Formulation improvements for critical collapse simulations

The precise tuning required to observe critical phenomena in gravitational collapse poses a challenge for most numerical codes. First, threshold estimation searches may be obstructed by the appearance of coordinate singularities, indicating the need for a better gauge choice. Second, the constraint violations to which simulations are susceptible may be too large and force searches to terminate prematurely. This is a particularly serious issue for first-order formulations. We want our adaptive pseudospectral code bamps to be a robust tool for the study of critical phenomena so, having encountered both of these difficulties in work on the vacuum setting, we turn here to investigate these issues in the classic context of a spherically symmetric massless scalar field. We suggest two general improvements. We propose a necessary condition for a gauge choice to respect discrete self-similarity (DSS). The condition is not restricted to spherical symmetry and could be verified with any 3+1 formulation. After evaluating common gauge choices against this condition, we suggest a DSS-compatible gauge source function in generalized harmonic gauge (GHG). To control constraint violations, we modify the constraint damping parameters of GHG, adapting them to collapse spacetimes. This allows us to improve our tuning of the critical amplitude for several families of initial data, even going from 6 up to 11 digits. This is the most precise tuning achieved with the first-order GHG formulation to date. Consequently, we are able to reproduce the well-known critical phenomena as well as competing formulations and methods, clearly observing up to three echoes.