xPert: Computer algebra for metric perturbation theory

TL;DR

The paper addresses the challenge of performing high-order metric perturbations in general relativity by introducing xPert, a Mathematica/xAct package that uses explicit nonrecursive formulas for curvature perturbations and efficient index canonicalization. It provides a modular toolkit to construct and manipulate nth-order perturbations, including gauge changes, with demonstrated performance up to high orders. The main contributions are the implementation of nonrecursive expansion formulas, integration with the xAct suite, and practical timing demonstrations that show feasibility on standard hardware. This work enables accurate, nonlinear perturbative analyses in GR, cosmology, and quantum-field-in-curved-space contexts, while remaining freely available as GPL software.

Abstract

We present the tensor computer algebra package xPert for fast construction and manipulation of the equations of metric perturbation theory, around arbitrary backgrounds. It is based on the combination of explicit combinatorial formulas for the n-th order perturbation of curvature tensors and their gauge changes, and the use of highly efficient techniques of index canonicalization, provided by the underlying tensor system xAct, for Mathematica. We give examples of use and show the efficiency of the system with timings plots: it is possible to handle orders n=4 or n=5 within seconds, or reach n=10 with timings below 1 hour.

Figures (4)

• Figure 1: The second-order perturbation of the Einstein tensor is constructed and canonicalized in less than one second. The blue labels of the h tensors denote the perturbative order.
• Figure 2: Canonicalization timings (in seconds) for the perturbation of the Riemann tensor at perturbative orders $n=1...10$ (lower, red line). Also shown number of terms in the expression (upper, blue dashed line). Both lines are clear exponentials in $n$, with the timings growing slightly faster because terms with larger $n$ are harder to canonicalize due to their larger average number of indices.
• Figure 3: Timings of expansion (in seconds; red lines, stars) and number of terms (blue dashed lines, diamonds) of the perturbation of the product of $m$ factors, for different perturbative orders $n=1...10$ [formula (\ref{['product']})]. Different lines correspond to increasing values of $m$, from $m=2$ to $m=10$ starting from below. All practical cases stay below one second.
• Figure 4: Timings of expansion (in seconds; red lines, stars) and number of terms (blue dashed lines, diamonds) of the perturbation of a scalar function of $m$ scalar arguments, for different perturbative orders $n=1...10$. Different lines correspond to increasing values of $m$, from $m=1$ to $m=10$ starting from below. We include only those cases which can be handled with 2Gb of RAM memory, corresponding to a few tens thousand terms. For example, for $n=10$ we can only handle up to $m=4$.