[Paper Review] Bianchi type-VI model with cosmic strings in the presence of a magnetic field

Bijan Saha, Mihai Visinescu|arXiv (Cornell University)|May 27, 2010

Cosmology and Gravitation Theories2 references4 citations

TL;DR

This paper proposes a Bianchi type-VI cosmological model incorporating cosmic strings and a magnetic field, using a specific spacetime metric with spatial anisotropy. By assuming a z-dependent magnetic field and simplified string parametrization, exact analytical solutions are derived, showing asymptotic isotropization and stable expansion under certain conditions, offering insights into early-universe anisotropy and magnetic field evolution.

ABSTRACT

A Bianchi type-VI cosmological model in the presence of a magnetic flux together with a cloud of cosmic strings is considered. In general, the presence of a magnetic field imposes severe restrictions regarding the consistency of the field equations. These difficulties could be overtaken working either in a Bianchi type-VI$_0$ spacetime or assuming a particular coordinate-dependence of the magnetic field. Using a few plausible assumptions regarding the parametrization of the cosmic strings, some exact analytical solutions are presented. Their asymptotic behavior for large time is exhibited.

Motivation & Objective

• To develop a consistent cosmological model incorporating cosmic strings and a magnetic field within the Bianchi type-VI spacetime framework.
• To address the challenge of field equation consistency in anisotropic spacetimes with electromagnetic and string sources.
• To explore the dynamical behavior of the universe under combined effects of cosmic strings and a magnetic field using exact solutions.
• To analyze the asymptotic evolution of the scale factors and assess isotropization tendencies in the long-time limit.

Proposed method

• Adopts a Bianchi type-VI metric with time-dependent scale factors $a_1(t), a_2(t), a_3(t)$ and spatial dependence via $e^{-2mz}, e^{2nz}$, modeling anisotropic expansion.
• Uses an energy-momentum tensor combining contributions from cosmic strings ($ ho = ho_p + ar{ ho}$), four-velocity $u^ u$, and string direction $x^ u$, with $u_ u x^ u = 0$.
• Models the magnetic field via the dual electromagnetic tensor $F_{ ueta}$ with only $F_{12} eq 0$, and defines the magnetic flux vector $h_ u$ with $h_3 eq 0$ along the z-axis.
• Imposes infinite conductivity ($F_{0i} = 0$) and comoving coordinates ($u^0 = 1, u^i = 0$) to simplify field equations.
• Derives the Einstein field equations from the Ricci tensor and scalar, using Christoffel symbols and Riemann tensor components specific to the BVI metric.
• Applies simplifying assumptions on string parametrization and magnetic field dependence to obtain exact analytical solutions of the field equations.

Experimental results

Research questions

• RQ1 How can a consistent cosmological model be constructed for Bianchi type-VI spacetime with both cosmic strings and a magnetic field?
• RQ2 What are the dynamical behaviors of the scale factors $a_1(t), a_2(t), a_3(t)$ under the combined influence of cosmic strings and a magnetic field?
• RQ3 Does the model exhibit asymptotic isotropization as $t \to \infty$, and under what conditions?
• RQ4 How does the spatial dependence of the magnetic field ($z$-dependence) affect the consistency and solvability of the field equations?
• RQ5 What role does the tension density $ar{ ho}$ of cosmic strings play in the long-term evolution of the universe in this anisotropic model?

Key findings

• Exact analytical solutions are obtained for the scale factors $a_1(t), a_2(t), a_3(t)$ under specific assumptions on string parametrization and magnetic field dependence.
• The model exhibits asymptotic isotropization: the expansion rates $rac{ ilde{a}_1}{a_1}, rac{ ilde{a}_2}{a_2}, rac{ ilde{a}_3}{a_3}$ approach equality as $t \to \infty$, indicating a tendency toward isotropy.
• The magnetic field's spatial dependence via $e^{-2mz}, e^{2nz}$ is essential for maintaining consistency in the field equations, particularly in the $T_3^0$ component.
• The shear tensor components $ ilde{ ho}_1, ilde{ ho}_2, ilde{ ho}_3$ decay over time, confirming the system evolves toward a more isotropic state.
• The Ricci scalar and field equations are consistent under the chosen ansatz, validating the model’s physical plausibility.
• The expansion scalar $ heta = rac{ ilde{a}_1}{a_1} + rac{ ilde{a}_2}{a_2} + rac{ ilde{a}_3}{a_3}$ evolves smoothly, with no singularities in the derived solutions.