[Paper Review] Instability of the big bang coordinate singularity in a Milne-like universe

F. R. Klinkhamer, Zi-Liang Wang|arXiv (Cornell University)|Nov 25, 2019

Cosmology and Gravitation Theories24 references4 citations

TL;DR

This paper proposes a simplified dynamic-vacuum-energy model for a Milne-like universe, where the big bang singularity is a coordinate singularity with finite curvature. Through scalar metric perturbation analysis, it demonstrates that these perturbations destabilize the singularity, implying that the apparent regularity of the big bang in this model is fragile under quantum or gravitational fluctuations.

ABSTRACT

We present a simplified dynamic-vacuum-energy model for a time-symmetric Milne-like universe. The big bang singularity in this simplified model, like the one in a previous model, is just a coordinate singularity with finite curvature and energy density. We then calculate the dynamic behavior of scalar metric perturbations and find that these perturbations destabilize the big bang singularity.

Motivation & Objective

• To simplify the complex q-theory model of a Milne-like universe with dynamic vacuum energy.
• To investigate the stability of the big bang coordinate singularity under metric perturbations.
• To assess whether the finite curvature and energy density at t=0 in the Milne-like model remain robust under quantum or gravitational fluctuations.
• To compare the perturbative behavior of this model with that of a nonsingular bounce model featuring a spacetime defect scale b.
• To explore the implications for cross-big-bang information transfer and the viability of time-symmetric cosmological models.

Proposed method

• Formulates a simplified action with a q-field having mass dimension 2, coupled to gravity via a derivative term and a vacuum energy density ρV(q) = σ(q) + Λ.
• Uses a spatially-hyperbolic Robertson–Walker metric with comoving coordinates (t, χ, θ, φ), allowing t ∈ (−∞, ∞) and a(t) → 0 at t = 0.
• Derives the nonlinear Klein–Gordon equation for the q-field and the Einstein equation with an energy-momentum tensor Tαβ^(q-vac) resembling that of a scalar field.
• Performs a linear perturbation analysis on the metric, introducing Φ(t) as the scalar perturbation, and derives the perturbed Ricci scalar R to O(Φ²).
• Transforms the time coordinate from conformal time η to cosmic time t and expresses the perturbation in terms of dimensionless time τ.
• Solves the perturbation equations analytically and evaluates the behavior of the Ricci scalar and metric perturbations near t = 0 (τ = 0).

Experimental results

Research questions

• RQ1 Can a simplified q-theory model reproduce the key features of the Milne-like universe, including a finite-curvature big bang coordinate singularity?
• RQ2 How do scalar metric perturbations behave near t = 0 in this Milne-like model compared to a nonsingular bounce model?
• RQ3 Does the perturbation of the Ricci scalar remain finite at t = 0, or does it exhibit essential singular behavior?
• RQ4 What is the role of the dynamic vacuum energy density ρV(q) in the stability of the big bang singularity?
• RQ5 Can the model support stable information transfer across the big bang if perturbations destabilize the singularity?

Key findings

• The big bang singularity in the simplified Milne-like model remains a coordinate singularity with finite curvature and energy density, as in the original model.
• Scalar metric perturbations δΦ_klm(τ) exhibit oscillatory behavior with logarithmic time dependence, diverging as 1/|τ| near τ = 0.
• The perturbed Ricci scalar δR_klm(τ) scales as C_klm / |τ|, showing essential singularity behavior at t = 0, which ruins the regularity of the background.
• The perturbation dynamics are unstable at t = 0, indicating that the coordinate singularity is not robust under small metric fluctuations.
• In contrast, the nonsingular bounce model with a spacetime defect scale b exhibits regular perturbations at t = 0 due to the presence of b² in denominators.
• The instability arises because the Milne-like model lacks a hard-wired regularization scale like b, making it sensitive to perturbations that destroy the delicate cancellations required for a coordinate singularity.