Topological vacuum structures are investigated in models with spontaneous breaking of exact and approximate global symmetries. The cosmological. evolution of the structure is discussed. A spontaneous breaking of an exact global U|, '1) symmetry gives rise to vacuum strings which can produce cosmological density fluctuations leading to gal. axy formation. In a simplified axion model, the vacuum structures are strings connected by domain walls. They decay before they can dominate the universe. PAGS numbers: 98.80.Bp, 12.10.-g, 11. 30.c Gauge theories of elementary particles with spontaneous symmetry breaking predict that the early universe passes through a sequence of phase transitions as it cools after the big bang. These phase transitions can give rise to vacuum structures vacuum domain walls, strings, or monopoles, depending on the topology of the gauge theory. ' Domain walls are produced if M, the manifold of degenerate vacuums after symmetry breaking, consists of two or more disconnected components. Strings are formed if iM contains incontractible loops (that is, if M is not simply connected), and monopoles are formed if M contains incontractible two-dimensional surf aces. The cosmological evolution of these structures has been discussed previously. ' 'Besides gauge symmetries, some particle models have exact or approximate global symmetries which can also be spontaneously broken.Well-known examples are the U(1) 8-L symmetry of SU(5), ' and the approximate U(1) chiral symmetry in models with axions. " In the present paper, we shall discuss the vacuum structures arising in such models and their cosmological evolution. We shall concentrate on the most interesting case of a global U(1) symmetry.Let us first consider the case of an exact U(1) symmetry. The character of vacuum structures is insensitive to the details of the model, and all relevant physics is present in a simple Goldstone model described by the Lagrangian I, = 8 C+8t C -'hs(~C~s &s)2 where 4 is a complex scalar field. The manifold M of degenerate vacuum states in this model is a circle. It is not simply connected, and thus we must have strings. The strings can be classified according to their winding number, n= b, 6/2v, where 68 is the change of phase of the vacuum expectation value (VEV) (4) as we go around the string. The lowest energy strings have n = 1.

The "warp drive" metric recently presented by Alcubierre has the problem that an observer at the center of the warp bubble is causally separated from the outer edge of the bubble wall. Hence such an observer can neither create a warp bubble on demand nor control one once it has been created. In addition, such a bubble requires negative energy densities. One might hope that elimination of the first problem might ameliorate the second as well. We analyze and generalize a metric, originally proposed by Krasnikov for two spacetime dimensions, which does not suffer from the first difficulty. As a consequence, the Krasnikov metric has the interesting property that although the time for a one-way trip to a distant star cannot be shortened, the time for a round trip, as measured by clocks on Earth, can be made arbitrarily short. In our four dimensional extension of this metric, a "tube" is constructed along the path of an outbound spaceship, which connects the Earth and the star. Inside the tube spacetime is flat, but the light cones are opened out so as to allow superluminal travel in one direction. We show that, although a single Krasnikov tube does not involve closed timelike curves, a time machine can be constructed with a system of two non-overlapping tubes. Furthermore, it is demonstrated that Krasnikov tubes, like warp bubbles and traversable wormholes, also involve unphysically thin layers of negative energy density, as well as large total negative energies, and therefore probably cannot be realized in practice. *

Some spontaneously broken gauge theories can give rise to stringlike vacuum structures (vortices). It has been pointed out by Vilenkin that in grand unified theories these can be sufficiently massive to have cosmological implications, e.g., in explaining the formation of galaxies.