Constraint-Consistent Numerical Tests of Delayed Trapped-Region Formation in Saturating Scalar Field Collapse

Abstract

We investigate the nonlinear evolution of a self-gravitating scalar field with a bounded saturating potential using constraint-consistent Cauchy data and a horizon-penetrating BSSN evolution. The study revisits configurations that, in polar–areal evolution, enter a highly compact regime in which the final spacetime outcome cannot be reliably determined. Compatible Cauchy data are reconstructed and evolved independently in a formulation capable of continuing into the near-marginal regime. For two investigated amplitudes, A_0=0.0475 and A_0=0.0500, the evolution produces a future trapped region. The result is identified geometrically through marginal surfaces with vanishing outgoing null expansion and negative ingoing null expansion, with \theta_+<0 and \theta_-<0 between the two marginal surfaces. Trapped-region formation is reproduced across a three-level resolution hierarchy. For the baseline amplitude A_0=0.0475, the result additionally survives preregistered variations of numerical dissipation and Gamma-driver damping. For A_0=0.0500, trapped-region formation is independently replicated at the second amplitude and the ingoing-expansion condition is confirmed at the finest resolution. The two evolutions spend finite coordinate-time intervals in a near-marginal regime before trapped-region formation. Using the common diagnostic> \Delta t_{0.998} > =t_{\rm cross}-t_{\rm first}(C_{\max}\ge0.998), >we obtain 0.7360 for A_0=0.0475 and 2.7304 for A_0=0.0500. These coordinate-time intervals are reported as diagnostics and are not interpreted as invariant lifetimes or as establishing an amplitude trend from two data points. The numerical campaign includes explicit solver qualification, convergence testing, sparse/dense derivative validation, preregistered decision criteria, and reporting of failed as well as successful validation tests. In particular, an earlier validation protocol that failed as a test is retained in the audit trail rather than removed retrospectively. Within the investigated initial-data family, the results show that extreme compactness can precede rather than coincide with trapped-region formation. They do not establish a universal collapse threshold, formulation independence, or behavior for arbitrary initial-data families.