Cosmology in the Laboratory

One of the outstanding puzzles of modern cosmology is the structure of the universe on large distance scales. Current sky surveys have revealed striking patterns in the distribution of galaxies and galaxy clusters. The galaxies appear to form an interconnected sponge-like network permeated with vast voids. Several theoretical models or mechanisms for generating this large-scale structure have been proposed in recent years. A fascinating example of such a model arises from the interplay between the physics of the very small (elementary particles) and the physics of the very large (cosmology).

In the very successful Big Bang model the universe evolves from a hot, homogeneous, structureless gas of elementary particles. In the tiny fractions of a second after the Big Bang all the elementary particles interact according to the fundamental laws of a Grand Unified Theory: the distinct strong, weak, and electromagnetic forces we observe at low energies are united into one force. Grand Unified Theories also hypothesize the existence of a field or particle known as the Higgs particle, necessary to explain the observed masses of the particles detected in laboratory experiments.

As the universe expands and cools, it may undergo an abrupt transition to a state (phase) with very different physical properties and symmetries - much as water cooling below freezing point turns to solid ice. Phase transitions are usually marked by a change in the symmetry of the physical system - water is a fluid whereas ice is a solid. This change may be abrupt, in a first-order transition like freezing, which is accompanied by a liberation or absorption of latent heat, or gradual, in a so-called continuous transition. Both cases are treated in this article. In many cases these phase transitions unavoidably generate defects, or irregularities, in the fields describing the location of elementary particles in space, somewhat analogous to cracks in the ice. These objects - known technically as topological defects - prevent the system from being in its lowest possible energy state and, in the case of phase transitions in the very early universe, are extremely massive. This mass is a seed for gravitational effects which can ultimately lead to the clumping of matter on the enormous scales of galaxies and galaxy clusters.

A more accurate description of topological defects is obtained by constructing the space of possible low-temperature ground states in a phase transition. Suppose the high-temperature phase has a symmetry group G. The low-temperature phase, in almost every case, has less symmetry - it is more ordered. Suppose it has a symmetry group H, some subgroup of G. Now given a ground state we can construct another ground state by applying a symmetry transformation from the group G. Even though the symmetry G is not respected by any given ground state it is still a symmetry of the system. Not all ground states obtained this way are genuinely different. Any ground state obtained by acting on with an element of H is, by definition, equivalent to . The space of inequivalent ground states is the coset . When is a space with non-trivial topology then topological defects are possible.

The most thoroughly investigated class of topological defects relevant to cosmology are line-like configurations of the Higgs field known as cosmic strings. They occur whenever the space admits loops that are not continuously contractible to points. Cosmic strings are thought to have formed about seconds after the Big Bang - when the universe had a temperature of K and was no bigger than a mango. One centimeter of cosmic string weighs about kg - as much as the Rocky Mountains - and is times thinner than the atomic nucleus. In the cosmic string model of the formation of large-scale structure this huge energy density leads to gravitational perturbations that result in the clumping of the initially homogeneous universe into galaxies and galaxy clusters.

While experimental tests of theories for the formation of large-scale structure are difficult, there are many examples of phase transitions, with associated topological defects, in more accessible condensed matter systems such as liquid crystals and superfluid helium. Recently both these systems have been investigated as laboratory models of certain aspects of the formation and dynamics of topological defects in the universe and it has proven possible to verify some of the theoretical predictions of particle astrophysics. The strength of the analogy between phase transitions in the early universe and those in more conventional laboratory systems rests on the powerful concept of universality - physics is essentially the same everywhere. If a physical process occurs in one setting it is very likely that it will also play some role in analogous settings. This is widely seen within the theory of phase transitions itself - many phase transitions in very different systems have essentially the same physical description.

Condensed Matter Cosmology:

Nematic Liquid Crystals. A simple and accessible laboratory analog of defect-producing phase transitions in the cooling universe may be observed in a heated sample of the simplest kind of liquid crystal, known as a nematic liquid crystal. Nematic liquid crystals (NLCs) are materials made of many individual rod-like organic molecules. A snapshot of such a liquid crystal at high temperatures would reveal that the rods are randomly located and also randomly oriented. In other words, the rods have no preferred positions and no preferred axis of alignment. This high-temperature phase is known as the isotropic phase. As a liquid crystal cools it undergoes an abrupt change of phase at a specific temperature, usually between C and C, depending on the structure of the liquid crystal. Below this temperature the constituent rods are, on average, aligned with a particular axis in space, like a field of wild sunflowers inclined towards the setting sun. The centers of mass of the rods, however, are still distributed randomly. Such a system is said to be orientationally ordered, as opposed to the translational order of a crystal. This low-temperature, orientationally ordered phase of a liquid crystal is called the nematic phase. The isotropic-nematic transition is first order.

