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Relativistic perfect fluids near Kasner singularities

Florian Beyer, Todd A. Oliynyk

TL;DR

This work analyzes the compressible relativistic Euler equations on Kasner spacetimes near the big‑bang singularity without symmetry or isotropy restrictions. By transforming the Euler system into a symmetric hyperbolic Fuchsian form and applying a new singular initial value problem theory, the authors establish the existence of a stable family of solutions carrying the full asymptotic data and prove nonlinear stability under small perturbations. The analysis employs a forward Fuchsian method to capture the singular dynamics and a backward framework to relate forward and singular limits, yielding precise decay rates and co-moving behavior under a regime where the sound speed satisfies $0< c_s^2 < 1$ and $c_s^2 > \max\{p_1,p_2,p_3\}$. The results extend to Kasner–scalar field backgrounds and illuminate stable fluid asymptotics near Kasner big‑bang singularities, with implications for early-universe cosmology and beyond.

Abstract

We establish the existence of a stable family of solutions to the Euler equations on Kasner backgrounds near the singularity with the full expected asymptotic data degrees of freedom and no symmetry or isotropy restrictions. Existence is achieved through transforming the Euler equations into the form of a symmetric hyperbolic Fuchsian system followed by an application of a new existence theory for the singular initial value problem. Stability is shown to follow from the existence theory for the (regular) global initial value problem for Fuchsian systems that was developed in arXiv:1907.04071. In fact, for each solution in the family, we prove the existence of an open set of nearby solutions with the same qualitative asymptotics and show that any such perturbed solution agrees again with another solution of the singular initial value problem. All our results hold in the regime where the speed of sound of the fluid is large in comparison to all Kasner exponents. This is interpreted as the regime of stable fluid asymptotics near Kasner big bang singularities.

Relativistic perfect fluids near Kasner singularities

TL;DR

This work analyzes the compressible relativistic Euler equations on Kasner spacetimes near the big‑bang singularity without symmetry or isotropy restrictions. By transforming the Euler system into a symmetric hyperbolic Fuchsian form and applying a new singular initial value problem theory, the authors establish the existence of a stable family of solutions carrying the full asymptotic data and prove nonlinear stability under small perturbations. The analysis employs a forward Fuchsian method to capture the singular dynamics and a backward framework to relate forward and singular limits, yielding precise decay rates and co-moving behavior under a regime where the sound speed satisfies and . The results extend to Kasner–scalar field backgrounds and illuminate stable fluid asymptotics near Kasner big‑bang singularities, with implications for early-universe cosmology and beyond.

Abstract

We establish the existence of a stable family of solutions to the Euler equations on Kasner backgrounds near the singularity with the full expected asymptotic data degrees of freedom and no symmetry or isotropy restrictions. Existence is achieved through transforming the Euler equations into the form of a symmetric hyperbolic Fuchsian system followed by an application of a new existence theory for the singular initial value problem. Stability is shown to follow from the existence theory for the (regular) global initial value problem for Fuchsian systems that was developed in arXiv:1907.04071. In fact, for each solution in the family, we prove the existence of an open set of nearby solutions with the same qualitative asymptotics and show that any such perturbed solution agrees again with another solution of the singular initial value problem. All our results hold in the regime where the speed of sound of the fluid is large in comparison to all Kasner exponents. This is interpreted as the regime of stable fluid asymptotics near Kasner big bang singularities.

Paper Structure

This paper contains 24 sections, 13 theorems, 370 equations.

Table of Contents

  1. Introduction
  2. Symmetric hyperbolic Fuchsian systems and the singular initial value problem
  3. Preliminaries
  4. Spatial manifolds, coordinates, indexing and partial derivatives
  5. Vector Bundles
  6. Inner-products and operator inequalities
  7. Constants, inequalities and order notation
  8. Sobolev spaces
  9. The class of symmetric hyperbolic Fuchsian systems
  10. The singular initial value problem
  11. Proof of Theorem \ref{['thm:mainresult']}
  12. Transformation to a canonical Fuchsian system
  13. Existence and uniform estimates on time intervals $[T_0,{t_*}]$ with ${t_*}<0$
  14. Singular initial value problem existence and uniqueness for canonical Fuchsian systems
  15. Completion of the proof of Theorem \ref{['thm:mainresult']}
  16. ...and 9 more sections

Key Result

Theorem 1.1

Suppose $p_1, p_2, p_3$ are Kasner exponents satisfying and $W^0_*>0$, $W^1_*$, $W^2_*$ and $W^3_*$ are given functions on $\mathbb{T}{}^3$ at $\tilde{t}=0$ that together define the asymptotic data for the Euler equations. Then near the singularity at $\tilde{t}=0$ of the Kasner vacuum solution with exponents $p_1$, $p_2$ and $p_3$ given by Kasner-k-orig for some uniformly bounded functions $W^0$

Theorems & Definitions (27)

  • Theorem 1.1: Informal version
  • Definition 2.1: Symmetric hyperbolic Fuchsian systems
  • Theorem 2.2: The singular initial value problem for symmetric hyperbolic Fuchsian systems
  • Remark 2.3
  • Definition 2.4: Canonical symmetric hyperbolic Fuchsian systems
  • Lemma 2.5
  • Proposition 2.6
  • proof
  • Proposition 2.7
  • proof
  • ...and 17 more