GFiRe—Gauge Field integrator for Reheating

Abstract

We present a new numerical algorithm and code, GFiRe, for solving the non-linear evolution of Abelian gauge fields coupled to complex scalar fields in homogeneous and isotropic spacetimes. We adopt a hybrid approach to solving the system: the spatial derivatives are discretized using standard Lattice Gauge Field Theory techniques, whereas the time evolution of the fields and scale factor is implemented with explicit, composite, symplectic integrators. An important property of our compound algorithm is that the discretized Gauss constraint is respected exactly, regardless of the order of the symplectic integrator. This remains true even when the background expansion is computed “self-consistently”; that is, when the expansion history is computed using spatial averaged components of the energy momentum tensor in the simulation volume. Hence, our code can also be used in cases where the fields dominate the energy density of the universe, for example, during reheating after inflation. We test the algorithm in scenarios of reheating where the inflaton is a complex scalar field with a potential ∝(2|φ|2−v2)2 and is coupled to an Abelian gauge field. Tracing the evolution of the system through complex dynamics (including resonant excitation of fields, backreaction, formation of solitons, and changes in the equation of state) in a self-consistently expanding universe, we find the energy conservation violation (<10−4) to be very stable and the Gauss constraint violation (<10−6) to be dominated by differencing noise.

The First Three Seconds: a Review of Possible Expansion Histories of the Early Universe

“…For a given model, the full nonlinear dynamics of fields after inflation can now be explored numerically using a number of codes which evolve scalar and vector fields on a lattice [179,180,181,182,183,149,184,185,186,187]. After such nonlinear evolution, however, the spectrum of decay products is typically far from an equilibrium distribution.…”

Gravitational wave background from non-Abelian reheating after axion-like inflation

“…[8]), the other on classical field theory simulations (cf., e.g., refs. [45][46][47]). While simulations should account for the full non-perturbative dynamics of momentum modes with large occupation numbers, they are not sensitive to phenomena where the occupation number is of order unity.…”

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

“…• The creation of topological defects, like cosmic string networks [72,114,[157][158][159], and their evolution during the scaling regime [160][161][162][163][164][165][166] and corresponding emission of GWs [167,168].…”

“…Only in this way, we will achieve a certain robustness in the predictions of the potentially observational implications from non-linear high energy phenomena. Furthermore, the techniques developed for studying nonlinear dynamics of classical fields are common to many other non-linear problems in the early universe, like the dynamics of phase transitions [74,75,102,[195][196][197][198][199] and their emission of gravitational waves [200][201][202][203][204][205][206], cosmic defect formation [114,159,[207][208][209][210][211][212][213][214], their later evolution [160-166, 215, 216] and gravitational wave emission [114,167,168,217], axion-like field dynamics [172,175,[218][219][220][221], moduli dynamics [222,223], etc. These techniques can also be used in applications of interest not only to cosmology, but also to other high energy physics areas.…”

“…Of course, exploring numerically the nonlinear dynamics of interacting fields during the early universe is not a new idea, as witnessed by the increasing number of lattice codes dedicated to this purpose that have appeared within the last years. With the exception of the recent GFiRe code [159], that includes an integrator for Abelian gauge theories, previous public packages were dedicated only to interacting scalar fields, either with finite difference techniques in real space, like Latticeeasy [251], Clustereasy [252], Defrost [253], CUDAEasy [254], HLattice [255], PyCOOL [256] and GABE [257], or pseudo-spectral codes like PSpectRe [258] and Stella [171]. In most of the mentioned codes, metric perturbations (whenever present) are sourced passively, neglecting backreaction effects on the dynamics of the scalar fields.…”