TOPOLOGICAL DEFECTS IN COSMOLOGY Pijushpani Bhat tacharjee ' Indian Institute of Astrophysics Koramangala Bangalore 560 034, India Abstract Topological Defects like magnetic monopoles, cosmic strings, domain walls, etc., could have formed during symmetry breaking phase transitions in the early universe. We give an overview of the implications of topological defects for cosmology, with particular emphasis on their possible role as sources of the extremely high energy cosmic rays detected in several recent cosmic ray experiments; these experiments may indeed provide the signature of existence of topological defects in the Universe. Ie-mail: [email protected] ... still I .cannot close without stating the simple fact that I and everyone of my col- leagues who share the title of astrophysicist is personally in your debt. It is a very happy event to see a portion of that debt recognized in this fashion. - A.A.PENZIAS. Please accept my warmest congratulations. It is hard to imagine a Nobel Award that would be as widely and enthusiastically approved by physicists and astronomers. - E.M.PURCELL. 1 Introduction Extended particle-like non-dissipative solutions of classical non-linear field theories, variously called solitons, lumps, etc., have been known and studied for a long time. For a comprehensive review of the subject, see for example, Refs. [1, 2J. The importance of these solutions came into sharp focus with the discovery in the early seventies that spontaneously broken gauge theories (SBGTs) possessed these finite-energy, non-singular, soliton-like solutions, namely, the vortex solution of Nielsen and Olesen simplest of spontaneously broken gauge theories, and the magnetic monopole solution of 't Hooft [4] and Polyakov [5] in the simplest spontaneously broken non-obelion gauge theory. A fundamental notion in a SBGT is the existence of more than one degenerate vacua (indeed a whole continuum of them, in general), which allows one to choose non-trivial boundary conditions for the fields and thereby construct topologically non-trivial solutions. Typically, these solutions represent extended objects with a 'core', at the centre of which the symmetry under consideration is unbroken, while outside the core the symmetry is spontaneously broken. It is in this sense that these solutions are generally called topological defects (TDs)- they represent regions of space within which the underlying fields are constrained, due to the special topological properties of these solutions, to remain in the 'false' vacuum of unbroken symmetry as opposed to the 'true' vacua representing the broke'n symmetry outside the regions. The 'center' of a defect could be a point, a line; or a 2-dimensional surface, corresponding respectively to point-like, line-like and surface-like defects; in the cosmological context, these are usually referred to as monopole, cosmic strings, and domain walls, respectively. Topological Defects as physical objects can be formed in a system that exhibits the phenomenon of spontaneous symmetry breaking (SSB). Often, the phenomenon of SSB in a physical system is manifested in the form of a phase transition as the system is cooled through a critical temperature. Laboratory examples of TDs include vortex filaments associated with superfluid phase transition in liquid 4 He, magnetic flux tubes associated with superconducting phase transition in Type-II su- perconductors, disclinations and dislocations associated with isotropic-to-nematic phase transition in liquid crystals, and so on. Similarly, in the cosmological context, TDs could have been formed during SSB phase transitions in the early Universe as the Universe expanded and cooled through certain critical temperatures. The link between temperature and symmetries is well known. One well-known example is the Ferromagnet: A piece of iron cooled below the Curie temperature becomes a ferromagnet; the full rotation symmetry of the piece of iron (in which the spins are randomly aligned) is broken below the Curie temperature - the spins get aligned along a particular direction which defines the magnetization vector. So, a higher temperature in general corresponds to more symmetry, and symmetry can be 'broken' as temperature is reduced. Conversely, a symmetry can also be restored at high temperatures- the ferromagnet loses its magnetization when heated above the Curie temperature, and the full rotation symmetry is restored. When a symmetry is broken, it need not always be completely broken, however; in the above case of ferromagnet, there is a residual symmetry left unbroken, namely the invariance under rotation about the direction of magnetization. Perhaps we should also mention here that there are counterexamples, at least theoretical models[6),in which a symmetry broken at a high temperature can be restored at a lower temperature. These tnodels are, however, rather special. In the example of ferromagnet above, the relevant symmetry is the rotation symmetry in the physical space- it is ~o space-time symmetry associated with a transformation of the space-time coordinates. It was shown in the mid-seventies [7, 8, 9] that