arXiv subjects

Andrés Franco-Grisales

Publications and source records attributed to Andrés Franco-Grisales.

3 recordsLinked to original sources

arXiv2026

Localized formation of quiescent big bang singularities

We prove a localized big bang formation result, which does not require proximity of the initial data to any background solution. Suppose that we are given initial data for the Einstein--nonlinear scalar field equations on an open set $U \subset \mathbb{R}^3$. We identify a general condition on the initial data such that if the condition is satisfied in a large enough neighborhood of $x \in U$, then the corresponding maximal globally hyperbolic development has a local quiescent big bang singularity with curvature blow up to the past of $x$. We achieve the localization by introducing a new kind of foliation by spacelike hypersurfaces, given by the level sets of a time function satisfying a certain second order differential equation. This time function allows us to synchronize the singularity while at the same time yielding a symmetric hyperbolic formulation of Einstein's equations. Our new formulation also has two key advantages over previous localized big bang stability results. First, it is independent of the matter model, so it is possible that it could be used to prove big bang formation with matter models different from a scalar field. And second, it allows us to conclude that our solutions induce geometric initial data on the singularity, thus giving a complete description of the asymptotics towards the big bang.

gr-qc↗

arXiv2026

Complete asymptotics in the formation of quiescent big bang singularities

There are three categories of mathematical results concerning quiescent big bang singularities: the derivation of asymptotics in a symmetry class; the construction of spacetimes given initial data on the singularity; and the proof of big bang formation in the absence of symmetries, including the proof of stable big bang formation. In a recent article, the first author demonstrated the existence of developments corresponding to a geometric notion of initial data on a big bang singularity. Moreover, this article, combined with previous articles by the second author, gives a unified and geometric perspective on large classes of seemingly disparate results in the first two categories. Concerning the third category, Oude Groeniger et al recently formulated a general condition on initial data ensuring big bang formation, including curvature blow up. This result, among other things, generalises previous results on stable big bang formation. However, it does not include a statement saying that the solutions induce initial data on the singularity. Here we tie all three categories of results together by demonstrating that the solutions of Oude Groeniger et al induce data on the singularity. However, the results are more general and can potentially be used to derive similar conclusions in other gauges.

gr-qc↗

arXiv2025

Developments of initial data on big bang singularities for the Einstein-nonlinear scalar field equations

In a recent work, Ringström proposed a geometric notion of initial data on big bang singularities. Moreover, he conjectured that initial data on the singularity could be used to parameterize quiescent solutions to Einstein's equations; that is, roughly speaking, solutions whose leading order asymptotics are convergent. We prove that given initial data on the singularity for the Einstein-nonlinear scalar field equations in 4 spacetime dimensions, as defined by Ringström, there is a corresponding unique development of the data. We do not assume any symmetry or analyticity, and we allow for arbitrary closed spatial topology. Our results thus present an important step towards resolving Ringström's conjecture. Furthermore, our results show that the Einstein-nonlinear scalar field equations have a geometric singular initial value problem formulation, which is analogous to the classical result by Choquet-Bruhat and Geroch for initial data on a Cauchy hypersurface. In the literature, there are two conditions which are expected to ensure that quiescent behavior occurs. The first one is an integrability condition on a special spatial frame. The second one is an algebraic condition on the eigenvalues of the expansion normalized Weingarten map associated with a foliation of the spacetime near the singularity. Our result is the first such result where both possibilities are allowed. That is, we allow for the first condition to ensure quiescence in one region of space and for the second condition to take over in the region where the first one is violated. This fact allows for our results to include the vacuum setting.

gr-qc↗