Cosmic Strings and Cosmic Superstrings

Cosmic Strings and Cosmic Superstrings

Mairi Sakellariadou a

aDepartment of Physics, King’s College, University of London, Strand, London WC2R 2LS, U.K.

In these lectures, I review the current status of cosmic strings and cosmic superstrings. I first discuss topo-

logical defects in the context of Grand Unified Theories, focusing in particular in cosmic strings arising as gauge

theory solitons. I discuss the reconciliation between cosmic strings and cosmological inflation, I review cosmic

string dynamics, cosmic string thermodynamics and cosmic string gravity, which leads to a number of interesting

observational signatures. I then proceed with the notion of cosmic superstrings arising at the end of brane infla-

tion, within the context of brane-world cosmological models inspired from string theory. I discuss the differences

between cosmic superstrings and their solitonic analogues, I review our current understanding about the evolution

of cosmic superstring networks, and I then briefly describe the variety of observational consequences, which may

help us to get an insight into the stringy description of our Universe.

1. Introduction

Provided our understanding about unificationof forces and big bang cosmology are correct,it is natural to expect that topological defects,appearing as solutions to many particle physicsmodels of matter, could have formed naturallyduring phase transitions followed by sponta-neously broken symmetries, in the early stagesof the evolution of the Universe. Certain types oftopological defects (local monopoles and local do-main walls) may lead to disastrous consequencesfor cosmology, hence being undesired, while oth-ers (cosmic strings) may play a useful role.

Cosmic strings [1] are linear topological defects,analogous to flux tubes in type-II superconduc-tors, or to vortex filaments in superfluid helium.These objects gained a lot of interest in the 1980’sand early 1990’s, since they offered a potential al-ternative to the cosmological inflation for the ori-gin of initial density fluctuations leading to theCosmic Microwave Background (CMB) temper-ature anisotropies and the observed structure inthe Universe. They however lost their appeal,when it was found that they lead to inconsisten-cies in the power spectrum of the CMB. It waslater shown [2] that cosmic strings are genericallyformed at the end of an inflationary era, withinthe framework of Supersymmetric Grand Unified

Theories (SUSY GUTs). Hence cosmic stringshave to be included as a sub-dominant partnerof inflation. This theoretical support gave a newboost to the field of cosmic strings, a boost whichhas been more recently enhanced when it wasshown that cosmic superstrings [3] (fundamentalor one-dimensional Dirichlet branes) can play therole of cosmic strings, in the framework of brane-world cosmologies.

A realistic cosmological scenario necessitatesthe input of high energy physics; any models de-scribing the early stages of the evolution of theUniverse have their foundations in general rela-tivity and high energy physics. Comparing thetheoretical predictions of such models against cur-rent astrophysical and cosmological data, resultsto either their acceptance or their rejection, whilein the first case it also fixes the free parametersof the models (see e.g., Ref. [4,5]). In particu-lar, by studying the properties of cosmic super-string networks and comparing their phenomeno-logical consequences against observational data,we expect to pin down the successful and natu-ral inflationary model and get some insight intothe stringy description of the Universe. Cosmicstrings/superstrings represent a beautiful exam-ple of the strong and fruitful link between cos-mology and high energy physics.

In what follows, I will summarise the material I

Nuclear Physics B (Proc. Suppl.) 192–193 (2009) 68–90

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had presented in my lectures during the summerschool at Cargese (June 2008) 1. I will highlightonly certain aspects of the subject, which I con-sider either more important due to their obser-vational consequences, or more recently obtainedresults.

2. Topological Defects in GUTs

In the framework of the hot big bang cosmolog-ical model, the Universe was originally at a veryhigh temperature, hence the initial equilibriumvalue of the Higgs field φ, which plays the roleof the order parameter, was at φ = 0. Since thePlanck time, the Universe has, through its expan-sion, steadily cooled down and a series of phasetransitions followed by Spontaneously SymmetryBreaking 2 (SSBs) took place in the frameworkof GUTs. Such SSBs may have left behind topo-logical defects as false vacuum remnants, via theKibble mechanism [6].

The formation or not of topological defectsand the determination of their type, depend onthe topology of the vacuum manifold Mn. Theproperties of Mn are described by the kth ho-motopy group πk(Mn), which classifies distinctmappings from the k-dimensional sphere Sk intothe manifold Mn. Consider the symmetry break-ing of a group G down to a subgroup H of G.If Mn = G/H has disconnected components —equivalently, if the order k of the non-trivial ho-motopy group is k = 0 — two-dimensional de-fects, called domain walls, form. The space-timedimension d of the defects is given in terms ofthe order of the non-trivial homotopy group byd = 4 − 1 − k. If Mn is not simply connected— equivalently, if Mn contains loops which can-not be continuously shrunk into a point — cosmicstrings form. A necessary, but not sufficient, con-dition for the existence of stable strings is thatthe fundamental group π1 of Mn, is non-trivial,or multiply connected. Cosmic strings are linear-like defects, d = 2. If Mn contains unshrinkablesurfaces, then monopoles form; k = 1, d = 1. IfMn contains non-contractible three-spheres, then

1http://www.lpthe.jussieu.fr/cargese/2The concept of spontaneous symmetry breaking has itsorigin in condensed matter physics.

event-like defects, textures, form; k = 3, d = 0.Depending on whether the symmetry is local

(gauged) or global (rigid), topological defects arerespectively, local or global. The energy of localdefects is strongly confined, while the gradientenergy of global defects is spread out over thecausal horizon at defect formation. Global de-fects having long range density fields and forces,can decay through long-range interactions, hencethey do not contradict observations, while localdefects may be undesirable for cosmology. Inwhat follows, I will discuss local defects, since weare interested in gauge theories, being the morephysical ones 3. Patterns of symmetry breakingwhich lead to the formation of local monopolesor local domain walls are ruled out, since theyshould soon dominate the energy density of theUniverse and close it, unless an inflationary eratook place after their formation. This is one ofthe reasons for which cosmological inflation — aperiod in the earliest stages of the evolution ofthe Universe, during which the Universe could bein an unstable vacuum-like state having high en-ergy density, which remained almost constant —was proposed. Local textures are insignificant incosmology since their relative contribution to theenergy density of the Universe decreases rapidlywith time.

Even in the absence of a non-trivial topologyin a field theory, it may still be possible to havedefect-like solutions, since defects may be em-bedded in such topologically trivial field theories.However, while stability of topological defects isguaranteed by topology, embedded defects are ingeneral unstable under small perturbations.

Let me discuss the genericity of cosmic stringformation in the context of SUSY GUTs, whichcontain a large number of SSB patterns leadingfrom a large gauge group GGUT to the Stan-dard Model (SM) gauge group GSM ≡ SU(3)C×SU(2)L× U(1)Y. The minimum rank of GGUT

has to be at least equal to 4, to contain the GSM

as a subgroup; we set the upper bound on therank r of the group to be r ≤ 8. The embeddingsof GSM in GGUT must be such that there is an

3Note that when we say cosmic strings we refer to localone-dimensional topological defects.

M. Sakellariadou / Nuclear Physics B (Proc. Suppl.) 192–193 (2009) 68–90 69

agreement with the SM phenomenology and es-pecially with the hypercharges of the known par-ticles. The large gauge group GGUT must includea complex representation, needed to describe theSM fermions, and it must be anomaly free. A de-tailed investigation [2] has concluded that GGUT

could be either one of SO(10), E6, SO(14), SU(8),SU(9); flipped SU(5) and [SU(3)]3 are includedwithin this list as subgroups of SO(10) and E6,respectively. The formation of domain walls ormonopoles, necessitates an era of supersymmet-ric hybrid inflation to dilute them. ConsideringGUTs based on simple gauge groups, the type ofsupersymmetric hybrid inflation will be of the F-type. The baryogenesis mechanism will be ob-tained via leptogenesis, either thermal or non-thermal leptogenesis. Finally, to ensure the sta-bility of proton, the discrete symmetry Z2, whichis contained in U(1)B−L, must be kept unbrokendown to low energies; the successful SSB schemesshould end at GSM× Z2. Taking all these con-siderations into account, a detailed study of allSSB schemes leading from a GGUT down to theGSM, by one or more intermediate steps, showsthat cosmic strings are generically formed at theend of hybrid inflation.

The results [2] can be summarised as follows: Ifthe large gauge group GGUT is the SO(10), thencosmic strings formation is unavoidable. Thegenericity of string formation in the case thatthe large gauge group is the E6, depends uponwhether one considers thermal or non-thermalleptogenesis. More precisely, for non-thermalleptogenesis, cosmic string formation is unavoid-able, while for thermal leptogenesis, cosmic stringformation arises in 98% of the acceptable SSBschemes. If the requirement of having Z2 un-broken down to low energies is relaxed and ther-mal leptogenesis is considered as being the mech-anism for baryogenesis, then cosmic string for-mation accompanies hybrid inflation in 80% ofthe SSB schemes. The SSB schemes of eitherSU(6) or SU(7), as the large gauge group, downto the GSM, which could accommodate an infla-tionary era with no defect (of any kind) at latertimes are inconsistent with proton lifetime mea-surements, while minimal SU(6) and SU(7) donot predict neutrino masses, implying that these

models are incompatible with high energy physicsphenomenology. Higher rank groups, namelySO(14), SU(8) and SU(9), should in general leadto cosmic string formation at the end of hybridinflation. In all these schemes, cosmic string for-mation is sometimes accompanied by the forma-tion of embedded strings. The strings which format the end of hybrid inflation have a mass whichis proportional to the inflationary scale.

