Eric Hirschmann's Research

Steve and I have also worked on texture collapse. The original motivation was an extension of the nonlinear sigma models we had been considering in gravitational collapse. We have looked primarily at a single model of texture collapse, that of SO(3)/SO(2). Some nice evolutions have been done by Sornborger, Carroll and Pyne. We have tried to increase the resolution of the numerical integration by reducing the dimensionality of the problem. In particular, we have tried to exploit the fact that the above authors, in performing a 3D calculation, observed that the evolution remained axisymmetric. So we write the equations assuming axisymmetry. We then use a variety of initial data. Some of the more interesting evolutions resulted from using "toroidal" initial data. This would appear to lead rather dramatically to monopole-antimonopole production together with their subsequent annihiliation.
Some background first. We are considering a model of three scalar fields in flat space which interact through the usual quartic symmetry breaking potential. The equations are invariant under a global SO(3) symmetry. The initial data is such that the fields are initially on the vacuum manifold. However, the instability of the texture (due to Derrick's theorem) causes it to collapse and hence drives the fields off the vacuum manifold. We see this in the movies by the fact that the potential energy density does not remain zero. There are essentially two parameters in the model with the toroidal initial data which we can vary. One is the scale of symmetry breaking (or via a rescaling the coupling constant for the interacting fields). The other arises because we use toroidal coordinates in defining the initial data. This second parameter is the size of the degenerate torii. Roughly speaking, as the coupling constant increases and/or the size of the degenerate torii decreases, the volume of the region within which the main dynamics occurs (pair production and subsequent annihilation) decreases.
The following 2D, axisymmetric evolutions are all on a spatial grid in rho and z with rho extending from 0 to 40 and z extending from -40 to 40. Most of the runs are at a resolution of 129 by 257. In general, we plot the total energy density and the potential energy density. Our code is a second order in space and time Crank-Nicholson code. We have checked the code for convergence and that the total energy is conservered. For both of these tests, the code performed well.
In the first set of data we maintain a constant radius of the degenerate torus while varying the coupling constant (or symmetry breaking scale). Each of the evolutions has been rescaled to aid in viewing them with the potential energy density being rescaled by a factor approximately five times larger than that for the corresponding total energy density.
radius of the degenerate torus = 4
- coupling constant = 0.0316 total energy potential energy
- coupling constant = 0.1 total energy potential energy
- coupling constant = 0.316 total energy potential energy
  [We have also included a longer evolution of these same parameter values to show that the scheme looks to be stable: total energy potential energy ]
  [We have also included a "closeup" of the evolution of the potential energy: potential energy ]
- coupling constant = 1.0 total energy potential energy
For the second set of data we maintain the same coupling constant while varying the radius of the degenerate torus. We again rescale the energy densities to make things a little easier to look at.
coupling constant = 0.316
- radius of the denegerate torus = 2 total energy potential energy
- radius of the denegerate torus = 4 total energy potential energy
- radius of the denegerate torus = 12 total energy potential energy
Updated another time August 12, 1998.

Eric Hirschmann ehirsch@dirac.ph.utexas.edu