( R ) denotesthe vector of the M first components of Φ and K ( R ) the M by M submatrix of K obtained by taking only thefirst M lines and columns. The vector Φ ( R ) then obeysthe equation: ∂ t Φ ( R ) = − K ( R ) · Φ ( R ) . (A9)Let us assume the “restricted” problem has an instability,i.e., that K ( R ) has a strictly negative eigenvalue λ ( R ) .Then there exists a configuration Φ ( R )0 (cid:54) = 0 such that K ( R ) · Φ ( R )0 = λ ( R ) Φ ( R )0 . (A10)Our goal is to show that K also has a strictly negativeeigenvalue λ , such that λ ≤ λ ( R ) . This will prove thatthe “full” problem is also unstable, with a growth ratelarger than or equal to that of the “restricted” problem.We proceed by contradiction. Let us assume for a mo-ment that all eigenvalues λ i of K are strictly larger than λ ( R ) . Since K is a symmetric real matrix, it can be di-agonalized in an orthonormal basis. Let Φ be any non-vanishing vector. Let us expand it asΦ = (cid:88) i a i Φ i , (A11)where (Φ i ) ≤ i ≤ N is an orthonormal basis of eigenvectorsof K , such that K Φ i = λ i Φ i , and the a i are real numbers.We have:Φ T · K · Φ = (cid:88) i a i λ i > (cid:88) i a i λ ( R ) (A12)Φ T · K · Φ > λ ( R ) Φ T · Φ . (A13) To get a contradiction, we thus only have to find a vectorfor which this inequality is not satisfied. One example ofsuch a vector, Φ , is found by taking the first M compo-nents of Φ ( R )0 and N − M zeros. We write it schematicallyas Φ ≡ (cid:18) Φ ( R )0 (cid:19) . (A14)Then, K · Φ = (cid:18) K ( R ) · Φ ( R )0 ∗ (cid:19) = (cid:18) λ ( R ) Φ ( R )0 ∗ (cid:19) , (A15)where the star represents N − M coefficients which playno role in the following, so thatΦ T · K · Φ = λ ( R ) Φ ( R ) T · Φ ( R )0 = λ ( R ) Φ T · Φ . (A16)We obtain a contradiction, which shows that K has atleast one eigenvalue smaller than or equal to λ ( R ) .
3. Generalization to a field theory
Let us consider a theory with N ∈ N ∗ real fields ψ i , i ∈ [[1 , N ]], in ( d + 1) dimensions. For all i ∈ [[1 , N ]], wedenote by φ i a perturbation of the field ψ i . We definethe vector Φ ≡ φ φ ... φ N . (A17)Let us assume that the quadratic action S (2) may bewritten as S (2) = (cid:90) d t d d x (cid:112) | g | (cid:18) ∂ t Φ T · ∂ t Φ −
12 Φ T · K · Φ (cid:19) , (A18)where K is a real matrix of differential operators whichdoes not involve ∂ t and is independent of t , and where g is the determinant of the metric, assumed to be be every-where nonvanishing. We assume all functions and theirderivatives are bounded. As above, a superscript “ T ” in-dicates vector transposition. As above also, without lossof generality, one can assume K is symmetric for the L scalar product. Let us further assume that g is indepen-dent on time. The linear equation on perturbations isthen ∂ t Φ = − K · Φ . (A19)For each negative eigenvalue λ of K , there is thus a grow-ing and a decaying mode in time, as e ±√− λt . Conversely,any mode growing or decaying exponentially in time withrate ν corresponds to a negative eigenvalue − ν of K .8Let us assume that the “restricted” equation ∂ t φ = K , φ , (A20)where K , denotes the (1 ,