gauge theories involving internal symmetries (i.e., symmetries associated with transformations of the fields themselves) may also possess similar behaviour- the internal symmetries broken at low temperatures can be restored at higher temperatures. Thus within the context of the Big-Bang model of the Universe, according to which the Universe was hotter (and denser) in the past than it is now, it is possible that the spontaneously broken (internal) symmetries incorporated in unified gauge theories (of which the electroweak theory is the most successful example) were actually unbroken at some sufficiently 117 early time. In particular, it is possible that the full symmetry of the so-called Grand Unified Theories (GUTs) that describe unified strong and electroweak interactions was unbroken at some epoch in the early Universe, and as the Universe expanded and cooled, a series of phase transitions took place by way of which the full GUT symmetry was successively broken to the presently observed fundamental unbroken symmetry of the Universe, namely, the symmetry under the group SU(3)c x U(I)em' where SU(3)c denotes the 'color' symmetry group that describes strong interaction of quarks and gluons, and U(l)em describes the electromagnetic interaction. Various kinds of TDs, namely, magnetic monopoles, cosmic strings, domain walls, textures, etc. [10] would have formed at some of these symmetry breaking phase transitions. As we shall see below, the typical mass (energy) scale of the TDs are roughly of same order as the energy (or temperature) scale of the symmetry breaking phase transition at which the particular kind of TD is formed. Thus, 16 TDs formed at a GUT scale phase transition (at an energy or temperature scale of >- 10 GeV) would be very massive and so they would have tremendous implications for cosmology mainly through their gravitational interactions with matter. For a nice summary of the subject of TDs, their formation, evolution and their (mainly gravitational) implications for cosmology in general, see the recent monograph by Vilenkin and Shellardj l l]. In this talk I will not deal much with those well-known gravitational implications of TDs, but will rather concentrate on some non- gravitationa.l implications of TDs; in particular, I shall discuss the possibility that TDs, under certain circumstances, may be sources of extremely energetic particles in the Universe, and that some of the currently operating and planned future detectors of extremely high energy cosmic rays (with energy;:;: 1020 eV) may provide direct signatures of TDs in the Universe. In section 2 we briefly discuss t.he nature of TDs in theories that exhibit SSB. For illustration we explicitly discuss three different kinds of TDs, namely, domain walls, cosmic strings and monopoles, by considering the simplest models of SSB in which these objects are possible, and also discuss the general topological classification of TDs. In section 3 we discuss the mechanism of cosmological formation of TDs during phase transitions in the early Universe, and briefly discuss the cosmological constraints on domain wall and magnetic monopole defects. Finally, in section 4, we discuss the possibility that TDs, under certain circumstances, may be the sources of the observed highest- energy cosmic rays. Unless otherwise stated, I use. natural units with h. = c = kB = 1, where kB denotes the Boltzmann constant. I should emphasize that this talk is not intended to be an exhaustive review of the subject of TDs---for this purpose, see Ref.[11]. 2 Topological defects and their classification 2.1 Spontaneous Symmetry Breaking (SSB) and Topological Defects (TDs) 2.1.1 Domain Wall Consider the theory of a single scalar field ¢ described by the Lagrangian density L = ~ (0i'¢) (oi'¢) - V(¢) , (1) where the "potential" V (¢) is chosen as (2) where 1'/ is a const.ant. The shape of this potential is shown schematically in Fig. 1. This Lagrangian with the above "double-well" potential is invariant under the symmetry ¢ -t -¢. For the case of a static field (¢ = 0) the equilibrium or the lowest-energy state of the field corresponds to the minimum of the potential V(¢). The above potential has two minima, and so there are two lowest-energy or ground states of the theory corresponding to ¢ == ¢+ = 1'/ and 118 .~~ ..~..-------------------------- Figure 1: The double-well potential ¢ == ¢_ = -7]. The symmetry ¢ -t -¢ is spontaneously broken because although the Lagrangian is invariant under the above symmetry, the ground states themselves are not (¢+ {-7 ¢_). Now suppose in one region of the universe we find the field to be in the state ¢ = ¢+ and in a neighbouring region ¢ = ¢_. (We shall discuss later why we may expect this to happen in general in a cosmological situation.) Then the spatial continuity of the field ¢ implies that along any line joining the two regions the field must pass through the value zero at least once. Indeed, in 3 spatial dimensions, the set of points where the field is zero forms a 2-dimensional surface -- a "domain wall" that separates a region where ¢ = ¢+ from a neighbouring region where ¢ = ¢_.