Difference between revisions of "Bianchi I Model"
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Revision as of 00:44, 27 December 2013
(after Esra Russell, Can Battal Kılınç, Oktay K. Pashaev, Bianchi I Models: An Alternative Way To Model The Present-day Universe, arXiv:1312.3502)
Theoretical arguments and indications from recent observational data support the existence of an anisotropic phase that approaches an isotropic one. Therefore, it makes sense to consider models of a Universe with an initially anisotropic background. The anisotropic and homogeneous Bianchi models may provide adequate description of anisotropic phase in history of Universe. One particular type of such models is Bianchi type I (BI) homogeneous models whose spatial sections are flat, but the expansion rates are direction dependent,
\[ds^2={c^2}dt^2-a^{2}_{1}(t)dx^2-a^{2}_{2}(t)dy^2-a^{2}_{3}(t)dz^2\]
where $a_{1}$, $a_{2}$ and $a_{3}$ represent three different scale factors which are a function of time $t$.
Problem 1
problem id: bianchi_01
Find the field equations of the BI Universe.
If we admit the energy-momentum tensor of a perfect fluid, then the field equations of the BI universe are found as, \begin{eqnarray} \label{feforgm}\frac{{\dot{a}_{1}}{\dot{a}_{2}}}{a_{1} a_{2}}+\frac{{\dot{a}_{1}}{\dot{a}_{3}}}{a_{1} a_{3}}+\frac{{\dot{a}_{2}}{\dot{a}_{3}}}{a_{2} a_{3}}&=&\rho,\\ \frac{{\ddot{a}_{1}}}{a_{1}}+\frac{{\ddot{a}_{3}}}{a_{3}}+\frac{{\dot{a}_{1}}{\dot{a}_{3}}}{a_{1} a_{3}}&=& -p,\\ \frac{{\ddot{a}_{2}}}{a_{2}}+\frac{{\ddot{a}_{1}}}{a_{1}}+\frac{{\dot{a}_{2}}{\dot{a}_{1}}}{a_{2} a_{1}}&=&-p,\\ \frac{{\ddot{a}_{3}}}{a_{3}}+\frac{{\ddot{a}_{2}}}{a_{2}}+\frac{{\dot{a}_{3}}{\dot{a}_{2}}}{a_{3} a_{2}}&=&-p. \end{eqnarray}
Problem 2
problem id: bianchi_02
Find the partial energy densities for the two components of the BI Universe dominated by radiation and matter in terms of volume element $V_{rm}$.
Equations of state for the considered components read: \begin{eqnarray}\label{eosrm} {p}_{m}=0,\phantom{a}{p}_{r}=\frac{1}{3}{\rho}_{r}. \end{eqnarray} \noindent As a result, the energy conservation equations in the radiation-matter period are \begin{eqnarray} \dot{\rho}_{r}=-4 \bar H_{rm} {\rho}_{r},\phantom{a} \dot{\rho}_{m}=-3 \bar H_{rm}{\rho}_{m}, \label{energyconsermatradzero} \end{eqnarray} \noindent Using the definition \[\bar H=\frac13\frac{\dot V}{V}\] one obtains \begin{eqnarray} {\rho}_{r}=\rho_{r,0}\left(\frac{V_{rm,0}}{V_{rm}}\right)^{4/3},\phantom{a} {\rho}_{m}=\rho_{m,0}\frac{V_{rm,0}}{V_{rm}}. \label{energyconsermatrad} \end{eqnarray}
Problem 3
problem id: bianchi_03
Obtain time evolution equation for the total volume $V_{rm}$ in the BI Universe dominated by radiation and matter.
In the considered case of radiation+matter dominated BI Universe \[\bar H +3H^2=\frac12(\rho-p) \to \dot{\bar H}_{rm}+3{\bar H}^2_{rm}=\frac12\left(\frac{2}{3}\rho_{r}+\rho_{m}\right).\] Substitution of \[\bar H=\frac13\frac{\dot V_{rm}}{V_{rm}}\] gives \[{\ddot{V}_{rm}}-\frac32\left(\rho_{m,0}+\frac{2}{3}\frac{\rho_{r,0}}{V_{rm}^{1/3}}\right)=0.\] Multiplying this equation with $\dot{V}_{rm}$, integrate it in terms of time, and substitute the normalized densities \[{\rho}_{r,0}=3\bar H^2_{0}\Omega_{r,0},\quad{\rho}_{m,0}=3\bar H^2_{0}\Omega_{m,0},\] we then obtain \[{{\dot V}_{rm}}^2-9\bar H^2_{0}\Omega_{m,0} V_{rm} - 9\bar H^2_{0}\Omega_{r,0} V_{rm}^{2/3}=0.\]
Problem 4
problem id: bianchi_04
Using result of the previous problem, obtain a relation between the mean Hubble parameter and the volume element.
\[\left(\frac{\bar H_{rm}}{\bar H_0}\right)^2=\frac{\Omega_{m,0}}{V_{rm}}+\frac{\Omega_{r,0}}{V_{rm}^{4/3}}.\]
