Abstract: 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(delta t(2)) accuracy like staggered leapfrog and Verlet integration, to Runge-Kutta methods up to O(delta t(4)) accuracy, and the Yoshida and Gauss-Legendre higher-order integrators, accurate up to O(delta t(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(delta t(2)) up to O(delta t(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, including when 'self-consistent' expansion is considered. As a numerical example we analyze the post-inflationary dynamics of an oscillating inflaton charged under SU(2) x U(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.cosrnolattice.net.

Abstract: Axion inflation, i.e. an axion-like inflaton coupled to an Abelian gauge field through a Chern-Simons interaction, comes with a rich and testable phenomenology. This is particularly true in the strong backreaction regime, where the gauge field production heavily impacts the axion dynamics. Lattice simulations have recently demonstrated the importance of accounting for inhomogeneities of the axion field in this regime. We propose a perturbative scheme to account for these inhomogeneities while maintaining high computational efficiency. Our goal is to accurately capture deviations from the homogeneous axion field approximation within the perturbative regime as well as self -consistently determine the onset of the nonperturbative regime.

Abstract: The value of the tensor-to-scalar ratio r in the region allowed by the latest Planck 2015 measurements can be associated to a large variety of inflationary models. We discuss here the potential of future Cosmic Microwave Background cosmological observations in disentangling among the possible theoretical scenarios allowed by our analyses of current Planck temperature and polarization data. Rather than focusing only on r, we focus as well on the running of the primordial power spectrum, alpha(s) and the running thereof, beta(s). If future cosmological measurements, as those from the COrE mission, confirm the current best-fit value for beta(s) greater than or similar to 10(-2) as the preferred one, it will be possible to rule-out the most favoured inflationary models.