[Paper Review] Quantum and classical cosmology with Born-Infeld scalar field

H. Q. Lu, Wei Fang+1 more|arXiv (Cornell University)|Apr 5, 2005

Cosmology and Gravitation Theories1 citations

TL;DR

This paper proposes a quantum cosmological model coupling gravity with a Born-Infeld scalar field, solving the Wheeler-DeWitt equation analytically for extreme scale factors. It shows that both Vilenkin's and Hartle-Hawking wave functions predict a positive cosmological constant—specifically, $\frac{1}{\lambda}$—suggesting the Born-Infeld scalar field as a source of dark energy driving accelerated expansion with $-1 < w < -\frac{1}{3}$, and further explores phantom-like behavior violating energy conditions.

ABSTRACT

In this paper, we consider a quantum model of gravitation interacting with a Born-Infeld(B-I) type scalar field $\phi$. The corresponding Wheeler-Dewitt equation can be solved analytically for both very large and small cosmological scale factor. In the condition that small cosmological scale factor tend to limit, the wave function of the universe can be obtained by applying the methods developed by Vilenkin, Hartle and Hawking. Both Vilenkin's and Hartle-Hawking's wave function predicts nonzero cosmological constant. The Vilenkin's wave function predicts a universe with a cosmological constant as large as possible, while the Hartle-Hawking's wave function predicts a universe with positive cosmological constant, which equals to $\frac{1}{\lambda}$. It is different from Coleman's result that cosmological constant is zero, and also different from Hawking's prediction of zero cosmological constant in quantum cosmology with linear scalar field. We suggest that dark energy in the universe might result from the B-I type scalar field with potential and the universe can undergo a phase of accelerating expansion. The equation of state parameter lies in the range of $-1<$w$<-{1/3}$. When the potential $V(\phi)=\frac{1}{\lambda}$, our Lagrangian describes the Chaplygin gas. In order to give a explanation to the observational results of state parameter w$<-1$, we also investigate the phantom model that posses negative kinetic energy. We find that weak and strong energy conditions are violated for phantom B-I type scalar field. At last, we study a specific potential with the form $V_0(1+\frac{\phi}{\phi_0})e^{-(\frac{\phi}{\phi_0})}$ in phantom B-I scalar field in detail. The attractor property of the system is shown by numerical analysis.

Motivation & Objective

• To develop a quantum cosmological model incorporating a Born-Infeld scalar field to explore the origin of dark energy.
• To analyze the wave function of the universe using Vilenkin and Hartle-Hawking boundary conditions in the context of Born-Infeld gravity.
• To investigate whether the Born-Infeld scalar field can generate a positive cosmological constant consistent with observational data.
• To examine the possibility of phantom behavior with negative kinetic energy and its implications for energy conditions and cosmic acceleration.
• To study the attractor dynamics of a specific non-linear potential $V_0(1 + \frac{\phi}{\phi_0})e^{-(\frac{\phi}{\phi_0})}$ in the phantom regime.

Proposed method

• Analytical solution of the Wheeler-DeWitt equation for both small and large scale factor limits in a Born-Infeld scalar field coupled to gravity.
• Application of Vilenkin's tunneling and Hartle-Hawking's no-boundary wave function approaches to derive quantum states of the universe.
• Derivation of the effective cosmological constant as $\frac{1}{\lambda}$ from the Hartle-Hawking wave function, differing from Hawking's zero prediction with linear fields.
• Construction of a phantom model with negative kinetic energy to explore $w < -1$ and its cosmological implications.
• Numerical analysis of the dynamical system governed by the potential $V_0(1 + \frac{\phi}{\phi_0})e^{-(\frac{\phi}{\phi_0})}$ to assess attractor behavior.
• Evaluation of energy conditions (weak and strong) under the phantom Born-Infeld scalar field to assess physical viability.

Experimental results

Research questions

• RQ1 Can the Hartle-Hawking and Vilenkin wave functions in a Born-Infeld scalar field model predict a positive cosmological constant?
• RQ2 Does the Born-Infeld scalar field with a specific potential reproduce the equation of state parameter $w$ in the range $-1 < w < -\frac{1}{3}$, consistent with dark energy?
• RQ3 How does the inclusion of negative kinetic energy (phantom behavior) affect energy conditions and the dynamics of the scalar field in quantum cosmology?
• RQ4 What is the role of the potential $V_0(1 + \frac{\phi}{\phi_0})e^{-(\frac{\phi}{\phi_0})}$ in stabilizing the system and inducing attractor behavior?
• RQ5 Can the Born-Infeld scalar field model explain the observed accelerating expansion of the universe without introducing a cosmological constant a priori?

Key findings

• The Hartle-Hawking wave function predicts a positive cosmological constant equal to $\frac{1}{\lambda}$, differing from Hawking's zero prediction in linear scalar field models.
• The Vilenkin wave function predicts a cosmological constant as large as possible, indicating a strong quantum preference for large vacuum energy.
• The equation of state parameter $w$ lies in the range $-1 < w < -\frac{1}{3}$, consistent with dark energy and accelerated expansion.
• When $V(\phi) = \frac{1}{\lambda}$, the model reduces to the Chaplygin gas, linking it to known dark energy parametrizations.
• The phantom model with negative kinetic energy violates both weak and strong energy conditions, indicating non-standard dynamics.
• Numerical analysis confirms attractor behavior for the potential $V_0(1 + \frac{\phi}{\phi_0})e^{-(\frac{\phi}{\phi_0})}$, suggesting long-term stability in the phantom regime.