Bianchi I Model - Revision history

Cosmo All at 22:00, 18 June 2015

2015-06-18T22:00:52Z

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Cosmo All at 01:57, 23 April 2014

2014-04-23T01:57:55Z

Cosmo All at 01:56, 23 April 2014

2014-04-23T01:56:31Z

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Cosmo All: Created page with "8 NOTOC (after Esra Russell, Can Battal Kılınç, Oktay K. Pashaev, Bianchi I Models: An Alternative Way To Model The Present-day Universe, arX..."

2013-12-27T00:44:51Z

Created page with "8 __NOTOC__ (after Esra Russell, Can Battal Kılınç, Oktay K. Pashaev, Bianchi I Models: An Alternative Way To Model The Present-day Universe, arX..."

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[[Category:Dark Energy|8]]

__NOTOC__

(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 $t$.

<div id="bianchi_01"></div>
<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 1 ===
<p style= "color: #999;font-size: 11px">problem id: bianchi_01</p>
Find the field equations of the BI Universe.
<div class="NavFrame collapsed">
<div class="NavHead">solution</div>
<div style="width:100%;" class="NavContent">
<p style="text-align: left;">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}</p>
</div>
</div></div>

<div id="bianchi_02"></div>
<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 2 ===
<p style= "color: #999;font-size: 11px">problem id: bianchi_02</p>
Find the partial energy densities for the two components of the BI Universe dominated by radiation and matter in terms of volume element $V_{rm}$.
<div class="NavFrame collapsed">
<div class="NavHead">solution</div>
<div style="width:100%;" class="NavContent">
<p style="text-align: left;">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}</p>
</div>
</div></div>

<div id="bianchi_03"></div>
<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 3 ===
<p style= "color: #999;font-size: 11px">problem id: bianchi_03</p>
Obtain time evolution equation for the total volume $V_{rm}$ in the BI Universe dominated by radiation and matter.
<div class="NavFrame collapsed">
<div class="NavHead">solution</div>
<div style="width:100%;" class="NavContent">
<p style="text-align: left;">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
\[{{\dot V}_{rm}}^2-9\bar H^2_{0}\Omega_{m,0} V_{rm} - 9\bar H^2_{0}\Omega_{r,0} V_{rm}^{2/3}=0.\]</p>
</div>
</div></div>

<div id="bianchi_04"></div>
<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 4 ===
<p style= "color: #999;font-size: 11px">problem id: bianchi_04</p>
Using result of the previous problem, obtain a relation between the mean Hubble parameter and the volume element.
<div class="NavFrame collapsed">
<div class="NavHead">solution</div>
<div style="width:100%;" class="NavContent">
<p style="text-align: left;">\[\left(\frac{\bar H_{rm}}{\bar H_0}\right)^2=\frac{\Omega_{m,0}}{V_{rm}}+\frac{\Omega_{r,0}}{V_{rm}^{4/3}}.\]</p>
</div>
</div></div>

