Difference between revisions of "New problems"

From Universe in Problems
Jump to: navigation, search
Line 193: Line 193:
  
  
<div id=""></div>
+
<div id="1301_1"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
Problem
+
'''Problem 1'''
<p style= "color: #999;font-size: 11px">problem id: </p>
+
<p style= "color: #999;font-size: 11px">problem id: 1301_1</p>
 
+
Show that for a spatially flat Universe consisting of one component with equation of state $p=w\rho$ the deceleration parameter is equal to $q=(1+3w)/2$.
 
<div class="NavFrame collapsed">
 
<div class="NavFrame collapsed">
 
   <div class="NavHead">solution</div>
 
   <div class="NavHead">solution</div>
 
   <div style="width:100%;" class="NavContent">
 
   <div style="width:100%;" class="NavContent">
     <p style="text-align: left;"></p>
+
     <p style="text-align: left;">\[\frac{\ddot a}{a}=-\frac{4\pi G}{3}\rho(1+3w).\]
 +
In the flat case \[\rho=\frac{3}{8\pi G}H^2\] and \[q=-\frac{\dot a}{aH^2}=\frac12(1+3w).\]</p>
 
   </div>
 
   </div>
 
</div></div>
 
</div></div>
  
  
<div id=""></div>
+
<div id="1301_2"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
'''Problem '''
+
'''Problem 2'''
<p style= "color: #999;font-size: 11px">problem id: </p>
+
<p style= "color: #999;font-size: 11px">problem id: 1301_2</p>
 
+
Find generalization of the relation $q=(1+3w)/2$ for the non-flat case.
 
<div class="NavFrame collapsed">
 
<div class="NavFrame collapsed">
 
   <div class="NavHead">solution</div>
 
   <div class="NavHead">solution</div>
 
   <div style="width:100%;" class="NavContent">
 
   <div style="width:100%;" class="NavContent">
     <p style="text-align: left;"></p>
+
     <p style="text-align: left;">\[\frac{\ddot a}{a}=-\frac{4\pi G}{3}\rho(1+3w).\]
 +
Substitution \[\rho=\frac{3}{8\pi G}\left(H^2+\frac k{a^2}\right)\] gives \[q=-\frac{\dot a}{aH^2}=\frac12(1+3w)\left(1+\frac k{a^2H^2}\right).\]</p>
 
   </div>
 
   </div>
 
</div></div>
 
</div></div>
  
  
<div id=""></div>
+
<div id="1301_3"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
'''Problem '''
+
'''Problem 3'''
<p style= "color: #999;font-size: 11px">problem id: </p>
+
<p style= "color: #999;font-size: 11px">problem id: 1301_3</p>
 
+
Find deceleration parameter for multi-component non-flat Universe.
 
<div class="NavFrame collapsed">
 
<div class="NavFrame collapsed">
 
   <div class="NavHead">solution</div>
 
   <div class="NavHead">solution</div>
 
   <div style="width:100%;" class="NavContent">
 
   <div style="width:100%;" class="NavContent">
     <p style="text-align: left;"></p>
+
     <p style="text-align: left;">In this case
 +
\[q=-\frac{\dot a}{aH^2}=\frac12(1+3\frac p\rho)\left(1+\frac k{a^2H^2}\right) =\] \[=\frac12\left[1+3\frac p\rho\left(1+\frac k{a^2H^2}\right)\right]+\frac k{2a^2H^2},\]
 +
\[1+\frac k{a^2H^2}=\sum\limits_i\Omega_i,\quad\frac p\rho=\frac{\sum\limits_iw_i\Omega_i}{\sum\limits_i\Omega_i},\]
 +
\[q=\frac12\left(1+3\sum\limits_i\Omega_i\right)+\frac k{2a^2H^2}.\]
 +
</p>
 
   </div>
 
   </div>
 
</div></div>
 
</div></div>
  
  
<div id=""></div>
+
<div id="1301_4"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
'''Problem '''
+
'''Problem 4'''
<p style= "color: #999;font-size: 11px">problem id: </p>
+
<p style= "color: #999;font-size: 11px">problem id: 1301_4</p>
 
+
Show that the expression for deceleration parameter obtained in the previous problem can be presented in the following form \[q=\frac{\Omega_{total}}2+\frac32\sum\limits_i w_i \Omega_i.\]
 
<div class="NavFrame collapsed">
 
<div class="NavFrame collapsed">
 
   <div class="NavHead">solution</div>
 
   <div class="NavHead">solution</div>
 
   <div style="width:100%;" class="NavContent">
 
   <div style="width:100%;" class="NavContent">
     <p style="text-align: left;"></p>
+
     <p style="text-align: left;">Using results of the previous problem one obtains
 +
\[q=\frac12\left(1+3\sum\limits_iw_i\Omega_i\right)+\frac k{2a^2H^2}=\frac12\left(1+\frac k{a^2H^2}\right)+\frac32\sum\limits_iw_i\Omega_i=\]
 +
\[=\frac12\sum\limits_i\Omega_i+\frac32\sum\limits_iw_i\Omega_i=\frac{\Omega_{total}}2+\frac32\sum\limits_i w_i \Omega_i.\]</p>
 
