Generalised Velocity-Dependent One-Scale Model for Current-Carrying Strings

Generalised velocity-dependent one-scale model for current-carrying strings

C. J. A. P. Martins,1, 2, ∗ Patrick Peter,3, 4, † I. Yu. Rybak,1, 2, ‡ and E. P. S. Shellard4, § 1Centro de Astrofísica da Universidade do Porto, Rua das Estrelas, 4150-762 Porto, Portugal 2Instituto de Astrofísica e Ciências do Espaço, CAUP, Rua das Estrelas, 4150-762 Porto, Portugal 3 GRεCO – Institut d’Astrophysique de Paris, CNRS & Sorbonne Université, UMR 7095 98 bis boulevard Arago, 75014 Paris, France 4Centre for Theoretical Cosmology, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom (Dated: November 20, 2020) We develop an analytic model to quantitatively describe the evolution of superconducting cosmic string networks. Speciﬁcally, we extend the velocity-dependent one-scale (VOS) model to incorporate arbitrary currents and charges on cosmic string worldsheets under two main assumptions, the validity of which we also discuss. We derive equations that describe the string network evolution in terms of four macroscopic parameters: the mean string separation (or alternatively the string correlation length) and the root mean square (RMS) velocity which are the cornerstones of the VOS model, together with parameters describing the averaged timelike and spacelike current contributions. We show that our extended description reproduces the particular cases of wiggly and chiral cosmic strings, previously studied in the literature. This VOS model enables investigation of the evolution and possible observational signatures of superconducting cosmic string networks for more general equations of state, and these opportunities will be exploited in a companion paper.

I. INTRODUCTION alternative methods. First, by modelling the full field theory equations, in principle, all the key physical prop- The early stage of the Universe’s evolution is thought erties of cosmic strings are reproduced, but at huge com- to have included phase transitions that may have led putational cost [16–18], although this can be mitigated to the production of cosmic strings, as originally sug- with highly efficient accelerated codes [19, 20] or by us- gested by Tom Kibble [1] (for exhaustive introductions ing adaptive mesh refinement [21]. Secondly, by relying see [2,3]). Scenarios including cosmic strings are ubiq- on the effective Nambu-Goto action, cosmic strings are uitous in grand unified theories (GUT) [4–6] and mod- approximated to be infinitely thin, thus considerably in- els of inflation [7–9]. A prominent feature of these one- creasing the dynamic range of simulations[22–24]. dimensional topological defects is their stability, i.e. once While most numerical simulations to date have been such a network is produced in the early universe, it will performed for the simplest Abelian-Higgs (or Nambu- generically survive until the present day. These networks Goto) model, it is expected that realistic cosmic strings lead to many potentially detectable observational signa- have non-trivial internal structure. A first example of tures, including anisotropies of the Cosmic Microwave such strings was discussed in [25], where the string su- Background (CMB) [10, 11], gravitational waves [12, 13] perconductivity is due to an additional charged scalar and lensing [14, 15]. Thus, astrophysical searches for cos- boson or fermion that is trapped in the string core. Cur- mic string networks also probe the high energy physics rent carrying cosmic strings were also found to be typ- of the early universe. ical outcomes of supersymmetric theories [26–28], some To obtain a precise connection between the high en- non-Abelian models [29–31] and other possible scenarios ergy stage of the universe and observational constraints [32–35]. Thus, analytic and numerical studies of these on cosmic strings, one needs to have an accurate model extended models are highly desired. for string network evolution. There are two approaches to To study the evolution of superconducting strings it is tackling this problem: on the one hand, high-resolution convenient to use an infinitely thin effective model, which numerical simulations of the detailed network evolution describes the original field theory in the same way as the arXiv:2011.09700v1 [astro-ph.CO] 19 Nov 2020 and, on the other, thermodynamical evolution models for Nambu-Goto action describes strings from the Abelian- the averaged network properties. These prove to be com- Higgs model. A first effective model for superconducting plementary because analytic models must be calibrated cosmic strings was given in [25], while a more realistic with numerical simulations, whose restricted dynamic description was developed in [36, 37]. The properties of range can then be extrapolated to cosmological scales. the current for such strings were also studied in various The simulations themselves can be performed using two works [38, 39]; a detailed review can be found in [40]. It is expected that the existence of a current flowing along cosmic strings impacts the evolution of the string ∗ [email protected] network and thus its observational predictions. A rel- † [email protected] evant example is a superconducting loop that, unlike a ‡ [email protected] standard currentless loop, can have an equilibrium con- § [email protected] figuration, known as a vorton [41, 42]. A more detailed 2 treatment of vortons was carried out through an effec- where γ is the determinant of the induced metric tive model [43, 44] and also by numerically studying its µ ν field theory [45, 46] leading to additional observational ∂X ∂X µ ν γab ≡ gµν = gµν X,aX,b, (2) constraints [47, 48]. ∂σa ∂σb While it is possible to numerically investigate particu- given in terms of the internal string coordinates σa, lar configurations of superconducting strings, it is more a ∈ [0, 1], with σ0 timelike and σ1 spacelike. The challenging to perform full simulations of a superconduct- string spans the two-dimensional worldsheet defined by µ a 0 0 1 0 1 ing cosmic string network. What is currently lacking is X (σ ) = X (σ , σ ), X(σ , σ ) . the extension of the thermodynamical approach, where We assume that the background metric g has signature the evolution of the string network is described by macro- −2: in the cosmological setup which will be the focus of scopic parameters. There are several approaches for an- our analysis, the relevant line element is given by the alytic description of a string network evolution [49, 50]. Friedmann-Lemaître-Robertson-Walker (FLRW) metric, We will follow the quantitative velocity-dependent one- namely scale (VOS) model [51–53], which has already been shown 2 2 2 2 2 2 2 to be extendable to include the effects of non-trivial inter- dsflrw = a (τ) dτ − dx = dt − a (t)dx , (3) nal structure. Specifically, such extensions have already been reported for elastic strings [54, 55], also known in terms of the cosmic (t) or conformal (τ) times. We as wiggly strings, to treat the small-scale structure on shall work below in terms of the conformal time τ only. strings, and for chiral superconducting strings [56]. In Eq. (1), the surface lagrangian L = −µ0f(κ) is nor- The purpose of this work is to fill the gap in the lit- malized by means of a parameter µ0, a constant with erature, by introducing and starting the exploration of units of mass squared (or mass per unit length, in nat- a further extension of the VOS model that applies to ural units with ~ = c = 1), and depends on a so-called a generic current-carrying string networks, using the pre- state parameter κ, defined through a scalar field ϕ (σ ) viously known results for the specific cases of wiggly and living on the string worldsheet; in practice, this is identi- chiral strings as validation of our methodology. The plan fied with the phase of a condensate [25, 39]. Specifically, for this paper is as follows. We start by reviewing the this scalar quantity of the worldsheet reads dynamical effects of currents on cosmic strings in Section ab ∂ϕ ∂ϕ ab 2, and by outlining our key assumptions in Section 3. κ = γ a b = γ ϕ,aϕ,b . (4) Both of these are then used to systematically derive the ∂σ ∂σ generalised VOS model for strings with currents, which To complete the macroscopic description, one needs the we do in Section 4. We then discuss the stability of gen- µν stress-energy tensor Tstring, which can be locally diago- eral scaling behaviours and some validating special cases nalized in Sections 5 and 6 respectively, and finally present our µν µ ν µ ν conclusions in section 7. Tstring = Uu u − T v v , (5)

