The art of simulating the early Universe. Part II

TL;DR

The paper extends lattice cosmology methods to non-canonical field theories and gravitational waves, addressing non-minimal couplings to gravity, non-trivial field-space metrics, and axion–gauge interactions. It details lattice discretizations that preserve gauge invariance, Bianchi identities, and energy conservation, while providing explicit schemes for initial-condition generation, chirality, and multi-dimensional simulations. Key contributions include non-symplectic RK integrators for non-canonical dynamics, chirality-preserving lattice formulations, and practical frameworks for simulating cosmic defects and ALP–gauge systems with backreaction. By offering a comprehensive theoretical foundation and concrete lattice implementations (to be released in CosmoLattice v2.0), the work enables robust, first-principles predictions of early-Universe observables, including gravitational-wave backgrounds and defect-driven phenomena, across varied dimensionalities and coupling structures.

Abstract

We present a discussion on lattice techniques for the simulation of non-canonical field theory circumstances, complementing our previous monograph (arXiv:2006.15122) on canonical cases. We begin by reviewing basic aspects of lattice field theory, including symplectic and non-symplectic evolution algorithms. We then introduce lattice implementations of non-canonical interactions, considering scalars with a non-minimal coupling to gravity, $φ^2R$, non-minimal scalar kinetic theories, $\mathcal{G}_{ab}(\lbraceφ_c\rbrace)\partial_μφ^a\partial^μφ^b$, and axion-like particle (ALP) interactions with Abelian gauge fields, $φF_{μν}\tilde F^{μν}$. Next, we discuss methods to set up special field configurations, including the creation of cosmic defect networks towards scaling (e.g. cosmic strings and domain walls), field configurations based on arbitrary power spectra or spatial profiles, and probabilistic methods as required e.g. for thermal configurations. We further extend the notion of non-canonical theories, discussing the discretization of scalar field dynamics in $d + 1$ dimensions, with $d \neq 3$. Unrelated to non-canonical aspects, we also discuss implementation(s) of gravitational wave (GW) dynamics on the lattice. This document represents the theoretical basis for the non-canonical field theory aspects (interactions, initial conditions, dimensionality) and GW dynamics implemented in ${\mathcal C}$osmo${\mathcal L}$attice v2.0, to be released in 2026.

Figures (7)

• Figure 1: Evolution of the kinetic (dotted), potential (solid), gradient (dashed-dotted) and electromagnetic (dashed) energy components for $\alpha_{\Lambda} = 12$, $\alpha_{\Lambda} = 14$ and $\alpha_{\Lambda} = 18$. The vertical grey lines indicate the moments at which $\epsilon_H = 1$ is reached from $\epsilon_H < 1$. We use dashed lines when this occurs more than once, in order to distinguish them from the solid line, which corresponds to the final crossing.
• Figure 2: Evolution of the gauge field for $\alpha_{\Lambda}=18$, from earlier to later times, represented by colder and warmer colors, respectively. We represent the period from $\mathcal{N} = -4.5$ to $\mathcal{N} = 4$ e-foldings. Left: Evolution of power spectra for each chirality and the longitudinal mode, plotted every 0.5 e-folds, for $\sigma = +$ (solid), $-$ (dashed), $\parallel$ (dotted), Right: chiral imbalance $\delta_{A}(\mathcal{N},k)$ defined in Eq. (\ref{['eq:chiralImb']}) plotted every 0.1 e-folds.
• Figure 3: Schematic representation of the loss-of-resolution problem for cosmic strings (top row) and the two main resolution-preserving techniques used in simulations: fattening (middle row) and the use of an initial phase of extra-fattening followed by physical evolution (bottom row). The blue circle represents the transverse section of the string.
• Figure 4: Snapshots of $|\varphi^2|=0.25v^2$ isosurfaces of from a simulation of global cosmic strings generated with $a(\tau)=\tilde{\tau}/70$, $\tilde{\ell}_\text{str}=5$ and $\tilde{\tau}_\text{end}=92$, in a lattice with $\tilde{L}=64$, $\delta\tilde{x}=0.25$. The snapshots correspond to the random initial conditions (left), the configuration after the diffusion evolution (center), and the network after the extra-fattening phase (right).
• Figure 5: Time-evolution of the comoving mean string separation of a network of global (left) and local (right) strings, averaged over multiple independent realizations. The initial conditions are generated as described in \ref{['subsubsec:GlobalStrings', 'subsubsec:LocalStrings']}, respectively, using $\tilde{\ell}_\text{str}=15$ and $\tilde{\tau}_\text{diff}=20$ units of diffusion, and simulations are performed in periodic lattices with $N=1024$ and $\delta\tilde{x}=0.25$. The vertical dotted line indicates the end of the extra-fattening phased, followed by physical evolution in a radiation-dominated (RD) background, and the shaded region represents one standard deviation.