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 the 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(δ t²) accuracy like staggered leapfrog and Verlet integration, to Runge-Kutta methods up to 𝒪(δ t⁴) accuracy, and the Yoshida and Gauss-Legendre higher-order integrators, accurate up to 𝒪(δ t¹⁰). 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. We present lattice formulations of canonical cases of: i) Interacting scalar fields, ii) Abelian (1) gauge theories, and iii) Non-Abelian (2) gauge theories. In all three cases we provide symplectic integrators, with accuracy ranging from 𝒪(δ t<sup>2</sup>) up to 𝒪(δ t<sup>10</sup>). For each algorithm we provide the form of relevant observables, such as energy density components, field spectra and the Hubble constraint. We note that all our algorithms for gauge theories always respect the Gauss constraint to machine precision, including when `self-consistent' expansion is considered. As a numerical example we analyze the post-inflationary dynamics of an oscillating inflaton charged under (2)×(1). We note that the present manuscript is meant to be part of the theoretical basis for the code 𝒞osmoℒattice, a multi-purpose MPI-based package for simulating the non-linear evolution of field theories in an expanding universe, publicly available at http://www.cosmolattice.net.

We study the nonperturbative dynamics of the standard model (SM) after inflation, in the regime where the SM is decoupled from (or weakly coupled to) the inflationary sector. We use classical lattice simulations in an expanding box in ($3+1$) dimensions, modeling the SM gauge interactions with both global and Abelian-Higgs analogue scenarios. We consider different postinflationary expansion rates. During inflation, the Higgs forms a condensate, which starts oscillating soon after inflation ends. Via nonperturbative effects, the oscillations lead to a fast decay of the Higgs into the SM species, transferring most of the energy into $Z$ and $W^{\pm }$ bosons. All species are initially excited far away from equilibrium, but their interactions lead them into a stationary stage, with exact equipartition among the different energy components. From there on, the system eventually reaches equilibrium. We have characterized in detail, in the different expansion histories considered, the evolution of the Higgs and of its dominant decay products until equipartition is established. We provide a useful mapping between simulations with different parameters, from which we derive a master formula for the Higgs decay time as a function of the coupling constants, Higgs initial amplitude and postinflationary expansion rate.

The extension of the adiabatic regularization method to spin-$1/2$ fields requires a self-consistent adiabatic expansion of the field modes. We provide here the details of such expansion, which differs from the WKB ansatz that works well for scalars, to firmly establish the generalization of the adiabatic renormalization scheme to spin-$1/2$ fields. We focus on the computation of particle production in de Sitter spacetime and obtain an analytic expression of the renormalized stress-energy tensor for Dirac fermions.