where the normalized eigenvectors u and v are respectively timelike and spacelike; the slightly different case of II. DYNAMICS OF CURRENT-CARRYING a chiral current, usually treated in a completely different COSMIC STRINGS IN EXPANDING UNIVERSES manner [59], appears in our framework as the limit when the current becomes lightlike, i.e. κ → 0. µν We start by providing the microscopic equations of It can be shown that the eigenvalues of Tstring appear- motions driving the cosmological dynamics of generic ing in Eq. (5), namely the energy per unit length U and current-carrying cosmic strings in the infinitely thin ap- tension T , can be expressed through the state parame- proximation. These will be the basis for the subsequent ter κ and the dimensionless lagrangian function f(κ) as development of the VOS model. [36, 58] df U = µ f − (1 − s) κ , 0 dκ A. Embedding in background spacetime (6) df T = µ f − (1 + s) κ , 0 dκ It is well known [36, 40, 43, 57] that the influence of the current on the motion of the string worldsheet can be de- where s = +1 when κfκ > 0 and s = −1 when κfκ < 0, scribed as in the usual Lorentz-invariant (Nambu-Goto) i.e. s = κfκ/|κfκ| = sign (κfκ). This ensures that U − case by integrating the transverse degrees of freedom, ex- T = 2sµ0κfκ > 0, in agreement with the definition and cept that one is now left with a more general action [58] meaning of U and T . Here and in what follows, we denote fκ ≡ df(κ)/dκ; we use this unusual notation instead of Z √ Z √ the traditional f 0 in order to avoid any confusion with S = L(κ) −γ d2σ = −µ f(κ) −γ dσ0dσ1, 0 (1) derivatives with respect to the spacelike coordinate σ1. 3

As a side remark, we shall also make use of another thereby defining the microscopic notation for derivatives useful notation, namely the Legendre transform f˜ of the of a quantity A by A0 ≡ ∂A/∂σ and A˙ ≡ ∂A/∂τ. With generating function f. It is defined through this choice and within the framework of the metric (3), the induced metric (2) becomes ˜ f ≡ f − 2κfκ. (7) h 2 ˙ 2 2 0 2i γab = diag a 1 − X , −a X , (14) It can be seen from Eqs. (6) that f and f˜ yield in turn the energy U and tension T depending on whether the leading to the determinant current is, respectively, timelike (electric regime, κ ≥ 0) r or spacelike (magnetic regime, κ ≤ 0). √ −γ = a2 X0 2 1 − X˙ 2 , (15) With the energy per unit length U and tension T defined above, one derives [36] the velocities of propagation for transverse (ct) and longitudinal (cl) perturbations. and the inverse metric They are given by   1 1 γab = diag  , − 2 0 2  , (16) 2 T f − (1 + s) κfκ a2 1 − X˙ 2 a X ct = = , (8) U f − (1 − s) κfκ

2 dT sfκ + κfκκ (s + 1) from which one obtains the state parameter as cl = − = . (9) dU sfκ + κfκκ (s − 1) ϕ˙ 2 ϕ02 κ = − 2 0 2 . (17) In order for any model to make sense, it should be causal, a2 1 − X˙ 2 a X and therefore these velocities must be less than unity. s κf From the definition of as the sign of κ, it is clear that In what follows, as in most previous literature on this c ≤ 1 the first of these condition, t , is trivially satisfied, topic, we will make use of the quantity defined by while the limit of the longitudinal perturbation velocity sets bounds on the equation of state, specifically X0 2 2 = . (18) 1 − X˙ 2 κfκκ ≤ 0. (10) It turns out that the equations of motion derived from 2 Moreover, string stability also demands both ct ≥ 0 the action (1) with equation of state (6) are also more 2 U¯ T¯ and cl ≥ 0. The first of these constraints in turn imposes tractable if one uses the dimensionless variables , that both U and T are positive [60]. This yields and Φ defined through the relations, derived in Ref. [58] df f > 2κfκ. (11) U¯ ≡ f − 2γ00 ϕ˙ 2 = f − 2q2f , dκ κ ¯ 11 df 0 2 2 Given the definitions (6), this justifies the global sign in T ≡ f − 2γ ϕ = f + 2j fκ, (19) (1) in order to include the Nambu-Goto and chiral lim- dκ iting cases for which κ → 0: the dimensionless function 2 df 0 Φ ≡ −√ ϕ ϕ˙ = −2qjfκ, f must be assumed positive definite, f > 0. Similarly, −γ dκ the other stability criterion, with respect to longitudinal 2 where for convenience we have defined perturbations cl ≥ 0, provides 2 2 00 2 ϕ˙ fκ > 2κfκκ if fκ > 0, q ≡ γ ϕ˙ = , (12) a2 1 − X˙ 2 fκ < 2κfκκ if fκ < 0. (20) ϕ02 j2 ≡ −γ11ϕ02 = = q2 − κ. The models are discussed below. a2X0 2

With our metric convention (3), q2 and j2 are positive B. Equations of motion deﬁnite. In terms of these variables, the state parameter (17) has the simple form

In what follows, we use the gauge in which the string 2 2 κ = q − j , (21) worldsheet timelike coordinate σ0 coincides with the con- σ0 = τ σ1 ≡ σ formal time, i.e., and , and restrict atten- and is thus easily seen to be a worldsheet Lorentz scalar. tion to the choice We are now in position to write down the equations ∂X ∂X of motion, again assuming the gauge choice (13). As in X0 = τ and · ≡ X˙ · X0 = 0, (13) Ref. [58], we ﬁnd the following explicit form ∂τ ∂σ 4

a˙ h i ∂ U¯ + U¯ + T¯ X˙ 2 + U¯ − T¯ = ∂ Φ, (22a) τ a σ ¯ ¨ ¯ a˙ ¯ ¯ ˙ 2 ˙ T 0 ˙ 0 0 ˙ a˙ XU + U + T 1 − X X = ∂σ X + 2ΦX + X Φ + 2 Φ , (22b) a a q p 2 0 2 2 ˙ 2 ∂τ fκa q X = ∂σ fκa j (1 − X ) . (22c)

