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The Instability of the Critical Friedmann Spacetime at the Big Bang as an Alternative to Dark Energy

Christopher Alexander, Blake Temple, Zeke Vogler

TL;DR

This work proves local instability of the critical Friedmann spacetime ($k=0$) to radial perturbations at the Big Bang by formulating the Einstein–Euler equations in self-similar variables and expanding solutions in even powers of $\xi=r/t$, yielding STV PDE/ODE systems. The critical spacetime is shown to be a saddle rest point $SM$ with a $(2n-2)$-dimensional unstable manifold; nonzero-$k$ Friedmann spacetimes lie on this manifold at every order, and an underdense subfamily $\\mathcal{F}$ accelerates at intermediate times before asymptotically returning to the leading-order Friedmann spacetime, suggesting a purely gravitational mechanism for accelerated expansion without a cosmological constant. The analysis introduces a solution-dependent gauge, time since the Big Bang, and defines families $\\mathcal{F}_n$ and $\\mathcal{F}_{\\infty}$ that capture formal and actual asymptotics, with $\\mathcal{F}$ as the maximal asymptotically stable set evolving from underdense perturbations. In cosmological terms, the results indicate that Einstein gravity alone can produce acceleration effects akin to dark energy, via the instability of the Big Bang in the self-similar Einstein–Euler framework, and motivate further work on higher-order corrections and observational implications.

Abstract

We characterize the local instability of pressureless Friedmann spacetimes to radial perturbation at the Big Bang. The analysis is based on a formulation of the Einstein-Euler equations in self-similar variables $(t,ξ)$, with $ξ=r/t$, conceived to realize the critical ($k=0$) Friedmann spacetime as a stationary solution whose character as an unstable saddle rest point $SM$ is determined via an expansion of smooth solutions in even powers of $ξ$. The eigenvalues of $SM$ imply the $k\neq0$ Friedmann spacetimes are unstable solutions within the unstable manifold of $SM$. We prove that all solutions smooth at the center of symmetry agree with a Friedmann spacetime at leading order in $ξ$, and with an eye toward Cosmology, we focus on $\mathcal{F}$, the set of solutions which agree with a $k<0$ Friedmann spacetime at leading order, providing the maximal family into which generic underdense radial perturbations of the unstable critical Friedmann spacetime will evolve. We prove solutions in $\mathcal{F}$ generically accelerate away from Friedmann spacetimes at intermediate times but decay back to the same leading order Friedmann spacetime asymptotically as $t\to\infty$. Thus instabilities inherent in the Einstein-Euler equations provide a natural mechanism for an accelerated expansion without recourse to a cosmological constant or dark energy.

The Instability of the Critical Friedmann Spacetime at the Big Bang as an Alternative to Dark Energy

TL;DR

This work proves local instability of the critical Friedmann spacetime () to radial perturbations at the Big Bang by formulating the Einstein–Euler equations in self-similar variables and expanding solutions in even powers of , yielding STV PDE/ODE systems. The critical spacetime is shown to be a saddle rest point with a -dimensional unstable manifold; nonzero- Friedmann spacetimes lie on this manifold at every order, and an underdense subfamily accelerates at intermediate times before asymptotically returning to the leading-order Friedmann spacetime, suggesting a purely gravitational mechanism for accelerated expansion without a cosmological constant. The analysis introduces a solution-dependent gauge, time since the Big Bang, and defines families and that capture formal and actual asymptotics, with as the maximal asymptotically stable set evolving from underdense perturbations. In cosmological terms, the results indicate that Einstein gravity alone can produce acceleration effects akin to dark energy, via the instability of the Big Bang in the self-similar Einstein–Euler framework, and motivate further work on higher-order corrections and observational implications.

Abstract

We characterize the local instability of pressureless Friedmann spacetimes to radial perturbation at the Big Bang. The analysis is based on a formulation of the Einstein-Euler equations in self-similar variables , with , conceived to realize the critical () Friedmann spacetime as a stationary solution whose character as an unstable saddle rest point is determined via an expansion of smooth solutions in even powers of . The eigenvalues of imply the Friedmann spacetimes are unstable solutions within the unstable manifold of . We prove that all solutions smooth at the center of symmetry agree with a Friedmann spacetime at leading order in , and with an eye toward Cosmology, we focus on , the set of solutions which agree with a Friedmann spacetime at leading order, providing the maximal family into which generic underdense radial perturbations of the unstable critical Friedmann spacetime will evolve. We prove solutions in generically accelerate away from Friedmann spacetimes at intermediate times but decay back to the same leading order Friedmann spacetime asymptotically as . Thus instabilities inherent in the Einstein-Euler equations provide a natural mechanism for an accelerated expansion without recourse to a cosmological constant or dark energy.
Paper Structure (17 sections, 10 theorems, 50 equations, 4 figures)

This paper contains 17 sections, 10 theorems, 50 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Mathematical Background
  3. The STV ODE
  4. The Instability of Friedmann Spacetimes
  5. Time Since the Big Bang
  6. The Families of Spacetimes $\mathcal{F}_n$ and $\mathcal{F}_{\infty}$
  7. Statement of Results
  8. A Self-Similar Expression of the Critical Friedmann Spacetime
  9. The STV PDE
  10. The STV ODE
  11. The STV ODE Phase Portrait of Order $n=1$
  12. The Trajectory Connecting $SM$ to $M$
  13. Time Since The Big Bang
  14. The STV ODE Phase Portrait of Order $n=2$
  15. The STV ODE Phase Portrait of Order $n\geq2$
  16. ...and 2 more sections

Key Result

Theorem 1

Let $\eta=\bar{r}/t$ and define the transformation $\Psi:(t,r)\to(\bar{t},\bar{r})$ by where Then that is, $\eta$ is an implicit function of $\xi$ alone, $\Psi$ is a bijective regular coordinate transformation for all $\xi<1$ and $\Psi$ transforms the $k=0$ Friedmann metric to the SSC metric form where noting that $B_\sigma=1$ for $r=0$. Moreover, so are the density and velocity variables r

Figures (4)

  • Figure 1: The phase portrait for the $2\times2$ system.
  • Figure 2: The space $\mathcal{F}$ of solutions which decay to $M$.
  • Figure 3: The phase portrait for the $4\times4$ system.
  • Figure 4: A vector field phase portrait for the $2\times2$ system.

Theorems & Definitions (10)

  • Theorem 1
  • Theorem 2: The STV PDE
  • Theorem 3: The STV ODE
  • Corollary 1
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Theorem 9