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Mixmaster Fluids Near the Big Bang

Elliot Marshall

TL;DR

We study the approach to the big-bang singularity in $\mathbb{T}^{2}$-symmetric spacetimes with a non-stiff fluid $p=K\rho$, $K\in[0,1)$. We formulate the $\beta$-normalised Einstein–Euler system and evolve it with a path-conservative finite-volume scheme that directly updates primitive fluid variables. The simulations reveal local, oscillatory mixmaster-like gravitational dynamics that drive tilt transitions in the fluid, with tilt oscillating between orthogonal and extremely tilted states in accord with generalized BKL triggers; these transitions generate significant local density inhomogeneities, suggesting a mechanism for early-universe structure formation. Overall, the results provide numerical support for the generalized BKL conjecture in inhomogeneous, fluid-filled cosmologies and highlight the role of fluid tilt in shaping near-singularity dynamics and matter inhomogeneities.

Abstract

We numerically study the approach to the singularity in $\mathbb{T}^{2}$-symmetric cosmological spacetimes containing a non-stiff perfect fluid satisfying a linear equation of state $p=Kρ$, $K \in [0,1)$. Near the singularity, the dynamics are found to be local and oscillatory. In particular, our results show, for the first time, that the fluid velocity in inhomogeneous cosmologies develops mixmaster-esque oscillations consistent with the generalised BKL conjecture of Uggla et al. Moreover, we find these fluid oscillations are responsible for the development of local inhomogeneities in the matter density of the early universe.

Mixmaster Fluids Near the Big Bang

TL;DR

We study the approach to the big-bang singularity in -symmetric spacetimes with a non-stiff fluid , . We formulate the -normalised Einstein–Euler system and evolve it with a path-conservative finite-volume scheme that directly updates primitive fluid variables. The simulations reveal local, oscillatory mixmaster-like gravitational dynamics that drive tilt transitions in the fluid, with tilt oscillating between orthogonal and extremely tilted states in accord with generalized BKL triggers; these transitions generate significant local density inhomogeneities, suggesting a mechanism for early-universe structure formation. Overall, the results provide numerical support for the generalized BKL conjecture in inhomogeneous, fluid-filled cosmologies and highlight the role of fluid tilt in shaping near-singularity dynamics and matter inhomogeneities.

Abstract

We numerically study the approach to the singularity in -symmetric cosmological spacetimes containing a non-stiff perfect fluid satisfying a linear equation of state , . Near the singularity, the dynamics are found to be local and oscillatory. In particular, our results show, for the first time, that the fluid velocity in inhomogeneous cosmologies develops mixmaster-esque oscillations consistent with the generalised BKL conjecture of Uggla et al. Moreover, we find these fluid oscillations are responsible for the development of local inhomogeneities in the matter density of the early universe.

Paper Structure

This paper contains 27 sections, 128 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Indexing Conventions
  3. Index Operations
  4. -Normalised Einstein-Euler Equations
  5. Orthonormal Frame Equations
  6. Symmetry Reduced Frame Equations
  7. Change of Variables
  8. -Normalised Equations
  9. Euler Equations
  10. Symmetry Reduced Euler Equations
  11. -Normalised Euler Equations
  12. Gauge Reduced Einstein Euler Equations
  13. Characteristic Speeds
  14. Primitive Form of the Euler Equations
  15. Spatially Homogeneous Asymptotics
  16. ...and 12 more sections

Figures (10)

  • Figure 1: Convergence plots of $\Sigma_{-}$ and $\nu^{1}$ at $t \approx -20$, $K=0.5$.
  • Figure 2: Convergence plots of $\log_{2}(\|(\mathcal{C}_{M})_{1}\|_{2})$, $K=0.5$.
  • Figure 3: Shock formation in $\nu^{1}$. $N=4000$, $K=0.5$.
  • Figure 4: Maximum values of $\log(|\tilde{T}^{00}|)$ and $\log(|E_{1}^{1}|)$ over time. $N=4000$, $K=0.5$.
  • Figure 5: Tilt transitions in $|\nu|$ at single point in space. $N=500$, $K=0.5$.
  • ...and 5 more figures

Theorems & Definitions (5)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 4.1