3. Cosmic Strings and Inflation

An appealing solution to the drawbacks of thestandard hot big bang model is to introduce,during the very early stages of the evolution ofthe Universe, a period of accelerated expansion,known as cosmological inflation [7]. The infla-tionary era took place when the Universe wasin an unstable vacuum-like state at a high en-ergy density, leading to a quasi-exponential ex-pansion. The combination of the hot big bangmodel and the inflationary scenario provides atpresent the most comprehensive picture of theUniverse at our disposal. Inflation ends when theHubble parameter H =

√8πρ/(3M2

Pl) (where ρdenotes the energy density and MPl stands forthe Planck mass) starts decreasing rapidly. Theenergy stored in the vacuum-like state gets trans-formed into thermal energy, heating up the Uni-verse and leading to the beginning of the standardhot big bang radiation-dominated era.

Inflation is based on the basic principles of gen-eral relativity and field theory, while when theprinciples of quantum mechanics are also con-sidered, it provides a successful explanation forthe origin of the large scale structure, associatedwith the measured temperature anisotropies inthe CMB spectrum. Despite its remarkable suc-cess, inflation still remains a paradigm in searchof model. An inflationary model should be in-spired from a fundamental theory, while its pre-dictions should be tested against current data. Inaddition, releasing the present Universe form itsacute dependence on the initial data, inflation isfaced with the challenging task of proving itselfgeneric [8], in the sense that inflation would takeplace without fine-tuning of the initial conditions.

Theoretically motivated inflationary models

M. Sakellariadou / Nuclear Physics B (Proc. Suppl.) 192–193 (2009) 68–9070

can be built in the context of supersymmetryor Supergravity (SUGRA). N=1 supersymmetrymodels contain complex scalar fields which of-ten have flat directions in their potential, thusoffering natural candidates for inflationary mod-els. In this framework, hybrid inflation driven byF-terms or D-terms is the standard inflationarymodel, leading generically to cosmic string for-mation at the end of inflation. Hybrid inflation isbased on Einstein’s gravity but is driven by thefalse vacuum. The inflaton field rolls down itspotential while another scalar field is trapped inan unstable false vacuum. Once the inflaton fieldbecomes much smaller than some critical value, aphase transition to the true vacuum takes placeand inflation ends. F-term inflation is potentiallyplagued with the Hubble-induced mass problem4 (η-problem), while D-term inflation avoids it.

F-term inflation can be naturally accommo-dated in the framework of GUTs, when a GGUT

is broken down to the GSM, at an energy scaleMGUT according to the scheme

GGUTMGUT−−−→ H1

Minfl−−−−→Φ+Φ−

H2−→GSM ,

where Φ+, Φ− is a pair of GUT Higgs super-fields in non-trivial complex conjugate represen-tations, which lower the rank of the group by oneunit when acquiring non-zero vacuum expectationvalue. The inflationary phase takes place at the

beginning of the symmetry breaking H1Minfl−→ H2.

The gauge symmetry is spontaneously broken byadding F-terms to the superpotential. The Higgsmechanism leads generically [2] to Abrikosov-Nielsen-Olesen strings, called F-term strings.

F-term inflation is based on the globally super-symmetric renormalisable superpotential

WFinfl = κS(Φ+Φ− − M2) , (1)

where S is a GUT gauge singlet left handed su-perfield and κ, M are two constants (M has di-mensions of mass) which can be taken positivewith field redefinition.

4In supergravity theories, the supersymmetry breaking istransmitted to all fields by gravity, and thus any scalarfield, including the inflaton, gets an effective mass of theorder of the expansion rate H during inflation.

The scalar potential, as a function of the scalarcomplex component of the respective chiral su-perfields Φ±, S, is

V (φ+, φ−, S) = |FΦ+ |2 + |FΦ−|2 + |FS |2

+1

2

∑a

g2aD2

a . (2)

The F-term is such that FΦi≡ |∂W/∂Φi|θ=0,

where we take the scalar component of the super-fields once we differentiate with respect to Φi =Φ±, S. The D-terms are Da = φi (Ta)i

j φj + ξa,with a the label of the gauge group generators Ta,ga the gauge coupling, and ξa the Fayet-Iliopoulosterm. By definition, in the F-term inflation thereal constant ξa is zero; it can only be nonzero ifTa generates an extra U(1) group. In the contextof F-term hybrid inflation the F-terms give riseto the inflationary potential energy density whilethe D-terms are flat along the inflationary trajec-tory, thus one may neglect them during inflation.

The potential, has one valley of local minima,V = κ2M4, for S > M with φ+ = φ− = 0, andone global supersymmetric minimum, V = 0, atS = 0 and φ+ = φ− = M . Imposing initiallyS � M , the fields quickly settle down the valleyof local minima. Since in the slow-roll inflation-ary valley the ground state of the scalar poten-tial is non-zero, supersymmetry is broken. In thetree level, along the inflationary valley the poten-tial is constant, therefore perfectly flat. A slopealong the potential can be generated by includingone-loop radiative corrections. Hence, the scalarpotential gets a little tilt which helps the inflatonfield S to slowly roll down the valley of minima.The one-loop radiative corrections to the scalarpotential along the inflationary valley lead to theeffective potential [4]

V Feff(|S|) = κ2M4

{1 +

κ2N32π2

[2 ln

|S|2κ2

Λ2

+ (z + 1)2 ln(1 + z−1)

+ (z − 1)2 ln(1 − z−1)

]},(3)

with z = |S|2/M2, and N stands for the dimen-sionality of the representation to which the com-plex scalar components φ+, φ− of the chiral su-perfields Φ+, Φ− belong. This implies that the


effective potential, Eq. (3), depends on the par-ticular symmetry breaking scheme considered.

D-term inflation can be easily implementedwithin high energy physics (e.g., SUSY GUTs,SUGRA, or string theories) and it avoids theη-problem. Within D-term inflation, the gaugesymmetry is spontaneously broken by introduc-ing Fayet-Iliopoulos (FI) D-terms. In standardD-term inflation, the constant FI term gets com-pensated by a single complex scalar field at theend of the inflationary era, which implies thatstandard D-term inflation ends always with theformation of cosmic strings, called D-term strings.A supersymmetric description of the standard D-term inflation is insufficient, since the inflatonfield reaches values of the order of the Planckmass, or above it, even if one concentrates onlyaround the last 60 e-folds of inflation. Thus, D-term inflation has to be studied in the context ofsupergravity [4,5]).

Standard D-term inflation requires a scheme

GGUT × U(1)MGUT−−−→ H × U(1)

Minfl−−−−→Φ+Φ−

H → GSM .

It is based on the superpotential

W = λSΦ+Φ− , (4)

where S, Φ+, Φ− are three chiral superfields andλ is the superpotential coupling. It assumesan invariance under an Abelian gauge groupU(1)ξ, under which the superfields S, Φ+, Φ−

have charges 0, +1 and −1, respectively. It alsoassumes the existence of a constant FI term ξ.In D-term inflation the superpotential vanishesat the unstable de Sitter vacuum (anywhere elsethe superpotential is non-zero), implying thatwhen the superpotential vanishes, D-term infla-tion must be studied within a non-singular for-mulation of supergravity. Various formulationsof effective supergravity can be constructed fromthe superconformal field theory. To constructa formulation of supergravity with constant FIterms from superconformal theory, one finds [9]that under U(1) gauge transformations in the di-rections in which there are constant FI terms ξα,the superpotential W must transform as δαW =ηαi∂

iW = −i(gξα/M2Pl)W ; one cannot keep any

longer the same charge assignments as in stan-

dard supergravity.D-term inflationary models can be built with

different choices of the Kahler geometry. Variouscases have been explored in the literature. Thesimplest case is that of D-term inflation withinminimal supergravity [4]. It is based on

Kmin =∑

i

|Φi|2 = |Φ−|2 + |Φ+|2 + |S|2 , (5)

with fab(Φi) = δab.Another example is that of D-term inflation

based on Kahler geometry with shift symmetry,

Kshift =1

2(S + S)2 + |φ+|2 + |φ−|2 , (6)

and minimal structure for the kinetic function [5].One can also consider consider [5] a Kahler po-

tential with non-renormalisable terms:

Knon−renorm = |S|2 + |Φ+|2 + |Φ−|2

+f+

( |S|2M2

Pl

)|Φ+|2

+f−

( |S|2M2

Pl

)|Φ−|2 + b

|S|4M2

Pl

, (7)

where f± are arbitrary functions of (|S|2/M2Pl)

and the superpotential is given in Eq. (4).Having the superpotential, one must proceed

in the same way as in F-term inflation and writedown the three level scalar potential and then in-clude the one-loop radiate corrections.