As a bulk sample of liquid crystal cools through the isotropic to nematic phase transition, nematic domains form spontaneously like droplets of early morning mist. The average orientation of molecules in each domain is roughly uniform. Different domains, however, orient independently. The average initial size of these nematic domains is called the correlation length and is determined by the cooling rate and the microscopic physics of liquid crystals. This domain formation is illustrated in the figure at right. In the cosmological analogy a domain is akin to a causally connected region of spacetime. Physical processes in two regions not causally connected are necessarily independent since there can never have been any communication between the regions.

With time the individual nematic domains grow and coalesce. Since it costs energy for molecular orientations to twist, bend or splay significantly, it is preferable for the entire sample to orient uniformly. It is a striking consequence of the mathematical discipline of topology that this is impossible for a nematic liquid crystal. It is inevitable that regions of rapidly changing orientation - called singularities - are trapped during the cooling to the nematic phase. At a singularity a liquid crystal is completely confused as to which orientation it should choose - it solves this dilemma by remaining in the high-temperature isotropic phase. These regions of isotropic phase are the topological defects. In nematic liquid crystals the most important topological defects are line-like, in exact analogy to cosmic strings, and are called disclinations. In Fig. 2(a), the field lines of nematic orientation have been illustrated for two planes cutting a disclination. At left is a photograph of disclinations observed in the laboratory.

With some further assumptions, it is even possible to calculate the probability that a disclination forms as randomly oriented nematic bubbles coalesce. First of all the space of orientations in three dimensions is the space of all straight lines through the origin. Imagine each line (orientation) as a long toothpick completely piercing an orange through its center. One sees that the space of all orientations is equally well described mathematically as the space of all points on the surface of a sphere, with each point and its antipodal point being regarded as identical. This space is known mathematically as the projective plane .

A loop on is either a loop on the two-sphere or a path from a point on to its antipodal point. The first kind of loop may be continuously contracted to a point and is thus topologically trivial. The second kind of loop may not be continuously contracted to a point - it has to be cut to do this and cutting is not a continuous operation - it is therefore topologically non-trivial. Superimposing two topologically non-trivial disclinations gives a configuration in which the lines of orientation wind 360 in a plane around the defect line. Such a configuration can relax into a topologically trivial configuration by a continuous bending of the lines into the third dimension. As a consequence disclinations are described by the group .

One can model coalescing domains by considering three domains of size meeting. Assign a random point on the manifold to each of the three domains. Orientations for points between domains may then be obtained by continuing the orientation from one domain to the next in as smooth a manner as possible. In this way one can see if a disclination results in a correlation volume of order . Repeating this process many times, typically numerically, yields a probability p - the expected number of disclinations per coalescing domain, or correlation volume . The total number of defects N in a volume V is then given by .

Experiments. This theoretical number has been compared with the following experiment. A droplet of the nematic liquid crystal 5CB was placed on a glass slide in a phase-contrast microscope equipped with a television camera and a video cassette recorder. The droplet was heated into the isotropic phase and allowed to cool through the isotropic-nematic transition at C. A still photograph of nematic-domain formation was extracted from the video recording and used to count the total number of domains N and their effective linear size. This size also determines the average length d of disclination produced in a domain coalescence. The total length of disclination formed (L) is then given by L = pNd.

A second photograph of disclinations at the moment they are formed can then be used to measure the total length of disclination in the sample. It is found that theory is in reasonable agreement with the experimental result. This lends support to the mechanism described above for the generation of topological defects in phase transitions and gives a theory of the density of defects at the time of their formation.

Defect Dynamics. The degree of large-scale structure generated by topological defects depends not just on their initial density but also on their evolution in time after formation. It is also possible to study directly the interactions of disclinations in a liquid crystal, as they evolve after formation, and to compare their behavior with that predicted for cosmic strings. There are remarkable similarities - disclinations shrink, straighten, cut and reconnect - just as expected for cosmic strings. Since small loops of excised cosmic string are thought to ultimately seed single objects such as galaxies, this is an important feature of the theory to verify experimentally, albeit in a dramatically different setting.