1) component of K , has astrictly negative eigenvalue λ . Then there exists a non-vanishing solution φ (0)1 such that K , φ (0)1 = λ φ (0)1 . (A21)Let us define the following vector of functions with onlyone nonvanishing component:Φ (0) ≡ φ (0)1 . (A22)We have: (cid:90) d d x (cid:112) | g | Φ (0) T · K · Φ (0) = (cid:90) d d x (cid:112) | g | φ (0)1 K , φ (0)1 = λ (cid:90) d d x (cid:112) | g | (cid:16) φ (0)1 (cid:17) < . (A23)Since K is real and symmetric, it is hermitian and thusdiagonalizable. From the above expression, using thesame argument as in the case of finite number of dimen-sion, one deduces that it has at least one strictly negativeeigenvalue (otherwise bracketing it with a L vector couldgive only positive or vanishing values). Moreover, for any λ > λ , the same argument applies to K − λ , where is the identity operator, showing that K − λ has(at least) one strictly negative eigenvalue, and thus that K has one eigenvalue strictly smaller than λ . Taking the limit λ → λ , one finds that K has (at least) oneeigenvalue smaller than or equal to λ .So, under the hypotheses of this subsection, the ex-istence of a strictly negative eigenvalue λ for the “re-stricted” problem (A20) implies that of (at least) onestrictly negative eigenvalue λ (cid:48) ≤ λ for the full prob-lem (A19).This argument can be made manifestly Lorentz-invariant in the (t,z) plane by considering an action ofthe form: S (2) = (cid:90) d t d d x (cid:112) | g | (cid:18) ∂ t Φ T · ∂ t Φ − ∂ z Φ T · ∂ z Φ −
12 Φ T · K · Φ (cid:19) , (A24)where K is symmetric for the L scalar product, inde-pendent of ( t, z ), and does not involve ( ∂ t , ∂ z ), then thelinear equation on perturbations reads ∂ t Φ − ∂ z Φ = − K · Φ . (A25)The same argument as above (with eventually a minussign) shows that if ∂ t − ∂ z has a strictly negative (respec-tively strictly positive) eigenvalue for the “restricted”problem, then it also has a strictly negative (resp. strictlypositive) eigenvalue for the full problem, with a larger orequal absolute value.
4. Application to the problem studied in the maintext
For simplicity, we work with the neutral model e = 0.We look for solutions where B = B = 0 and assumethat the metric readsd s = d t − d r − r d θ − d z . (A26)The Lagrangian density then becomes L = − G i G i − G i G i + 12 | ∂ t φ | − | ∂ z φ | + 12 | ∂ t σ | − | ∂ z σ | − G ij G ij + 12 ( D i φ )( D i φ ) ∗ + 12 ( ∂ i σ )( ∂ i σ ) ∗ − V ( φ, σ ) , (A27)where the indices i and j run from 1 to 2. Using that B = B = 0, this may be rewritten as L = −
12 ( ∂ t B i )( ∂ t B i ) + 12 ( ∂ z B i )( ∂ z B i ) + 12 | ∂ t φ | − | ∂ z φ | + 12 | ∂ t σ | − | ∂ z σ | − G ij G ij + 12 ( D i φ )( D i φ ) ∗ + 12 ( ∂ i σ )( ∂ i σ ) ∗ − V ( φ, σ ) . To go further, we restrict to solutions of the form (cid:40) φ : ( t, r, θ, z ) (cid:55)→ e i nθ ϕ ( t, r, z ) ,σ : ( t, r, θ, z ) (cid:55)→ e i( ω t − kz ) ξ ( t, r, z ) , (A28)9where ϕ and ξ are real-valued functions. The Lagrangian density may be rewritten as L = −
12 ( ∂ t B i )( ∂ t B i ) + 12 ( ∂ z B i )( ∂ z B i ) + 12 ( ∂ t ϕ ) −
12 ( ∂ z ϕ ) + 12 ( ∂ t ξ ) −