Let me finally note that different ap-proaches [10] have been proposed in order toavoid cosmic string formation in the context of D-term inflation. For example, one can add a non-renormalisable term in the potential, or add anadditional discrete symmetry, or consider GUTmodels based on non-simple groups, or finally in-troduce a new pair of charged superfields so thatcosmic string formation is avoided within D-terminflation.

4. Cosmic String Dynamics

The world history of a cosmic string can be ex-pressed by a two-dimensional surface in the four-dimensional string world-sheet:

xμ = xμ(ζa) , a = 0, 1 ; (8)


the world-sheet coordinates ζ0, ζ1 are arbitraryparameters, ζ0 is time-like and ζ1 (≡ σ) is space-like.

Over distances that are large compared to thewidth of the string, but small compared to thehorizon size, solitonic cosmic strings can be con-sidered as one-dimensional objects and their mo-tion can be well-described by the Nambu-Gotoaction. Thus, the string equation of motion, inthe limit of a zero thickness string, is derived fromthe Goto-Nambu effective action

S0[xμ] = −μ

∫ √−γd2ζ , (9)

where γ = det(γab) with γab = gμνxμ,axν

,b and μ

stands for the linear mass density, with μ ∼ T 2c ,

where Tc is the critical temperature of the phasetransition followed by SSB leading to cosmicstring formation. By varying the action, Eq. (9),with respect to xμ(ζa), and using dγ = γγabdγab,we get the string equation of motion:

xμ ;a,a + Γμ

νσγabxν,axσ

,b = 0 ; (10)

Γμνσ is the four-dimensional Christoffel symbol.We have neglected the friction [11], due to the

scattering of thermal particles off the string. Forstrings formed at the grand unification scale, fric-tion is important only for a very short period oftime. For strings formed at a later phase transi-tion (e.g., closer to the electroweak scale), frictionwould dominate their dynamics through most ofthe thermal history of the Universe.

By varying the action with respect to the met-ric, the string energy-momentum tensor reads

T μν√−g = μ

∫d2ζ

√−γγabxμ,axν

,bδ(4)(xσ−xσ(ζa)) .

In an expanding Universe, the cosmic stringequation of motion is most conveniently writtenin comoving coordinates, where the Friedmann-Lemaıtre-Robertson-Walker (FLRW) metrictakes the form ds2 = a2(τ)[dτ2 − dr2] ; a(τ) isthe cosmic scale factor in terms of conformal timeτ (related to cosmological time t, by dt = adτ).Under the gauge condition ζ0 = τ , the comovingspatial string coordinates, x(τ, σ), are writtenas a function of τ , and the length parameter σ.

For a string moving in a FLRW Universe, theequation of motion, Eq. (10), can be simplifiedin the gauge for which the unphysical parallelcomponents of the velocity vanish,

x · x′ = 0 ; (11)

overdots and primes denote derivatives with re-spect to τ and σ, respectively. In these coordi-nates, the Goto-Nambu action yields the follow-ing string equation of motion in a FLRW metric:

x + 2

(a

a

)x(1 − x2) =

(1

ε

)(x′

ε

)′

. (12)

The string energy per unit σ, in comoving units,is ε ≡

√x

′2/(1 − x2), implying that the stringenergy is μa

∫εdσ. One usually fixes entirely the

gauge by choosing σ so that ε = 1 initially.

The string equation of motion is much simplerin Minkowski space-time. Equation (10) for flatspace-time simplifies to

∂a(√−γγabxμ

,b) = 0 . (13)

We impose the conformal gauge

x · x′ = 0 , x2 + x′2 = 0 ; (14)

overdots and primes denote derivatives with re-spect to ζ0 and ζ1, respectively. In this gaugethe string equation of motion is just a two-dimensional wave equation:

x − x′′ = 0 . (15)

To fix entirely the gauge, we set t ≡ x0 = ζ0 ,which allows us to write the string trajectoryas the three-dimensional vector x(σ, t), whereζ1 ≡ σ, the space-like parameter along the string.Hence, the constraint equations, Eq. (14), and thestring equation of motion, Eq. (15), become

x · x′ = 0 , x2 + x′2 = 1 , x − x′′ = 0 . (16)

The above equations imply that the string movesperpendicularly to itself with velocity x, that σis proportional to the string energy, and thatthe string acceleration in the string rest frameis inversely proportional to the local string cur-vature radius. A curved string segment tends tostraighten itself, resulting to string oscillations.


The general solution to the string equation ofmotion in flat space-time, Eq. (16c), is

x =1

2[a(σ − t) + b(σ + t)] , (17)

where a(σ − t) and b(σ + t) are two continuousarbitrary functions which satisfy

a′2 = b′2 = 1 . (18)

Hence, σ is the length parameter along the three-dimensional curves a(σ),b(σ).

The Goto-Nambu action describes to a goodapproximation cosmic string segments which areseparated. However, it leaves unanswered the is-sue of what happens when strings cross; a studywhich necessitates full field theory. When twostrings of the same type collide, they may eitherpass simply through one another, or they mayreconnect (intercommute). A necessary, but notsufficient, condition for string reconnection is thatthe initial and final configurations be kinemati-cally allowed in the infinitely thing string approx-imation. Numerical simulations (and analyticalestimates) of type-II (and weakly type-I) stringsin the Abelian Higgs model suggest that the prob-ability that a pair of strings will reconnect, afterthey intersect, is close to unity. The results arebased on lattice simulations of the correspondingclassical field configurations in the Abelian Higgsmodel; the internal structure of strings is highlynon-linear, and thus difficult to treat via analyt-ical means. String-string and self-string intersec-tions lead to the formation of new long strings andloops. String intercommutations produce discon-tinuities, kinks, in x and x′ on the new stringsegments at the intersection point, composed ofright- and left-moving pieces travelling along thestring at the speed of light.

The first analytical studies of the evolution ofa cosmic string network have shown [12] the ex-istence of scaling, in the sense that the stringnetwork can be characterised by a single lengthscale, roughly the persistence length or the inter-string distance ξ which grows with the horizon.This important property of cosmic strings ren-ders them cosmologically acceptable, in contrastto local monopoles or domain walls. Early nu-merical simulations have shown [13] that the typ-

ical curvature radius of long strings and the char-acteristic distance between the strings are bothcomparable to the evolution time t. The energydensity of super-horizon 5 strings in the scal-ing regime is given (in the radiation-dominatedera) by ρlong = κμt−2 , where κ is a numeri-cal coefficient (κ = 20 ± 10). Assuming that thesuper-horizon strings are characterised by a singlelength scale ξ(t), implies

ξ(t) = κ−1/2t . (19)

The typical distance between the nearest stringsegments and the typical curvature radius of thestrings are both of the order of ξ. These resultsagree with the picture of the scale-invariant evo-lution of the string network and with the one-scale hypothesis. Further numerical investiga-tions however revealed dynamical processes, suchas the production of small sub-horizon loops, atscales much smaller than ξ [14]. In responseto these findings, a three-scale model was devel-oped [15] which describes the network in thermsof three scales: the energy density scale ξ, a corre-lation length ξ along the string, and a scale ζ re-lated to local structure on the string. The small-scale structure (wiggliness), which offers an expla-nation for the formation of the small sub-horizonsized loops, is basically developed through inter-sections of long string segments. Aspects of thethree-scale model have been checked [16] evolvinga cosmic string network in Minkowski space-time.

The sub-horizon strings (loops), their size dis-tribution, and the mechanism of their formationremained for years the least understood parts ofthe string evolution. Recently, numerical sim-ulations of cosmic string evolution in a FLRWUniverse (see, Fig. 1), found [17] evidence of ascaling regime for the cosmic string loops in theradiation- and matter-dominated eras down tothe hundredth of the horizon time. The scal-ing was found without considering any gravita-tional back reaction effect; it was just the result

5Often in the literature, strings are divided into twoclasses: string loops and infinite, or long, strings. How-ever, in numerical simulations strings are always loops, inthe sense that they do not have open ends. Hence, theterm loops corresponds to sub-horizon string loops, whilethe term infinite strings corresponds to super-horizonstring loops.


Figure 1. Snapshot of a network of long stringsand closed loops in the matter-dominated era.Figure taken from Ref. [17].

of string intercommutations. The scaling regimeof string loops appears after a transient relaxationera, driven by a transient overproduction of stringloops with length close to the initial correlationlength of the string network. Subsequently, nu-merical [18] and analytical [19] studies supportedthe results of Ref. [17].

Let me note that there are two approaches ofdeveloping numerical simulations of cosmic stringevolution. Either cosmic strings are modelled asidealised one-dimensional objects, or field theo-retic calculations have been considered. In partic-ular, for the field theoretic approach, the simplestexample of an underlying field theory contain-ing local U(1) strings, namely the Abelian Higgsmodel, has been recently employed [20].

