We study fermion number non-conservation (or chirality breaking) in Abelian gauge theories at finite temperature. We consider the presence of a chemical potential µ for the fermionic charge, and monitor its evolution with real-time classical lattice simulations. This method accounts for short-scale fluctuations not included in the usual effective magneto-hydrodynamics (MHD) treatment. We observe a self-similar decay of the chemical potential, accompanied by an inverse cascade process in the gauge field that leads to a production of long-range helical magnetic fields. We also study the chiral charge dynamics in the presence of an external magnetic field B, and extract its decay rate Γ 5 ≡ − d log µ dt . We provide in this way a new determination of the gauge coupling and magnetic field dependence of the chiral rate, which exhibits a best fit scaling as Γ 5 ∝ e 11/2 B 2 . We confirm numerically the fluctuation-dissipation relation between Γ 5 and Γ diff , the Chern-Simons diffusion rate, which was obtained in a previous study. Remarkably, even though we are outside the MHD range of validity, the dynamics observed are in qualitative agreement with MHD predictions. The magnitude of the chiral/diffusion rate is however a factor ∼ 10 times larger than expected in MHD, signaling that we are in reality exploring a different regime accounting for short scale fluctuations. This discrepancy calls for a revision of the implications of fermion number and chirality non-conservation in finite temperature Abelian gauge theories, though not definite conclusion can be made at this point until hard-thermal-loops (HTL) are included in the lattice simulations.

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(δt 2 ) accuracy like staggered leapfrog and Verlet integration, to Runge-Kutta methods up to O(δt 4 ) accuracy, and the Yoshida and Gauss-Legendre higher-order integrators, accurate up to O(δt 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. We present lattice formulations of canonical cases of: i) Interacting scalar fields, ii) Abelian U(1) gauge theories, and iii) Non-Abelian SU(2) gauge theories. In all three cases we provide symplectic integrators, with accuracy ranging from O(δt 2 ) up to O(δt 10 ). 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,

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The presence of scalar fields with non-minimal gravitational interactions of the form ξ|φ| 2 R may have important implications for the physics of the early universe. While many studies solve the dynamics of non-minimally coupled scalars in the Einstein frame, where gravity is simply described by the Einstein-Hilbert action, we instead propose a procedure to solve the dynamics directly in the original Jordan frame where the non-minimal couplings are maintained explicitly. Our algorithm can be applied to scenarios that include minimally coupled fields and an arbitrary number of non-minimally coupled scalars, with the expansion of the universe sourced by all fields present. This includes situations when the dynamics become fully inhomogeneous, fully non-linear (due to e.g. backreaction or mode rescattering effects), and/or when the expansion of the universe is dominated by non-minimally coupled species. As an example, we study geometric preheating with a non-minimally coupled scalar spectator field when the inflaton oscillates following the end of inflation. In the future, our technique may be used to shed light on aspects of the equivalence of the Jordan and Einstein frames at the quantum level.

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