   </div>
 
   </div>
 
</div></div>
 
</div></div>
  
  
<div id=""></div>
+
<div id="1301_5"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
'''Problem '''
+
'''Problem 5'''
 
<p style= "color: #999;font-size: 11px">problem id: </p>
 
<p style= "color: #999;font-size: 11px">problem id: </p>
 
+
Let us consider the model of two-component Universe [ J. Ponce de Leon, cosmological model with variable equations of state for matter and dark energy, arXiv:1204.0589]. Such approximation is sufficient to achieve good accuracy on each stage of its evolution. At the present time the two dark components - dark matter and dark energy - are considered to be dominating. We neglect meanwhile the interaction between the components and as a result they separately satisfy the conservation equation. Let us assume that the state equation parameter for each component depends on the scale factor
 +
\begin{align}
 +
\nonumber p_{de} & = W(a) \rho_{de};\\
 +
\nonumber p_m & = w(a)\rho_{m}.
 +
\end{align}
 +
Express the relative densities $\Omega_m$ and $\Omega_{de}$ in terms of the deceleration parameter and the state equation parameters $w(a)$ and $W(a)$.
 
<div class="NavFrame collapsed">
 
<div class="NavFrame collapsed">
 
   <div class="NavHead">solution</div>
 
   <div class="NavHead">solution</div>
 
   <div style="width:100%;" class="NavContent">
 
   <div style="width:100%;" class="NavContent">
     <p style="text-align: left;"></p>
+
     <p style="text-align: left;">In the considered case
 +
\[\Omega_m +\Omega_{de}=1+\frac{k}{a^2H^2},\]
 +
\[q=\frac12+\frac32\left[W\Omega_{de}+w\Omega_m\right]+\frac{k}{2a^2H^2}.\]
 +
These can, formally, be regarded as two equations for $\Omega_{de}$ and $\Omega_{m}$. Solving them we get
 +
\begin{align}
 +
\nonumber \Omega_{m} & = \frac{2q-1-3W}{3(w-W)}-\frac{k(1+3W)}{3a^2H^2(w-W)};\\
 +
\nonumber \Omega_{m} & = \frac{2q-1-3w}{3(w-W)}-\frac{k(1+3w)}{3a^2H^2(w-W)}.
 +
\end{align}</p>
 
   </div>
 
   </div>
 
</div></div>
 
</div></div>
  
  
<div id=""></div>
+
<div id="1301_6"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
'''Problem '''
+
'''Problem 6'''
<p style= "color: #999;font-size: 11px">problem id: </p>
+
<p style= "color: #999;font-size: 11px">problem id: 1301_6</p>
 
+
Find the upper and lower limits on the deceleration parameter using results of the previous problem.
 
<div class="NavFrame collapsed">
 
<div class="NavFrame collapsed">
 
   <div class="NavHead">solution</div>
 
   <div class="NavHead">solution</div>
 
   <div style="width:100%;" class="NavContent">
 
   <div style="width:100%;" class="NavContent">
     <p style="text-align: left;"></p>
+
     <p style="text-align: left;">We note that the denominator in these expressions is always positive because $W<0$ for dark energy. Thus, the fact that $\Omega_{m(de)}>0$ imposes an upper and lower limit on $q$,
 +
\[\frac{1+3W}{2}\left(1+\frac{k}{a^2H^2}\right)\le q\le \frac{1+3w}{2}\left(1+\frac{k}{a^2H^2}\right).\]</p>
 
   </div>
 
   </div>
 
</div></div>
 
</div></div>
  
  
<div id=""></div>
+
<div id="1301_7"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
'''Problem '''
+
'''Problem 7'''
<p style= "color: #999;font-size: 11px">problem id: </p>
+
<p style= "color: #999;font-size: 11px">problem id: 1301_7</p>
 
+
Find the relation between the total pressure and the deceleration parameter for the flat one-component Universe
 
<div class="NavFrame collapsed">
 
<div class="NavFrame collapsed">
 
   <div class="NavHead">solution</div>
 
   <div class="NavHead">solution</div>
 
   <div style="width:100%;" class="NavContent">
 
   <div style="width:100%;" class="NavContent">
     <p style="text-align: left;"></p>
+
     <p style="text-align: left;">It follows from the conservation equation that
 +
\[p=-\frac{\dot\rho}{3H}\frac{w}{1+w}.\]
 +
Using
 +
\[w = \frac{2q-1}3;\quad \dot\rho=\frac{3}{4\pi G}H\dot H,\quad \dot H=-H^2(1+q)\]
 +
one finds
 +
\[p=\frac{H^2}{8\pi G}(2q-1).\]</p>
 
