A regime of linear stability for the Einstein-scalar field system with applications to nonlinear Big Bang formation

Abstract

We linearize the Einstein-scalar field equations, expressed relative to constant mean curvature (CMC)-transported spatial coordinates gauge, around members of the well-known family of Kasner solutions on $(0,\infty) \times \mathbb{T}^3$. The Kasner solutions model a spatially uniform scalar field evolving in a (typically) spatially anisotropic spacetime that expands towards the future and that has a “Big Bang” singularity at $\lbrace t = 0 \rbrace$. We place initial data for the linearized system along $\lbrace t = 1 \rbrace \simeq \mathbb{T}^3$ and study the linear solution’s behavior in the collapsing direction $t \downarrow 0$. Our first main result is the proof of an approximate $L^2$ monotonicity identity for the linear solutions. Using it, we prove a linear stability result that holds when the background Kasner solution is sufficiently close to the Friedmann-Lemaître-Robertson-Walker (FLRW) solution. In particular, we show that as $t \downarrow 0$, various time-rescaled components of the linear solution converge to regular functions defined along $\lbrace t = 0 \rbrace$. In addition, we motivate the preferred direction of the approximate monotonicity by showing that the CMC-transported spatial coordinates gauge can be viewed as a limiting version of a family of parabolic gauges for the lapse variable; an approximate monotonicity identity and corresponding linear stability results also hold in the parabolic gauges, but the corresponding parabolic PDEs are locally well posed only in the direction $t \downarrow 0$. Finally, based on the linear stability results, we outline a proof of the following result, whose complete proof will appear elsewhere: the FLRW solution is globally nonlinearly stable in the collapsing direction $t \downarrow 0$ under small perturbations of its data at $\lbrace t = 1 \rbrace$.

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      fjournal = {Annals of Mathematics. Second Series},
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      mrclass = {83C75 (35L67 35L75 53C50 83C05)},
      mrnumber = {1680551},
      mrreviewer = {Alan D. Rendall},
      doi = {10.2307/121023},
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      ISSN = {1424-0637},
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      booktitle = {International {C}onference on {D}ifferential {E}quations, {V}ol. 1, 2},
      VENUE={{B}erlin, 1999},
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      issn = {0001-7701},
      mrclass = {83C05},
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      YEAR = {2009},
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      CODEN = {ANMAAH},
      MRCLASS = {83C75 (35L81 53C50 53C80 83C05)},
      MRNUMBER = {2600872 (2011d:83095)},
      MRREVIEWER = {Piotr T. Chru{\'s}ciel},
      DOI = {10.4007/annals.2009.170.1181},
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      author = {Rodnianski, Igor and Speck, Jared},
      title = {The nonlinear future stability of the {FLRW} family of solutions to the irrotational {E}uler-{E}instein system with a positive cosmological constant},
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      }
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      title = {Stable {Big} {Bang} Formation in Near-{FLRW} Solutions to the {Einstein}-Scalar Field and {Einstein}-Stiff Fluid Systems},
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Authors

Igor Rodnianski

Princeton University, Princeton, NJ

Jared Speck

Massachusetts Institute of Technology, Cambridge, MA