5. String Thermodynamics

It is well-known in string theory, that the de-generacy of string states increases exponentiallywith energy, namely

d(E) ∼ eβHE . (20)

Hence, there is a maximum temperature Tmax =1/βH, the Hagedorn temperature [21]. Let us con-

sider, in the microcanonical ensemble, a systemof closed string loops in a three-dimensional box.Intersecting strings intercommute, but otherwisethey do not interact and are described by theGoto-Nambu equation of motion. The statisticalproperties of a system of strings in equilibrium arecharacterised by only one parameter, the energydensity of strings, ρ, defined as ρ = E/L3, with Lthe size of the cubical box. The behaviour of thesystem depends on whether it is at low or highenergy densities, and it undergoes a phase tran-sition at a critical energy density, the Hagedornenergy density ρH. Quantisation implies a lowercutoff for the size of the string loops, determinedby the string tension μ. The lower cutoff on theloop size is roughly μ−1/2, implying that the massof the smallest string loops is m0 ∼ μ1/2.

For a system of strings at the low energy den-sity regime (ρ � ρH), all strings are choppeddown to the loops of the smallest size, while largerloops are exponentially suppressed. Thus, forsmall enough energy densities, the string equilib-rium configuration is dominated by the masslessmodes in the quantum description. The energydistribution of loops, given by the number dn ofloops with energies between E and E + dE perunit volume, is [21,22]

dn ∝ e−αEE−5/2dE (ρ � ρH) , (21)

where α = (5/2m0) ln(ρH/ρ).However, as the energy density increases, more

and more oscillatory modes of strings get excited.In particular, once a critical energy density, ρH,is reached, long oscillatory string states begin toappear in the equilibrium state. The density atwhich this happens corresponds to the Hagedorntemperature. The Hagedorn energy density —achieved when the separation between the small-est string loops is of the order of their sizes —is approximately μ2, and then the system under-goes a phase transition characterised by the ap-pearance of super-horizon (infintely long) strings.

At the high energy density regime, the energydistribution of string loops is [21,22]

dn = Am9/20 E−5/2dE (ρ � ρH) , (22)

where A is a numerical coefficient independentof m0 and of ρ. Equation (22) implies that the


mean-square radius R of the sub-horizon loops is

R ∼ m−3/20 E1/2 . (23)

Hence the large string loops are random walksof step approximately m−1

0 . Equations (22) and(23) imply

dn = A′R−4dR (ρ � ρH) , (24)

where A′ is a numerical constant. Thus, at thehigh energy density regime, the distribution ofclosed string loops is scale invariant, since it doesnot depend on the cutoff parameter. The totalenergy density in sub-horizon string loops is in-dependent of ρ. Increasing the energy density ρ ofthe system of strings, the extra energy E − EH,where EH = ρHL3, goes into the formation ofsuper-horizon long strings, implying

ρ − ρl = const (ρ � ρH) , (25)

where ρl denotes the energy density in super-horizon loops (often called in the literature as in-

finite strings).Clearly, the above analysis describes the be-

haviour of a system of strings of low or high en-ergy densities, while there is no analytic descrip-tion of the phase transition and of the interme-diate densities around the critical one, ρ ∼ ρH.An experimental approach to the problem hasbeen proposed in Ref. [23] and later extended inRef. [24].

The equilibrium properties of a system of cos-mic strings have been studied numerically inRef. [23]. The strings are moving in a three-dimensional flat space and the initial string statesare chosen to be a loop gas consisting of the small-est two-point loops with randomly assigned po-sitions and velocities. This choice is made justbecause it offers an easily adjustable string en-ergy density. Clearly, the equilibrium state is in-dependent of the initial state. The simulationsrevealed a distinct change of behaviour at a crit-ical energy density ρH = 0.0172 ± 0.002. Forρ < ρH, there are no super-horizon strings, theirenergy density, ρl, vanishes. For ρ > ρH, theenergy density in sub-horizon string loops is con-stant, equal to ρH, while the extra energy goes tothe super-horizon string loops with energy den-sity ρl = ρ − ρH. Thus, Eqs. (22) and (25) are

valid for all ρ > ρH, although they were de-rived only in the limit ρ � ρH. At the criticalenergy density, ρ = ρH, the system of stringsis scale-invariant. At bigger energy densities,ρ > ρH, the energy distribution of sub-horizonstring loops at different values of ρ were found[23] to be identical within statistical errors, andwell-defined by a line dn/dE ∝ E−5/2. Thus,for ρ > ρH, the distribution of sub-horizon stringloops is still scale-invariant, but in addition thesystem includes super-horizon string loops, whichdo not exhibit a scale-invariant distribution. Thenumber distribution for super-horizon loops goesas dn/dE ∝ 1/E, which means that the totalnumber of super-horizon string loops is roughlylog(E − EH). So, typically the number of longstrings grows very slowly with energy; for ρ > ρH

there are just a few super-horizon strings, whichtake up most of the energy of the system.

The above numerical experiment has been ex-tended [24] for strings moving in a higher dimen-sional box. The Hagedorn energy density wasfound for strings moving in boxes of dimensional-ity dB = 3, 4, 5 [24]:

ρH =

⎧⎨⎩

0.172 ± 0.002 for dB = 30.062 ± 0.001 for dB = 40.031 ± 0.001 for dB = 5

(26)

Moreover, the size distribution of sub-horizonstring loops at the high energy density regime wasfound to be independent of the particular value ofρ for a given dimensionality of the box dB. Thesize distribution of sub-horizon string loops wasfound [24] to be well defined by a line

dn

dE∼ E−(1+dB/2) , (27)

where the space dimensionality dB was takenequal to 3, 4, or 5 . The statistical errors indi-cated a slope equal to −(1 + dB/2) ± 0.2. Abovethe Hagedorn energy density the system is againcharacterised by a scale-invariant distribution ofsub-horizon string loops and a number of super-horizon string loops with a distribution which isnot scale invariant.


6. Cosmic String Gravity

The gravitational properties of cosmic stringsare very different than those of non-relativisticlinear mass distributions. Straight cosmic stringsproduce no gravitational force in the surround-ing matter: ∇2Φ = 0, where Φ stands for theNewtonian potential. Cosmic strings have rela-tivistic motion, implying that oscillating stringloops can be strong emitters of gravitational ra-diation. Since super-horizon cosmic strings havewiggles and small-scale structure due to string in-tercommutations, they are also sources of grav-itational radiation [25]. A gravitating string isdescribed by a coupled system of Einstein, Higgsand gauge field equations, for which no exact solu-tion is known. We thus usually make two simpli-fications: we consider the cosmic string thicknessto be much smaller than any other relevant di-mension and the cosmic string gravitational fieldto be sufficiently weak, so that linearised Einsteinequations can be used (for Gμ � 1).

The geometry around a straight cosmic string islocally identical to that of flat space-time, but thisgeometry is not globally Euclidean; the azimuthalcoordinate varies in the range [0, 2π(1 − 4Gμ)).Hence, the effect of a cosmic string is to intro-duce an azimuthal deficit angle Δ, whose mag-nitude is determined by the symmetry breakingscale Tc leading to the cosmic string formation,namely Δ = 8πGμ. Thus, the string metricds2 = dt2−dz2−dr2−(1−8πGμ)r2dθ2 describesa conical space leading to interesting observa-tional effects on the propagation of light (i.e.,double images of light sources located behind cos-mic strings) and of particles (i.e., discontinuousDoppler shift effects). The centre-of-mass veloc-ity u of two particles, moving towards a string atthe same velocity v reads

u = v sin(Δ/2)[1 − v2 cos2(Δ/2)]−1/2 . (28)

Considering that one of the particles carries alight source, while the other one is an observer,one realises that the observer will detect a discon-tinuous change in the frequency ω of light, givenby [26]

δω

ω=

v√1 − v2

Δ . (29)

This discontinuous change in the frequency has itsorigin in the Doppler shift: particles start mov-ing towards each other, decreasing their distance,once the line connecting the particles crosses thestring.

In the framework of gravitational instability,topological defects in general and cosmic stringsin particular, offered an alternative to the infla-tionary paradigm for the origin of the initial fluc-tuations leading to the observed large-scale struc-ture and the measured anisotropies of the CMBtemperature anisotropies. The angular powerspectrum of CMB is expressed in terms of thedimensionless coefficients C�, in the expansion ofthe angular correlation function in terms of theLegendre polynomials P� reads 6:⟨

0

∣∣∣∣δTT (n)δT

T(n′)

∣∣∣∣0⟩ ∣∣∣

(n·n′=cos ϑ)

=1

4π

∑�

(2� + 1)C�P�(cosϑ)W2� , (30)

where W� stands for the �-dependent windowfunction of the particular experiment. Equation(30) compares points in the sky separated by anangle ϑ. The value of C� is determined by fluctu-ations on angular scales of the order of π/�. Theangular power spectrum of anisotropies observedtoday is usually given by the power per logarith-mic interval in �, plotting �(� + 1)C� versus �.

