Conventions and Notation

Unless otherwise specified, we use the following conventions throughout the document:

  • We use natural units \(c=\hbar=1\) and metric signature \((-1,+1,+1,+1)\).

  • We use interchangeably the Newton constant \(G\), the full Planck mass \(M_p \simeq 1.22\cdot 10^{19}\) GeV, and the reduced Planck mass \(m_p \simeq 2.44\cdot 10^{18}\) GeV, all related through \(M_p^2 = 8\pi m_p^2 = 1/G\).

  • Latin indices \(i, j, k, ... = 1,2,3\) are reserved for spatial dimensions, and Greek indices \(\alpha, \beta, \mu, \nu,... = 0,1,2,3\) for space-time dimensions. We use the Einstein convention of summing over repeated indices only in the continuum. However, on the lattice, unless stated otherwise, repeated indices do not represent summation.

  • We consider a flat FLRW metric \(ds^2 = -a^{2\alpha}(\eta)d\eta^2 + a^2(\eta) \delta_{ij} dx^i dx^j\) with \(\alpha \in \mathbb{R}\) a constant chosen conveniently in each scenario. For \(\alpha = 0\), \(\eta\) denotes the coordinate time \(t\), whereas for \(\alpha = 1\), \(\eta\) denotes the conformal time \(\tau = \int {dt'\over a(t')}\). For arbitrary \(\alpha\), we will refer to the time variable as the \(\alpha\)-time.

  • We reserve the notation \(()^{\cdot}\) for derivatives with respect to cosmic time with \(\alpha = 0\), and \(()'\) for derivatives with respect to \(\alpha\)-time with arbitrary \(\alpha\).

  • Physical momenta are represented by \({\bf p}\), comoving momenta by \({\bf k}\), the \(\alpha\)-time Hubble rate is given by \(\mathcal{H} = a'/a\), whereas the physical Hubble rate is denoted by \(H = \mathcal{H}|_{\alpha = 0}\).

  • Our Fourier transform convention in the continuum is given by

    \[ \begin{align}\tag{1} f({\bf x}) = \frac{1}{(2 \pi)^3} \int d^3 {\bf k} f({\bf k}) e^{+i {\bf k} {\bf x}} \Longleftrightarrow f({\bf k}) = \int d^3 {\bf x} f ( {\bf x}) e^{-i {\bf k} {\bf x}} . \end{align} \]

  • Our discrete Fourier transform (DFT) is defined by

    \[ \begin{align}\tag{2} f({\bf n}) \equiv {1\over N^3}\sum_{\tilde n} e^{+i{2\pi\over N} {\bf \tilde n n}} f({\bf \tilde n}) \Leftrightarrow f({\bf \tilde n}) \equiv \sum_{n} e^{-i{2\pi\over N} {\bf n \tilde n} }f({\bf n}) . \end{align} \]

  • A scalar field living in a generic lattice site \(n = (n_0,{\bf n}) = (n_0,n_1,n_2,n_3)\), i.e. \(\phi_n = \phi(n)\), will be simply denoted as \(\phi\). If the point is displaced in the \(\mu\)-direction by one unit lattice spacing/time step, \(n + \hat\mu\), we will then use the notation \(n+\mu\) or simply \(+\mu\) to indicate this, so that the field amplitude in the new point is expressed as \(\phi_{+\mu} \equiv \phi(n+\hat\mu)\).

  • When representing gauge fields on the lattice, it is usually understood that they live in between lattice points, half step away from each lattice site, i.e. \(A_{\mu} \equiv A_{\mu}(n+{1\over2}\hat\mu)\). It follows then that e.g. \(A_{\mu,+\nu} \equiv A_{\mu}\big(n + {1\over2}\hat\mu + \hat\nu\big)\). In the case of links, we will use the notation \(U_\mu \equiv U_{\mu,n} \equiv U_\mu(n+{1\over2}\hat\mu)\), and hence \(U_{\mu,\pm\nu} = U_{\mu,n\pm\nu} \equiv U_\mu(n + {1\over2}\hat\mu \pm \hat\nu)\). In the very specific case of gauge fields coupled to a fluid, however, it is possible to consider an alternative formulation where the gauge fields are collocated, i.e. living at integer lattice sites \(A_\mu \equiv A_\mu(n)\).

  • Even though the lattice spacing \(\delta x\) and the time step \(\delta t\) do not need to be equal, we may often speak loosely of corrections of order \(\mathcal{O}(\delta x)\), independently of whether we are referring to the lattice spacing or the time step.

  1. N. Aghanim Planck and others. Planck 2018 results. vi. cosmological parameters. Astron. Astrophys., 641:A6, 2020. arXiv:1807.06209, doi:10.1051/0004-6361/201833910. ↩

  2. Y. Akrami Planck and others. Planck 2018 results. x. constraints on inflation. Astron. Astrophys., 641:A10, 2020. arXiv:1807.06211, doi:10.1051/0004-6361/201833887. ↩