12 ( ∂ z ξ ) + ( ω − k ) ξ − G ij G ij + 12 ( ∂ i ϕ )( ∂ i ϕ ) − e r ( B θ + n ) ϕ − e B r ϕ + 12 ( ∂ i ξ )( ∂ i ξ ) − V ( ϕ, ξ ) . Considering perturbations δB i of B i , δϕ of ϕ , and δξ of ξ from a stationary solution independent of ( t, z ). Thesecond-order Lagrangian density is: L (2) = −
12 ( ∂ t δB i )( ∂ t δB i ) + 12 ( ∂ z δB i )( ∂ z δB i ) + 12 ( ∂ t δϕ ) −
12 ( ∂ z δϕ ) + 12 ( ∂ t δξ ) −
12 ( ∂ z δξ ) + ( ω − k ) δξ − δG ij δG ij + 12 ( ∂ i δϕ )( ∂ i δϕ ) − e r ( B θ + n ) ( δϕ ) − e r δB θ ϕ − e r ( B θ + n ) ϕδB θ δϕ − e δB r ) ϕ − e B r ( δϕ ) − e B r ϕδB r δϕ + 12 ( ∂ i δξ )( ∂ i δξ ) − ∂ ϕ V ( ϕ, ξ )( δϕ ) − ∂ ξ V ( ϕ, ξ )( δξ ) − ∂ ϕ ∂ ξ V ( ϕ, ξ ) δϕδξ, where δG i,j ≡ ∂ i δB j − ∂ j δB i . Let us define Φ ≡ δξδϕδB r δB θ /r . (A29)We obtain: L (2) = 12 ( ∂ t Φ) T · ( ∂ t Φ) −
12 ( ∂ z Φ) T · ( ∂ z Φ) −
12 Φ T · K · Φ + · · · , (A30)where “ · · · ” denotes total derivatives obtained by inte-gration by parts to make all derivatives in r and θ act on Φ, and K is a differential operator involving only ∂ i and depending only on r and θ . The action may thus bewritten in the form (A24). [1] H. B. Nielsen and P. Olesen, Nucl. Phys. B61 , 45 (1973).[2] T. W. B. Kibble, J. Phys. A9 , 1387 (1976).[3] A. Vilenkin, Phys. Rept. , 263 (1985).[4] M. B. Hindmarsh and T. W. B. Kibble, Rept. Prog. Phys. , 477 (1995), arXiv:hep-ph/9411342 [hep-ph].[5] A. Vilenkin and E. P. S. Shellard, Cosmic strings andother topological defects (Cambridge University Press,2000).[6] T. Vachaspati, L. Pogosian, and D. Steer, Scholarpedia , 31682 (2015), arXiv:1506.04039 [astro-ph.CO].[7] P. A. R. Ade et al. (Planck), Astron. Astrophys. ,A13 (2016), arXiv:1502.01589 [astro-ph.CO].[8] P. A. R. Ade et al. (Planck), Astron. Astrophys. ,A25 (2014), arXiv:1303.5085 [astro-ph.CO].[9] R. Ciuca, O. F. Hern´andez, and M. Wolman, (2017),arXiv:1708.08878 [astro-ph.CO].[10] E. Allys, Phys. Rev. D93 , 105021 (2016),arXiv:1512.02029 [gr-qc].[11] E. Allys, JCAP , 009 (2016), arXiv:1505.07888 [gr-qc].[12] A.-C. Davis and P. Peter, Phys. Lett.
B358 , 197 (1995), arXiv:hep-ph/9506433 [hep-ph].[13] E. Witten, Nucl. Phys.
B249 , 557 (1985).[14] C. Ringeval, Phys. Rev.
D63 , 063508 (2001), arXiv:hep-ph/0007015 [hep-ph].[15] C. Ringeval, Phys. Rev.
D64 , 123505 (2001), arXiv:hep-ph/0106179 [hep-ph].[16] A. Babul, T. Piran, and D. N. Spergel, Phys. Lett.
B202 ,307 (1988).[17] P. Peter, Phys. Rev.
D45 , 1091 (1992).[18] P. Peter, Phys. Rev.
D46 , 3335 (1992).[19] P. Peter, Phys. Rev.
D47 , 3169 (1993).[20] B. Carter and P. Peter, Phys. Rev.
D52 , 1744 (1995),arXiv:hep-ph/9411425 [hep-ph].[21] B. Hartmann and B. Carter, Phys. Rev.
D77 , 103516(2008), arXiv:0803.0266 [hep-th].[22] C. Ringeval and F. R. Bouchet, Phys. Rev.
D86 , 023513(2012), arXiv:1204.5041 [astro-ph.CO].[23] I. Yu. Rybak, A. Avgoustidis, and C. J. A. P. Martins,(2017), arXiv:1709.01839 [astro-ph.CO].[24] B. Hartmann, F. Michel, and P. Peter, Phys. Lett.
B767 ,354 (2017), arXiv:1608.02986 [hep-th]. [25] A. Babul, T. Piran, and D. N. Spergel, Phys. Lett. B209 ,477 (1988).[26] P. Amsterdamski and P. Laguna-Castillo, Phys. Rev.
D37 , 877 (1988).[27] P. Peter and D. Puy, Phys. Rev.