   </div>
 
   </div>
 
</div></div>
 
</div></div>
  
  
<div id=""></div>
+
<div id="1301_8"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
'''Problem '''
+
'''Problem 8'''
<p style= "color: #999;font-size: 11px">problem id: </p>
+
<p style= "color: #999;font-size: 11px">problem id: 1301_8</p>
 
+
The expansion of pressure by the cosmic time is given by
 +
\[p(t)=\left.\sum\limits_{k=0}^\infty\frac1{k!}\frac{d^kp}{dt^k}\right|_{t=t_0}(t-t_0)^k.\]
 +
Using cosmography parameters , evaluate the derivatives up to the fourth order.
 
<div class="NavFrame collapsed">
 
<div class="NavFrame collapsed">
 
   <div class="NavHead">solution</div>
 
   <div class="NavHead">solution</div>
Line 297: Line 325:
  
  
<div id=""></div>
+
<div id="1301_9"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
'''Problem '''
+
'''Problem 9'''
<p style= "color: #999;font-size: 11px">problem id: </p>
+
<p style= "color: #999;font-size: 11px">problem id: 1301_9</p>
 
+
Show that for one-component flat Universe filled with ideal fluid of density $\rho$.
 
<div class="NavFrame collapsed">
 
<div class="NavFrame collapsed">
 
   <div class="NavHead">solution</div>
 
   <div class="NavHead">solution</div>
 
   <div style="width:100%;" class="NavContent">
 
   <div style="width:100%;" class="NavContent">
     <p style="text-align: left;"></p>
+
     <p style="text-align: left;">\[q=-1-\frac12\frac{d\ln\rho}{d\ln a}.\]</p>
 
   </div>
 
   </div>
 
</div></div>
 
</div></div>
  
  
<div id=""></div>
+
<div id="1301_10"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
'''Problem '''
+
'''Problem 10'''
<p style= "color: #999;font-size: 11px">problem id: </p>
+
<p style= "color: #999;font-size: 11px">problem id: 1301_10</p>
 
+
For what values of the state parameter $w$ the rate of expansion of a one-component flat Universe increases with time?
 
<div class="NavFrame collapsed">
 
<div class="NavFrame collapsed">
 
   <div class="NavHead">solution</div>
 
   <div class="NavHead">solution</div>
Line 323: Line 351:
  
  
<div id=""></div>
+
<div id="1301_11"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
'''Problem '''
+
'''Problem 11'''
<p style= "color: #999;font-size: 11px">problem id: </p>
+
<p style= "color: #999;font-size: 11px">problem id: 1301_11</p>
 
+
Express the age of the spatially flat Universe filled with a single component with equation of state $p=w\rho$ through the deceleration parameter.
 
<div class="NavFrame collapsed">
 
<div class="NavFrame collapsed">
 
   <div class="NavHead">solution</div>
 
   <div class="NavHead">solution</div>
 
   <div style="width:100%;" class="NavContent">
 
   <div style="width:100%;" class="NavContent">
     <p style="text-align: left;"></p>
+
     <p style="text-align: left;">For spatially flat one-component Universe with state equation $p=w\rho$ the scale factor is \[a\propto t^\frac{2}{3(1+w)}\] and therefore \[H=\frac{2}{3(1+w)}\frac1t.\]
 +
 
 +
Using $q=(1+3w)/2$ one can find a simple relation between the current age of the Universe and the DP
 +
\[t_0=\frac{H_0^{-1}}{1+q}.\]</p>
 
   </div>
 
   </div>
 
</div></div>
 
</div></div>
  
  
<div id=""></div>
+
<div id="1301_12"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
'''Problem '''
+
'''Problem 12'''
<p style= "color: #999;font-size: 11px">problem id: </p>
+
<p style= "color: #999;font-size: 11px">problem id: 1301_12</p>
 
+
Suppose we know the current values of the Hubble constant $H_0$ and the deceleration parameter $q_0$ for a closed Universe filled with dust only. How many times larger will  it ever become? Find lifetime of such a Universe.
 
<div class="NavFrame collapsed">
 
<div class="NavFrame collapsed">
 
   <div class="NavHead">solution</div>
 
   <div class="NavHead">solution</div>
Line 349: Line 380:
  
  
<div id=""></div>
+
<div id="1301_13"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
'''Problem '''
+
'''Problem 13'''
<p style= "color: #999;font-size: 11px">problem id: </p>
+
<p style= "color: #999;font-size: 11px">problem id: 1301_13</p>
 
+
In a closed Universe filled with non-relativistic matter the current values of the Hubble constant is $H_0$, the deceleration parameter is $q_0$. Find the current age of this Universe.
 