A fundamental idea of cosmic string theory that has been tested in this way is the scaling hypothesis. One assumes that the dynamics of string interactions is governed by a single length scale, the correlation length . This scale corresponds to both the mean distance between strings and their mean radius of curvature. According to this hypothesis the pattern of evolving defects remains the same statistically, with merely the overall scale of the structures growing. For disclinations in liquid crystals one can derive that should grow as the square root of the elapsed time. By dimensional analysis the length per unit volume of disclination should scale as and thus be expected to fall linearly with time. This result has been confirmed in experiments on rapid pressure-induced isotropic-nematic phase transitions in a cell of the nematic liquid crystal 5CB. The density of disclinations is measured by light transmission through the cell. One sees quite graphically then that initially small defects produced by the phase transition lead to progressively larger structures at later time, just as required for the generation of large-scale structure in the universe. Analogous experiments have also been performed on two-dimensional systems. Here the expected result is also found.

Superfluid Helium. Another beautiful laboratory system in which defects are produced in a phase transition is superfluid Helium-4. In a rapid (few-millisecond) adiabatic expansion from high pressure to low pressure, at a constant temperature of about 2K, normal liquid helium turns superfluid. This is a continuous phase transition. The decompression is performed with bellows mechanically linked to the top of the cryostat containing the liquid helium sample. The defects, in this case, are superfluid vortices - like microscopic tornadoes - with normal fluid trapped inside and superfluid helium on the outside. Helium-4 is described phenomenologically by a many-body complex Bose-condensate wavefunction . In the superfluid state the magnitude of is non-zero but its phase is arbitrary. The manifold of superfluid states is thus the set of all possible phases, namely the circle . The associated group is the symmetry group of the circle U(1), corresponding to rotations around a fixed axis. Around a vortex configuration the phase increases from zero to some integer multiple of 2 . Vortices are thus characterized by the integer winding number n.

Superfluid helium vortices are detected in the experiment by their attenuation of heat pulses (more technically, second sound) propagated across a small space of a few millimeters between a gold film heater and a carbon bolometer situated within the helium cell. Second-sound is an entropy-temperature wave in which the normal fluid and the superfluid oscillate 180 out of phase. In contrast ordinary, or first sound, is a pressure-density wave in which the normal fluid and superfluid oscillate in phase.

>From the degree of attenuation of second sound one finds that the length per unit volume of vortices formed in the transition is roughly 10 million per square centimeter. To produce the same density of vortices by spinning a vessel of liquid helium, the classic method of creating vortices, the vessel would have to be spun at the enormous rate of 630 revolutions per second. As with disclinations, vortices decay away with time. Although a fascinating and fundamental system, superfluid helium suffers from the disadvantage, compared to liquid crystals, that the vortices cannot be directly visualized. In this respect the liquid crystal systems are clearly superior.

Future. There are a rich variety of directions in which the developments described here may be extended. To begin with there are many types of liquid crystals beyond the nematics discussed in this article. Some of the translation symmetry of the nematic may also be broken, in addition to the orientational symmetry, leading to chiral nematics and smectics. There may also be a second symmetry axis, giving rise to a biaxial nematic. Biaxial nematics are particularly fascinating because they possess three fundamentally different types of disclination which should interact among themselves according to the algebra of the group of quaternions.

The liquid helium experiment could be carried out with an annular geometry rather than in the bulk. As domains form statistically in a quench to the superfluid state the phase of the wavefunction will vary randomly from one domain to the next. This will generate a spatial gradient of the phase which corresponds physically to a superfluid rotation around the annulus. Such spontaneously generated rotation would indeed be a graphic and dramatic consequence of domain formation giving rise to topological defects.

Finally all the defects discussed in this article arise from the breaking of global symmetries. A global symmetry transformation is the same at all points in space. In both particle physics and condensed matter systems there also arise local (or gauge) symmetries - these are symmetry transformations which can change from point to point in space and yet still describe the identical physical system. Indeed the electromagnetic field itself is best understood as being a consequence of a local (U(1)) symmetry.

Local cosmic strings may play as important a role in the early universe as global cosmic strings. Ideas about the formation of local cosmic strings might well be tested in a rapid temperature-quench of a type-II superconductor in a vanishing external magnetic field. One would look for spontaneously generated flux lines in this system - the analogue of the local cosmic strings. These are line-like defects that contain a normal core surrounded by superconducting material and a spatially decaying magnetic field. Experiments with this system are hard to perform as the quench to the superconducting state must be done thermally and the specific heat diverges at the phase transition. This means it is difficult to perform the quench rapidly enough to generate a significant density of defects.

• References