Quantitative blow-up estimates for spacelike singularities in gravitational-collapse cosmological spacetimes

Under spherical symmetry, with double-null coordinates $(u,v)$, we study the gravitational collapse of the Einstein--scalar field system with a positive cosmological constant. The spacetime singularities arise when area radius $r$ vanishes and they are spacelike. We derive new quantitative estimates, obtain polynomial blow-up rates $O(1/r^N)$ for various quantities, and extend the results in [5] by the first author and Zhang and the arguments in [3] by the first author and Gajic to the cosmological settings. In particular, we sharpen the estimates of $r\partial_u r$ and $r\partial_v r$ in [5] and prove that the spacelike singularities where $r(u,v)=0$ are $C^{1,1/3}$ in $(u,v)$ coordinates. As an application, these estimates also give quantitative blow-up upper bounds of fluid velocity and density for the hard-phase model of the Einstein-Euler system under irrotational assumption. Near the timelike infinity, we also generalize the theorems in [3] by linking the precise blow-up rates of the Kretschmann scalar to the exponential Price's law along the event horizon. In cosmological settings, this further reveals the mass-inflation phenomena along the spacelike singularities for the first time.