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

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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, y I. Yu. Rybak,1, 2, z and E. P. S. Shellard4, x 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 GR"CO – 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 incorpo- rate 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 contribu- tions. 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- y [email protected] evant example is a superconducting loop that, unlike a z [email protected] standard currentless loop, can have an equilibrium con- x [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 2 [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 ' (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 @' @' ab 2, and by outlining our key assumptions in Section 3. κ = γ a b = γ ';a';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 respec- tively 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.
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