The dynamical equation (22b) must be solved under the the worldsheet coordinates,1 denoted by hOi, through constraint (22a), while Eq. (22c) completes the system Z (22) with the dynamics of the current. O dσ It is worth mentioning that Eqs. (22) were obtained hOi ≡ Z . (24) without any assumption. They describe the dynamics dσ of individual strings. Below, we discuss an approach, based on Refs. [51, 52, 56], thanks to which one can av- In particular, we deﬁne the total charge Q and current J erage equations Eqs. (22) and obtain a thermodynamical energy densities through the relations description which extends the currently available VOS model. In order to implement this scheme however, we Q2 ≡ hq2i,J 2 ≡ hj2i, (25) will need two speciﬁc assumptions, pertaining to boundary conditions and uncorrelated variables, which we now and the RMS velocity discuss. q v ≡ hX˙ 2i. (26)

Since over distances larger than the correlation length III. KEY MODELLING ASSUMPTIONS the string parameters become, by definition, uncorrelated, we can understand the above averaging process as In order to obtain the macroscopic variables for the a sum of uncorrelated segments with length ≈ ξ [23, 61]. VOS network evolution model, one needs to integrate In the current-carrying case, we expect the current corre- over the spacelike variable σ along all the strings in the lation length to be comparable to that of the uncharged network [51, 52]. In practice, this means that we average strings, although in full generality, one could consider a over boxes of strings, since this also assumes a similar different correlation length for the current. Indeed, in summation, as illustrated in Fig.1. the case of the wiggly model, which has been previously studied in [54, 55], the VOS model effectively describes the small-scale structure on the strings through an average at a mesoscopic scale, which is intermediate be- + + Sum over = all strings tween the microscopic scale (where the RMS velocity is defined) and the correlation length scale. A similar situation could occur for the charge and current densities

- correlation length introduced above. (independent: > )

Figure 1. A schematic illustration of our averaging process, which is calculated by integration over σ and represents a summation over segments that are uncorrelated on distances larger than ξ. A. Assumption 1: Uncorrelated variables Tours - 24th October 2019

We start by introducing two macroscopic parameters, Our first assumption is that for microscopic variables namely the total energy E and bare (currentless) energy with a definite sign (generally positive definite), we can E0 Z Z ¯ 1 E = aµ0 U dσ and E0 = aµ0 dσ (23) In Refs. [54, 55], the notation hOi refers to the weighted average involving the measure Udσ instead of our choice dσ only. These choices of weighting averages are of course not equivalent unless and define the average value of any generic function O of one assumes uncorrelated variables as will be discussed presently. 5

0.14 0.14

0.12 0.12

0.10 0.10

0.08 0.08

0.06 0.06

0.04 0.04

0.02 0.02

0.00 0.00

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

0.14 0.14

0.12 0.12

0.10 0.10

0.08 0.08

0.06 0.06

0.04 0.04

0.02 0.02

0.00 0.00

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Figure 2. Normal density probability function N for string velocities averaged over length scales ` = 0.005ξ (solid lines) and ` = ξ (dashed lines), ﬁtted to data from Nambu-Goto simulations [23] of radiation, matter and Λ-dominated universes, and as well as the averaged values for all three epochs. The derived best-ﬁt parameters are shown in the accompanying tableI.

Table I. Best-ﬁt parameters from simulations [23]. ` = 0.005ξ ` = ξ

D 4E D 2E2 2 2 D 4E D 2E2 2 2 X˙ X˙ var X˙ , X˙ X˙ X˙ var X˙ , X˙ Radiation-domination 0.14 0.09 0.05 0.024 0.016 0.008 Matter-domination 0.15 0.10 0.05 0.025 0.016 0.009 Λ-domination 0.12 0.07 0.05 0.023 0.014 0.009 Average 0.14 0.09 0.05 0.024 0.015 0.009 use the approximation to normal and the relative corrections due to the vari- 2 2 ance are consistent across the two different lengthscales, hO i ≈ hOi , (27) implying that this may be a good approximation provided one is interested in the scaling or proportionality or even, as we shall do later, hF (O)i ≈ F (hOi), for any of the quantities being averaged (rather than their abso- given arbitrary function F of the function O. lute values). The main conclusion is that this appears to We can discuss how fair this assumption actually is, by 4 2 2 be reasonable approximation on correlation length scales considering the specific example hX˙ i = h(X˙ ) i, which 4 2 2 in light of these caveats. we therefore claim can be approximated by v = (v ) . We note that this is already the case in the standard Given that the one scale modeling assumption under- VOS model and was discussed extensively in [51]. Thus lies the successful VOS approach, we therefore adopt this q2 = 0 j2 = 0 averaging Eq. (22b) (with zero current , ) assumption in what follows, bearing in mind its potential yields terms of the form weakness. This can in principle be mitigated by incor- hX˙ 4i = hX˙ 2i2 + cov(X˙ 2, X˙ 2) = v4 + var(X˙ 2). (28) porating the effect of the actual velocity distributions into the VOS model [51], specifically by allowing for dif- Lacking other theoretical input, we resort to numerical ferences between hX˙ 4i and hX˙ 2i2 in the standard VOS simulations results to check how small the variance of model [51]. One possible such treatment is outlined in the RMS velocity is. Using the results from the simula- AppendixA, where we show that this cannot be done in tions reported in [23], we fitted normal distributions to closed form without resorting to some additional assump- the RMS velocity for both small scales (` = 0.005ξ) and tion. This is not surprising given the underlying one-scale correlation length scales (` = ξ); this provides the re- context (which would clearly have to be extended to al- sults shown in Fig.2. The distributions are clearly close low for velocity distributions), but in any case this as- 6 sumption should not be a strong limiting factor in our It rests on the following two assumptions, both of which analysis. are central to the VOS model:

