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

Abstract

We prove a stable singularity formation result, without symmetry assumptions, for solutions to the Einstein-scalar field and Einstein-stiff fluid systems. Our results apply to small perturbations of the spatially flat Friedmann-Lemaître-Robertson-Walker (FLRW) solution with topology (0, ∞)×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 {0} × T 3 , where its curvature blows up. We place "initial" data on a Cauchy hypersurface Σ ′ 1 that are close, as measured by a Sobolev norm, to the FLRW data induced on {1} × 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, we construct constant mean curvature-transported spatial coordinates for the perturbed solution covering (t, x) ∈ (0, 1] × T 3 and show that it also has a Big Bang at {0} × T 3 , where its curvature blows up. The blow-up confirms Penrose's Strong Cosmic Censorship hypothesis for the "past-half" of near-FLRW solutions. Furthermore, we show that the Einstein equations are dominated by kinetic (time derivative) terms that induce approximately monotonic behavior near the Big Bang, and consequently, various time-rescaled components of the solution converge to functions of x as t ↓ 0.The most difficult aspect of the proof is showing that the solution exists for (t, x) ∈ (0, 1] × T 3 , and to this end, we derive a hierarchy of energy estimates that are allowed to mildly blow-up as t ↓ 0. To close these estimates, it is essential that we are able to rule out more singular energy blow-up, which is in turn tied to the most important ingredient in our analysis: an L 2 −type energy approximate monotonicity inequality that holds for near-FLRW solutions. In the companion article [73], we used the approximate monotonicity to prove a stability result for solutions to linearized versions of the equations. The present article shows that the linear stability result can be upgraded to control the nonlinear terms.1 Globally hyperbolic spacetimes are those containing a Cauchy hypersurface. 2 The assumptions needed in Hawking's theorem are i) the matter model verifies the strong energy condition, which is T µν − 1 2 (g −1 ) αβ T αβ T µν X µ X ν ≥ 0 whenever X is timelike; ii) k a a < −C everywhere on the initial Cauchy hypersurface, where k is the second fundamental form of Σ t (see (3.6)) and C > 0 is a constant. More precisely, Hawking's theorem shows that no past-directed timelike curve can have length greater than 3 C . In our main results, we derive a slightly sharper version of this estimate that is tailored to the solutions addressed in this article (see inequality (15.6)). 3 As we describe below, the relevant time coordinate t for the perturbed solution has level sets with mean curvature −(1 3)t −1 . 4 Roughly, this conjecture asserts that the maximal globally hyperbolic development...

Contracting asymptotics of the linearized lapse-scalar field sub-system of the Einstein-scalar field equations

“…We first notice that all these Cauchy problems are well-posed. This can be proved using the information provided in [31,32] and with standard arguments incorporating Proposition 5.1. Given this, we proceed to obtain "rough (a-priori) estimates" for solutions of the Cauchy problem which are independent of the value of σ and the particular choice of F .…”

“…Investigating these questions for the full Einstein-scalar field system with its large number of degrees of freedom appears out of reach in light of the technical arguments required for the results of [31,32]. The main simplification which we introduce in this paper is therefore to freeze some of the degrees of freedom.…”

“…The removal of these terms is key to the derivation of useful estimates for both evolutions towards the singular time t = 0 (the decreasing time direction) as well as away from the singular time (the increasing time direction). As opposed to the works [31,32], where only estimates in the decreasing time direction required, both time directions are necessary for constructing the homeomorphism determined above.…”

Well-posedness of cubic Horndeski theories

“…where the expressions for the Hamiltonian and momentum constraints (H and M i ) are given by equations (13) and (14). When the generalized Bianchi-identity (37) holds, the evolution equations for the Hamiltonian and momentum constraints are…”

Mathematical general relativity

“…A stable singularity formation result for expanding cosmological solutions to the Einstein-scalar field and Einstein-stiff fluid systems was proven in [208] for small perturbations of the spatially flat FLRW solution with topology R × T 3 . It was shown that the basic features of the perturbed solution in the collapsing direction closely resemble those of the FLRW solution, where its curvature blows up at a "big bang" singularity at T 3 , confirming Penrose's strong cosmic censorship hypothesis for the "past-half" of near-FLRW solutions.…”