← Older revision		Revision as of 01:57, 23 April 2014
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	</div>		</div>
	</div></div>		</div></div>
		+
		+	==Radiation dominated BI model ==
		+
		+
		+	<div id=""></div>
		+	<div style="border: 1px solid #AAA; padding:5px;">
		+	=== Problem 10 ===
		+	<p style= "color: #999;font-size: 11px">problem id: </p>
		+	Find the energy density of the radiation dominated BI Universe in terms of volume element $V_r$.
		+	<div class="NavFrame collapsed">
		+	<div class="NavHead">solution</div>
		+	<div style="width:100%;" class="NavContent">
		+	<p style="text-align: left;">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.</p>
		+	</div>
		+	</div></div>
		+
		+
		+	<div id=""></div>
		+	<div style="border: 1px solid #AAA; padding:5px;">
		+	=== Problem 11 ===
		+	<p style= "color: #999;font-size: 11px">problem id: </p>
		+	Find the mean Hubble parameter of the radiation dominated case.
		+	<div class="NavFrame collapsed">
		+	<div class="NavHead">solution</div>
		+	<div style="width:100%;" class="NavContent">
		+	<p style="text-align: left;">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
		+	\[\bar H=\frac13\frac{\dot V_r}{V}=\frac1{2t}.\]</p>
		+	</div>
		+	</div></div>
		+
		+
		+	<div id=""></div>
		+	<div style="border: 1px solid #AAA; padding:5px;">
		+	=== Problem 12 ===
		+	<p style= "color: #999;font-size: 11px">problem id: </p>
		+	Find the directional expansion rates of the radiation dominated model.
		+	<div class="NavFrame collapsed">
		+	<div class="NavHead">solution</div>
		+	<div style="width:100%;" class="NavContent">
		+	<p style="text-align: left;">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
		+	\[H_{r,i}t_0=\alpha_{r,i}\left(\frac{t_0}{t}\right)^{3/2}+\frac12\frac{t_0}{t};\quad \alpha_{r,i}\equiv\frac{K_{r,i}}{t_0}.\]</p>
		+	</div>
		+	</div></div>
		+
		+
		+	<div id=""></div>
		+	<div style="border: 1px solid #AAA; padding:5px;">
		+	=== Problem 13 ===
		+	<p style= "color: #999;font-size: 11px">problem id: </p>
		+	Find time dependence for the scale factors $a_i$ in the radiation dominated BI Universe.
		+	<div class="NavFrame collapsed">
		+	<div class="NavHead">solution</div>
		+	<div style="width:100%;" class="NavContent">
		+	<p style="text-align: left;">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.</p>
		+	</div>
		+	</div></div>
		+
		+
		+	<div id="bianchi_02"></div>
		+	<div style="border: 1px solid #AAA; padding:5px;">
		+	=== Problem 14 ===
		+	<p style= "color: #999;font-size: 11px">problem id: bianchi_02</p>
		+	Find the partial energy densities for the two components of the BI Universe dominated by radiation and matter in terms of volume element $V_{rm}$.
		+	<div class="NavFrame collapsed">
		+	<div class="NavHead">solution</div>
		+	<div style="width:100%;" class="NavContent">
		+	<p style="text-align: left;">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}</p>
		+	</div>
		+	</div></div>
		+
		+
		+	<div id="bianchi_03"></div>
		+	<div style="border: 1px solid #AAA; padding:5px;">
		+	=== Problem 15 ===
		+	<p style= "color: #999;font-size: 11px">problem id: bianchi_03</p>
		+	Obtain time evolution equation for the total volume $V_{rm}$ in the BI Universe dominated by radiation and matter.
		+	<div class="NavFrame collapsed">
		+	<div class="NavHead">solution</div>
		+	<div style="width:100%;" class="NavContent">
		+	<p style="text-align: left;">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
		+	\[{{\dot V}_{rm}}^2-9\bar H^2_{0}\Omega_{m,0} V_{rm} - 9\bar H^2_{0}\Omega_{r,0} V_{rm}^{2/3}=0.\]</p>
		+	</div>
		+	</div></div>
		+
		+
		+	<div id="bianchi_04"></div>
		+	<div style="border: 1px solid #AAA; padding:5px;">
		+	=== Problem 16 ===
		+	<p style= "color: #999;font-size: 11px">problem id: bianchi_04</p>
		+	Using result of the previous problem, obtain a relation between the mean Hubble parameter and the volume element.
		+	<div class="NavFrame collapsed">
		+	<div class="NavHead">solution</div>
		+	<div style="width:100%;" class="NavContent">
		+	<p style="text-align: left;">\[\left(\frac{\bar H_{rm}}{\bar H_0}\right)^2=\frac{\Omega_{m,0}}{V_{rm}}+\frac{\Omega_{r,0}}{V_{rm}^{4/3}}.\]</p>
		+	</div>
		+	</div></div>
		+
		+
		+	----

Bianchi I Model - Revision history

Cosmo All at 22:00, 18 June 2015

Cosmo All at 01:57, 23 April 2014

Cosmo All at 01:56, 23 April 2014

Cosmo All: Created page with "8 __NOTOC__ (after Esra Russell, Can Battal Kılınç, Oktay K. Pashaev, Bianchi I Models: An Alternative Way To Model The Present-day Universe, arX..."

Cosmo All: Created page with "8 NOTOC (after Esra Russell, Can Battal Kılınç, Oktay K. Pashaev, Bianchi I Models: An Alternative Way To Model The Present-day Universe, arX..."