<div class="NavFrame collapsed">
 
<div class="NavFrame collapsed">
 
   <div class="NavHead">solution</div>
 
   <div class="NavHead">solution</div>
Line 362: Line 393:
  
  
<div id=""></div>
+
<div id="1301_14"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
'''Problem '''
+
'''Problem 14'''
<p style= "color: #999;font-size: 11px">problem id: </p>
+
<p style= "color: #999;font-size: 11px">problem id: 1301_14</p>
 +
In a closed Universe filled with dust the current value of the Hubble constant is $H_0$ and of the deceleration parameter $q_0$.
 +
 
 +
a) What is the total  proper volume of the Universe at present time?
  
 +
b) What is the total  current proper volume of space occupied by matter which we are presently observing?
 +
 +
c) What is the total  proper volume of space which we are directly observing?
 
<div class="NavFrame collapsed">
 
<div class="NavFrame collapsed">
 
   <div class="NavHead">solution</div>
 
   <div class="NavHead">solution</div>
Line 375: Line 412:
  
  
<div id=""></div>
+
<div id="1301_15"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
'''Problem '''
+
'''Problem 15'''
<p style= "color: #999;font-size: 11px">problem id: </p>
+
<p style= "color: #999;font-size: 11px">problem id: 1301_15</p>
 
+
For the closed ($k=+1$) model of Universe, filled with non-relativistic matter, show that solutions of the Friedmann equations can be represented in terms of the two parameters $H_0$ and $q_0$. [Y.Shtanov, Lecture Notes on theoretical cosmology, 2010 ]
 
<div class="NavFrame collapsed">
 
<div class="NavFrame collapsed">
 
   <div class="NavHead">solution</div>
 
   <div class="NavHead">solution</div>
 
   <div style="width:100%;" class="NavContent">
 
   <div style="width:100%;" class="NavContent">
     <p style="text-align: left;"></p>
+
     <p style="text-align: left;">The Friedman equations can then be represented in the form
 +
\begin{align}
 +
\label{background_2_29} H^2 & +\frac1{a^2} =\frac{8\pi G}3\rho,\\
 +
\nonumber 2\frac{\ddot a}{a} & +H^2+\frac1{a^2} = 0.
 +
\end{align}
 +
Using this equations one can find the relations between the current values of the Universe's parameters
 +
\begin{align}
 +
\label{background_2_30} H_0^2 & =\frac1{a_0^2(2q_0-1)},\\
 +
\nonumber q_0 & = \frac{4\pi G}{3H_0^2}\rho_0.
 +
\end{align}
 +
Note that in general
 +
\[q_0=\frac{4\pi G}3\frac{\rho_0+3p_0}{H_0^2}=\frac{\rho_0+3p_0}{2\rho_{0,crit}},\quad \rho_{0,crit}=\frac{3H_0^2}{8\pi G}.\]
 +
For Universe filled only with non-relativistic matter one has $\Omega_m = 2q_0$. It is easy to see that $q_0>1/2$ and $\Omega_m>1$ as was expected in the closed model. Using (\ref{background_2_30}) one can rewrite the equation for scale factor in the form
 +
\begin{equation}\label{background_2_31}\dot a^2=\frac\alpha a-1,\quad \alpha\equiv\frac{2q_0}{H_0(2q_0-1)^{3/2}}.\end{equation}
 +
It is easy to see that the considered model includes both parameters $H_0$ and $q_0$. integration of (\ref{background_2_31}) gives \[t=\int \sqrt{\frac{a}{\alpha-a}}\,da.\]
 +
Substitution \[a=\frac\alpha2(1-\cos\tau)\] leads to \begin{equation}\label{background_2_33}t=\frac\alpha2(\tau-\sin\tau).\end{equation}
 +
Because of the relation $dt=ad\tau$ it is evident that the variable $\tau$ is the conformal time. Taking the constants of integration so that $a=0$ as $t=0$ (and $\tau=0$) we can see that $a=a_0$ at $\tau=\tau_0$. Consequently, \[\cos\tau_0=\frac{1-q_0}{q_0},\quad \sin\tau_0=\frac{\sqrt{2q_0-1}}{q_0}.\]
 +
From (\ref{background_2_33}) it follows that the age of Universe in close model is \[t_0=\frac\alpha2(\tau_0-\sin\tau_0)=\frac{q_0}{H_0(2q_0-1)^{3/2}}\left(\arccos{\frac{1-q_0}{q_0}} - \frac{\sqrt{2q_0-1}}{q_0}\right).\]
 +
The maximum of scale factor reaches at $\tau=\pi$,
 +
\[a_{\rm max}=\alpha=\frac{2q_0}{H_0(2q_0-1)^{3/2}}.\]
 +
As one can see, all parameters of the model can be expressed in terms of parameters $H_0$ and $q_0$.</p>
 
   </div>
 
   </div>
 
</div></div>
 
</div></div>
  
  
<div id=""></div>
+
<div id="1301_16"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
'''Problem '''
+
'''Problem 16'''
<p style= "color: #999;font-size: 11px">problem id: </p>
+
<p style= "color: #999;font-size: 11px">problem id: 1301_16</p>
 
+
Do the same as in the previous problem for the case of open ($k=-1$) model of Universe.
 
