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Publication# 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(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.

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José Roberto Canivete Cuissa, Daniel Garcia Figueroa

We present a lattice formulation of an interaction phi/Lambda F (F) over tilde between an axion and some U(1) gauge sector with the following properties: it reproduces the continuum theory up to O(dx(mu)(2)) corrections, it preserves exact gauge invariance and shift symmetry on the lattice, and it is suitable for self-consistent expansion of the Universe. The lattice equations of motion can no longer be solved by explicit methods, but we propose an implicit method to overcome this difficulty, which preserves the relevant system constraints down to arbitrary (tunable) precision. As a first application we study, in a comoving grid in (3 +1) dimensions, the last efolds of axion-inflation with quadratic potential and the preheating stage following afterwards. We fully account for the inhomogeneity and non-linearity of the system, including the gauge field contribution to the expansion rate of the Universe and its backreaction into the axion dynamics. We characterize in detail, as a function of the coupling, the energy transfer from the axion to the gauge field. Two coupling regimes are identified, sub- and super-critical, depending on whether the final energy fraction stored in the gauge field is below or above similar to 50% of the total energy. The Universe is very efficiently reheated for super-critical couplings, rapidly entering in a radiation dominated stage. Our results on preheating confirm previously published results.

Solitons are stable, non-singular solutions to the classical equations of motion of non-linear field theory. Their energy is localized and finite and their shape remains unaltered during propagation. For this reason they represent particle-like states in field theory. Their mass and their size can be very large compared to those of the elementary particles in the theory. Therefore, a soliton can be viewed as a single particle-like object containing a large number of individual particles. The chiral Abelian Higgs model contains an interesting class of non-topological solitons, that carry a non-zero fermion number NF or Chern-Simons number NCS, which is the same because of the chiral anomaly. They consist of a bosonic configuration of gauge and Higgs fields characterized by NCS and are stable for sufficiently large NCS. In the first part of this thesis we study the properties of these anomalous solitons. We find that their energy-versus-fermion-number ratio is given by E ∼ NCS3/4 or E ∼ NCS2/3 depending on the structure of the scalar potential. For the former case we prove, using some inequalities from functional analysis, that there is a lower bound on the soliton energy, which reads E ≥ c NCS3/4 , where c is some parameter expressed through the masses and coupling constants of the theory. We construct the anomalous solitons numerically for two different choices for the potential accounting both for Higgs and gauge dynamics. Solutions are obtained as a function of NCS and the Higgs mass mH and we find that they are not spherically symmetric. In addition, we outline a relation between the structure of anomalous Abelian solitons and the intermediate state observed in type-I superconductors in external magnetic fields. In the limit of large NCS anomalous solitons can be described in the thin wall approximation, which allows us to remove the Higgs field from consideration. For absolute stability of anomalous solitons, it is essential that the gauge group is Abelian. If the gauge group is non-Abelian, fermions can always be converted to a gauge vacuum configuration with an arbitrary integer NCS. Therefore, if anomalous non-Abelian solitons exist, they could only be metastable. Interestingly, anomalous solitons can potentially exist in the electroweak theory, because this theory contains all necessary ingredients, namely chiral fermions and an Abelian gauge symmetry. In the second part of this thesis we investigate this possibility. Using the numerical solutions for anomalous Abelian solitons as a starting point, we construct the corresponding numerical solutions in electroweak theory. These solutions have a similar structure as the Abelian solitons with the Abelian gauge field replaced by the Z boson field. The charged boson fields W± vanish identically. However, for weak mixing angle θω > 0, the solutions have an associated magnetic field as well, that can be characterized by a magnetic dipole moment mem. Furthermore, the shape of the solutions and the structure of the gauge fields depend on θω. In the last part of this work we analyze the classical stability of the numerical solutions in the electroweak case. It is clear that the solutions are stable in the semilocal limit sin θω → 1, where the Abelian case is reproduced exactly. For arbitrary θω, we consider perturbations in the Higgs field and in the gauge fields Z and A and show that the solutions are stable with respect to these perturbations. For small θω however, the solutions are unstable with respect to the formation of a condensate of charged boson fields W± in the centre of the solution. This W-condensation instability is essentially the same, which also destabilizes the Z-string solution of electroweak theory.

This thesis is concerned with gauge theories, their complicated vacuum and resulting effects. After an introduction to the subject, it is divided into four parts.
Firstly, we treat the problem of chiral charge dynamics at finite temperature. Quantum field theory predicts a possibility for massless fermions to be transferred into electromagnetic fields with non-zero helicity and vice-versa. This phenomenon has applications ranging from cosmology to heavy-ions physics. We present a numerical investigation from first principles of the resulting complex dynamics and find a qualitative agreement with previous studies based on hydrodynamical approaches but measure rates that differ by up to an order of magnitude. We interpret this effect as contributions coming from small scales not previously taken into account.
Secondly, we present a study of open-boundary conditions in lattice QCD at finite temperature. They were designed to ease up the problem of "topological freezing", which plagues numerical simulations close to the continuum limit. In particular, we determine the length of the "boundary zone" for two different temperatures. We also use the boundary effects to extract screening masses.
Thirdly, we move on to present a compendium of lattice techniques, including some new algorithms, to perform real-time classical simulations of bosonic matter, Abelian and non-Abelian gauge fields in an expanding universe. We also briefly introduce CosmoLattice, a numerical software designed to perform such simulations, which are particularly interesting to study the reheating phase of our universe.
Finally, we study yet another technique to probe non-perturbative sectors of field theories. Namely, we show that one can reconstruct the Schwinger pair production rate, which is the rate of production of particles due to the presence of a strong electric field, using only a few terms of the weak magnetic field expansion. This surprising result is obtained by using techniques coming from the field of resurgence and the analysis of asymptotic expansions.
We conclude this work by presenting some general outlooks, sharing aspects of all these different yet related topics.