D48 , 5546 (1993).[28] I. Moss and S. J. Poletti, Phys. Lett.
B199 , 34 (1987).[29] B. Linet, Class. Quant. Grav. , 435 (1989).[30] P. Peter, Class. Quant. Grav. , 131 (1994).[31] M. Christensen, A. L. Larsen, and Y. Verbin, Phys. Rev. D60 , 125012 (1999), arXiv:gr-qc/9904049 [gr-qc].[32] Y. Brihaye and M. Lubo, Phys. Rev.
D62 , 085004 (2000),arXiv:hep-th/0004043 [hep-th].[33] B. Hartmann and F. Michel, Phys. Rev.
D86 , 105026(2012), arXiv:1208.4002 [hep-th].[34] A. L. Fetter and A. A. Svidzinsky, Journal of PhysicsCondensed Matter , R135 (2001), cond-mat/0102003.[35] W. G. Unruh, Phys. Rev. Lett. , 1351 (1981).[36] C. Barcelo, S. Liberati, and M. Visser, Living Rev.Rel. , 12 (2005), [Living Rev. Rel.14,3(2011)], arXiv:gr-qc/0505065 [gr-qc].[37] S. Weinfurtner, E. W. Tedford, M. C. J. Penrice, W. G.Unruh, and G. A. Lawrence, Phys. Rev. Lett. ,021302 (2011), arXiv:1008.1911 [gr-qc].[38] L. P. Euv´e, F. Michel, R. Parentani, T. G. Philbin,and G. Rousseaux, Phys. Rev. Lett. , 121301 (2016),arXiv:1511.08145 [physics.flu-dyn].[39] J. Steinhauer, Nature Phys. , 959 (2016),arXiv:1510.00621 [gr-qc].[40] T. Jacobson, Phys. Rev. D44 , 1731 (1991).[41] B. Carter, Phys. Lett.
B228 , 466 (1989).[42] F. W. Olver, D. W. Lozier, R. F. Boisvert, and C. W.Clark,
NIST Handbook of Mathematical Functions , 1sted. (Cambridge University Press, New York, NY, USA,2010). [43] B. Carter, in
Formation and interactions of topologicaldefects. Proceedings, NATO Advanced Study Institute,Cambridge, UK, August 22 - September 2, 1994 (1994)pp. 303–348.[44] P. Forgacs and A. Luk´acs, Phys. Lett.
B762 , 271 (2016),arXiv:1603.03291 [hep-th].[45] P. Forg´acs and A. Luk´acs, Phys. Rev.
D94 , 125018(2016), arXiv:1608.00021 [hep-th].[46] A. Gangui, P. Peter, and C. Boehm, Phys. Rev.
D57 ,2580 (1998), arXiv:hep-ph/9705204 [hep-ph].[47] J. Garriga and P. Peter, Class. Quant. Grav. , 1743(1994), arXiv:gr-qc/9403025 [gr-qc].[48] N. Kaiser and A. Stebbins, Nature , 391 (1984).[49] D. P. Bennett and F. R. Bouchet, Phys. Rev. D41 , 2408(1990).[50] D. P. Bennett and F. R. Bouchet, Phys. Rev. Lett. ,2776 (1989).[51] J. J. Blanco-Pillado, K. D. Olum, and B. Shlaer,Phys. Rev. D83 , 083514 (2011), arXiv:1101.5173 [astro-ph.CO].[52] G. Vincent, N. D. Antunes, and M. Hindmarsh, Phys.Rev. Lett. , 2277 (1998), arXiv:hep-ph/9708427 [hep-ph].[53] J. N. Moore, E. P. S. Shellard, and C. J. A. P. Martins,Phys. Rev. D65 , 023503 (2002), arXiv:hep-ph/0107171[hep-ph].[54] M. Hindmarsh, S. Stuckey, and N. Bevis, Phys. Rev.
D79 , 123504 (2009), arXiv:0812.1929 [hep-th].[55] M. Hindmarsh,
Cosmology - the next generation. Proceed-ings, Yukawa International Seminar, YKIS 2010, Kyoto,Japan, June 28-July 2, 2010, and Long-Term Workshopon Gravity and Cosmology 2010, GC 2010, Kyoto, Japan,May 24-July 16, 2010 , Prog. Theor. Phys. Suppl.190