Cosmic Strings and Cosmic Superstrings
Mairi Sakellariadou a
aDepartment of Physics, Kings 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 rst discuss topo-
logical defects in the context of Grand Unied Theories, focusing in particular in cosmic strings arising as gauge
theory solitons. I discuss the reconciliation between cosmic strings and cosmological ination, 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 ina-
tion, within the context of brane-world cosmological models inspired from string theory. I discuss the dierences
between cosmic superstrings and their solitonic analogues, I review our current understanding about the evolution
of cosmic superstring networks, and I then briey describe the variety of observational consequences, which may
help us to get an insight into the stringy description of our Universe.
Provided our understanding about unicationof 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  are linear topological defects,analogous to ux tubes in type-II superconduc-tors, or to vortex laments in superuid helium.These objects gained a lot of interest in the 1980sand early 1990s, since they oered a potential al-ternative to the cosmological ination for the ori-gin of initial density uctuations 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  that cosmic strings are genericallyformed at the end of an inationary era, withinthe framework of Supersymmetric Grand Unied
Theories (SUSY GUTs). Hence cosmic stringshave to be included as a sub-dominant partnerof ination. This theoretical support gave a newboost to the eld of cosmic strings, a boost whichhas been more recently enhanced when it wasshown that cosmic superstrings  (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 rst case it also xes 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 inationary 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.) 192193 (2009) 6890
0920-5632/$ see front matter 2009 Elsevier B.V. All rights reserved.
had presented in my lectures during the summerschool at Carge`se (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 eld , which plays the roleof the order parameter, was at = 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 .
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 k(Mn), which classies distinctmappings from the k-dimensional sphere Sk intothe manifoldMn. 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 sucient, con-dition for the existence of stable strings is thatthe fundamental group 1 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, then1http://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 conned, while the gradientenergy of global defects is spread out over thecausal horizon at defect formation. Global de-fects having long range density elds 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 inationary eratook place after their formation. This is one ofthe reasons for which cosmological ination 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 insignicant incosmology since their relative contribution to theenergy density of the Universe decreases rapidlywith time.
Even in the absence of a non-trivial topologyin a eld theory, it may still be possible to havedefect-like solutions, since defects may be em-bedded in such topologically trivial eld 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)CSU(2)L U(1)Y. The minimum rank of GGUThas to be at least equal to 4, to contain the GSMas 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.) 192193 (2009) 6890 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  has concluded that GGUTcould be either one of SO(10), E6, SO(14), SU(8),SU(9); ipped 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 ination to dilute them. ConsideringGUTs based on simple gauge groups, the type ofsupersymmetric hybrid ination 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)BL, 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 ination.
The results  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 ination 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 ina-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 hybridination. 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 ination have a mass whichis proportional to the inationary scale.
3. Cosmic Strings and Ination
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 ination . The ina-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 inationary scenario provides atpresent the most comprehensive picture of theUniverse at our disposal. Ination ends when theHubble parameter H =
denotes 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.
Ination is based on the basic principles of gen-eral relativity and eld 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, ination still remains a paradigm in searchof model. An inationary 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, ination isfaced with the challenging task of proving itselfgeneric , in the sense that ination would takeplace without ne-tuning of the initial conditions.
Theoretically motivated inationary models
M. Sakellariadou / Nuclear Physics B (Proc. Suppl.) 192193 (2009) 689070
can be built in the context of supersymmetryor Supergravity (SUGRA). N=1 supersymmetrymodels contain complex scalar elds which of-ten have at directions in their potential, thusoering natural candidates for inationary mod-els. In this framework, hybrid ination driven byF-terms or D-terms is the standard inationarymodel, leading generically to cosmic string for-mation at the end of ination. Hybrid ination isbased on Einsteins gravity but is driven by thefalse vacuum. The inaton eld rolls down itspotential while another scalar eld is trapped inan unstable false vacuum. Once the inaton eldbecomes much smaller than some critical value, aphase transition to the true vacuum takes placeand ination ends. F-term ination is potentiallyplagued with the Hubble-induced mass problem4 (-problem), while D-term ination avoids it.
F-term ination can be naturally accommo-dated in the framework of GUTs, when a GGUTis broken down to the GSM, at an energy scaleMGUT according to the scheme
GGUTMGUT H1 Min
where +, is a pair of GUT Higgs super-elds in non-trivial complex conjugate represen-tations, which lower the rank of the group by oneunit when acquiring non-zero vacuum expectationvalue. The inationary 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  to Abrikosov-Nielsen-Olesen strings, called F-term strings.
F-term ination is based on the globally super-symmetric renormalisable superpotential
WFin = S(+ M2) , (1)where S is a GUT gauge singlet left handed su-pereld and , M are two constants (M has di-mensions of mass) which can be taken positivewith eld redenition.
4In supergravity theories, the supersymmetry breaking istransmitted to all elds by gravity, and thus any scalareld, including the inaton, gets an eective mass of theorder of the expansion rate H during ination.
The scalar potential, as a function of the scalarcomplex component of the respective chiral su-perelds , S, is
V (+, , S) = |F+ |2 + |F |2 + |FS |2
g2aD2a . (2)
The F-term is such that Fi |W/i|=0,where we take the scalar component of the super-elds once we dierentiate with respect to i =, S. The D-terms are Da = i (Ta)
j + a,with a the label of the gauge group generators Ta,ga the gauge coupling, and a the Fayet-Iliopoulosterm. By denition, in the F-term ination thereal...