Bianchi I Model
(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: bi_2
Reformulate the field equations of the BI Universe in terms of the directional Hubble parameters. \[H_1\equiv\frac{\dot{a_1}}{a_1},\ H_2\equiv\frac{\dot{a_2}}{a_2},\ H_3\equiv\frac{\dot{a_3}}{a_3}.\]
Inserting the directional Hubble parameters and their time derivatives \[\dot H_1=\frac{\ddot a_1}{a_1}-\left(\frac{\dot a_1}{a_1}\right)^2,\ \dot H_2=\frac{\ddot a_2}{a_2}-\left(\frac{\dot a_2}{a_2}\right)^2,\ \dot H_3=\frac{\ddot a_3}{a_3}-\left(\frac{\dot a_3}{a_3}\right)^2\] into the modified Friedmann equations we obtain \begin{align} \nonumber H_1H_2+H_1H_3+H_2H_3 & =\rho,\\ \nonumber \dot H_1+ H_1^2 +\dot H_3+ H_3^2 +H_1H_3& =-p,\\ \nonumber \dot H_1+ H_1^2 +\dot H_2+ H_2^2 +H_1H_2& =-p,\\ \nonumber \dot H_2+ H_2^2 +\dot H_3+ H_3^2 +H_2H_3& =-p. \end{align}
Problem 3
problem id:
The BI Universe has a flat metric, which implies that its total density is equal to the critical density. Find the critical density.
\[\rho_{cr}=H_1H_2+H_1H_3+H_2H_3.\]
Problem 4
problem id:
Obtain an analogue of the conservation equation $\dot\rho+3H(\rho+p)=0$ for the case of the BI Universe.
The energy conservation equation $T^\mu_{\nu;\mu}=0$ yields \[\dot\rho+3\bar H(\rho+p)=0,\quad \bar H\equiv \frac13(H_1+H_2+H_3)=\frac13\left(\frac{\dot a_1}{a_1} +\frac{\dot a_2}{a_2} +\frac{\dot a_3}{a_3}\right),\] where $\bar H$ represents the mean of the three directional Hubble parameters in the BI Universe.
Problem 5
problem id:
Obtain the evolution equation for the mean of the three directional Hubble parameters $\bar H$.
Adding the three latter Friedmann equations (see Problem \ref{bi_2}) one obtains \begin{equation}\label{bi_5_1} 2\frac{d}{dt}\sum\limits_{i=1}^3 H_i+2(H_1^2+H_2^2+H_3^2)+H_1H_2+H_1H_3+H_2H_3=-3p. \end{equation} where $\bar H$ represents the mean of the three directional Hubble parameters in the BI Universe. Substituting \[\sum\limits_{i=1}^3 H_i^2=\left(\sum\limits_{i=1}^3 H_i\right)^2-2(H_1H_2+H_1H_3+H_2H_3)\] and \[H_1H_2+H_1H_3+H_2H_3=\rho\] into equation (\ref{bi_5_1}), we then obtain \[\frac{d}{dt}\sum\limits_{i=1}^3 H_i+\left(\sum\limits_{i=1}^3 H_i\right)^2=\frac32(\rho-p).\] Using the mean of the three directional Hubble parameters $\bar H$ we obtain a nonlinear first order differential equation \[\dot{\bar H}+3\bar H^2=\frac12(\rho-p).\]
Problem 6
problem id:
Show that the system of equations for the BI Universe \begin{align} \nonumber H_1H_2+H_1H_3+H_2H_3 & =\rho,\\ \nonumber \dot H_1+ H_1^2 +\dot H_3+ H_3^2 +H_1H_3& =-p,\\ \nonumber \dot H_1+ H_1^2 +\dot H_2+ H_2^2 +H_1H_2& =-p,\\ \nonumber \dot H_2+ H_2^2 +\dot H_3+ H_3^2 +H_2H_3& =-p, \end{align} can be transformed to the following \begin{align} \nonumber H_1H_2+H_1H_3+H_2H_3 & =\rho,\\ \nonumber \dot H_1+ 3H_1\bar H & =\frac12(\rho-p),\\ \nonumber \dot H_2+ 3H_2\bar H & =\frac12(\rho-p),\\ \nonumber \dot H_3+ 3H_3\bar H & =\frac12(\rho-p). \end{align}
Problem 7
problem id:
Show that the mean of the three directional Hubble parameters $\bar H$ is related to the elementary volume of the BI Universe $V\equiv a_1a_2a_3$ as \[\bar H=\frac13\frac{\dot V}{V}.\]
\[\bar H=\frac13\frac{d}{dt}\ln(a_1a_2a_3)=\frac13\frac{d}{dt}\ln V=\frac13\frac{\dot V}{V}.\]
Problem 8
problem id:
Obtain the volume evolution equation of the BI model.
Using the relation between volume $V$ and the mean Hubble parameter $\bar H$, obtained in the previous problem, one finds \[\dot{\bar H}=\frac13\frac{\ddot V}{V}-3\bar H^2.\] As \[\dot{\bar H}+3\bar H^2=\frac12(\rho-p),\] we obtain \[\ddot V-\frac32(\rho-p)V=0.\]
Problem 9
problem id:
Find the generic solution of the directional Hubble parameters.