<div class="NavFrame collapsed">
 
<div class="NavFrame collapsed">
 
   <div class="NavHead">solution</div>
 
   <div class="NavHead">solution</div>
 
   <div style="width:100%;" class="NavContent">
 
   <div style="width:100%;" class="NavContent">
     <p style="text-align: left;"></p>
+
     <p style="text-align: left;">In the case of open Universe the formulae (\ref{background_2_29})-(\ref{background_2_31}) transform into
 +
\begin{align}
 +
\label{background_2_37} H^2 & -\frac1{a^2} =\frac{8\pi G}3\rho,\\
 +
\nonumber 2\frac{\ddot a}{a} & +H^2-\frac1{a^2} = 0,\\
 +
\label{background_2_38} H_0^2 & =\frac1{a_0^2(1-2q_0)},\\
 +
\nonumber q_0 & = \frac{4\pi G}{3H_0^2}\rho_0,
 +
\end{align}
 +
\begin{equation}\label{background_2_39}\dot a^2=\frac\beta a+1,\quad \beta\equiv\frac{2q_0}{H_0(1-2q_0)^{3/2}}.\end{equation}
 +
Now $q_0<1/2$, $0\le\Omega_m< 1$. The solution of (\ref{background_2_39}) in the conformal time parametrization is \[a=\frac\beta2(\cosh\tau-1),\quad t=\frac\beta2(\sinh\tau-\tau).\] Current value $\tau_0$ of the conformal time is defined by the relation \[\cosh\tau_0=\frac{1-q_0}{q_0}.\]
 +
The age of the Universe is
 +
\[t_0=\frac\beta2(\sinh\tau_0-\tau_0) = \frac{q_0}{H_0(1-q_0)^{3/2}}\left(\frac{\sqrt{1-2q_0}}{q_0} - \ln\frac{1-q_0+\sqrt{1-2q_0}}{q_0}\right).\]
 +
In this case again all characteristics of the model are expressed in terms of the parameters $H_0$ and $q_0$.</p>
 
   </div>
 
   </div>
 
</div></div>
 
</div></div>
  
  
<div id=""></div>
+
=Classification of models of Universe based on the deceleration parameter=
 +
 
 +
<div id="1301_17"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
'''Problem '''
+
'''Problem 17'''
<p style= "color: #999;font-size: 11px">problem id: </p>
+
<p style= "color: #999;font-size: 11px">problem id: 1301_17</p>
 
+
When the rate of expansion never changes, and $\dot a$ is constant, the scaling factor is proportional to time $t$, and the deceleration term is zero. When the Hubble parameter is constant, the deceleration parameter $q$ is also constant and equal to $-1$, as in the de Sitter model. All models can be characterized by whether they expand or contract, and accelerate or decelerate. Build a classification of such type using the signature of the Hubble parameter and the deceleration parameter.
 
<div class="NavFrame collapsed">
 
<div class="NavFrame collapsed">
 
   <div class="NavHead">solution</div>
 
   <div class="NavHead">solution</div>
 
   <div style="width:100%;" class="NavContent">
 
   <div style="width:100%;" class="NavContent">
     <p style="text-align: left;"></p>
+
     <p style="text-align: left;">\begin{description}
 +
  \item[(a)]$H>0$, $q>0$: expanding and decelerating
 +
  \item[(b)] $H>0$, $q<0$: expanding and accelerating
 +
  \item[(c)] $H<0$, $q>0$: contracting and decelerating
 +
  \item[(d)]$H<0$, $q<0$: contracting and accelerating
 +
  \item[(e)] $H>0$, $q=0$: expanding, zero deceleration
 +
  \item[(f)] $H<0$, $q=0$: contracting, zero deceleration
 +
  \item[(g)]$H=0$, $q=0$: static
 +
\end{description}
 +
 
 +
There is little doubt that we live in an expanding Universe, and hence only (a), (b), and (e) are possible.  Evidences in favor of the fact that the expansion is presently accelerating continuously grows in number and therefore the current dynamics belongs to type (b).</p>
 
   </div>
 
   </div>
 
</div></div>
 
</div></div>

Revision as of 22:56, 29 January 2015



New from 26-12-2014


Problem 1

problem id: 2612_1

Find dimension of the Newtonian gravitational constant $G$ and electric charge $e$ in the $N$-dimensional space.