• the cosmic string network is Brownian, i.e., B. Assumption 2: Vanishing boundary terms µ0V µ0V E = 2 2 and E0 = 2 2 (30) The second assumption which is required in order to Lc a ξc a integrate the macroscopic equations of motion concerns the terms expressed as a total derivative and averaged inside the volume V ; this defines the current- over the string network: endowing the entire network carrying characteristic length Lc and the bare (or Nambu-Goto) correlation length ξc. Given the defi- with periodic boundary conditions, such integrals auto- ¯ matically become closed loop integrals and thus vanish nition (23) of the energies and that of U in Eq. (19), identically. In practice, we will set one finds that

Z I 2 E 2 0 ∂ {F [X (σ, τ)]} dσ → ∂ {F [X (σ, τ)]} dσ ≈ 0. E = E0hf − 2q fκi =⇒ = F − 2Q F , (31) σ σ E0 (29) Condition (29) should be valid at all times provided we where we have introduced the macroscopic notation hfi assume that all strings, even the so-called inﬁnite ones, for and its derivatives through the following are actually loops on larger scales. This assumption also F ≡ hfi,F 0 ≡ hf i,F 00 ≡ hf i; holds when a phenomenological loop chopping parameter κ κκ (32) f is introduced. we also assume that since the function depends κ = q2 − j2 It should be mentioned that one can imagine a situa- on , its averaged counterpart similarly F = F (Q2 − J 2) tion in which the integral (29) does not vanish when it is satisﬁes . calculated for one string. This happens for instance when With (30), in turn, we obtain the required connec- the function F is discontinuous, i.e., when the string has tion between the two characteristic lengths, namely kink or cusp like structures. However, when we integrate p 2 0 and sum over the entire string network, it is reasonable to Lc F − 2Q F = ξc , (33) assume that the sum (29) will contain both positive and negative contributions that one can expect, on average, which are clearly not independent. In what follows to compensate, yielding an overall vanishing result. we will mostly work with Lc, but will occasionally also discuss the behaviour of ξc.

IV. VOS MODEL FOR STRINGS WITH • the average comoving radius of curvature for the CURRENTS string network coincides with its physical characteristic length Rc = ξc (see [51, 52] for further de- Under the assumptions discussed in the previous sec- tails). tion and after some algebraic manipulations, it is possible to obtain a system of diﬀerential equations for the rele- One ends up with the following system of evolution vant macroscopic variables describing the string network. equations, the derivation of which is outlined inB,

dLc a˙ Lc 2 2 2 0 2 2 0 = v F − Q − J F − Q + J F , (34a) dτ a F − 2Q2F 0 2 ( ) dv 1 − v k(v) 2 0 a˙ 2 2 0 = F + 2J F − 2v F − Q − J F , (34b) 2 0 p 2 0 dτ F − 2Q F Lc F − 2Q F a " # dJ 2 vk(v) a˙ = 2J 2 − , (34c) p 2 0 dτ Lc F − 2Q F a " # dQ2 F 0 + 2J 2F 00 vk(v) a˙ = 2Q2 − , (34d) 0 2 00 p 2 0 dτ F + 2Q F Lc F − 2Q F a

where we have used notation for F 0,F 00 deﬁned in (32). In Eqs. (34b), (34c) and (34d), the momentum parameter 7

where the parameters k0, α and β have been obtained 1.4 from a robust statistical analysis and have the values 1.2 k0 ≈ 1.3, α ≈ 2.3, β ≈ 1.5; different models are expected to yield a similar functional dependence in the velocity, 1.0 with potentially different values of the macroscopic pa- 0.8 rameters k0, α and β. For comparison, the relevant momentum parameter functions (36) and (37) are depicted 0.6 in Fig.3. 0.4 Eqs. (34) do not include energy loss terms, which should thus be added phenomenologically. String inter- 0.2 commutings can lead to the production of loops, which eventually shrink and radiate their energy away, thereby 0.0 0.2 0.4 0.6 0.8 reducing the energy contained in the string network, hence increasing the correlation length. To incorporate the energy loss terms into Eqs. (34), we follow the argu- Figure 3. Momentum parameter functions k (v) [Eq. (36)] ng ments of [56]. We first write down the energy loss for ξc and ksim(v) [Eq. (37)] used in the macroscopic equations of in the conventional form [51] motion (34).

˙ ξc c˜ v k(v) = ··· + , (38) is defined through ξc 2 ξc D E (1 − X˙ 2)(X˙ · u) c˜ k(v) ≡ , (35) where the chopping efficiency function is a constant v(1 − v2) for the simplest string models [18] but could conceivably depend on the charge and current carried by the network where u is a unit vector parallel to the curvature radius [54, 55]. vector; it is a velocity-dependent function2, whose form, The corresponding energy loss term for the overall in the Nambu-Goto case, is given by [52] characteristic length Lc has an analogous form, but √ 6 √ should generically include a bias function g(J, Q), 2 2 1 − 8v 2 3 k (v) = 1 − v 1 + 2 2v . (36) ng π 1 + 8v6 L˙ c c˜ v Eq. (36) is obtained in a semi-analytic way by estimating = ··· + g(J, Q) . (39) L 2 ξ its shape in two different regimes (non-relativistic and c c ultra relativistic) and then smoothly connecting those regimes, relying on comparisons with Nambu-Goto simu- This bias function phenomenologically allows for the pos- lations [52]. This is the reason why this phenomenological sibility that regions with different amounts of charge or function contains no free parameter. current may be subject to intercommutings and be incor- An alternative form of the momentum parameter was porated into loops with different probabilites (bearing in suggested and discussed in details in [18], relying on mind, for example, that they are likely to have different numerical simulations of Abelian-Higgs string networks. velocities). This has the form Upon using Eq. (33) between the characteristic length 1 − (αv2)β ξc associated with the bare energy E0 and that associated k (v) = k0 , (37) with the total E, namely Lc, we obtain sim 1 + (αv2)β

˙ 2• 2• 0 2 2• 2• 00 L˙ c v ξc v Q + J F + 2Q Q − J F 2 + ··· + gc˜ = 2 + ··· +c ˜ + 2 0 (40) Lc ξc ξc ξc F − 2Q F and

0 2 00 p 2 0 2• F + 2Q F 2• v F − 2Q F Q 0 2 00 + J = ··· +c ˜ 0 2 00 (g − 1) . (41) F − 2Q F Lc F − 2Q F

2 It might be the case that the momentum parameter also depends has standard velocity-dependent form and leave the question of on the current, or alternatively that it has a diﬀerent velocity de- charge dependence for future studies. pendence, hereinafter we will assume that momentum parameter 8

