The art of simulating the early Universe -- Part I

Understand

We present a comprehensive discussion on lattice techniques for the simulation of scalar and gauge field dynamics in an expanding universe.

• After reviewing the continuum formulation of scalar and gauge field interactions in Minkowski and FLRW backgrounds, we introduce basic tools for the discretization of field theories, including lattice gauge invariant techniques.
• Following, we discuss and classify numerical algorithms, ranging from methods of $O(dt^2)$ accuracy like $staggered~leapfrog$ and $Verlet$ integration, to $Runge-Kutta$ methods up to $O(dt^4)$ accuracy, and the $Yoshida$ and $Gauss-Legendre$ higher-order integrators, accurate up to $O(dt^{10})$.
• We adapt these methods for their use in classical lattice simulations of the non-linear dynamics of scalar and gauge fields in an expanding grid in $3+1$ dimensions, including the case of `self-consistent' expansion sourced by the volume average of the fields' energy and pressure densities.