Localized past stability of the subcritical Kasner-scalar field spacetimes

We prove the nonlinear stability, in the contracting direction, of the entire subcritical family of Kasner-scalar field solutions to the Einstein-scalar field equations in four spacetime dimensions. Our proof relies on a zero-shift, orthonormal frame decomposition of a conformal representation of the Einstein-scalar field equations. To synchronise the big bang singularity, we use the time coordinate $\tau = \exp\bigl(\frac{2}{\sqrt{3}}\phi\bigr)$, where $\phi$ is the scalar field, which coincides with a conformal harmonic time slicing. We show that the perturbed solutions are asymptotically pointwise Kasner, geodesically incomplete to the past and terminate at quiescent, crushing big bang singularities located at $\tau=0$, which are characterised by curvature blow up. Specifically, we establish two stability theorems. The first is a global in-space stability result where the perturbed spacetimes are of the form $M =\bigcup_{t\in (0,t_0]} \tau^{-1}(\{t\}) \cong (0,t_0] \times \mathbb{T}^{3}$. The second is a localised version where the perturbed spacetimes are given by $M=\bigcup_{t\in (0,t_0]}\tau^{-1}(\{t\})\cong \bigcup_{t\in (0,t_0]} \{t\}\times\mathbb{B}_{\rho(t)}$ with time-dependent radius function $\rho(t)=\rho_0+(1-\vartheta)\rho_0\bigl(\bigl(\frac{t}{t_0}\bigr)^{1-\epsilon}-1\bigr)$. Spatial localisation is achieved through our choice of zero-shift, harmonic time slicing that leads to hyperbolic evolution equations with a finite propagation speed.