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Complete asymptotics in the formation of quiescent big bang singularities

Andrés Franco-Grisales, Hans Ringström

TL;DR

This work advances the understanding of quiescent big bang singularities in Einstein--nonlinear scalar field spacetimes by integrating three strands: asymptotics in symmetry-reduced settings, geometric initial data on the singularity, and stability results for broad data. It develops a local, gauge-agnostic framework using expansion normalised data and a Fermi-Walker framed formulation (the FRS equations), to derive complete $C^\ell$-type asymptotics, and to show that solutions constructed in prior work induce data on the singularity with Kasner-type constraints. The paper proves that under admissible potentials and small initial expansion-normalised data, the spacetime admits a past crushing big bang with a CMC foliation and curvature blow-up, while the expansion normalised data converge to robust nondegenerate quiescent initial data on the singularity. This unifies stability results and data-at-singularity constructions, and provides a gauge-robust pathway for deriving data on the singularity in broader contexts.

Abstract

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.

Complete asymptotics in the formation of quiescent big bang singularities

TL;DR

This work advances the understanding of quiescent big bang singularities in Einstein--nonlinear scalar field spacetimes by integrating three strands: asymptotics in symmetry-reduced settings, geometric initial data on the singularity, and stability results for broad data. It develops a local, gauge-agnostic framework using expansion normalised data and a Fermi-Walker framed formulation (the FRS equations), to derive complete -type asymptotics, and to show that solutions constructed in prior work induce data on the singularity with Kasner-type constraints. The paper proves that under admissible potentials and small initial expansion-normalised data, the spacetime admits a past crushing big bang with a CMC foliation and curvature blow-up, while the expansion normalised data converge to robust nondegenerate quiescent initial data on the singularity. This unifies stability results and data-at-singularity constructions, and provides a gauge-robust pathway for deriving data on the singularity in broader contexts.

Abstract

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.
Paper Structure (32 sections, 31 theorems, 345 equations)

This paper contains 32 sections, 31 theorems, 345 equations.

Table of Contents

  1. Introduction
  2. The Einstein--nonlinear scalar field equations
  3. Initial data on the singularity
  4. The main result
  5. The FRS equations
  6. Higher order derivative estimates for solutions to the FRS equations
  7. Deriving asymptotics, given higher order derivative bounds
  8. Outline of the arguments
  9. Acknowledgements
  10. Higher order energy estimates for solutions to the FRS equations
  11. Elementary consequences of the basic supremum assumptions
  12. Energies and basic estimates
  13. Estimating the potential
  14. Algorithm
  15. Estimating the lapse function
  16. ...and 17 more sections

Key Result

Theorem 11

Fix $3\leq n\in\mathbb{N}^{}$ and admissibility thresholds $\sigma_V$, ${\sigma_p} \in (0,1)$. Let $(\Sigma, h_\mathrm{ref})$ be a closed Riemannian manifold of dimension $n$ and let $V\in C^{\infty}(\mathbb{R}^{})$ be a non-negative $\sigma_V$--admissible potential. Then there is a $\kappa\in\mathb $|p_I-p_J|>\zeta_{0}^{-1}$ for $I\neq J$, where $p_I$ are the eigenvalues of $\mathcal{K}_0$; and t

Theorems & Definitions (88)

  • Definition 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Definition 5
  • Definition 6
  • Remark 7
  • Definition 8
  • Definition 9
  • Remark 10
  • ...and 78 more