To find the power spectrum induced by topo-logical defects, one has to solve in Fourier space,for each given wave vector k, a system of linearperturbation equations with random sources:

DX = S , (31)

where D denotes a time dependent linear dif-ferential operator, X is a vector which containsthe various matter perturbation variables, and Sis the random source term, consisting of linearcombinations of the energy momentum tensor ofthe defect. For given initial conditions, Eq. (31)can be solved by means of a Green’s function,G(τ, τ ′). To compute power spectra or, more gen-erally, quadratic expectation values of the form

6Equation (30) holds only if the initial state for cosmo-logical perturbations of quantum-mechanical origin is thevacuum [27].


〈Xj(τ0,k)X∗m(τ0,k

′)〉, one has to calculate

〈Xj(τ0,k)Xl (τ0,k

′)〉 =

∫ τ0

τin

dτGjm(τ,k)

×∫ τ0

τin

dτ ′Gln(τ ′,k′)〈Sm(τ,k)S

n(τ ′,k′)〉 .(32)

To compute power spectra, one shouldknow the unequal time two-point correlators〈Sm(τ,k)S

n(τ ′,k′)〉 in Fourier space, calculatedby means of heavy numerical simulations.

The first determinations of the CMB powerspectrum from cosmic strings were based on theassumption that cosmic strings are of infinites-imal width. They were thus realised by eitheremploying Nambu-Goto simulations of connectedstring segments, or a model involving a stochas-tic ensemble of unconnected segments. However,a number of questions there have been later raisedregarding the accuracy of these approaches. Morerecently, the CMB power spectrum contributionfrom cosmic strings has been addressed [28] usingfield-theoretic simulations of the Abelian Higgsmodel, the simplest example of an underlyingfield theory with local U(1) strings. All ap-proaches agree on the basic form of the cosmicstring power spectrum, namely a power spectrumwith a roughly constant slope at low multipoles,rising up to a single peak, and consequently de-caying at small scales. This result is common ofall models, in which fluctuations are generatedcontinuously and evolve according to inhomoge-neous linear perturbation equations.

In topological defects models, fluctuations areconstantly generated by the non-linear defect evo-lution. This characteristic, combined with thefact that the random initial conditions of thesource term of a given scale leak into other scales,destroy perfect coherence. The incoherent aspectof active perturbations affects the structure ofsecondary oscillations, namely secondary oscilla-tions may get washed out. Thus, in topologicaldefects models, incoherent fluctuations lead to asingle bump at smaller angular scales (larger �),than those predicted within any inflationary sce-nario.

The cosmic string CMB power spectrum wasfound [28] to have a broad peak at � ≈ 150− 400.Decomposing the power spectrum into scalar,

vector and tensor modes, it was shown [28] thatthe origin of this broad peak lies in both the vec-tor and scalar modes, which peak at � ≈ 180and � ≈ 400, respectively. This analysis con-cluded [28] that the cosmic string power spec-trum is dominated by vector modes for all butthe smallest scales.

The position and amplitude of the acousticpeaks, as found by the CMB measurements, are indisagreement with the predictions of topologicaldefect models. As a consequence, CMB measure-ments rule out pure topological defect models ingeneral, and cosmic strings in particular, as theorigin of initial density perturbations leading tothe observed structure formation.

Since cosmic strings are expected to be gener-ically formed in the context of SUSY GUTs, oneshould consider mixed perturbation models wherethe dominant role is played by the inflaton fieldbut cosmic strings have also a contribution, smallbut not negligible. Restricting ourselves to theangular power spectrum, we can remain in thelinear regime. In this case,

C� = αCI

� + (1 − α)CS

� , (33)

where CI

� and CS

� denote the (COBE normalised)Legendre coefficients due to adiabatic inflatonfluctuations and those stemming from the cosmicstring network, respectively. The coefficient α inEq. (33) is a free parameter giving the relativeamplitude for the two contributions. Comparingthe C�, calculated using Eq. (33) – where CI

� istaken from a generic inflationary model and CS

�

from numerical simulations of cosmic string net-works – with data obtained from the most re-cent CMB measurements, one gets that a cosmicstring contribution to the primordial fluctuationshigher than 14% is excluded up to 95% confidencelevel [29].

Let us now return to F- and D-term hybridinflation and investigate the constraints on thefree parameters of the model (namely massesand couplings) so that the cosmic string con-tribution to the CMB data is within the al-lowed limits imposed from recent CMB measure-ments. Considering only large angular scales onecan get the contributions to the CMB temper-ature anisotropies analytically. The quadrupole


anisotropy has one contribution coming from theinflaton field, and one contribution coming fromthe cosmic string network. Fixing the number ofe-foldings to 60, the inflaton and cosmic stringcontribution to the CMB depend on the parame-ters of the model. For F-term inflation the cosmicstring contribution to the CMB data is consistentwith CMB measurements provided [4]

M <∼ 2 × 1015GeV ⇔ κ <∼ 7 × 10−7 . (34)

The superpotential coupling κ is also subject tothe gravitino constraint which imposes an up-per limit to the reheating temperature, to avoidgravitino overproduction. Within the frameworkof SUSY GUTs and assuming a see-saw mecha-nism to give rise to massive neutrinos, the inflatonfield decays during reheating into pairs of right-handed neutrinos. This constraint on the reheat-ing temperature can be converted to a constrainton the parameter κ. The gravitino constraint onκ reads [4] κ <∼ 8× 10−3, which is rather weaker.

The tuning of κ can be softened if one allows forthe curvaton mechanism. The curvaton is a scalarfield that is sub-dominant during the inflationaryera as well as at the beginning of the radiationdominated era following inflation. In the con-text of supersymmetric theories such scalar fieldsare expected to exist, and in addition, if embed-ded strings accompany the formation of cosmicstrings, they may offer a natural curvaton can-didate, provided the decay product of embeddedstrings gives rise to a scalar field before the onsetof inflation. Assuming the existence of a curva-ton field there is an additional contribution to thetemperature anisotropies and the CMB measure-ments impose [4] the following limit on the initialvalue of the curvaton field

ψinit <∼ 5×1013( κ

10−2

)GeV for κ ∈ [10−6, 1] .

D-term inflation can also be compatible withCMB measurements, provide we tune its free pa-rameters. In the case of minimal SUGRA, con-sistency between CMB measurements and the-oretical predictions impose [4,5] that g <∼ 2 ×10−2 and λ <∼ 3× 10−5, which can be expressedas a single constraint on the Fayet-Iliopoulos termξ, namely

√ξ <∼ 2 × 1015 GeV. The results are

1.�10�8 1.�10�6 0.0001Λ

0.1

1

10

100Cosmic strings contribution �%�

g=10−3

g=10−2

g=10−1

g=2x10−2

−4g=10

Figure 2. For D-term inflation in minimalSUGRA, cosmic string contribution to CMBquadrupole anisotropies as a function of the su-perpotential coupling constant λ, for various val-ues of the gauge coupling g. Figure taken fromRef. [4].

illustrated in Fig.2. The fine tuning on the cou-plings can be softened if one invokes the curvatonmechanism and constrains the initial value of thecurvaton field to be [5]

ψinit <∼ 3×1014( g

10−2

)GeV for λ ∈ [10−1, 10−4] .

For D-term inflation based on Kahler geometrywith shift symmetry, the cosmic string contribu-tion to the CMB anisotropies is dominant, in con-tradiction with the CMB measurements, unlessthe superpotential coupling is [5] λ <∼ 3 × 10−5.Finally, in the case of D-term inflation based on aKahler potential with non-renormalisable terms,the contribution of cosmic strings dominates if thesuperpotential coupling λ is close to unity. Theconstraints on λ read [5]

(0.1 − 5) × 10−8 ≤ λ ≤ (2 − 5) × 10−5

or, equivalently√ξ ≤ 2 × 1015 GeV ,

implying Gμ ≤ 8.4 × 10−7. Thus, higher orderKahler potentials do not suppress cosmic stringcontribution.


Apart the temperature power spectrum, impor-tant constraints on cosmic string scenarios mightalso arise in the future from measurements of thepolarisation of the CMB photons. More precisely,the B-polarisation spectrum offers an interestingwindow on cosmic strings [30] since inflation hasonly a weak contribution. Scalar modes maycontribute to the B-mode only via the gravita-tional lensing of the E-mode signal, with a secondinflationary contribution coming from the sub-dominant tensor modes.

Cosmic strings can also become apparentthrough their contribution in the small-angleCMB temperature anisotropies. More precisely,at high multipoles � (small angular resolution),the mean angular power spectrum of string-induced CMB temperature anisotropies can bedescribed [31] by �−α, with α ∼ 0.889. Thus,a non-vanishing cosmic string contribution to theoverall CMB temperature anisotropies may dom-inate at high multipoles � (small angular scales).In an arc-minute resolution experiment, stringsmay be observable [31] for Gμ down to 2× 10−7.

Cosmic strings should also induce deviationsfrom Gaussianity. On large angular scales suchdeviations are washed out due to the low stringcontribution, however on small angular scales, op-timal non-Gaussian string-devoted statistical es-timators may impose severe constraints on a pos-sible cosmic string contribution to the CMB tem-perature anisotropies.

