We prove the nonlinear stability of the asymptotic behaviour of perturbations of subfamilies of Kasner solutions in the contracting time direction within the class of polarized [Formula: see text]-symmetric solutions of the vacuum Einstein equations with arbitrary cosmological constant [Formula: see text]. This stability result generalizes the results proven in Ames E et al. (2022 Stability of AVTD Behavior within the Polarized [Formula: see text]-symmetric vacuum spacetimes. Ann. Henri Poincaré. (doi:10.1007/s00023-021-01142-0)), which focus on the [Formula: see text] case, and as in that article, the proof relies on an areal time foliation and Fuchsian techniques. Even for [Formula: see text], the results established here apply to a wider class of perturbations of Kasner solutions within the family of polarized [Formula: see text]-symmetric vacuum solutions than those considered in Ames E et al. (2022 Stability of AVTD Behavior within the Polarized [Formula: see text]-symmetric vacuum spacetimes. Ann. Henri Poincaré. (doi:10.1007/s00023-021-01142-0)) and Fournodavlos G et al. (2020 Stable Big Bang formation for Einstein's equations: the complete sub-critical regime. Preprint. (http://arxiv.org/abs/2012.05888)). Our results establish that the areal time coordinate takes all values in [Formula: see text] for some [Formula: see text], for certain families of polarized [Formula: see text]-symmetric solutions with cosmological constant. This article is part of the theme issue 'The future of mathematical cosmology, Volume 1'.
Keywords: big bang asymptotics; mathematical cosmology; stability.