论文标题
模拟早期宇宙的艺术 - 第一部分
The art of simulating the early Universe -- Part I
论文作者
论文摘要
我们对晶格技术进行了全面讨论，以在不断扩展的宇宙中模拟标量和量规场动力学。在回顾了Minkowski和FLRW背景中标量和量规场相互作用的连续公式之后，我们引入了基本工具，以实现现场理论的离散化，包括晶格量规不变技术。 Following, we discuss and classify numerical algorithms, ranging from methods of $O(dt^2)$ accuracy like $staggered~leapfrog$ and $Verlet$ integration, to $Runge-Kutta$ methods up to $O(dt^4)$ accuracy, and the $Yoshida$ and $Gauss-Legendre$ higher-order integrators, accurate up to $ O（dt^{10}）$。我们将这些方法适应了它们在$ 3+1 $尺寸的扩展网格中的经典晶格模拟中的经典晶格模拟，包括田间能量和压力密度的量平均值所产生的“自符势”扩展的情况。我们提出了以下规范案例的晶格配方：$ i）$互动标量字段，$ ii）$ abelian $ u（1）$量规理论和$ iii）$ non-abelian $ su（2）$ gauge理论。在所有三种情况下，我们提供符号积分器，精度从$ o（dt^2）$最高到$ o（dt^{10}）$。对于每种算法，我们提供相关可观察物的形式，例如能量密度成分，田间光谱和哈勃约束。值得注意的是，我们所有用于计量理论的算法都尊重高斯对机器精度的约束，包括当考虑“自洽”扩展时。作为数值示例，我们分析了在$ su（2）\ times u（1）$下收取的振荡的充气的通货膨胀后动力学。本手稿是$ COSMOLATTICE $的理论基础的一部分，$ COSMOLATTICE $，这是一种现代的基于C ++ MPI的软件包，用于在不断扩展的宇宙中模拟标量 - 钢板田间理论的非线性动态，并在www.cosmolattice.net上公开获得
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 $O(dt^2)$ accuracy like $staggered~leapfrog$ and $Verlet$ integration, to $Runge-Kutta$ methods up to $O(dt^4)$ accuracy, and the $Yoshida$ and $Gauss-Legendre$ higher-order integrators, accurate up to $O(dt^{10})$. 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 $3+1$ 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 $O(dt^2)$ up to $O(dt^{10})$. 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 $SU(2)\times U(1)$. The present manuscript is meant as part of the theoretical basis for $CosmoLattice$, 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
