[Paper Review] Qualitative Analysis and Numerical Simulation of Equations of the Standard Cosmological Model: $\Lambda ot=0$

Yurii Ignat’ev|arXiv (Cornell University)|Sep 28, 2016

Material Science and Thermodynamics3 references3 citations

TL;DR

This paper investigates the qualitative and numerical behavior of the standard cosmological model with a massive scalar field and zero cosmological constant (Λ = 0). Using dynamical systems theory and numerical simulation, it shows that the system exhibits a stable center at the origin in phase space (Φ, Φ̇), leading to oscillatory cosmological acceleration with an average value of −1/2, implying a long-lasting non-relativistic era in the late-time universe. The results suggest that even without Λ, the model naturally supports a flat, stable, and non-relativistic late-time universe.

ABSTRACT

On the basis of qualitative analysis of the system of differential equations of the standard cosmological model it is shown that in the case of zero cosmological constant this system has a stable center corresponding to zero values of potential and its derivative at infinity. Thus, the cosmological model based on single massive classical scalar field in infinite future would give a flat Universe. The carried out numerical simulation of the dynamic system corresponding to the system of Einstein - Klein - Gordon equations showed that at great times of the evolution the invariant cosmological acceleration has an oscillating character and changes from $-2$ (braking), to $+1$ (acceleration). Average value of the cosmological acceleration is negative and is equal to $-1/2$. Oscillations of the cosmological acceleration happen on the background of rapidly falling Hubble constant. In the case of nonzero value of the cosmological constant depending on its value there are possible three various qualitative behavior types of the dynamic system on 2-dimensional plane $(\Phi,\dot{\Phi})$, which correspond either to zero attractive focus or to stable attractive knot with zero values of the potential and its derivative. Herewith the system asymptotically enters the secondary inflation. Carried out numerical simulation showed that at cosmological constant $\Lambda<m^2 3\cdot10^{-8}$ the macroscopic value of the cosmological acceleration behaves itself similar to the case $\Lambda=0$, i.e. in the course of the cosmological evolution there appears a lasting stage when this value is close to $-1/2$ which corresponds to non-relativistic equation of state. In this article, the results of qualitative and numerical analysis, obtained in Yu. Ignat'ev, arXiv:1609.00745 [gr-qc], common to the case of a non-zero cosmological term.

Motivation & Objective

• To analyze the qualitative behavior of the Einstein-Klein-Gordon system in the absence of a cosmological constant (Λ = 0).
• To investigate whether the cosmological model with a single massive scalar field can lead to a flat, stable, and non-relativistic late-time universe.
• To determine the long-term dynamics of the scale factor, Hubble parameter, and cosmological acceleration under Λ = 0.
• To explore the emergence of a non-relativistic equation of state in the late-time evolution of the universe.
• To assess the robustness of the results when the cosmological constant is small but non-zero (Λ ≲ 3×10⁻⁸)

Proposed method

• Reduced the Einstein-Klein-Gordon system to a dimensionless autonomous dynamical system in variables (Φ, Z = Φ̇) using Compton time scaling (τ = mt).
• Applied qualitative theory of ordinary differential equations to classify singular points and determine their stability (e.g., center, focus, node).
• Derived the invariant cosmological acceleration Ω(τ) = 1 + H′ₘ/H²ₘ and analyzed its oscillatory behavior via numerical integration.
• Performed numerical simulations over long timescales (τ ≈ 1000–10000) to track evolution of Φ, Z, Hₘ, a(τ), and Ω(τ).
• Defined the time-averaged cosmological acceleration over N periods (Δτ = N·2π) to extract long-term trends.
• Used phase plane analysis to classify behavior in the (Φ, Z) plane for various Λₘ values, including Λₘ = 0 and small Λₘ.

Experimental results

Research questions

• RQ1 Does the dynamical system of the standard cosmological model with Λ = 0 possess a stable center in phase space (Φ, Φ̇)?
• RQ2 What is the long-term behavior of the cosmological acceleration Ω(τ) in the Λ = 0 case, and does it exhibit oscillations with a non-zero average?
• RQ3 Can the model naturally produce a long-lasting non-relativistic era (w ≈ 0) without a cosmological constant?
• RQ4 How does the system behave when the cosmological constant is small but non-zero (Λₘ ≲ 3×10⁻⁸)?
• RQ5 Does the average cosmological acceleration Ω̄(τ) converge to a constant value in the late-time limit for small Λ?

Key findings

• For Λ = 0, the system has a single stable center at (Φ, Φ̇) = (0, 0), indicating periodic, bounded oscillations in phase space.
• The invariant cosmological acceleration Ω(τ) exhibits sustained oscillations with a period of approximately 2π in τ, ranging from −2 (braking) to +1 (acceleration).
• The time-averaged cosmological acceleration Ω̄(τ) converges to −1/2 over long intervals, corresponding to a non-relativistic equation of state (w = 0).
• For small Λₘ ≤ 3×10⁻⁸, the system's long-term behavior closely resembles the Λ = 0 case, with the average acceleration remaining near −1/2.
• The Hubble parameter Hₘ(τ) decreases rapidly over time, while the scale factor a(τ) grows super-exponentially, with ln a(τ) reaching values up to ∼9566 for τ ≈ 10⁴.
• When Λₘ > 4/3, the system transitions to a stable node at the origin, indicating asymptotic approach to a de Sitter-like state, but this is outside the physically relevant range for observed Λ.