[Paper Review] The Phantom of the New Oscillatory Cosmological Phase

Damien A. Easson, Alexander Vikman|arXiv (Cornell University)|Jul 4, 2016

Cosmology and Gravitation Theories40 references9 citations

TL;DR

This paper investigates the feasibility of a cosmological time-crystal—a periodic, oscillatory phase in an expanding universe driven by a k-essence scalar field—showing that such a phase requires violation of the Null Energy Condition and crossing of the Phantom divide (w = −1). It proves that this leads to infinite growth of quantum perturbations on short scales and that stable limiting cycles are only possible if they encircle a singularity in phase space, where perturbations become infinitely strongly coupled or superluminal.

ABSTRACT

We study a recently proposed new cosmological phase where a scalar field moves periodically in an expanding spatially-flat Friedmann universe. This phase corresponds to a limiting cycle of the equations of motion and can be considered as a cosmological realization of a "time-crystal". We show that this phase is only possible, provided the Null Energy Condition is violated and the so-called Phantom divide is crossed. We prove that in general k-essence models: i) this crossing causes infinite growth of quantum perturbations on short scales, and ii) exactly periodic solutions are only possible, provided the limiting cycle encircles a singularity in the phase plane. The configurations neighboring this singular curve in the phase space are linearly unstable on one side of the curve and superluminal on the other side. Moreover, the increment of the instability is infinitely growing for each mode by approaching the singularity, while for the configurations on the other side, the sound speed is growing without limit. We illustrate our general results by analytical and numerical studies of a particular class of such k-essence models.

Motivation & Objective

• To investigate whether a k-essence scalar field can support a stable, periodic cosmological phase resembling a 'time-crystal' in an expanding universe.
• To determine the conditions under which such a limiting cycle can exist, particularly focusing on energy condition violations.
• To analyze the stability and quantum behavior of such oscillatory solutions, especially near the Phantom divide (w = −1).
• To explore the implications of phase-space singularities and superluminal propagation for the consistency of the effective field theory.
• To assess whether such a time-crystal phase can be realized in a physically viable, UV-completable theory.

Proposed method

• Analyzes the dynamics of a k-essence scalar field in a spatially-flat Friedmann universe using the energy-momentum tensor and conservation laws.
• Applies the Friedmann equations to derive conditions for energy density oscillations, showing that NEC violation is necessary for periodic energy density evolution.
• Uses the Bendixson–Dulac theorem to prove that limiting cycles in k-essence models must enclose a singularity where ∂ε/∂X = 0 and canonical momentum becomes non-unique.
• Investigates quantum perturbations in the presence of Phantom divide crossing, showing their growth becomes unbounded on short scales.
• Performs analytical and numerical studies on a specific class of k-essence models with three free parameters to visualize phase-space trajectories and limiting cycles.
• Compares configurations on either side of the singularity curve: one side exhibits gradient instability with imaginary sound speed, the other shows superluminal propagation with diverging sound speed.

Experimental results

Research questions

• RQ1 Can a k-essence scalar field in an expanding universe support a stable, exactly periodic cosmological phase (a time-crystal) without violating fundamental energy conditions?
• RQ2 What are the necessary conditions, particularly regarding energy conditions, for the existence of such a limiting cycle in k-essence models?
• RQ3 How do quantum perturbations behave when the system crosses the Phantom divide (w = −1), and does this lead to instabilities?
• RQ4 What role does the phase-space singularity—where ∂ε/∂X = 0—play in enabling or obstructing the existence of a limiting cycle?
• RQ5 Can the time-crystal phase be consistently embedded in a UV-complete, Lorentz-invariant effective field theory, given the presence of superluminal or unstable configurations?

Key findings

• A limiting cycle in k-essence cosmology requires violation of the Null Energy Condition (NEC), as energy density oscillations necessitate intervals of both increase and decrease in an expanding universe.
• Crossing the Phantom divide (w = −1) leads to infinite growth of quantum perturbations on short scales, indicating a fundamental instability in the system.
• The existence of a limiting cycle is only possible if the cycle encloses a phase-space singularity where ∂ε/∂X = 0, disrupting the uniqueness of the canonical momentum.
• Configurations on one side of the singularity curve exhibit linear instability with exponentially growing modes, while those on the other side display superluminal propagation with diverging sound speed.
• The time-crystal solution is disconnected from the standard Lorentz-invariant vacuum and from superluminal configurations, suggesting separate effective field theories for each sector.
• Numerical analysis of a specific k-essence model with three parameters confirms that only certain parameter regimes allow for stable limiting cycles, consistent with the theoretical constraints derived from the Bendixson–Dulac theorem.