The equations \begin{align} \nonumber \dot H_1+ 3H_1\bar H & =\frac12(\rho-p),\\ \nonumber \dot H_2+ 3H_2\bar H & =\frac12(\rho-p),\\ \nonumber \dot H_3+ 3H_3\bar H & =\frac12(\rho-p), \end{align} allow us to write the generic solution of the directional Hubble parameters, \[H_i(t)=\frac1{\mu(t)}\left[K_i+\frac12\int\mu(t)(\rho(t)-p(t))dt\right],\quad i=1,2,3,\] where $K_i$s are the integration constants. The integration factor $\mu$ is defined as, \[\mu(t)=\exp\left(3\int\bar H(t)dt\right).\] As can be seen, the initial values (integration constants) determine the solution of each directional Hubble parameter. These values are the origin of the anisotropy. Note that the generic solution of the directional Hubble parameters is incomplete. To obtain exact solutions of the Hubble parameters and therefore the Einstein equations, one has to know the state equation for the component which fills the Universe.
Radiation dominated BI model
Problem 10
problem id:
Find the energy density of the radiation dominated BI Universe in terms of volume element $V_r$.
By using the energy conservation equation \[\dot\rho+3\bar H(\rho+p)=0\to\dot\rho+4\bar H\rho=0,\] and the volume representation of the mean Hubble parameter \[\bar H=\frac13\frac{d}{dt}\ln(a_1a_2a_3)=\frac13\frac{d}{dt}\ln V=\frac13\frac{\dot V}{V}.\] we obtain (with $\rho\to\rho_r$, $V\to V_r$): \[\rho_r=\rho_{r0}\left(\frac{V_{r0}}{V_r}\right)^{4/3}.\] Here the density and the volume element is normalized to the present time $t_0$. The parameters $\rho_{r0}$ and $V_{r0}$ are the normalized density and normalized volume elements.
Problem 11
problem id:
Find the mean Hubble parameter of the radiation dominated case.
For the radiation dominated case \[\ddot V_r-\frac32(\rho-p)V_r=0\to\ddot V_r-V_r\rho_r=0.\] (see problem 8). Using \[\rho_r=\rho_{r0}\left(\frac{V_{r0}}{V_r}\right)^{4/3}\] we obtain (for $V_{r0}=1$) \[\ddot V_r-\rho_{r0}V_r^{-1/3}=0.\] Multiplying this equation with the $\dot V_r$ and integrating it, yields, \[\dot V_r^2-3\rho_{r0}V_r^{2/3}=0.\] Hence, the exact solution of the volume evolution equation is \[V_r=(2H_0t)^{3/2}.\] The mean Hubble parameter of the radiation dominated case is \[\bar H=\frac13\frac{\dot V_r}{V}=\frac1{2t}.\]
Problem 12
problem id:
Find the directional expansion rates of the radiation dominated model.
The generic solution of the directional Hubble parameters (see problem 9) is \[H_i(t)=\frac1{\mu(t)}\left[K_i+\frac12\int\mu(t)(\rho(t)-p(t))dt\right],\quad i=1,2,3,\] Using the expression for the mean Hubble parameter obtained in the previous problem, one finds \[\mu_r(t)=\exp(3\int\bar H(t)dt)\] By direct substitution of the integration factor $\mu_r$ and the equation of state $p_r=\rho_r/3$ of the radiation dominated case we obtain for the directional Hubble parameters that are normalized to the present-day time $t_0$ the following results \[H_{r,i}t_0=\alpha_{r,i}\left(\frac{t_0}{t}\right)^{3/2}+\frac12\frac{t_0}{t};\quad \alpha_{r,i}\equiv\frac{K_{r,i}}{t_0}.\]
Problem 13
problem id:
Find time dependence for the scale factors $a_i$ in the radiation dominated BI Universe.
The normalized scale factors $a_i$ can be obtained from the directional Hubble parameters \[H_{r,i}=\alpha_{r,i}\frac1{t_0}\left(\frac{t_0}{t}\right)^{3/2}+\frac12\frac{1}{t},\] with a direct integration in terms of cosmic time, \[a_{r,i}=\exp\left[-2\alpha_{r,i}\left(\sqrt{\frac{t_0}{t}}-1\right)\right]\left(\frac{t_0}{t}\right)^{1/2}.\] The scale factors of the BI radiation dominated model has the contribution from anisotropic expansion/contraction \[\exp\left[-2\alpha_{r,i}\left(\sqrt{\frac{t_0}{t}}-1\right)\right]\] as well as the standard matter dominated FLRW contribution $(t/t_0)^{1/2}$. These two different dynamical behaviors in three directional scale factors of the BI universe indicate that the FLRW part of the scale factor becomes dominant when time starts reaching the present-day. On the other hand, in the early times of the BI model, the expansion is completely dominated by the anisotropic part.
Problem 14
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 15
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 16
problem id: bianchi_04
Using result of the previous problem, obtain a relation between the mean Hubble parameter and the volume element.
\[\left(\frac{\bar H_{rm}}{\bar H_0}\right)^2=\frac{\Omega_{m,0}}{V_{rm}}+\frac{\Omega_{r,0}}{V_{rm}^{4/3}}.\]