To be consistent with Eq. (41) — in other words, to ensure energy conservation — one must also add analogous phenomenological terms for the charge and current loss through chopping in the following way

p 2 0 2• v F − 2Q F J = ··· + ρc˜ 0 2 00 (g − 1) , Lc F − 2Q F (42) p 2 0 2• v F − 2Q F Q = ··· + (1 − ρ)˜c 0 2 00 (g − 1) , Lc F + 2Q F where ρ is an arbitrary constant. We emphasize that g and ρ are phenomenologial bias parameters, whose ﬁducial values in the unbiased Nambu-Goto case are respectively g = 1 and ρ = 1/2. In particular, note that if ρ is biased then g must also be biased, but the opposite need not be true. As a result, one obtains the generalized evolution equation for the correlation length 1 a˙ v ξ˙ = ξ v2 W + Q2 + J 2 F 0 − vk(v)(J 2 + Q2)F 0 +c ˜ , c W 2 a c 2 given here for comparison convenience, together with the full system of our generalized VOS model including arbitrary charges and currents

˙ a˙ Lc 2 2 2 2 0 2 2 0 g c˜ Lc = v W + Q + J F − Q + J F + v, (43a) a W 2 W 2 2 1 − v k(v) 2 2 2 0 a˙ 2 2 2 0 v˙ = 2 W + 2 Q + J F − 2v W + Q + J F , (43b) W LcW a 2• 2 vk(v) a˙ v (g − 1) W J = 2J − + ρc˜ 0 2 00 , (43c) LcW a Lc F − 2Q F 0 2 00 2• 2 F + 2J F vk(v) a˙ v (g − 1) W Q = 2Q 0 2 00 − + (1 − ρ)˜c 0 2 00 . (43d) F + 2Q F LcW a Lc F + 2Q F p where for simplicity we have used the notation W = F − 2Q2F 0, assuming it to be positive-deﬁnite.

Having introduced the general evolution equations for One can extend this model to include higher order the VOS model, we now discuss specific examples of mod- terms at the expense of introducing another mass pa- els already considered in the literature, both at the mi- rameter mσ, as we shall see below, but before doing so, croscopic levels, which our formalism accommodated. one can consider another self-dual model, based on the generating function √ 1 V. MICROSCOPIC MODELS AND THEIR f kk = 1 − κ¯ =⇒ f˜kk = √ , (45) STABILITY 1 − κ¯ and resulting from the motion of a Nambu-Goto string in There are many ways to introduce a current along a a 5 dimensional Kaluza-Klein spacetime with the extra- cosmic string and to describe it in terms of the timelike dimension curled into a circle [63, 64]. This model also κ or spacelike state parameter ; their general properties describes the so-called wiggly case for which one inte- are discussed in Ref. [36], which we summarize and adapt grates over the small-scale structure of the string, which to our notations below. becomes effectively current-carrying [65, 66]. The simplest option, which we shall refer to as the A field theoretical model for superconducting strings linear model, consists in assuming the current to be small was proposed by Witten in Ref. [25] and explored numer- and the correction it induces on the equation of state to ically in full depth in Refs. [38, 39, 60]. Analytic approx- represent a perturbation, that is, we only allow for a first imations were proposed, based on asymptotic properties order term and write of the classical fields making up the internal string struc- κ κ¯ κ¯ ture, which yield two distinct phenomenological models. f lin = 1 − = 1 − =⇒ f˜lin = 1 + , (44) 2µ0 2 2 This “realistic” current-carrying string model is based on a string-forming Higgs field, breaking a U(1) symmetry, in which we factored out the string scale µ0 and intro- and a bosonic or fermionic charge-carrier that condenses duced the dimensionless degree of freedom κ¯ through in the vortex core at an energy scale mσ. This conden- κ = µ0κ¯. This possibly over-simplified equation of state sate may or may not [38, 67] be coupled to a long-range also has the advantage of being self-dual [36, 62]. gauge field, whose influence on the worldsheet dynamics 9 is mostly negligible [39, 60]; it is such a coupling with a long-range field, when the latter is identified to electro- magnetism, that led to the name “superconducting” for such strings. Figure4 illustrates the kind of equation of state, i.e. the energy per unit length and tension as functions of the state parameter κ¯ that can be obtained in a realistic, though simplified, Witten neutral model. Once the current-carrier degree of freedom is inte- grated over a cross-section of the string, the latter becomes effectively two-dimensional, and for a spacelike current [68], one finds that increasing the equation of state parameter leads first to an increase of the current, followed by a saturation effect. After that limit, any further increase of the state parameter (phase gradient of the current carrier) leads to a decrease of the corresponding current. While the energy per unit length always increases, the tension decreases for increasing current until saturation is reached, and then increases. This implies an instability with respect to longitudinal perturbations 2 (cl ≤ 0). A generic feature derivable from the Witten field the- Figure 4. Schematic equation of state obtained for the neu- ory model for superconducting strings is that the relevant tral Witten model showing the energy per unit length U and macroscopic model is supersonic, in other words that it tension T as functions of the square root of the state param- satisfies [38] ct > cl. Assuming this result to hold, the eter κ¯, which is proportional to the phase gradient itself, i.e. constraint p|κ| = ω for κ > 0 and p|κ| = −k for κ < 0. The hatched region is unstable withDAMTP respect – to3/12/2018 longitudinal perturbations κffκκ ≤ 0 (46) 2 (cl < 0). Adapted from [38] using arbitrary units. should also apply. The Witten model provides two separate (although numerically very close) equations of state. The first pro- − Linear equation of state vides the best approximation in the magnetic regime for The constraints on (44) reduce to which the current is spacelike. It is derived from the lin function f ≥ 0 =⇒ κ¯ < 2, (50) 1 κ¯ (1 + ακ¯) mag ˜mag lin f = 1 − =⇒ f = 1 + 2 . f˜ ≥ 0 =⇒ κ¯ > −2 , 2 1 − ακ¯ 2 (1 − ακ¯) (51) (47) lin 1 the derivatives being constant, namely fκ = − 2 The electric regime for which the current is timelike, on and f˜lin = 1 . the other hand, is better described by κ 2 − Kaluza-Klein equation of state ln (1 − ακ¯) f elec = 1 + For this other self-dual model, Eq. (45) yields 2α (48) κ¯ ln (1 − ακ¯) kk ˜kk =⇒ f˜elec =1 + + . f ≥ 0 and f ≥ 0 =⇒ κ¯ < 1, (52) 1 − ακ¯ 2α and this is the only requirement since In Eqs. (47) and (48), we introduced the dimensionless 1 parameter α f kk = − √ ≤ 0 ∀κ¯ < 1, (53) κ 2 1 − κ¯ µ m 2 α = 0 = Higgs , 2 (49) 1 mσ mσ ˜kk fκ = ≥ 0 ∀κ¯ < 1, (54) 2 (1 − κ¯)3/2 given by the ratio of the string-forming Higgs field mass mHiggs to that of the current-carrier mσ. In order for the − Witten model magnetic equation of state current to condense in the string, this ratio must be less than unity [25]; it is however hardly bounded from below This behavior is encoded in the phenomenological [42]. function (47). For the tension and the energy per Applying the constraints derived earlier to the specific unit length to be positive, one first needs to enforce models discussed above, we find the following limits on 2 f mag ≥ 0 =⇒ κ¯ ≤ . (55) either the state parameter κ¯ or the extra parameter α: 1 + 2α 10

