Difference between revisions of "The role of curvature in the dynamics of the Universe"

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[[Category:Dynamics of the Universe in the Big Bang Model]]
 
[[Category:Dynamics of the Universe in the Big Bang Model]]
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__NOTOC__
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<div id="dyn38"></div>
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<div style="border: 1px solid #AAA; padding:5px;">
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=== Problem 1. ===
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Derive $\rho(t)$ in a spatially open Universe filled with dust for the epoch when the curvature term in the first Friedman equation is dominating.
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<div class="NavFrame collapsed">
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  <div class="NavHead">solution</div>
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  <div style="width:100%;" class="NavContent">
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    <p style="text-align: left;"></p>
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  </div>
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</div>
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</div>
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<div id="dyn3"></div>
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<div style="border: 1px solid #AAA; padding:5px;">
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=== Problem 2. ===
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Show that in the early Universe the curvature term is negligibly small.
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<div class="NavFrame collapsed">
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  <div class="NavHead">solution</div>
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  <div style="width:100%;" class="NavContent">
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    <p style="text-align: left;"></p>
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  </div>
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</div>
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</div>
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<div id="dyn33"></div>
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<div style="border: 1px solid #AAA; padding:5px;">
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=== Problem 3. ===
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Show that $k =\text{sign}(\Omega-1)$ and express the current value of the scale factor $a_{0}$ through the observed quantities $\Omega_{0}$ and $H_{0}$.
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<div class="NavFrame collapsed">
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  <div class="NavHead">solution</div>
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  <div style="width:100%;" class="NavContent">
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    <p style="text-align: left;"></p>
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  </div>
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</div>
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</div>
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<div id="dyn_curv4"></div>
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<div style="border: 1px solid #AAA; padding:5px;">
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=== Problem 4. ===
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Find the lower bound for $a_{0}$, knowing that the Cosmic background (CMB) data combined with SSNIa data imply
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\[-0.0178<(1-\Omega)<0.0063.\]
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<div class="NavFrame collapsed">
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  <div class="NavHead">solution</div>
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  <div style="width:100%;" class="NavContent">
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    <p style="text-align: left;"></p>
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  </div>
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</div>
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</div>
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<div id="dyn43"></div>
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<div style="border: 1px solid #AAA; padding:5px;">
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=== Problem 5. ===
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Fnd the time dependence of $\left|\Omega-1\right|$ in a Universe with domination of
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\begin{description}
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    \item[a)] radiation,
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    \item[b)]  matter.
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\end{description}
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<div class="NavFrame collapsed">
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  <div class="NavHead">solution</div>
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  <div style="width:100%;" class="NavContent">
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    <p style="text-align: left;"></p>
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  </div>
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</div>
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</div>
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<div id="dyn4"></div>
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<div style="border: 1px solid #AAA; padding:5px;">
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=== Problem 6. ===
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Estimate the upper bound of the curvature term in the first Friedman equation during the electroweak epoch ($t\sim 10^{-10}$~s) and the nucleosynthesis epoch ($t\sim 1-200$~s).
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<div class="NavFrame collapsed">
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  <div class="NavHead">solution</div>
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  <div style="width:100%;" class="NavContent">
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    <p style="text-align: left;"></p>
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  </div>
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</div>
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</div>
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<div id="dyn-Kumar1"></div>
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<div style="border: 1px solid #AAA; padding:5px;">
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=== Problem 7. ===
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Derive and analyze the conditions of accelerated expansion for a one-component Universe of arbitrary curvature with the component's state parameter [http://arxiv.org/abs/1109.6924 S.Kumar arXiv: 1109.6924] $w$.
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<div class="NavFrame collapsed">
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  <div class="NavHead">solution</div>
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  <div style="width:100%;" class="NavContent">
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    <p style="text-align: left;"></p>
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  </div>
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</div>
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</div>

Revision as of 19:57, 19 July 2012


Problem 1.

Derive $\rho(t)$ in a spatially open Universe filled with dust for the epoch when the curvature term in the first Friedman equation is dominating.


Problem 2.

Show that in the early Universe the curvature term is negligibly small.


Problem 3.

Show that $k =\text{sign}(\Omega-1)$ and express the current value of the scale factor $a_{0}$ through the observed quantities $\Omega_{0}$ and $H_{0}$.


Problem 4.

Find the lower bound for $a_{0}$, knowing that the Cosmic background (CMB) data combined with SSNIa data imply \[-0.0178<(1-\Omega)<0.0063.\]


Problem 5.

Fnd the time dependence of $\left|\Omega-1\right|$ in a Universe with domination of \begin{description} \item[a)] radiation, \item[b)] matter. \end{description}


Problem 6.

Estimate the upper bound of the curvature term in the first Friedman equation during the electroweak epoch ($t\sim 10^{-10}$~s) and the nucleosynthesis epoch ($t\sim 1-200$~s).


Problem 7.

Derive and analyze the conditions of accelerated expansion for a one-component Universe of arbitrary curvature with the component's state parameter S.Kumar arXiv: 1109.6924 $w$.

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