Problem 2

problem id: 2612_2

In 1881, Johnson Stoney proposed a set of fundamental units involving $c$, $G$ and $e$. Construct Stoney's natural units of length, mass and time.


Problem 3

problem id: 2612_3

Show that the fine structure constant $\alpha=e^2/(\hbar c)$ can be used to convert between Stoney and Planck units.


Problem 4

problem id: 2612_4

Gibbons [G W Gibbons, The Maximum Tension Principle in General Relativity, arXiv:0210109] formulated a hypothesis, that in General Relativity there should be a maximum value to any physically attainable force (or tension) given by \begin{equation}\label{2612_e_1} F_{max}=\frac{c^4}{4G}. \end{equation} This quantity can be constructed with help of the Planck units: \[F_{max}=\frac14M_{Pl}L_{Pl}T_{Pl}^{-2}.\] The origin of the numerical coefficient $1/4$ has no deeper meaning. It simply turns out that $1/4$ is the value that leads to the correct form of the field equations of General Relativity. The above made assumption leads to existence of a maximum power defined by \begin{equation}\label{2612_e_2} P_{max}=\frac{c^5}{G}. \end{equation} There is a hypothesis [C. Schiller, General relativity and cosmology derived from principle of maximum power or force, arXiv:0607090], that the maximum force (or power) plays the same role for general relativity as the maximum speed plays for special relativity.

The black holes are usually considered as an extremal solution of GR equations, realizing the limiting values of the physical quantities. Show that the value $c^4/(4G)$ of the force limit is the energy of a Schwarzschild black hole divided by twice its radius. The maximum power $c^5/(4G)$ is realized when such a black hole is radiated away in the time that light takes to travel along a length corresponding to twice the radius.


Problem 5

problem id: 2612_5

E. Berti, A Black-Hole Primer: Particles, Waves, Critical Phenomena and Superradiant instabilities (arXiv:1410.4481[gr-qc])

A Newtonian analog of the black hole concept is a so-called "dark star". If we consider light as a corpuscle traveling at speed $c$, light cannot escape to infinity whenever \[V_{esc}>c \quad \left(V_{esc}^2=\frac{2GM}{R}\right)/\] Therefore the condition for existence of "dark stars" in Newtonian mechanics is \[\frac{2GM}{c^2R}\ge1.\] Can this condition be satisfied in the Newtonian mechanics?


Problem 6

problem id: 2612_6

The lookback time is defined as the difference between the present day age of the Universe and its age at redshift $z$, i.e. the difference between the age of the Universe at observation $t_0$ and the age of the Universe, $t$, when the photons were emitted. Find the lookback time for the Universe filled by non-relativistic matter, radiation and a component with the state equation $p=w(z)\rho$.


Problem 7

problem id: 2612_7

Find the solutions corrected by LQC Friedmann equations for a matter dominated Universe.


Problem 8

problem id: 2612_8

Show that in the case of the flat Friedmann metric, the third power of the scale factor $\varphi=a^3$ satisfies the equation \[\frac{d^2t\varphi}{dt^2}=\frac32(\rho-p)\varphi,\quad 8\pi G=1.\] Check validity of this equation for different cosmological components: non-relativistic matter, cosmological constant and a component with the state equation $p=w\rho$.


Problem 9

problem id:

It is of broad interest to understand better the nature of the early Universe especially the Big Bang. The discovery of the CMB in 1965 resolved a dichotomy then existing in theoretical cosmology between steady-state and Big Bang theories. The interpretation of the CMB as a relic of a Big Bang was compelling and the steady-state theory died. Actually at that time it was really a trichotomy being reduced to a dichotomy because a third theory is a cyclic cosmology model. Give a basic argument of the opponents to the latter model.


Problem 10

problem id: 2612_10

Express the present epoch value of the Ricci scalar $R$ and its first derivative in terms of $H_0$ and its derivatives (flat case).


New from 13-01-2015


Problem 1

problem id: 1301_1

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


Problem 2

problem id: 1301_2

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


Problem 3

problem id: 1301_3

Find deceleration parameter for multi-component non-flat Universe.


Problem 4

problem id: 1301_4

Show that the expression for deceleration parameter obtained in the previous problem can be presented in the following form \[q=\frac{\Omega_{total}}2+\frac32\sum\limits_i w_i \Omega_i.\]


Problem 5

problem id:

Let us consider the model of two-component Universe [ J. Ponce de Leon, cosmological model with variable equations of state for matter and dark energy, arXiv:1204.0589]. Such approximation is sufficient to achieve good accuracy on each stage of its evolution. At the present time the two dark components - dark matter and dark energy - are considered to be dominating. We neglect meanwhile the interaction between the components and as a result they separately satisfy the conservation equation. Let us assume that the state equation parameter for each component depends on the scale factor \begin{align} \nonumber p_{de} & = W(a) \rho_{de};\\ \nonumber p_m & = w(a)\rho_{m}. \end{align} Express the relative densities $\Omega_m$ and $\Omega_{de}$ in terms of the deceleration parameter and the state equation parameters $w(a)$ and $W(a)$.


