Simulating the Early Universe: Techniques and Challenges

This lecture discusses various numerical techniques used to simulate aspects of the early universe, focusing on the challenges posed by gauge invariance, fermions, and quantum behavior. The instructor outlines the necessity of numerical approaches in physics, especially when analytic solutions are difficult to obtain. The lecture introduces lattice gauge theory as a method to preserve gauge invariance while simulating finite volumes of space-time. The instructor explains how gauge fields and matter fields are represented on a lattice, emphasizing the importance of holonomies in this context. The discussion extends to the role of helicity in baryogenesis, linking it to Chern-Simons numbers and the dynamics of magnetic fields in the early universe. The lecture also covers the simulation of fermions, addressing the complexities of their quantum nature and the challenges of coupling them with gauge fields. The instructor concludes by highlighting the potential of both lattice and partial differential equation approaches in advancing our understanding of the universe's fundamental physics.