**4.1.1. Singularity Theory in Galaxy Formation and
Ring Galaxies**

In the field of galaxy and large-scale structure formation the Zeldovich approximation "has become a ubiquitous tool in analytical studies of clustering ..." (Coles, Melott and Shandarin 1993). The Zeldovich approximation is a kinematic approximation to the trajectories of particles within density perturbations in the expanding universe. In overdense regions it describes the initial collapse of (collisional) baryons and the collapse and non-linear clustering of collisionless dark matter (Shandarin and Zeldovich 1984). The kinematic Zeldovich equation has the same form as the equation for the epicyclic motions of stars in ring galaxies (see Eq. (4.1) below). The physical character of the solutions of the two equations is very different, since initially there is no significant angular momentum or centrifugal barrier in the Zeldovich problem, and no expanding background in ring galaxies. Nonetheless, the formal development of the two equations is almost exactly the same.

Solutions to the Zeldovich equation give the position of particles as a function of time, but we get a better idea of what the solution looks like from the density function. The initial cosmological collapse occurs primarily in one dimension, forming thin sheets, or "pancakes". At early times material near the plane collapses into it, with the Zeldovich approximation predicting a formally infinite density there. This is analogous to the stellar orbit crowding in the first ring discussed above. Later, in the case of collisionless dark matter, particles pass through the pancake plane, rise to a maximum distance (apoapse) on the other side, and fall back through the plane again. Between the particles at apoapse on either side of the central plane is a region of orbit crossing, much like that described above for second rings. In the cosmological case, particles falling from apoapse through the central plane move out to a second apoapse in their original half-plane. This second passage through the central plane leads to more orbit crossings in the pancake. This is analogous to a ring galaxy with the first, second, and subsequent rings all superimposed. However, since the rings propagate out through the disk the analogy breaks down at this point.

In the collisionless pancakes, the particles at apoapse also have a formally infinite density. They lie on a fold in phase space, and so they are all crowded into the same positional volume with velocities near zero relative to the pancake center. This fold is the simplest caustic or singularity or catastrophe (see Arnold, Shandarin and Zeldovich 1982). Because of the finite number of particles, and the fact that each has a random velocity component, the density on the fold caustic is not really infinite. The same is true at stellar ring edges, which are also fold caustics. In the pancakes, successive crossings of the central plane lead to the formation of successive sets of fold caustics, like the formation of new rings from the successive radial epicycles.

Singularity theory is a well-developed subject (see e.g., Poston and Stewart 1978; Arnold 1986). In two and three spatial dimensions it offers a complete classification of the generic, non-linear waveforms and their possible evolution. In the cylindrically symmetric case there are only two possibilities - either the compression wave is too weak to form caustics, or paired fold caustics marking the inner and outer ring edges form at some radius. A variety of ring widths and spacings are possible. In an off-center collision, several additional types of two dimensional waveform are possible within the disk. These include cusps, swallowtails, and pockets or purses (see Arnold 1986), and overlapping combinations.

The first attempt to understand waves in stellar disks as caustics was Hunter's (1974) modeling of spiral waves in non-interacting disks. Kalnajs (1973) also discussed the orbit crossing zones that developed in his kinematic spirals. Hunter pointed out the analogy between disk orbit crowding/crossing and the formation of optical caustics from ray crowding/crossing. This study showed that a spiral could consist of a crescent made up of fold and cusp caustics. Initially unaware of Hunter's work, Struck-Marcell and Lotan (1990) developed the caustics theory for cylindrically symmetric waves. Struck-Marcell (1990) presented some exploratory semi-analytic calculations showing that collisional disturbances could generate all of the two-dimensional caustics, and that the multiple streams of crossing orbits would provide distinct kinematic signatures. With limited spectral resolution these might not be recognizable, but would almost certainly appear as regions of high velocity dispersion. This is one important, observable difference between nonlinear caustic waves and linear density waves.

Donner et al. (1991) applied analytic caustics models to the study of more general tidal interactions. In addition to considering a broader class of interactions, this paper also presented some comparisons between the simple caustics models and self-consistent N-body simulations. The comparison was found to be good, which indicates the dominance of kinematic motions in the particular cases studied. Independently of Donner et al. (1991), Gerber (1993) carried out detailed comparisons of kinematic and tu-body (and hydrodynamical) calculations of disk waves created by asymmetric collisions. In these collisions the perturbation was also quite impulsive, and there was excellent agreement between the kinematic and self-gravitating models at the early stages of wave development and propagation through the disk. Gerber and Lamb (1994), Gerber (1993) also gave a very clear discussion of the dimensionless orbital and structural parameters in the kinematic equations, which determine the structure of the waves.