Scattering towards the singularity for the wave equation and the linearized Einstein-scalar field system in Kasner spacetimes
Warren Li
TL;DR
The paper establishes a sharp scattering theory for two linear models on Kasner spacetimes: (i) the scalar wave equation and (ii) the linearized Einstein–scalar field system. By reformulating the equations as first-order systems and performing a Fourier decomposition on $\mathbb{T}^D$, the authors prove modewise high- and low-frequency energy estimates and construct a frequency-adapted scattering framework that maps Cauchy data at $t=1$ to asymptotic data at the singularity $t=0$, with a precise half-derivative gain in regularity for the asymptotic data. The analysis reveals how Kasner anisotropy, via the symbol $\mathcal{T}_*^{\,-p_i+p_j}$, governs derivative losses and gains and necessitates gauge changes between Cauchy and asymptotic data. The results are sharp in the subcritical Kasner regime and lay the groundwork for nonlinear extensions, offering insight into velocity-dominated behaviour and BKL-type instability in cosmological singularities. Overall, the work provides a rigorous, frequency-adapted scattering construction for linearized gravity on anisotropic cosmological backgrounds and highlights the intricate interplay between gauge choices, anisotropy, and asymptotic data.
Abstract
We consider the scalar wave equation $\square_g φ$ and the linearized Einstein-scalar field system around generalized Kasner spacetimes with spatial topology $\mathbb{T}^D$. In suitable regimes for the Kasner exponents, it is known that solutions to such equations arising from regular Cauchy data (e.g. at $t = 1$) have certain quantitative blow-up asymptotics near the initial time (i.e. $t = 0$) singularity of Kasner. For instance, solutions to the wave equation behave as $φ(t, x) \approx ψ_{\infty}(x) \log t + \varphi_{\infty}(x)$ near $t = 0$. This article provides a description, and proof, of a scattering theory for the above equations, linking Cauchy data at $t = 1$ and suitable asymptotic data at $t = 0$ in Kasner. For the scalar wave equation, this means a Hilbert space isomorphism between $(φ, \partial_t φ)$ at $t = 1$ and the functions $(ψ_{\infty}, \varphi_{\infty})$. A curious detail is that certain quantities e.g. $ψ_{\infty}$, feature a gain of 1/2 a derivative when compared to $\partial_t φ$ at t = 1. The study of the linearized Einstein-scalar field system reveals further interesting phenomena, including differences between diagonal and off-diagonal components of certain tensors in the scattering theory, and that the losses of derivatives feature a sensitive dependence on the anisotropy of the background Kasner spacetime. In fact, though our result holds for the entire subcritical regime of background Kasner exponents, the number of derivatives lost and gained in the scattering theory can become unbounded as one nears the boundary of this regime.