[Paper Review] Formation of quiescent big bang singularities

Hans Oude Groeniger, Oliver Lindblad Petersen+1 more|arXiv (Cornell University)|Sep 20, 2023

Cosmology and Gravitation TheoriesPhysics and Astronomy3 citations

TL;DR

This paper establishes conditions under which quiescent big bang singularities with curvature blow-up form in the Einstein-non-linear scalar field equations. By analyzing expansion-normalized geometric quantities and imposing a large mean curvature condition combined with distinct, positive eigenvalues of the Weingarten map, the authors prove that curvature inevitably diverges toward the singularity, ensuring stable big bang formation without requiring proximity to background solutions.

ABSTRACT

Hawking's singularity theorem says that cosmological solutions arising from initial data with positive mean curvature have a past singularity. However, the nature of the singularity remains unclear. We therefore ask: If the initial hypersurface has sufficiently large mean curvature, does the curvature necessarily blow up towards the singularity? In case the eigenvalues of the expansion-normalized Weingarten map are everywhere distinct and satisfy a certain algebraic condition (which in 3+1 dimensions is equivalent to them being positive), we prove that this is the case in the CMC Einstein-non-linear scalar field setting. More specifically, we associate a set of geometric expansion-normalized quantities to any initial data set with positive mean curvature. These quantities are expected to converge, in the quiescent setting, in the direction of crushing big bang singularities. Our main result says that if the mean curvature is large enough, relative to an appropriate Sobolev norm of these geometric quantities, and if the algebraic condition is satisfied, then a quiescent (as opposed to oscillatory) big bang singularity with curvature blow-up forms. This provides a stable regime of big bang formation without requiring proximity to any particular class of background solutions. An important recent result by Fournodavlos, Rodnianski and Speck demonstrates stable big bang formation for all the spatially flat and spatially homogeneous solutions to the Einstein-scalar field equations satisfying the algebraic condition. Here we obtain analogous stability results for any solution inducing data at the singularity, in the sense introduced by the third author, in particular generalizing the aforementioned result. Moreover, we are able to prove both future and past global non-linear stability of a large class of spatially locally homogeneous solutions.

Motivation & Objective

• To determine whether large mean curvature in initial data leads to curvature blow-up at a big bang singularity.
• To clarify the nature of cosmological singularities beyond Hawking's theorem, which only guarantees incomplete geodesics.
• To establish a stable regime for quiescent (non-oscillatory) big bang formation without requiring smallness or proximity to specific background solutions.
• To extend recent results on stable big bang formation to a broader class of spatially locally homogeneous solutions.
• To prove both past and future global non-linear stability for a large class of solutions in the Einstein-non-linear scalar field setting.

Proposed method

• Introduces geometric expansion-normalized quantities derived from initial data with positive mean curvature.
• Imposes an algebraic condition on the eigenvalues of the expansion-normalized Weingarten map, equivalent to positivity in 3+1 dimensions.
• Uses Sobolev norms to control the size of geometric quantities relative to the mean curvature.
• Applies a bootstrap argument to prove global existence of solutions to the FRS (Frame, Reference, Scalar) equations.
• Employs energy estimates and decay estimates for frame coefficients, structure coefficients, second fundamental form, and scalar field gradients.
• Leverages regularity and separation estimates for eigenvalues of symmetric matrices to control spectral behavior near the singularity.

Experimental results

Research questions

• RQ1 Under what conditions does a big bang singularity with curvature blow-up form in the Einstein-non-linear scalar field equations?
• RQ2 Does a sufficiently large initial mean curvature guarantee curvature divergence, even without smallness assumptions?
• RQ3 Can quiescent big bang formation be proven stable for a broad class of solutions beyond spatially homogeneous models?
• RQ4 What role do the eigenvalues of the Weingarten map play in determining the nature of the singularity?
• RQ5 Can both past and future global non-linear stability be established for spatially locally homogeneous solutions in this setting?

Key findings

• If the initial mean curvature is sufficiently large relative to an appropriate Sobolev norm of the geometric expansion-normalized quantities, curvature blow-up necessarily occurs at the big bang singularity.
• The algebraic condition on the eigenvalues of the Weingarten map—specifically, their positivity and separation—ensures quiescent (non-oscillatory) singularity formation.
• The authors establish a stable regime for big bang formation without requiring proximity to any particular background solution, extending results by Fournodavlos, Rodnianski, and Speck.
• For solutions with induced data on a quiescent big bang singularity (in the sense of Ringström), the paper proves stable big bang formation for large classes of spatially locally homogeneous solutions.
• By combining the results with an analysis of Bianchi class A solutions, the paper proves both past and future global non-linear stability for a large class of spatially locally homogeneous solutions.
• The eigenvalue separation condition is preserved under evolution, and the eigenvalues remain $ C^ ho $-regular with controlled Hölder-type bounds, ensuring the validity of the bootstrap argument.