Finally, let me emphasise that one should keepin mind that all string-induced CMB temperatureanisotropies were performed for Abelian strings inthe zero thickness limit with reconnection prob-ability equal to unity and winding number equalto one. Even though in any model where fluctu-ations are constantly induced by sources (seeds)having a non-linear evolution, the perfect coher-ence which characterises the inflationary inducedspectrum of perturbations gets destroyed [32],there is still no reason to expect that quantita-tively the results found for conventional cosmicstring models will hold in more general cases.

7. Superconducting Cosmic Strings

Before finishing this brief review on cosmicstrings, let me mention the case of superconduct-ing strings [33], in the sense that in a large class ofhigh energy physics theories, strings have similarelectromagnetic properties as those of supercon-ducting wires. Such objects carry large electriccurrents and hence their interaction with the cos-mic plasma can lead to a variety of distinct as-trophysical effects.

Cosmic strings are characterised as supercon-ductors if electromagnetic gauge invariance is bro-ken inside the strings, a situation which can oc-cur for instance when a charged scalar field de-velops a non-zero expectation value in the vicin-ity of the string core 7. Superconducting stringsappear also in models with fermions, which ac-quire masses through a Yukawa coupling to theHiggs field of the strings. Thus, depending onthe considered model we can have bosonic orfermionic string superconductivity. In the firstcase, bosons can condensate and acquire a non-vanishing phase gradient, while in the second one,fermions may propagate in the form of zero modesalong the string.

Applying an electric field on a superconductingstring, the string will develop a growing electriccurrent according to

dJ

dt∼ ce2

�E , (35)

where E stands for the field component along thestring and e denotes the elementary charge.

In the case of fermionic superconductivity,fermions are massless inside the string, whereasthey have a finite mass, m, outside the string 8.Under the effect of an electric field, a current Jresults, growing in time until it reaches a criticalvalue Jc ∼ emc2/�, when the particles inside thestring, moving at relativistic speeds, have suffi-cient energy to leave the string. Thus, when thestring current reaches its critical value, Jc, parti-

7Note that also a vector field, whose flux is trapped insidea non-Abelian string can lead to a superconducting string.8The fermion mass is model-dependent, but it is boundedfrom above by the symmetry breaking scale of the string.


cles get produced at a rate

n ∼ eE/� , (36)

where n stands for the number of fermions perunit length. Note that Jc, even though model-dependent, it does not exceed a maximum value,given by Jmax ∼ e(μc3/�)1/2. Depending on theirenergy scale, superconducting strings may carryhuge currents. In the case of bosonic supercon-ductivity, the model-dependent critical current Jc

— determined by the energy scale at which scaleinvariance is broken — is again bounded by Jmax,defined as above.

Superconducting strings can also develop grow-ing currents in magnetic fields, according to

dJ

dt∼ e2

�vB , (37)

where v is the speed of the moving string segmentin a magnetic field B. For a string loop carryingsufficiently large currents, electromagnetic radia-tion can overtake gravitational radiation, becom-ing the dominant energy loss mechanism.

Superconducting string loops may be problem-atic in cosmology. For a current-carrying loop,the energy per unit length is not equal to thetension, their difference equals the string cur-rent. Such a loop can rotate and the resultingcentrifugal force — which can be expected to bevery much stronger than the inefficient magneticspring repulsion effect — may balance the ten-sion. When a rotating string loop reaches an equi-librium state — defined by the balance betweenthe string tension and the centrifugal force — it iscalled a vorton [34]. Vortons can be formed at, orsoon after, the phase transition followed by SSBleading to the string formation, and they possessa net charge as well as a current. If vortons arestable (certainly a model-dependent issue [35]),they will scale as matter in the Universe, domi-nating over its energy density. In this sense, or-tons may constrain models for superconductingstrings.

8. Cosmic Superstrings

The recent interplay between superstring the-ory and cosmology has led to the notion of cosmic

superstrings [3], providing the missing link be-tween superstrings and their classical analogues.

The possible astrophysical role of superstringshas been advocated already more than twentyyears ago. More precisely, it has been pro-posed [36], that superstrings of the O(32) andE8×E8 string theories are likely to generatestring-like stable vortex lines and flux tubes.However, in the context of perturbative stringtheory, the high tension (close to the Planck scale)of fundamental strings ruled them out [36] as po-tential cosmic string candidates. Luckily, thispicture has changed in the framework of brane-world cosmology, which offers an elegant realisa-tion of nature within string theory. Within thebrane-world picture, all standard model particlesare open string modes. Each end of an openstring lies on a brane, implying that all standardmodel particles are stuck on a stack of Dp-branes,while the remaining p − 3 of the dimensions arewrapping some cycles in the bulk. Closed stringmodes (e.g., dilaton, graviton) live in the high-dimensional bulk. Brane interactions lead to un-winding and thus evaporation of higher dimen-sional Dp-branes. We are eventually left withD3-branes — one of which could indeed play therole of our Universe [37] — embedded in a (9+1)-dimensional bulk and cosmic superstrings (one-dimensional D-branes, called D-strings, and Fun-damental strings, called F-strings).

Brane annihilations provide a natural mecha-nism for ending inflation. To illustrate the forma-tion of cosmic superstrings at the end of brane in-flation, let us consider a Dp-Dp brane-anti-branepair annihilation to form a D(p− 2) brane. Eachparent brane has a U(1) gauge symmetry andthe gauge group of the pair is U(1)×U(1). Thedaughter brane possesses a U(1) gauge group,which is a linear combination, U(1)−, of theoriginal two U(1)’s. The branes move towardseach other and as their inter-brane separation de-creases below a critical value, the tachyon field,which is an open string mode stretched betweenthe two branes, develops an instability. Thetachyon couples to the combination U(1)−. Therolling of the tachyon field leads to the decay ofthe parent branes. Tachyon rolling leads to spon-taneously symmetry breaking, which supports de-


fects with even co-dimension. So, brane annihi-lation leads to vortices, D-strings; they are cos-mologically produced via the Kibble mechanism.The other linear combination, U(1)+, disappearssince only one brane remains after the brane colli-sion. The U(1)+ combination is thought to disap-pear by having its fluxes confined by fundamen-tal closed strings. Such strings are of cosmolog-ical size and they could play the role of cosmicstrings [38]; they are referred to in the literatureas cosmic superstrings [39].

9. Differences between Cosmic Strings and

Cosmic Superstrings

Cosmic superstrings [3], even though cosmolog-ically extended, are quantum objects, in contrastto solitonic cosmic strings which are classical ob-jects. Hence, one expects a number of differencesto arise as regarding the properties of the twoclasses of objects. As I have earlier discussed,the probability that a pair of cosmic strings willreconnect, after having intersect, equals unity.The reconnection probability for cosmic super-strings is however smaller (often much smaller)than unity. The corresponding intercommutationprobabilities are calculated in string perturbationtheory. The result depends on the type of stringsand on the details of compactification. For fun-damental strings, reconnection is a quantum pro-cess and takes place with a probability of order g2

s

(where gs denotes the string tension). It can thusbe much less than one, leading to an increaseddensity of strings [40], implying an enhancementof various observational signatures. The recon-nection probability is a function of the relativeangle and velocity during the collision. One maythink that strings can miss each other, as a resultof their motion in the compact space. Dependingon the supersymmetric compactification, stringscan wander over the compact dimensions, thusmissing each other, effectively decreasing their re-connection probability. However, in realistic com-pactification schemes, strings are always confinedby a potential in the compact dimensions [41].The value of gs and the scale of the confining po-tential will determine the reconnection probabil-ity. Even though these are not known, for a large

number of models it was found [41] that the recon-nection probability for F-F collisions lies in therange between 10−3 and 1. The case of D-D col-lisions is more complicated; for the same modelsthe reconnection probability is anything between0.1 to 1. Finally, the reconnection probability forF-D collisions can vary from 0 to 1.

Brane collisions lead not only to the formationof F- and D-strings, they also produce boundstates, (p, q)-strings, which are composites of pF-strings and q D-strings [42]. The presence ofstable bound states implies the existence of junc-tions, where two different types of string meetat a point and form a bound state leading awayfrom that point. Thus, when cosmic superstringsof different types collide, they can not intercom-mute, instead they exchange partners and forma junction at which three string segments meet.This is just a consequence of charge conserva-tion at the junction of colliding (p, q)-strings. Forp = np′ and q = nq′, the (p, q) string is neutrallystable to splitting into n bound (p′, q′) strings.The angles at which strings pointing into a ver-tex meet, is fixed by the requirement that therebe no force on the vertex. In general, a (p, q) anda (p′, q′) string will form a trilinear vertex witha (p + p′, q + q′) or a (p − p′, q − q′) string. Thisleads certainly to the crucial question of whethera cosmic superstring network will reach scaling,or whether it freezes leading to predictions incon-sistent with our observed Universe universe.

