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The Semiclassical Einstein-Klein-Gordon System: Asymptotic Analysis of Minkowski Spacetime

Stefano Galanda, Paolo Meda, Simone Murro, Nicola Pinamonti, Gabriel Schmid

Abstract

We establish the linear instability of the semiclassical Einstein-Klein-Gordon system linearised about the Minkowski vacuum spacetime. The proof relies on formulating a forcing problem for both metric and state perturbations within the space of past-compact sections. This geometric framework admits a unique tensor decomposition which, in conjunction with the quantum Møller operator, enables the decoupling of the linearised system into two distinct Cauchy problems. Consequently, the metric perturbations are shown to be governed by a higher-order, nonlocal hyperbolic partial differential equation. By relegating the nonlocal contributions to subleading order, we establish the well-posedness of this forcing problem. Furthermore, we provide a rigorous asymptotic analysis for physically admissible choices of the renormalisation constants. We prove that the system exhibits a late-time linear instability: the metric perturbations grow exponentially, bounded strictly by a universal scale H, thereby indicating a quantum backreaction-driven transition toward a de Sitter cosmological spacetime. Provided the parameters governing the system are restricted to a physically relevant regime, this universal scale is compatible with the measured expansion of our universe.

The Semiclassical Einstein-Klein-Gordon System: Asymptotic Analysis of Minkowski Spacetime

Abstract

We establish the linear instability of the semiclassical Einstein-Klein-Gordon system linearised about the Minkowski vacuum spacetime. The proof relies on formulating a forcing problem for both metric and state perturbations within the space of past-compact sections. This geometric framework admits a unique tensor decomposition which, in conjunction with the quantum Møller operator, enables the decoupling of the linearised system into two distinct Cauchy problems. Consequently, the metric perturbations are shown to be governed by a higher-order, nonlocal hyperbolic partial differential equation. By relegating the nonlocal contributions to subleading order, we establish the well-posedness of this forcing problem. Furthermore, we provide a rigorous asymptotic analysis for physically admissible choices of the renormalisation constants. We prove that the system exhibits a late-time linear instability: the metric perturbations grow exponentially, bounded strictly by a universal scale H, thereby indicating a quantum backreaction-driven transition toward a de Sitter cosmological spacetime. Provided the parameters governing the system are restricted to a physically relevant regime, this universal scale is compatible with the measured expansion of our universe.

Paper Structure

This paper contains 35 sections, 25 theorems, 397 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The quantisation of a Klein-Gordon field
  3. The Cauchy problem for the semiclassical Einstein-Klein-Gordon system
  4. The linearised semiclassical Einstein equations
  5. Summary of the main results and structure of the paper
  6. Future outlook
  7. Historical survey
  8. General notation and conventions:
  9. Acknowledgements and Funding:
  10. Past-compact metric perturbations
  11. Degrees of freedom and metric perturbations
  12. Decomposition of past-compact symmetric tensor fields
  13. The four-dimensional case and decomposition of some tensors
  14. Linearised semiclassical Einstein equations
  15. Background solution
  16. ...and 20 more sections

Key Result

Theorem 1.1

Let $\bar{h}_{ab}\in\Gamma_{\mathrm{pc}}(T^{\ast}\mathbb{R}^{n}\otimes_{s}T^{\ast}\mathbb{R}^{n})$ be such that $\partial^{a}\bar{h}_{ab}=0$. Then, there are unique smooth and past-compact fields $\bar{w}\in C^{\infty}_{\mathrm{pc}}(\mathbb{R}^{n})$ and $\bar{h}_{{ab}}^{\mathrm{T}\mathrm{T}}\in\Gamm All the solutions of the constrained Cauchy problems are also solution of the Cauchy problem eq:FP

Figures (1)

  • Figure 1: The blue line denotes the zero set of the real part of $F(z)$, while the red line denotes the zero set the imaginary part of $F$ for $a=-1$, $b_0=-1$, $b_2=-10$ and $b_2$ respectively in $\{3,4,5,5.4\}$

Theorems & Definitions (52)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1: Past-compact tensor decomposition
  • Remark 2.2
  • proof : Proof of Proposition \ref{['prop:decomposition']}.
  • Corollary 2.3: Trace-reversal and decomposition
  • Proposition 2.4: Complete gauge fixing
  • proof
  • Proposition 2.5: Vanishing of the vector component
  • proof
  • ...and 42 more