Problem 6

problem id: 1301_6

Find the upper and lower limits on the deceleration parameter using results of the previous problem.


Problem 7

problem id: 1301_7

Find the relation between the total pressure and the deceleration parameter for the flat one-component Universe


Problem 8

problem id: 1301_8

The expansion of pressure by the cosmic time is given by \[p(t)=\left.\sum\limits_{k=0}^\infty\frac1{k!}\frac{d^kp}{dt^k}\right|_{t=t_0}(t-t_0)^k.\] Using cosmography parameters , evaluate the derivatives up to the fourth order.


Problem 9

problem id: 1301_9

Show that for one-component flat Universe filled with ideal fluid of density $\rho$.


Problem 10

problem id: 1301_10

For what values of the state parameter $w$ the rate of expansion of a one-component flat Universe increases with time?


Problem 11

problem id: 1301_11

Express the age of the spatially flat Universe filled with a single component with equation of state $p=w\rho$ through the deceleration parameter.


Problem 12

problem id: 1301_12

Suppose we know the current values of the Hubble constant $H_0$ and the deceleration parameter $q_0$ for a closed Universe filled with dust only. How many times larger will it ever become? Find lifetime of such a Universe.


Problem 13

problem id: 1301_13

In a closed Universe filled with non-relativistic matter the current values of the Hubble constant is $H_0$, the deceleration parameter is $q_0$. Find the current age of this Universe.


Problem 14

problem id: 1301_14

In a closed Universe filled with dust the current value of the Hubble constant is $H_0$ and of the deceleration parameter $q_0$.

a) What is the total proper volume of the Universe at present time?

b) What is the total current proper volume of space occupied by matter which we are presently observing?

c) What is the total proper volume of space which we are directly observing?


Problem 15

problem id: 1301_15

For the closed ($k=+1$) model of Universe, filled with non-relativistic matter, show that solutions of the Friedmann equations can be represented in terms of the two parameters $H_0$ and $q_0$. [Y.Shtanov, Lecture Notes on theoretical cosmology, 2010 ]


Problem 16

problem id: 1301_16

Do the same as in the previous problem for the case of open ($k=-1$) model of Universe.


Classification of models of Universe based on the deceleration parameter

Problem 17

problem id: 1301_17

When the rate of expansion never changes, and $\dot a$ is constant, the scaling factor is proportional to time $t$, and the deceleration term is zero. When the Hubble parameter is constant, the deceleration parameter $q$ is also constant and equal to $-1$, as in the de Sitter model. All models can be characterized by whether they expand or contract, and accelerate or decelerate. Build a classification of such type using the signature of the Hubble parameter and the deceleration parameter.


Problem

problem id:


Problem

problem id:


Problem

problem id:

Exactly Integrable n-dimensional Universes

Exactly Integrable n-dimensional Universes

Problem 1

problem id: gnd_1

Derive Friedmann equations for the spatially n-dimensional Universe.


Problem 2

problem id: gnd_2

Obtain the energy conservation law for the case of n-dimensional Universe.


Problem 3

problem id: gnd_3

Obtain relation between the energy density and scale factor in the case of a two-component n-dimensional Universe dominated by the cosmological constant and a barotropic fluid.


Problem 4

problem id: gnd_4

Obtain equation of motion for the scale factor for the previous problem.


To integrate (\ref{14}), we recall Chebyshev's theorem:

For rational numbers $p,q,r$ ($r\neq0$) and nonzero real numbers $\alpha,\beta$, the integral $\int x^p(\alpha+\beta x^r)^q\,d x$ is elementary if and only if at least one of the quantities \begin{equation}\label{cd} \frac{p+1}r,\quad q,\quad \frac{p+1}r+q, \end{equation} is an integer.

Another way to see the validity of the Chebyshev theorem is to represent the integral of concern by a hypergeometric function such that when a quantity in (\ref{cd}) is an integer the hypergeometric function is reduced into an elementary function. Consequently, when $k=0$ or $\Lambda=0$, and $w$ is rational, the Chebyshev theorem enables us to know that, for exactly what values of $n$ and $w$, the equation (\ref{14}) may be integrated.


Problem 5

problem id: gnd_4_0

Obtain analytic solutions for the equation of motion for the scale factor of the previous problem for spatially flat ($k=0$) Universe.


