Difference between revisions of "Solutions of Friedman equations in the Big Bang model"

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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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<div id="dyn5"></div>
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=== Problem 1. ===
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Derive $\rho(a)$, $\rho(t)$ and $ a(t)$ for a spatially flat\footnote{The ``spatial'' part is often omitted when there cannot be any confusion (and even when there can be), generally and hereatfer.} ($k=0$) Universe \label{r_m} that consists of only
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\begin{description}
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    \item[a)] radiation,
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    \item[b)] non-relativistic matter\footnote{In this context and below also quite often called just ``matter'' or dust.}.
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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 id="dyn6"></div>
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=== Problem 1. ===
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Consider two spatially flat Universes. One is filled with radiation, the other with dust. The current energy density is the same. Compare energy densities when both of them are twice as old.
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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 id="dyn11"></div>
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=== Problem 1. ===
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Find the scale factor and density of each component as functions of time in a flat Universe which consists of dust and radiation, for the case one of the components is dominating. Present the results graphically.
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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 id="dyn17"></div>
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=== Problem 1. ===
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Derive the time dependence of the Hubble parameter for a flat Universe in which either matter or radiation 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 id="dyn12"></div>
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=== Problem 1. ===
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Derive the exact solutions of the Friedman equations for the Universe filled with matter and radiation. Plot the graphs of scale factor and both densities as functions of time.
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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 id="dyn20"></div>
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=== Problem 1. ===
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At what moment after the Big Bang did matter's density exceed that of radiation for the first time?
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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 id="dyn21"></div>
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=== Problem 1. ===
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Determine the age of the Universe in which either matter or radiation has always been 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 id="dyn14"></div>
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=== Problem 1. ===
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Derive the dependence  $a(t)$ for a spatially flat Universe that consists of matter with equation of state $p=w\rho$, assuming that the parameter $w$ does not change throughout the evolution.
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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 id="dyn18"></div>
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=== Problem 1. ===
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Find the Hubble parameter as function of time for the previous problem.
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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 id="dyn15"></div>
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=== Problem 1. ===
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Using the first Friedman equation, construct the effective potential \label{V(a)} $V(a)$, which governs the one-dimensional motion of a fictitious particle with coordinate $a(t)$, for the Universe filled with several non-interacting components.
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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 id="dyn16"></div>
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=== Problem 1. ===
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Construct the effective one-dimensional potential $V(a)$ (using the notation of the previous problem) for the Universe consisting of non-relativistic matter and radiation. Show that motion with $\dot{a}>0$ in such a potential can only be decelerating.
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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 id="dyn39"></div>
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=== Problem 1. ===
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Derive the exact solutions of the Friedman equations for the Universe with arbitrary curvature, filled with radiation and matter.
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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 id="dyn41"></div>
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=== Problem 1. ===
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Show that for a spatially flat Universe consisting of one component with equation of state $p = w\rho$ the deceleration parameter $q\equiv -\ddot{a}/(aH^2)$ is equal to $\frac{1}{2}(1 + 3w)$.
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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 id="dyn21_1"></div>
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=== Problem 1. ===
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Express the age of the Universe through the deceleration parameter $q=-\ddot{a}/(aH^2)$ for a spatially flat Universe filled with single component with equation of state $p=w\rho$.
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<div class="NavFrame collapsed">
 +
  <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 id="dyn41nn"></div>
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=== Problem 1. ===
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Find the generalization of relation $q = \frac{1}{2}(1 + 3w)$ for the non-flat case.
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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>

Revision as of 08:44, 1 July 2012


Problem 1.

Derive $\rho(a)$, $\rho(t)$ and $ a(t)$ for a spatially flat\footnote{The ``spatial part is often omitted when there cannot be any confusion (and even when there can be), generally and hereatfer.} ($k=0$) Universe \label{r_m} that consists of only \begin{description} \item[a)] radiation, \item[b)] non-relativistic matter\footnote{In this context and below also quite often called just ``matter'' or dust.}. \end{description}

Problem 1.

Consider two spatially flat Universes. One is filled with radiation, the other with dust. The current energy density is the same. Compare energy densities when both of them are twice as old.

Problem 1.

Find the scale factor and density of each component as functions of time in a flat Universe which consists of dust and radiation, for the case one of the components is dominating. Present the results graphically.

Problem 1.

Derive the time dependence of the Hubble parameter for a flat Universe in which either matter or radiation is dominating.

Problem 1.

Derive the exact solutions of the Friedman equations for the Universe filled with matter and radiation. Plot the graphs of scale factor and both densities as functions of time.

Problem 1.

At what moment after the Big Bang did matter's density exceed that of radiation for the first time?

Problem 1.

Determine the age of the Universe in which either matter or radiation has always been dominating.

Problem 1.

Derive the dependence $a(t)$ for a spatially flat Universe that consists of matter with equation of state $p=w\rho$, assuming that the parameter $w$ does not change throughout the evolution.

Problem 1.

Find the Hubble parameter as function of time for the previous problem.

Problem 1.

Using the first Friedman equation, construct the effective potential \label{V(a)} $V(a)$, which governs the one-dimensional motion of a fictitious particle with coordinate $a(t)$, for the Universe filled with several non-interacting components.

Problem 1.

Construct the effective one-dimensional potential $V(a)$ (using the notation of the previous problem) for the Universe consisting of non-relativistic matter and radiation. Show that motion with $\dot{a}>0$ in such a potential can only be decelerating.

Problem 1.

Derive the exact solutions of the Friedman equations for the Universe with arbitrary curvature, filled with radiation and matter.

Problem 1.

Show that for a spatially flat Universe consisting of one component with equation of state $p = w\rho$ the deceleration parameter $q\equiv -\ddot{a}/(aH^2)$ is equal to $\frac{1}{2}(1 + 3w)$.

Problem 1.

Express the age of the Universe through the deceleration parameter $q=-\ddot{a}/(aH^2)$ for a spatially flat Universe filled with single component with equation of state $p=w\rho$.

Problem 1.

Find the generalization of relation $q = \frac{1}{2}(1 + 3w)$ for the non-flat case.

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