For the dual function, one ﬁnds that for α > 1/16, �� f˜mag(¯κ) > 0 provided κ¯ satisﬁes the constraint �� ˜mag above. Indeed, fκ = 0 for κ¯ = −1/(3α), and ˜mag f [−1/(3α)] = 1 − 1/(16α). If α ≤ 1/16, the �� equation f˜mag = 0 has two solutions, both negative (corresponding to a spacelike current), namely ��

4 �� κ¯± = √ with |κ¯+| ≥ |κ¯−|. (56) 4α − 1 ± 1 − 16α �� In actual model calculations, however, one ﬁnds that the current–carrier mass mσ must be suﬃ- �� ciently smaller than the Higgs mass in order for -� -� -� � � the condensate to occur, and this means one can safely assume α > 1/16 in what follows.

− Witten model electric equation of state Figure 5. Analytic equations of state constructed from the generating functions f lin [Eq. (44)], f kk [Eq. (45)], f mag The magnetic equation of state provides an ac- [Eq. (47)] and f elec [Eq. (48)]. For the magnetic and electric curate description for a spacelike current-carrying Witten models, we set α = 0.6 for representation purposes. cosmic string and for the most part of the timelike current case, but numerical simulations and asymptotic expansions also showed another effect in the electric regime, namely that of phase frequency threshold [38]. This amounts to a simple pole in the classical calculation of the condensed A. The wiggly case charge (as opposed to a second order pole aris- ing from f mag), and thus to a logarithmic generating function (48). The situation is roughly the For the wiggly model, we define same as for the magnetic Witten case, but the log- arithm prevents analytic calculations to be made elec p throughout. Again, the positiveness of f yields F = 1 − Q2 ≡ µ−1, (58) the phase frequency threshold 1 − e−2α f elec ≥ 0 =⇒ κ¯ ≤ (57) α where J = 0. Additionally, one can manage to be consistent with the notations in Refs. [54, 55] by introducing and that of its dual f˜elec leads to two negative solu- the parameters η and D such that tions κ¯± which must be calculated numerically. As above, f˜elec is however positive definite provided α exceeds a limiting value, numerically estimated to c˜ = 1 + D 1 − µ−2 c, αnum ' 0.156. We shall restrict attention to this 1 + η 1 − µ−1/2 physically motivated situation. g = , (59) 1 + D (1 − µ−2) Since, as with the previous models, one finds that elec −1 ρ = 0, fκ −[2(1−ακ¯)] < 0, the stability is then estab- ˜elec lished provided fκ > 0, which leads to κ¯ > −1/α.

These equations of state are shown graphically in where c is a constant loop chopping efficiency [51], while Fig.5. η quantifies how much energy the long string network loses to small-scale loops, which are produced due to the presence of wiggles when a large (typically correlation VI. MACROSCOPIC MODELS length-sized) loop is produced; the parameter D quantifies the energy transferred from the bare string into wig- Here we first show how the system consisting of gles as a result of any intercommuting (whether or not Eqs. (43) can be reduced to already known wiggly loops are produced). In other words, it quantifies the [54, 55, 65, 66] and chiral [56, 59] models, and then in- energy transfer from large to small scales. troduce a macroscopic version of the linear model which will be the starting point for our exploitation of the new Plugging Eqs. (58) and (59) into Eqs. (43), one recov- VOS model in the companion paper. ers the wiggly model described in Refs. [54, 55], namely 11

The set of equations to describe the chiral model is then obtained in a straightforward manner from a˙ 1 − v2 L˙ = L 1 + v2 − + Eqs. (43). They read c 2a c µ2 −1/2 ˙ a˙ Lc 2 gc˜ 1 + η 1 − µ c Lc = v + Y + √ v, (65a) √ v, (60a) a 1 + Y 2 1 + Y 2 µ 2 1 − v (1 − Y )k(v) a˙ 2 k(v) a˙ 1 v˙ = √ − 2v , (65b) v˙ = 1 − v − v 1 + , (60b) 1 + Y L 1 + Y a L µ5/2 a µ2 c c vk(v) a˙ v √ µ˙ v 1 Y˙ = 2Y √ − − c˜(g − 1) 1 + Y, = √ k(v) 1 − + L 1 + Y a L µ L µ µ2 c c c (65c) 1 c η 1 − µ−1/2 − D 1 − − µ2 which correspond to Eqs. (105-107) of Ref. [58] provided s → 0 c˜ → c a˙ 1 one sets and , and if one assumes the bias 1 − , (60c) g in (39) compensates exactly the diﬀerence between the a µ2 total√ and bare correlation lengths (33), in other words if g = 1 + Y . This can therefore be seen as the minimally which are in agreement with Eqs. (78–80) of Ref. [58]. biased model. Note that since ρ = 0, this is a maximally biased model (in the previously discussed sense), while the rest of Eqs (59) can be written C. The linear case c˜ = 1 + DQ2 c, The linear case is the natural macroscopic version of 1 (61) g ∼ 1 + η − D Q2 ; the microscopic linear model and is therefore obtained by 4 setting the fact the small-scale structure is seen to grow in K 1 F (K) = 1 − =⇒ F 0 = − and F 00 = 0, (66) Nambu-Goto simulations indicates that D > η/4 and 2 2 therefore g < 1 in this case. with the macroscopic state parameter