Problem 6

problem id: gnd_5

Use the analytic solutions obtained in the previous problem to study cosmology with $w>-1$ and $a(0)=0$.


Problem 7

problem id: gnd_6

Use results of the previous problem to calculate the deceleration $q=-a\ddot a/\dot a^2$ parameter.


Problem 8

problem id: gnd_4_1

Obtain the exact solvability conditions for the case $\Lambda=0$ in equation (\ref{14}) (see problem #gnd_4 ).


Problem 9

problem id: gnd_4_2

Obtain the explicit solutions for the case $n=3$ and $w=-5/9$ in the previous problem.


Problem 10

problem id: gnd_10

Rewrite equation of motion for the scale factor (\ref{14}) in terms of the conformal time.


Problem 11

problem id: gnd_11

Find the rational values of the equation of state parameter $w$ which provide exact integrability of the equation (\ref{48}) with $k=0$.


Problem 12

problem id: gnd_12

Obtain explicit solutions for the case $w=\frac1n-1$ in the previous problem.


Problem 13

problem id: gnd_13

Obtain exact solutions for the equation (\ref{48}) with $\Lambda=0$.


Problem 14

problem id: gnd_14

Obtain inflationary solutions using the results of the previous problem.


UNSORTED NEW Problems

The history of what happens in any chosen sample region is the same as the history of what happens everywhere. Therefore it seems very tempting to limit ourselves with the formulation of Cosmology for the single sample region. But any region is influenced by other regions near and far. If we are to pay undivided attention to a single region, ignoring all other regions, we must in some way allow for their influence. E. Harrison in his book Cosmology, Cambridge University Press, 1981 suggests a simple model to realize this idea. The model has acquired the name of "Cosmic box" and it consists in the following.

Imaginary partitions, comoving and perfectly reflecting, are used to divide the Universe into numerous separate cells. Each cell encloses a representative sample and is sufficiently large to contain galaxies and clusters of galaxies. Each cell is larger than the largest scale of irregularity in the Universe, and the contents of all cells are in identical states. A partitioned Universe behaves exactly as a Universe without partitions. We assume that the partitions have no mass and hence their insertion cannot alter the dynamical behavior of the Universe. The contents of all cells are in similar states, and in the same state as when there were no partitions. Light rays that normally come from very distant galaxies come instead from local galaxies of long ago and travel similar distances by multiple re?ections. What normally passes out of a region is reflected back and copies what normally enters a region.

Let us assume further that the comoving walls of the cosmic box move apart at a velocity given by the Hubble law. If the box is a cube with sides of length $L$, then opposite walls move apart at relative velocity $HL$.Let us assume that the size of the box $L$ is small compared to the Hubble radius $L_{H} $ , the walls have a recession velocity that is small compared to the velocity of light. Inside a relatively small cosmic box we use ordinary everyday physics and are thus able to determine easily the consequences of expansion. We can even use Newtonian mechanics to determine the expansion if we embed a spherical cosmic box in Euclidean space.


Problem 1

problem id:

As we have shown before (see Chapter 3): $p(t)\propto a(t)^{-1} $ , so all freely moving particles, including galaxies (when not bound in clusters), slowly lose their peculiar motion and ultimately become stationary in expanding space. Try to understand what happens by considering a moving particle inside an expanding cosmic box


Problem 2

problem id:

Show that at redshift $z=1$ , when the Universe is half its present size, the kinetic energy of a freely moving nonrelativistic particle is four times its present value, and the energy of a relativistic particle is twice its present value.


Problem 3

problem id:

Let the cosmic box is filled with non-relativistic gas. Find out how the gas temperature varies in the expanding cosmic box.


Problem 4

problem id:

Show that entropy of the cosmic box is conserved during its expansion.


Problem 5

problem id:

Consider a (cosmic) box of volume V, having perfectly reflecting walls and containing radiation of mass density $\rho $. The mass of the radiation in the box is $M=\rho V$ . We now weigh the box and find that its mass, because of the enclosed radiation, has increased not by M but by an amount 2M. Why?


Problem 6

problem id:

Show that the jerk parameter is \[j(t)=q+2q^{2} -\frac{\dot{q}}{H} \]


Problem 7

problem id:

We consider FLRW spatially flat Universe with the general Friedmann equations \[\begin{array}{l} {H^{2} =\frac{1}{3} \rho +f(t),} \\ {\frac{\ddot{a}}{a} =-\frac{1}{6} \left(\rho +3p\right)+g(t)} \end{array}\] Obtain the general conservation equation.


Problem 8

problem id:

Show that for extra driving terms in the form of the cosmological constant the general conservation equation (see previous problem) transforms in the standard conservation equation.


Problem 9

problem id:

Show that case $f(t)=g(t)=\Lambda /3$ corresponds to $\Lambda (t)CDM$ model.