Cosmic Strings, Domain Walls

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COSMIC STRINGS, DOMAIN WALLS

Certain models of elementary particle physics predict the existence of extended objects, such as cosmic strings or domain walls, in addition to the usual pointlike particles, such as quarks, leptons, and gauge bosons. Cosmic strings and domain walls are filamentlike and sheetlike structures, respectively, of microscopic thickness, typically less than 10^-18 meters, but of arbitrary, and possibly astronomical, length (and width). No definitive experimental evidence for either of these objects currently exists, but, if detected, their properties could help determine the correct theory of elementary particles. They might also play an important role in structure formation in the early universe. Cosmic strings and domain walls are examples of topological defects, so called because their existence is determined by the topology of the set of ground states of the theory.

Spontaneous Symmetry Breaking

Topological defects can only arise in theories with a feature called spontaneously broken symmetry. The ground state (state of lowest energy) in such theories is different at high and low temperatures, with the high-temperature ground state having a greater degree of symmetry than the low-temperature ground state. The early universe was much hotter than the current universe, and so was in a stable, symmetric phase. As the universe expanded and cooled, this symmetric phase became unstable, and the universe made a phase transition to a state of reduced symmetry. This process of spontaneous symmetry breaking can lead to the formation of topological defects, as explained below.

The Standard Model of elementary particle physics, which describes the strong, weak, and electromagnetic forces, has a symmetry that is spontaneously broken at a critical temperature of T = 10¹⁵ Kelvin; nevertheless, this model predicts neither cosmic strings nor domain walls. The Standard Model, however, is not believed to be a complete description of reality but only part of a larger theory. Some candidates for this enveloping theory, called grand unified theories, have additional symmetries broken at much higher temperatures, typically around T = 10²⁹ Kelvin. Depending on their symmetries and how these are broken, grand unified models may predict cosmic strings and/or domain walls, whose thickness would typically be about 10^-32 meters.

The Higgs Field

In particle physics models, spontaneous symmetry breaking is usually caused by a Higgs field, described here through an analogy. Consider a pencil suspended from its tip. Its gravitational potential energy is minimized when it hangs straight down. The vertical pencil is rotationally symmetric, since rotating the configuration does not change it. This maximum-symmetry, minimum-energy configuration is analogous to the ground state of the Higgs field above the critical temperature.

Next consider a pencil balancing vertically on its tip on a table. This configuration also has rotational symmetry but is unstable because the potential energy is a maximum. A configuration of minimum energy, with the pencil lying on its side, breaks the rotational symmetry. Moreover, because the pencil could fall in any direction, a continuum of minimum energy configurations exists. These broken-symmetry, minimum-energy states are analogous to the ground states of the Higgs field below the critical temperature.

Domain Walls

The existence of more than one ground state with broken symmetry is the key feature of spontaneous symmetry breaking that allows the possibility of topological defects. To see why, consider an infinite two-dimensional array of pencils balancing vertically on a table, with adjacent pencils connected by springs. Assume that the tips of the pencils are hinged so that each can only fall in either of two directions: to the left (L) or to the right (R). Let all the pencils be released simultaneously. If one of the pencils begins to fall to the right, the springs will cause nearby pencils to fall in the same direction, creating a region of the plane (or domain) in which all the pencils have fallen to the right. In another region of the plane, all the pencils might fall to the left. The plane is thus divided up into L and R domains. At the boundary between an L domain and an R domain will be a swath of standing or leaning pencils, supported by the springs connecting them and interpolating between the left-pointing and right-pointing fallen pencils. This swath of pencils separating two domains is termed a domain wall; it characterizes a region in space where the potential energy is not minimized. If one tries to reduce the energy further by pushing the standing pencils down in one direction or the other, the springs will force nearby pencils to pop up; the domain wall will move. Thus, a domain wall represents trapped energy density, which can move but cannot spread out or dissipate.

