Theory Framework Guide

Welcome to the Theory Framework Tab. Here we discuss the theoretical aspects behind \(\mathcal{C}\mathtt{osmo}\mathcal{L}\mathtt{attice}\), which we separate in four sections. The first two sections, Continuum Field Theory and Field Discretization Primer, are rather introductory, meant as a basic summary of continuum and lattice aspects of field theories, respectively, that serve to set up vocabulary, concepts, field variables, operators, etc. We recommend users without previous experience with lattice simulations to read these two sections before diving into the \(\mathcal{C}\mathtt{osmo}\mathcal{L}\mathtt{attice}\) Manual, or into the code itself. In the third section, Lattice-Cosmology Reviews, we present in-depth discussions on the rationale behind the equations implemented in \(\mathcal{C}\mathtt{osmo}\mathcal{L}\mathtt{attice}\), from the evolution equations, to the initializers and output observables of the code. As the theoretical framework of \(\mathcal{C}\mathtt{osmo}\mathcal{L}\mathtt{attice}\) is quite broad, we present this material split in a series of monographic reviews on lattice-cosmology techniques. The user interested in understanding well the lattice formulations behind \(\mathcal{C}\mathtt{osmo}\mathcal{L}\mathtt{attice}\) should read these monographs. Finally, in the last section, Technical Notes, we expand on a few technical aspects of lattice field theory.

In Section Continuum Field Theory, we describe briefly the formulation of scalar and gauge field theories in the continuum. We describe the dynamics of scalar and gauge fields living in a (spatially-flat) Friedmann–Lemaître–Robertson–Walker (FLRW) background, and the dynamics of the background itself, as sourced by the fields that live within it.

In Section Field Discretization Primer, we introduce the basic notions of Lattice Field Theory, including

  • The characterization of a lattice as a discrete set of grid points.
  • The introduction of finite-difference operators for reproducing continuum differential operations.
  • The definition of the lattice momentum associated with finite-difference operators.
  • The definition of lattice gauge-invariant techniques.
  • The notion of power spectra on a lattice.
  • Evolution algorithms for solving partial differential equations (PDEs).

When all these techniques are put together to solve problems characterized by a set of PDEs that describe the field dynamics of early Universe scenarios (as \(e.g.\) during inflation, preheating, phase transitions, topological defect dynamics, etc), we collectively refer to them as Lattice Cosmology Techniques (LCT), or simply Lattice Cosmology (LC).

In Section Lattice-Cosmology Reviews, we present the LCT basis for the equations implemented in \(\mathcal{C}\mathtt{osmo}\mathcal{L}\mathtt{attice}\), \(i.e.\) the rationale behind initializers, output observables and evolution equations in the code. As the theoretical framework of these aspects is quite broad, we present the material split in a series of monographic reviews that we have written over the years: The art of simulating the early Universe, Part I (Ref. 1), Part II (Ref. 2), and Part III (Ref. 3). These reviews, downloadable in PDF format, are colloquially referred to as \(\mathtt{The~Art-I}\), \(\mathtt{The~Art-II}\) and \(\mathtt{The~Art-III}\) monographs, respectively. They provide comprehensive discussions on the Lattice-Cosmology methods used in \(\mathcal{C}\mathtt{osmo}\mathcal{L}\mathtt{attice}\) to simulate non-linear field theory dynamics in an expanding universe of

   I) Singlet scalar, and Abelian-U(1) & Non-Abelian-SU(2) scalar-gauge field theories \(\longrightarrow\) discussed in \(\mathtt{The~Art-I}\) (Ref. 1)

   II) Gravitational waves (GWs) and non-canonical aspects of interacting field theories \(\longrightarrow\) discussed in \(\mathtt{The~Art-II}\) (Ref. 2)

   III) Fluid dynamics as a description of an ensemble of scalar, gauge & fermion fields \(\longrightarrow\) discussed in \(\mathtt{The~Art-III}\) (Ref. 3)

In Section Technical Notes, we provide further discussion on technical aspects of lattice field theory. This material, presented in the form of notes downloadable in PDF format, expands on some of the notions introduced in Section Lattice-Cosmology Reviews, dwelling in particular on the concepts of power spectrum and gravitational waves on the lattice.

  1. D. G. Figueroa, A. Florio, F. Torrenti, and W. Valkenburg. The art of simulating the early universe – part i: integration techniques and canonical cases. JCAP, 04:035, 2021. arXiv:2006.15122, doi:10.1088/1475-7516/2021/04/035. ↩↩

  2. J. Baeza-Ballesteros, D. G. Figueroa, A. Florio, J. Lizarraga, N. Loayza, K. Marschall, T. Opferkuch, B. A. Stefanek, F. Torrentí, and A. Urio. The art of simulating the early universe. part ii. non-canonical cases & gravitational waves. JCAP, 06:087, 2026. arXiv:2512.15627, doi:10.1088/1475-7516/2026/06/087. ↩↩

  3. D. G. Figueroa, K. Marschall, A. Midiri, and A. Roper Pol. The art of simulating the early universe. part iii. scalar-gauge-fluid dynamics. JCAP (submitted ), XX:YYY, 2026. arXiv:2606.XYZYZXY, doi:ZZZZ. ↩↩