The art of simulating the early universe. Part I. Integration techniques and canonical cases

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(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,

Characterizing the post-inflationary reheating history, Part I: single daughter field with quadratic-quadratic interaction

“…Simulations have been carried out with Velocity-Verlet integration [32] in CosmoLattice [50], a recent package for lattice simulations of interacting fields in an expanding universe. We have run simulations in (3 + 1) dimensions, but also in (2 + 1) dimensions, as this reduces the simulation time by a factor ∼ 10 2 − 10 3 in comparison to three-dimensional ones (see footnote 1 for an explanation of the meaning of "(2 + 1)-dimensional simulations").…”

“…[41]), so it becomes increasingly difficult to study very large values of q * . We have partially alleviated this issue by simulating the system with higher-order Velocity-Verlet algorithms [32], which are implemented in CosmoLattice up to O(dt −10 ).…”

“…The late time regime, on the other hand, is governed by non-linear dynamics. Thus, to investigate the dynamics of the (p)reheating phase in its full extent, it is necessary to resort to lattice field theory techniques (for a review see [32]). Post-inflationary dynamics have been studied with the help of such techniques in the past, in particular for models with monomial inflaton potentials.…”

“…There we studied the evolution of the energy distribution and equation of state of the universe as a function the model parameters. The simulations were carried out with the CosmoLattice package [32,50], which implements various evolution techniques that allow to decrease the simulation time and to execute the code parallellized in multi-core systems. In particular, we took advantage of the higher-order Velocity-Verlet evolution algorithms implemented in the package, which allowed us to simulate the late-time regime of the system while preserving energy conservation sufficiently well.…”

"Stairway to Heaven" -- Spectroscopy of Particle Couplings with Gravitational Waves

“…( 19), but in terms of a lattice momentum k (L) i that depends on the choice of spatial-derivative, see [143] for a discussion. We use the nearest-neighbor derivative of equation ( 71) in [145], for which the lattice momenta is given by k…”