The above analogy describes a theory with discrete symmetry breaking, one in which the Higgs field has a finite number of ground states (two, in this case: L and R) below the critical temperature. In such a theory, when the universe cools below the critical temperature (corresponding to the release of the pencils), the Higgs field at each point evolves from the symmetric state to one of the ground states; this is called a cosmological phase transition. The gradient energy of the Higgs field causes the transitions at nearby points to be correlated (just as the springs cause nearby pencils to fall in the same direction). Because the symmetry breaking occurs simultaneously and randomly, and because information can travel no faster than the speed of light, distantly separated regions will not necessarily be in the same ground state. Since the Higgs field fills space, the L and R domains are three-dimensional, so the boundaries separating them are two-dimensional domain walls. The mismatch of the ground states in different domains forces the Higgs field at the boundaries to remain in the higher-energy symmetric state. This trapped energy represents the mass of the domain wall.

Cosmological phase transitions are studied using numerical simulations, in which each spatial region is randomly assigned one of the broken-symmetry states. Typically, a wall that spans the universe will form. Such a wall would disrupt the observed homogeneity of the cosmic microwave background radiation unless the symmetry-breaking scale is much less than T 10¹⁵ Kelvin. This constraint rules out grand unified theories that predict domain walls (unless an era of inflation occurs during or after domain wall formation).

Cosmic strings

Domain walls can only form in theories with discrete symmetry breaking. Theories with a continuously broken symmetry, those in which the set of ground states below the critical temperature forms a connected continuum, do not give rise to domain walls, but can host other types of topological defects, for example, cosmic strings.

A cosmic string can be described by another analogy. Again imagine a two-dimensional array of pencils standing on a table and connected by springs, but now suppose that the pencils may fall in any direction. Each pencil now has a continuum of ground states, characterized by the angle of the fallen pencil (measured counterclockwise from the east). Because of this multiplicity of ground states, pencils in different regions of the plane will randomly fall in different directions when released. The springs cause nearby pencils to fall in nearly the same direction, and because the ground states are continuously connected, some realignment of the pencils can occur after they have fallen to minimize the stretching of the springs. Suppose, however, that the pencils all fall outward from some arbitrary origin: The pencils to the east fall in the direction θ = 0, those to the north in the direction θ = 90, those to the west in the direction θ = 180, and those to the south in the direction θ = 270, with the pencils at intermediate points of the compass interpolating smoothly between these angles. Since the springs do not allow the directions of nearby pencils to differ greatly, the pencils near the origin of the configuration must remain standing or partially standing. This core of standing pencils constitutes a region of stable, trapped potential energy density; although it can be moved around, no amount of realignment can eliminate it.

This analogy describes a theory with a continuously broken symmetry. Since the Higgs field fills three dimensions, the core of trapped energy density extends along one dimension and is called a cosmic string. The two-dimensional configuration of pencils above represents a cross section of the string (see Figure 1). A cosmic string can curve or form loops, and it can move in space.

As with domain walls, the formation and evolution of cosmic strings are studied using numerical simulations. These simulations show that when the universe cools below the critical temperature, the majority of cosmic strings that form stretch across the universe, with the remainder being closed loops. When two strings intersect, the ends can break and reconnect differently. If a single string intersects itself, it can reconnect so that a loop breaks off from

FIGURE 1

the rest of the string. Closed loops can lose energy through gravitational or other radiation, shrinking away to nothing. Simulations show, however, that in theories that predict cosmic strings, some of the strings would persist to the present day.

Because of their mass (typically 10²¹ kg/m), cosmic strings would curve space-time and act as gravitational lenses. They would also cause anisotropy in the cosmic microwave background but typically not so much as to conflict with observations. The distribution of matter was extremely homogeneous in the early universe. Through gravitational attraction, cosmic strings may have acted as seeds for matter to clump onto, leading to the formation of galaxies or quasars.

String Theory

Cosmic strings should not be confused with the fundamental strings of superstring theory, an ambitious framework for describing all the fundamental forces of nature, including gravity. If superstring theory is true, fundamental strings are the building blocks of all particles and fields; they have no thickness and typically microscopic length. Cosmic strings, on the other hand, are made out of the Higgs field, with their thickness depending on the details of the theory and with typically astronomical lengths. Superstring theories, however, can also predict the existence of cosmic strings and domain walls, as well as other types of extended objects called D-branes.