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Stable Big Bang formation in near-FLRW solutions to the Einstein-scalar field and Einstein-stiff fluid systems

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Abstract

We prove a stable blowup result for solutions to the Einstein-scalar field and Einstein-stiff fluid systems. Our result applies to small perturbations of the spatially flat Friedmann–Lemaître–Robertson–Walker (FLRW) solution with topology \((0,\infty ) \times \mathbb {T}^3\). The FLRW solution models a spatially uniform scalar-field/stiff fluid evolving in a spacetime that expands towards the future and that has a “Big Bang” singularity at \(\lbrace 0 \rbrace \times \mathbb {T}^3\), where various curvature invariants blow up. We place “initial” data on a Cauchy hypersurface \(\Sigma _1'\) that are close, as measured by a Sobolev norm, to the FLRW data induced on \(\lbrace 1 \rbrace \times \mathbb {T}^3\). We then study the asymptotic behavior of the perturbed solution in the collapsing direction and prove that its basic qualitative and quantitative features closely resemble those of the FLRW solution. In particular, for the perturbed solution, we construct constant mean curvature-transported spatial coordinates covering \((t,x) \in (0,1] \times \mathbb {T}^3\) and show that it also has a Big Bang at \(\lbrace 0 \rbrace \times \mathbb {T}^3\), where its curvature blows up. The blowup confirms Penrose’s Strong Cosmic Censorship hypothesis for the “past-half” of near-FLRW solutions. Furthermore, we show that the equations are dominated by kinetic (that is, time-derivative-involving) terms that induce approximately monotonic behavior near the Big Bang. As a consequence of the monotonicity, we also show that various time-rescaled components of the solution converge to regular functions of x as \(t \downarrow 0\). The most difficult aspect of the proof is showing that the solution exists for \((t,x) \in (0,1] \times \mathbb {T}^3\), and to this end, we derive a hierarchy of energy estimates that allow for the possibility of mild energy blowup as \(t \downarrow 0\). To close these estimates, it is essential that we are able to rule out more singular energy blowup, a step that is in turn tied to the most important ingredient in our analysis: a collection of integral identities that, when combined in the right proportions, yield an \(L^2\)-type approximate monotonicity inequality, a key point being that the error terms are controllable up to the singularity for near-FLRW solutions. In a companion article, we derived similar approximate monotonicity inequalities for linearized versions of the Einstein-scalar field equations and used them to prove linear stability results for a family of spatially homogeneous background solutions. The present article shows that the linear stability of the FLRW background solution can be upgraded to a full proof of the nonlinear stability of its singularity.

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Acknowledgements

The authors thank Mihalis Dafermos for offering enlightening comments that helped them improve the exposition. They also thank David Jerison for providing insights that aided their proof of Proposition 14.4. Finally, they thank the anonymous referee for their careful reading of the manuscript and for providing helpful feedback.

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  1. Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ, 08544-1000, USA

    Igor Rodnianski

  2. Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Ave, Room 2-265, Cambridge, MA, 02139-4307, USA

    Jared Speck

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  1. Igor Rodnianski
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  2. Jared Speck
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Correspondence to Jared Speck.

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IR gratefully acknowledges support from NSF Grant # DMS-1001500. JS gratefully acknowledges support from NSF Grant # DMS-1162211 and from a Solomon Buchsbaum grant administered by the Massachusetts Institute of Technology.

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Rodnianski, I., Speck, J. Stable Big Bang formation in near-FLRW solutions to the Einstein-scalar field and Einstein-stiff fluid systems. Sel. Math. New Ser. 24, 4293–4459 (2018). https://doi.org/10.1007/s00029-018-0437-8

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