K ≡ Q2 − J 2 , (67) B. The chiral case effectively√ measuring the distance to chirality and leading For the chiral model, we set to W = 1 + Y , where the average current amplitude Y is defined by Eq. (64). This transforms Eqs. (43) into 1 F = 1,F 0 = − ,F 00 = 0 and J 2 = Q2, (62) (65) for the variables Lc, v and Y , together with 2 vk(v) a˙ K˙ = 2K √ − so as to ensure that the current is everywhere lightlike, a κ → 0 Q2 J 2 Lc 1 + Y i.e., . The evolution equations for and are v √ (68) then identical provided we choose − 2 c˜(g − 1) (1 − 2ρ) 1 + Y, Lc 1 ρ = , (63) for the chirality parameter K. 2 For an initially very small current with K 1 and 2 2 Y 1 and it therefore suffices to fix J (τini) = Q (τini) for some , the linear model applies whatever the true model, initial time τini to ensure the current remains chiral at all and it becomes possible to figure out the conditions un- times. The only physically relevant variable to describe der which the current might grow, at least in the case the effect of the current is thus its amplitude, namely when the source term in Eq. (68) vanishes (ρ = 1/2). In this special case, supposing the other quantities reach a 1 2 2 scaling solution in which the linear regime is still a valid Y ≡ Q + J . (64) 2 approximation, the averaged 4-current magnitude K(τ) 2α behaves as Kρ=1/2 = τ , with The other macroscopic parameter, g, can be chosen in many different ways, depending on the model that we v k(v ) v k(v ) α = sc√ sc − n ≈ sc sc − n, wish to reproduce in Ref. [56]. It should be also pointed (69) ζsc 1 + Ysc ζsc out that we reproduce the model of Ref. [56] with vanishing s (s = 0), due to our previously discussed assumption to zeroth order in Ysc 1, where vsc, ζscand Ysc are the of vanishing boundary terms. scaling values of the relevant functions. 12

If α > 0, the average 4-current K(τ) grows so the “dis- previous work on the wiggly and chiral cases, that the be- tance” to chirality increases and the non-linear regime haviour of the additional degrees of freedom (in our case, may be reached to yield another, non-trivial and current- the charges and currents) will depend on a competition carrying, scaling solution. When α < 0 on the other between the cosmological expansion rate and available hand, the string network is dragged back to its original physical mechanisms determining how these charges and condition, approaching the chiral conditions K(τ) → 0 at currents are produced (through reconnection or say pri- late times, although perhaps with a non-vanishing cur- mordial magnetic fields) and removed from the network rent amplitude Ysc 6= 0. It is interesting to note that (through reconnection and loop production). Naturally, such a non-linear current is more probable to build dur- such physical mechanisms are expected to be different for ing the radiation dominated era (n = 1) than during the different models. subsequent matter dominated era (n = 2). What our preliminary analysis already suggests is that a faster expansion rate (say the matter era, as opposed to the radiation era) facilitates the evolution towards the VII. CONCLUSION chiral limit, with equal amounts of charge and current. One can therefore envisage significantly different proper- We have proposed a natural extension of the veloc- ties of superconducting string networks in the radiation ity one-scale VOS model, originally aimed at describing and matter eras, leading to correspondingly different ob- Nambu-Goto cosmic string networks through the evolu- servational signatures That said, one must also bear in tion of their most salient statistical properties, namely a mind that the Nambu-Goto limit, with zero charge and characteristic length scale or correlation length and a root current, is a (trivial) case of this chiral limit. While mean square velocity, to include superconducting current it is natural that such a Nambu-Goto limit exists for properties that are, in principle, expected in many par- some parameter range within these models (indeed, all ticle physics scenarios. the more so in the linear model), the interesting ques- An arbitrary equation of state supposedly derivable tion is whether or not there is also a parameter range from the microscopic structure of the string yields a non- for which a chiral solution with a non-trivial charge and linear σ−model description, enabling the identification current also exists. of a single Lagrangian function of a state parameter, itself leading unambiguously to dynamical charge and current densities along the string network. Averaging, ACKNOWLEDGMENTS one obtains a generalization of the VOS model, namely Eqs. (43) which, among others, applies to the wiggly and chiral cases examined in earlier studies. As in these two This work was financed by FEDER—Fundo Eu- specific cases, previously discussed in some detail, such ropeu de Desenvolvimento Regional funds through the extended models include two different length scales, de- COMPETE 2020—Operational Programme for Com- noted Lc and ξc, that are related through the remaining petitiveness and Internationalisation (POCI), through degrees of freedom. Broadly speaking, the former en- grants POCI-01-0145-FEDER-028987 and POCI-01- codes the total energy while the latter (which retains the 0145-FEDER-031938 and by Portuguese funds through physical interpretation of a correlation length) encodes FCT - Fundação para a Ciência e a Tecnologia in the the energy in the bare string. This is to be contrasted framework of the projects PTDC/FIS-AST/28987/2017 with the original one-scale model, where the string corre- and PTDC/FIS-PAR/31938/2017. lation length, inter-string separation and string curvature PP wishes to thank Churchill College, Cambridge, radius are all assumed to coincide. where he was partially supported by a fellowship funded We have also started the exploration of our new formal- by the Higher Education, Research and Innovation Dpt of ism by briefly considering the simplest non-trivial case— the French Embassy to the United-Kingdom during this the small-current limit described by Eqs. (68). This very research. PS acknowledges funding from STFC Consoli- preliminary analysis confirms expectations, informed by dated Grant ST/P000673/1. 13

Appendix A: hX˙ 4i and hX˙ 2i2 diﬀerences for standard VOS model

The difference between hX˙ 4i and hX˙ 2i2 can be studied in the standard VOS model perturbatively. We first set all charge and current terms to zero and study equation (22b), which can be rewritten as 1 dX˙ 2 a˙ X00 · X˙ + 2X˙ 2 1 − X˙ 2 = , 2 dτ a (A1) 1 dX˙ 4 a˙ X00 · X˙ + 2X˙ 4 1 − X˙ 2 = X˙ 2 , 4 dτ a and similar expressions for the time derivative of higher powers of X˙ 2. Averaging (A1), one obtains an infinite series of equations dv2 1 (2) a˙ h 4 4 i 2 a˙ 4 a˙ k[v(2)] h 2 i + v(2) − v(4) + 2v(2) − 2v(4) = 1 − v(2) , 2 dτ a a a Rc dv4 1 (4) 1 a˙ h 6 4 2 i 4 a˙ 6 a˙ k[v(4)] h 2 4 i + v(6) − v(4)v(2) + 2v(4) − 2v(6) = v(4) v(2) − v(4) , 4 dτ 2 a a a Rc (A2) dv6 1 (6) 1 a˙ h 8 6 2 i 6 a˙ 8 a˙ k[v(6)] h 4 6 i + v(8) − v(6)v(2) + 2v(6) − 2v(8) = v(6) v(4) − v(6) , 6 dτ 3 a a a Rc ··· , where the vector u is a unit normal vector oriented towards the radius of curvature and Rc is the averaged comoving D E1/2 D E1/(2p) ˙ 2 ˙ 2p radius of curvature, v(2) = X = v as defined in the main text, Eq. (26), and similarly, v(2p) = X for D E2 D E ˙ 2 ˙ 4 any p ∈ N, and we need to distinguish products of the form X 6= X , which merely mean that v(2p) 6= v(2q) for p 6= q. The set of equations (A2) for the averages of various powers of X˙ 2 is, in theory, an infinite hierarchy. In practice n n−2 2 however, since the velocities are always less than unity, the difference v(n) − v(n−2)v(2) is getting increasingly smaller with larger values of n so that it is reasonable to end the series for some finite value of n: truncating these equations at a given n point merely amounts to choosing the required precision. Let us in what follows truncate the chain of 8 6 2 h 8 6 2 i equations (A2) on the third step, and therefore assume v(8) ≈ v(6)v(2) + O v(8) − v(6)v(2) . We wish to compare the standard VOS model [51, 52] and the model with distinguished v(2), v(4) and v(6). Intro- ducing the chopping efficiency c and combining Eq. (A2) with the average energy relation (22b) we get

