At the time matter forms, particles can be made either due to quantum decay of the inflaton, or the inflaton can resonantly decay by pumping energy into fields that it is coupled to (this is a more or less classical process). Thus, we are faced with a driven-dissipative system.

Matt Parry and I have been able to show that pattern formation is one of the possible outcomes at this stage in the evolution of the universe. Here are some movies of what the patterns can look like: Below, on the left is a snapshot of the actual spatial pattern (this is from a two-dimensional simulation). If you click on the images, you will see a movie of the evolution of the field energy as it develops in time. On the right, is the fourier transform of the field energy (snapshot, and click for movies). NB: The pictures of the field energy in configuration space have fixed normalization, so that you can see the initial growth, then decay of the field. The pictures of the fourier transform are normalized to the largest mode amplitude at each timestep, so even though the field amplitude grows, then decays, the fourier transform pictures don't show it. I set it up this way so that the actual pattern that is there would be visible even when the field decays.

The first two pictures are of a simulation setup with symmetric initial conditions: all of the resonant modes are populated. You can see that the initial set of modes grows, then the modes begin to interact with each other, creating patterns. A dodecagonal pattern shows up about five times intermixed with intermediate states, then goes over to a square pattern, which is also intermixed with intermediate states.

In the next set pictures and movies, you can see that an initial random set of modes (a more likely configuration in this physical scenario) placed on the resonant band grows, then fragmentation occurs due to nonlinear effects, forming a pattern. In this case, the pattern also changes in time, with mostly square wave patterns being evoked.

The above simulations had initial conditions where the initial spectrum all had the same phases. In these results, the initial amplitudes are the same, but the initial phases of the waves are random.

And here, are the most realistic initial conditions: random phase, gaussian spectrum noise.

Below are realistic initial conditions and patterns forming in an expanding universe gamma = 0.002.

Below are realistic initial conditions and patterns forming in an expanding universe gamma = 0.00005, here we are pushing the extent of the weakly nonlinear regime.

These are patterns that form in 3-dimensions.