English

The art of simulating the early Universe -- Part I

Cosmology and Nongalactic Astrophysics 2021-11-08 v3 General Relativity and Quantum Cosmology High Energy Physics - Lattice High Energy Physics - Phenomenology High Energy Physics - Theory

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 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(dt2)O(dt^2) accuracy like staggered leapfrogstaggered~leapfrog and VerletVerlet integration, to RungeKuttaRunge-Kutta methods up to O(dt4)O(dt^4) accuracy, and the YoshidaYoshida and GaussLegendreGauss-Legendre higher-order integrators, accurate up to O(dt10)O(dt^{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+13+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)i) Interacting scalar fields, ii)ii) Abelian U(1)U(1) gauge theories, and iii)iii) Non-Abelian SU(2)SU(2) gauge theories. In all three cases we provide symplectic integrators, with accuracy ranging from O(dt2)O(dt^2) up to O(dt10)O(dt^{10}). For each algorithm we provide the form of relevant observables, such as energy density components, field spectra and the Hubble constraint. Remarkably, all our algorithms for gauge theories 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)×U(1)SU(2)\times U(1). The present manuscript is meant as part of the theoretical basis for CosmoLatticeCosmoLattice, a modern C++ MPI-based package for simulating the non-linear dynamics of scalar-gauge field theories in an expanding universe, publicly available at www.cosmolattice.net

Keywords

Cite

@article{arxiv.2006.15122,
  title  = {The art of simulating the early Universe -- Part I},
  author = {Daniel G. Figueroa and Adrien Florio and Francisco Torrenti and Wessel Valkenburg},
  journal= {arXiv preprint arXiv:2006.15122},
  year   = {2021}
}

Comments

Minor corrections to match published version, and one more algorithm added. Still 79 pages, 8 figures, 1 appendix, and many algorithms