The tension of solitonic strings is set from theenergy scale of the phase transition, followed bya spontaneously broken symmetry, which left be-hind these defects as false vacuum remnants. Cos-mic superstrings however span a whole range oftensions, set from the particular brane inflationmodel. The tension of F-strings in 10 dimensionsis μF = 1/(2πα′), and the tension of D-strings isμD = 1/(2πα′gs), where gs stands for the stringcoupling. In 10 flat dimensions, supersymmetrydictates that the tension of the (p, q) bound statesreads

μ(p,q) = μF

√p2 + q2/g2

s . (38)

Individually, the F- and D-strings are 12 -

BPS (Bogomol’nyi-Prasad-Sommerfield) objects,which however break a different half of the su-


persymmetry each. Equation (38) represents theBPS bound for an object carrying the chargesof p F-strings and q D-strings. Note that theBPS bound is saturated by the F-strings, (p, q) =(1, 0), and the D-strings, (p, q) = (0, 1). Let mealso make the remark that the string tension forstrings at the bottom of a throat is different fromthe (simple) expression given in Eq. (38) and itdepends on the choice of flux compactification.

Consider an F- and a D-string, both lying alongthe same axis. The total tension of this config-uration is (g−1

s + 1)/(2πα′), which exceeds theBPS bound; thus the configuration is not super-symmetric. It can however lower its energy, if theF-string breaks, its ends being attached to the D-string. Since the end points can then move off atinfinity, leaving only the D-string behind, a fluxwill run between the end points of the F-string.Now the tension reads (g−1

s +O(gs))/(2πα′), andhence the final state represents a D-string with aflux, which is a supersymmetric state.

10. Cosmic Superstring Evolution

The evolution of cosmic superstring networks,is a complicated issue, which has been addressedby numerical [40,43–45], as well as analytical [46]approaches.

The first numerical attempt [40], studying inde-pendent stochastic networks of D- and F-stringsin a flat space-time, has shown that the charac-teristic length scale ξ, giving the typical distancebetween the nearest string segments and the typ-ical curvature of strings, grows linearly with time

ξ(t) ∝ ζt ; (39)

the slope ζ depends on the reconnection proba-bility P , and on the energy of the smallest al-lowed loops (i.e., the energy cutoff). For recon-nection (or intercommuting) probability in therange 10−3 <∼ P <∼ 0.3, it was shown [40] that

ζ ∝√P ⇒ ξ(t) ∝

√Pt . (40)

One can find in the literature (e.g., Ref. [[38]c])statements claiming that ξ(t) should be insteadproportional to Pt. If this were correct, then theenergy density of cosmic superstrings, of giventension, could be considerably higher than that

of their field theory analogues. However, the au-thors of Ref. [38] have missed out in their analy-sis that intersections between two long strings isnot the most efficient mechanism for energy lossof the string network. The findings of Ref. [40]cleared the misconception about the behaviour ofthe scale ξ, and shown that the cosmic super-string energy density may be higher than in thefield theory case, but at most only by one orderof magnitude 9.

As I have already discussed, in a realistic case(p, q) strings come in a large number of dif-ferent types, while a (p, q) string can decay toa loop only if it self-intersects of collide withanother (p, q) or (−p,−q) string. A collision be-tween (p, q) and (p′, q′) strings will lead to a new(p ± p′, q ± q′) string, provided the end points ofthe initial two strings are not attached to otherthree-string vertices, thus they are not a partof a web. If the collision between two stringscan lead to the formation of one new string, ona timescale much shorter than the typical colli-sion timescale, then the creation of a web maybe avoided, and the resulting network is com-posed by strings which are on the average non-intersecting. Then one can imagine the follow-ing configuration: A string network, composedby different types of (p, q) strings undergoes col-lisions and self-intersections. Energy considera-tions imply the production of lighter daughterstrings, leading eventually to one of the followingstrings: (±1, 0), (0,±1),±(1, 1),±(1,−1). Theseones may then self-intersect, form loops and scaleindividually. Provided the relative contributionof each of these strings to the energy density ofthe Universe is small enough, the Universe willnot be overclosed.

Let us now study the dynamics of a three-stringjunction in a simple model. The solutions of theBPS saturated formula

μ(p,q) =√

[pμ(1,0)]2 + [qμ(0,1)]2 , (41)

read

μ(p,q) sinα = qμ(0,1) ; μ(p,q) cosα = pμ(1,0) ,(42)

9A discussion and explanation of this misconception canbe found in Ref. [[1]c].


where tanα = q/(pgs). The balance conditionsfor three strings imply that when an F-stringends on a D-string, it causes it to bend at anangle set by the string coupling; on the otherside of the junction there is a (1, 1) string. Con-sider a junction of three strings, with coordi-nates x(σ, t) , tension μ and parameter lengthsL1(t), L2(t), L3(t),which are joined at a junctionand whose other end terminate on parallel branes.The action for this configuration reads 10 [46]

S = −3∑

α=1

μα

∫dt

∫ Lα(t)

0

dσ√

−γ(α)

+

3∑α=1

∫dtlα · (x(t, Lα(t))) − xjunc(t)) , (43)

where the first part stands for the Nambu-Gototerms for the three strings, and lα denote the La-grange multipliers to describe the junction, lo-cated at position xjunc. From Eq. (43) one canderive the equations of motion as well as the en-ergy conservation. One can easily check that

μ1(1 − L1)

μ1 + μ2 + μ3=

M1(1 − c23)

M1(1 − c23) + M2(1 − c13) + M3(1 − c12),

μ1L1 + μ2L2 + μ3L3 = 0 , (44)

and cyclic permutations. Note that M1 =μ2

1 − (μ2 − μ3)2 (and cyclic permutations giving

M2, M3) and cij = a′i(t − Li(t)) · a′

j(t − Lj(t)).Equation (44) implies that the rate of creation ofnew string must balance the disappearance of oldone. Thus, for an F-string with μ1 = 1, and aD-string with μ2 = 1/gs, the FD-bound state hastension μ3 =

√1 + 1/g2

s = 1/gs + gs/2 + O(g2s ).

Since the angle α goes to π/2 in the limit of zerostring coupling, we conclude that in the small gs-limit, the length of the F-string remains constant,while the length of the D-string decreases and thelength of the FD-bound state increases. This re-sult has been recently confirmed from numericalexperiments [45].

10Note that in principle, cosmic superstring dynamicsought to be studied using the Dirac-Born-Infeld action,the low-energy effective action for many varieties of stringsarising in the context of string theory.

To shed some light on the evolution of cosmicsuperstring networks, a number of numerical ex-periments have been conducted, each of them ata different level of approximation. One shouldkeep in mind that the initial configuration de-pends on the particular brane inflation scenario,while a realistic network should contain stringswith junctions and allow for a spectrum of possi-ble tensions.

I will briefly describe the approach and findingsof one of these numerical approaches [44], whichI consider more realistic than others. The aimof that study was to build a simple field theorymodel of (p, q) bound states, in analogy with theAbelian Higgs model used to investigate the prop-erties of solitonic cosmic string networks, and tostudy the overall characteristics of the networkusing lattice simulations. Two models were in-vestigated, one in which both species of stringhave only short-range interactions and anotherone in which one species of string features long-range interactions. We thus modelled the networkwith no long-range interactions using two sets offields, complex scalars coupled to gauge fields,with a potential chosen such that the two types ofstrings will form bound states (see, Fig. 3). In thisway junctions of 3 strings with different tensionwere successfully modelled. In order to introducelong-range interactions we considered a networkin which one of the scalars forms global strings.This is important if the strings are of a non-BPSspecies. For example, for cosmic superstrings atthe bottom of a Klebanov-Strassler throat the F-string is not BPS while the D-string is.

More precisely, the (p, q) string network wasmodelled [44] using two sets of Abelian Higgsfields, φ, χ. In the case that both species of cos-mic strings are BPS, the model is described bythe action [44]:

S =

∫d3xdt

[− 1

4F 2 − 1

2(Dμφ) (Dμφ)∗

− λ1

4

(φφ∗ − η2

1

)2

−1

4H2 − 1

2(Dμχ) (Dμχ)

∗

− λ2

4φφ∗

(χχ∗ − η2

2

)2]

, (45)


where the covariant derivative Dμ is defined by

Dμφ = ∂μφ − ie1Aμφ ,

Dμχ = ∂μχ − ie2Cμχ . (46)

For clarity, we label the φ field as “Higgs” andthe χ field as “axion”, even though both fieldsare Higgs-like. The scalars are coupled to theU(1) gauge fields Aμ and Cμ, with coupling con-stants e1 and e2 and field strength tensors Fμν =∂μAν − ∂νAμ and Hμν = ∂μCν − ∂νCμ, respec-tively. The scalar potentials are parametrised bythe positive constants λ1, η1 and λ2, η2, respec-tively. In the case that one species of string isnon-BPS, we remove the second gauge field bysetting e2 = 0. In this way, this species of string isrepresented by the topological defect of a complexscalar field with a global U(1) symmetry. Notethat such defects are characterised by the exis-tence of long-range interactions — as opposed tolocal strings in which all energy density is con-fined within the string, so that local strings haveonly gravitational interactions — implying differ-ent consequences for the evolution of the network.

