[Paper Review] A Schrodinger formulation of Bianchi I scalar field cosmology

Jennie D’Ambroise|ArXiv.org|Nov 25, 2007

Cosmology and Gravitation Theories4 references3 citations

TL;DR

This paper establishes a direct correspondence between solutions of the Bianchi I scalar field cosmology equations and solutions of a linear Schrödinger equation, enabling exact solutions via quantum mechanical methods. By mapping the anisotropic Einstein field equations to a time-independent Schrödinger equation with a specific potential, the approach provides a novel, alternative method for solving complex cosmological models through quantum mechanical techniques.

ABSTRACT

We show that the Bianchi I Einstein field equations in a perfect fluid scalar field cosmology are equivalent to a linear Schrodinger equation. This is achieved through a special case of the recent FLRW Schrodinger-type formulation, and provides an alternate method of obtaining exact solutions of the Bianchi I equations.

Motivation & Objective

• To establish a direct mathematical correspondence between solutions of the Bianchi I Einstein field equations with a scalar field and solutions of a linear Schrödinger equation.
• To extend the previously known FLRW-Schrödinger correspondence to anisotropic cosmologies, specifically the Bianchi I model.
• To provide an alternative method for obtaining exact solutions of the Bianchi I equations by leveraging solvable quantum mechanical systems.
• To demonstrate the utility of the correspondence through explicit examples, including vacuum and non-vacuum solutions with constant and non-constant potentials.
• To validate the correspondence by showing that solutions derived from the Schrödinger equation satisfy the original Einstein and Klein-Gordon equations.

Proposed method

• The method begins by expressing the Bianchi I metric and scalar field equations in terms of scale factors $X(t), Y(t), Z(t)$ and scalar field $φ(t)$, with energy density and pressure derived from the scalar field Lagrangian.
• The Einstein equations are recast using the expansion rate $\theta$ and shear scalars $\eta_1, \eta_2, \eta_3$, leading to a form that matches a special case of the FLRW-Schrödinger correspondence from prior work.
• A key step is identifying that the combination $XYZ\eta_i$ being constant implies the existence of a time-reparametrization that allows the system to be mapped to a Schrödinger equation.
• The core transformation involves defining a new time variable $\sigma(t)$ such that $\dot{\sigma}(t) = u(\sigma(t))$, where $u(x)$ is a solution of the linear Schrödinger equation $u''(x) + [E - P(x)]u(x) = 0$.
• The scale factors $X(t), Y(t), Z(t)$ are reconstructed from $R(t) = u(\sigma(t))^{-1/3}$ and exponential terms involving constants $c_1, c_2$ satisfying $c_1^2 + c_1c_2 + c_2^2 = -4E/3$.
• The scalar field $\phi(t)$ and potential $V(\phi)$ are derived via $\phi(t) = \psi(\sigma(t))$ and $V = \frac{1}{3K^2}[(u')^2 + u^2(E - P)] \circ \psi^{-1}$, with $\psi'$ defined by $\psi'^2 = \frac{2}{3K^2}P(x)$.

Experimental results

Research questions

• RQ1 Can the Einstein field equations for a Bianchi I universe with a scalar field be reformulated as a linear Schrödinger equation?
• RQ2 What is the precise mathematical correspondence between solutions of the Bianchi I equations and solutions of a linear Schrödinger equation?
• RQ3 How can the known FLRW-Schrödinger correspondence be extended to anisotropic cosmologies like Bianchi I?
• RQ4 What class of exact solutions to the Bianchi I equations can be generated using solvable Schrödinger potentials?
• RQ5 Under what conditions does the inverse mapping from Schrödinger solutions back to physical cosmological solutions remain valid and well-defined?

Key findings

• The paper establishes a direct, invertible correspondence between solutions of the Bianchi I scalar field cosmology and solutions of a linear time-independent Schrödinger equation with a potential $P(x)$ and negative energy $E < 0$.
• For any solution $u(x)$ of the Schrödinger equation $u''(x) + [E - P(x)]u(x) = 0$, a corresponding solution $(X,Y,Z,\phi,V)$ of the Bianchi I equations can be explicitly constructed via the transformations in equations (3.8)–(3.11).
• When $P(x) = 0$, the Schrödinger solution $u(x) = Ae^{-\sqrt{-E}x}$ yields a vacuum Bianchi I solution with power-law scale factors, such as $X(t) \propto (\sqrt{-E}(t - c_0))^{c_1/(2\sqrt{-E}) + 1/3}$, and constant potential $V = 0$.
• For $P(x) = 2/x^2 + E^2x^2$, the solution $u(x) = (1/x)e^{Ex^2/2}$ leads to a non-trivial potential $V(\phi)$ that depends on the inverse of $\psi$, with explicit expressions for $X(t), Y(t), Z(t)$ involving logarithmic and power-law terms in $t - c_0$.
• The method successfully recovers known vacuum solutions: for constant $u(x) = u_0$, the inverse mapping yields $X,Y,Z$ with exponential time dependence, and $V = 0$, consistent with vacuum Bianchi I models.
• The correspondence is validated by showing that all derived solutions satisfy the original Einstein equations (i)–(iv), the Klein-Gordon equation for $\phi$, and the energy conditions via equations (3.12) and (3.13).