dv2 √ k[v ] h i h i h i 1 (2) 2 (2) 2 a˙ 4 4 a˙ 2 4 = v 1 − v(2) + v(4) − v(2) − 2 v(2) − v(4) , 2 dτ Lc a a dv4 1 (4) k[v(4)] h 2 4 i 1 a˙ h 6 4 2 i a˙ h 4 6 i = v(4) v(2) − v(4) + v(6) − v(4)v(2) − 2 v(4) − v(6) , 4 dτ Lc 2 a a (A3) dv6 1 (6) k[v(4)] h 4 6 i 1 a˙ h 8 6 2 i a˙ h 6 8 i = v(6) v(4) − v(6) + v(8) − v(6)v(2) − 2 v(6) − v(8) , 6 dτ Lc 3 a a dL a˙ c c = L v2 + v , dτ a c (2) 2 (2) 2 where we assumed that the network is Brownian, so that E = µ0V/(aLc) in the volume V [Eq. (30)], we also 8 6 2 approximated that Rc = Lc, and defined the truncation at v(8) ≈ v(6)v(2). The initial conditions are chosen to be different for v(2), v(4) and v(6), with differences of order ≈ 0.1. The numerical solution of the system (A3) is compared to the standard VOS model, and the result is shown on figure6. It is seen that even if we try to impose the difference between hX˙ 4i and hX˙ 2i2 in the VOS model, by introducing the new variables v(2), v(4), and so forth, the variance eventually goes to zero as the system (A3) evolves. It might be an illustration of the fact that VOS model works on large scale (distances larger than ξ) and cannot properly grasp small-scale structure dynamics. This issue should be addressed to the studies of the models that are legitimate on different scales. Similar assumption for terms as hX˙ qi should be valid even with higher accuracy, due to smaller correlations between the current and the velocity. For any variable O satisfying hO2i= 6 hOi2 and leading to an expansion in the average, we anticipate a behavior similar to that obtained for the velocities. 14

10 1

10 3

10 5

10 7

10 9

10 11

0.02 0.04 0.06 0.08 0.10

2 2 4 4 6 6 Figure 6. Numerical solution of (A3) for a ∝ τ, with initial conditions v(2)0 = 0.4 + 0.1, v(4)0 = 0.4 + 0.1, v(6)0 = 0.4 + 0.1, Lc0 = 0.001, c = 0.23, and v? is the velocity for the standard VOS model with the assumption 1.

Appendix B: Derivation of the macroscopic equations

The system of Eqs. (22) can be rewritten with a supplementary equation, which arises from the combinations of Eqs. (22b) and (22c), in the following form

a˙ n o ∂ f − 2q2f + 2 X˙ 2 f − f (q2 − j2) − f j2 + q2 = − 2∂ (f qj) , (B1a) τ κ a κ κ σ κ a˙ f + 2f p X¨ f − 2q2f + 2X˙ 1 − X˙ 2 f − f (q2 − j2) = ∂ κ X0 − 4X˙ 0f qj κ a κ σ κ (B1b) a˙ − 2X0 2 f qj + ∂ (f qj) , a κ τ κ " 00 2 # q a˙ X · X˙ • 2q f + f • f q ˙ 2 2 κκ κ 2 κκ ˙ 2 fκq (1 − X ) − 0 2 + q 2 − j = ∂σ fκj (1 − X ) , (B1c) a X 2q fκ fκ 2 ( 00 ˙ 2 ) j a˙ X · X 2q fκκ + fκ 2• 2• j 1 − 2 fκ − 0 2 + 2 2 q − j = ∂σ fκ , (B1d) q a X 2fκ(q − j ) q where we used relations from the gauge condition (13)

X˙ 0 · X˙ = −X¨ · X0, X˙ 0 · X0 = −X˙ · X00, (B2) and

0 X0 · X00 X¨ · X0 = − , (B3) X0 2 1 − X˙ 2 ˙ X¨ · X˙ X˙ · X00 = − . (B4) 1 − X˙ 2 X0 2

Let us use the macroscopic variables deﬁned in (23)–(26) to obtain a thermodynamical description of the system (B1). This provides a connection between the bare and total energies of the string network, namely Z 2 E = aµ0 f − 2q fκ dσ Z R fdσ R q2f dσ = aµ dσ − 2 κ (B5) 0 R dσ R dσ 2 2 2 2 2 0 = E0 Q − J F − 2Q Q − J F , 15 where we used the notations (32) together with assumption 1. This leads to

˙ a˙ Lc 2 2 2 0 0 Lc = v F − Q − J F − F (Q + J) , (B6) a F − 2Q2F 0 once the Brownian assumption (30) is also used. Similarly, the average equation of motion for the velocity (B1b) can be written as

2 ( ) (1 − v ) k(v) 2 0 a˙ 2 2 0 v˙ = F + 2J F − 2v F − Q − J F , (B7) 2 0 p 2 0 F − 2Q F Lc F − 2Q F a

p 2 0 where we assumed Rc = ξc (Rc = Lc F − 2Q F ) and used both assumptions 1 and 2. Finally, the average Eqs. (B1c) and (B1d) for the charge and current take the form " # • k(v) a˙ J 2 = 2J 2 v − , p 2 0 Lc F − 2Q F a (B8) 2 00 0 " # • 2J F + F k(v) a˙ Q2 = 2Q2 v − . 2 00 0 p 2 0 2Q F + F Lc F − 2Q F a

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