Figure 3. Left: Bound states for local-local (p, q)strings. Right: Bound states for local-global (p, q)strings. Figure taken from Ref. [44].

Thus, different components of the (p, q) stateare expected to exhibit different types of long-range interactions. The evolution of the stringnetworks suggested that the long-range interac-tions have a much more important role in thenetwork evolution than the formation of bound

states. In the local-global networks the boundstates tend to split as a result of the long-rangeinteractions, resulting in two networks that evolvealmost independently. The formation of short-lived bound states and their subsequent splittingonly increases the small-scale wiggliness of the lo-cal strings. In the case of a local-local network,the absence of long-range interactions allows thebound states to be much longer-lived and signif-icantly influences the evolution of the string net-work [44]. The most convincing evidence comesfrom analysing the reverse problem [44], namelythat of a bound state splitting as a result ofthe long-range interactions between strings, pre-sented in Fig. 4. Only in the absence of long-rangeinteractions the strings remain in the (1, 1) statethroughout their entire evolution.

0.001

0.01

0.1

1

0 20 40 60 80 100

Fra

ctio

ns

Time0.001

0.01

0.1

1

0 20 40 60 80 100

Fra

ctio

ns

Time

0.001

0.01

0.1

1

0 20 40 60 80 100

Fra

ctio

ns

Time0.001

0.01

0.1

1

0 20 40 60 80 100

Fra

ctio

ns

Time

Figure 4. The total physical volume of the simu-lation box occupied by Higgs strings (green), ax-ion strings (red), and their bound states (blue).The left panels refer to local-global networks,while the right ones to local-local networks start-ing from the same initial conditions as the local-global ones. Figure taken from Ref. [44].

Let us now investigate more thoroughly the is-sue of scaling. The evolution of F-, D-strings and


their bound states is a rather complicated prob-lem, which necessitates both numerical as well asanalytical investigations. As I have already men-tioned, junctions may prevent the network fromachieving a scaling solution, invalidating the cos-mological model leading to their formation. Fol-lowing the approach of Ref. [44], numerical simu-lations [45], achieving control over the initial pop-ulation of bound states, found clear evidence forscaling of all three components — p F-strings, qD-strings and their (p, q) bound states — of thenetwork, independently of the chosen initial con-figurations, while they concluded that the exis-tence of bound states effects the evolution of thenetwork. In Fig. 5 we show the string correla-tion length for the Higgs and axion fields, as wellas for their bound states, as a function of time.The initial configuration is a local-global networkwith a large amount of bound states. The corre-

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Figure 5. The Higgs (left), axion (middle), andbound state (right) string correlation length as afunction of time. The network is a local-globalone. The data and linear fits for the two regimesare shown. Figure taken from Ref. [45].

sponding plots for local-local networks are drawnin Fig. 6. Clearly, there is convincing evidence forscaling of the three components of the network for

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Figure 6. The same as Fig. 5, but for a local-localnetwork. Figure taken from Ref. [45].

both networks. This scaling is characterised witha distinct change of the correlation length slopeduring the network evolution. Note that the re-sult holds even in the case of networks with smallamounts of bound states.

Moreover, these numerical experiments haveshown that for (p, q) strings there is a supplemen-tary energy loss mechanism, in addition to thechopping off of loops; it is this new mechanismthat allows the network to scale. More precisely,the additional energy loss mechanism is the for-mation of bound states, whose length increases,lowering the overall energy of the network.

11. Cosmic Superstrings: A window into

String Theory

Cosmic superstrings have gained a lot of in-terest, the main reason being that they can of-fer a large (and possibly unique) window intostring theory, and in particular shed some lighton the appropriate (if any) stringy description ofthe Universe. Since they interact with StandardModel particles only via gravity, their detectioninvolves their gravitational interactions. Cosmicsuperstrings, in an analogy to their solitonic ana-logues, can lead to a variety of astrophysical sig-natures, like gravitational waves, ultra high en-


ergy cosmic rays, and gamma ray bursts.At this point, let me however emphasise that

given the complexity of the dynamics of a cosmicstring network, which we certainly do not fullyunderstand, and the model-dependent initial con-figuration, any theoretical estimations of the ob-servational signatures of cosmic superstrings haveto be taken with caution. Note that even the su-perstring tension depends on the considered infla-tionary scenario within a particular brane-worldcosmological model.

Gravitational waves is one of the main exploredavenues [47], in which case three channels of emis-sion have been identified. Radiation can be emit-ted by cusps, kinks, and/or from the reconnec-tion process itself. Cusps, where momentarilythe string moves relativistically, have played acrucial role in discussing the radiation emittedfrom (ordinary) cosmic strings 11. Kinks, result-ing from cosmic string collisions and subsequentreconnection, are basically replaced in the caseof cosmic superstrings by junctions. Finally, theradiation emitted from the reconnection processitself, which a sub-dominant process in the case ofcosmic strings, may not be negligible in the caseof cosmic superstrings because of the small re-connection probability P . In the scaling regime,the density of long strings goes like 1/P [40], im-plying that the number of reconnection attemptsgoes like 1/P2, and hence the number of success-ful reconnections is approximately 1/P . Very re-cently, the gravitational waveform produced bycosmic superstring reconnections has been calcu-lated [49]. Comparing the obtained result to thedetection threshold for current and future grav-itational wave detectors, it was concluded [49]that neither bursts nor the stochastic gravita-tional background, produced during the cosmicsuperstring reconnection process, would be de-tectable by Advanced LIGO. Thus, the most rele-vant process for gravitational waves emitted fromcosmic superstrings turns out to be through theircusps. Hence, one should estimate the abundancyof cusps in cosmic superstrings with junctions.

11Even though one has to keep in mind that the number ofcusps in a realistic cosmi string network has not been esti-mated, while preliminary numerical studies indicate thatit may be rather low [48].

Following simple geometric arguments, it hasbeen recently shown [50] that strings ending on D-branes can indeed lead to cusps, in an analogousway as cusps in ordinary cosmic strings. In par-ticular, cusps would be a generic feature of an F-string ending on two (parallel and stationary) D-strings. Hence, pairs of FD-string junctions, suchas those that they would form after intercommu-tations of F- and D-strings, generically containcusps. This result opens up a new energy lossmechanism for the network, in addition to the for-mation and subsequent decay of closed loops andthe formation of bound states [45]. Phenomeno-logical consequences of cusps from junctions oncosmic superstrings will be most significant atearly times, namely towards the end of brane in-flation, since then the typical separation of heavystrings is small as compared to the length of F-strings stretched between them [50].

12. Cosmic Superstring Thermodynamics

One has to extend previous studies of stringthermodynamics in the case of cosmic superstringnetworks, characterised by the existence of (p, q)bound states and different string tensions. Re-cently, the Hagedorn transition of strings withjunctions has been investigated [51], in the con-text of a simple model with three different typesand tensions of string, following an effective fieldtheory approach. More precisely, the authors ofRef. [51] translated the thermodynamics of stringnetworks with junctions into the thermodynamicsof a set of interacting dual fields. Thus, the Hage-dorn transition of the strings becomes a transitionof the fields.

In this approach, the equilibrium statisticalmechanics of cosmic superstring networks havebeen studied [51], by extending known meth-ods for describing quark deconfinement. It wasfound [51] that as the system is heated, thelightest strings are the first ones to undergo aHagedorn transition; the existence of junctionsdoes not affect the occurrence of the transition.The system is also characterised by a second,higher, critical temperature above which longstring modes of all tensions and junctions, do ex-ist. The existence of multiple tensions indicates


the appearance of multiple Hagedorn transitions.

13. Conclusions

In these lectures, I have summarised our cur-rent understanding on the physics of cosmicstrings and cosmic superstrings. I have discussedtheir formation, evolution, statistical mechan-ics and astrophysical/cosmological consequences.This is a topic of active research at present, re-lating fundamental theoretical ideas with experi-mental and observational facts.

On the one hand, any successful cosmologicalscenario, such as the inflationary paradigm, mustbe inspired from a fundamental theory. On theother hand, any successful high energy physicstheory, such as string theory or supersymmet-ric grand unified theories, must be tested againstdata; the only available laboratory for the re-quired energy scales, is indeed the early Universe.

Inflation within brane-world cosmological mod-els leads naturally to cosmic superstrings. Infla-tion within supersymmetric grand unified theo-ries leads generically to cosmic strings, the soli-tonic analogues of cosmic superstrings. The studyof these objects is interesting by itself. In addi-tion, cosmic (super)strings may provide an ex-planation for the origin of a variety of astrophysi-cal/cosmological observations; they may also offera test (often a unique one) of fundamental theo-ries of physics, thus shedding some light about theappropriate stringy description of the Universe.

Acknowledgments

It is a pleasure to thank the organisers of theESF Summer School in High Energy Physics andAstrophysics “Theory and Particle Physics: theLHC perspective and beyond”, which took placeat the Cargese Institute of Scientific Studies inCorsica, for inviting me to present these lecturesin such a beautiful and stimulating environment.

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Cosmic Strings and Cosmic Superstrings