Difference between revisions of "New Cosmography"

From Universe in Problems
Jump to: navigation, search
Line 138: Line 138:
 
'''Problem 11'''
 
'''Problem 11'''
 
<p style= "color: #999;font-size: 11px">problem id: cs-11</p>
 
<p style= "color: #999;font-size: 11px">problem id: cs-11</p>
 
+
Use result of the previous problem to find $dC_{n} /dt$.
 
<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;">\[\frac{dC_{n} }{dt} =-H\frac{dC_{n} }{d\ln (1+z)} =H\left[\frac{\gamma _{n} }{\gamma _{n+1} } C_{n+1} -C_{n} +nC_{n} (1+q)\right]\]</p>
 
   </div>
 
   </div>
 
</div></div>
 
</div></div>
  
  
<div id=""></div>
+
<div id="cs12"></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: cs-12</p>
 
+
Use the general formula for $dC_{n} /dt$ obtained in the previous problem to obtain time derivatives of the cosmographic parameters $q,j,s,l$.
 
<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;">\[\begin{array}{l} {\dot{q}=-H\left(j-2q^{2} -q\right),} \\ {\dot{j}=H\left[s+j(2+3q)\right],} \\ {\dot{s}=H\left[l+s(3+4q)\right],} \\ {\dot{l}=H\left[m+l(4+5q)\right]} \end{array}\]</p>
 
   </div>
 
   </div>
 
</div></div>
 
</div></div>
  
<div id=""></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
'''Problem '''
 
<p style= "color: #999;font-size: 11px">problem id: </p>
 
  
 +
<div id="cs-13"></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
'''Problem 13'''
 +
<p style= "color: #999;font-size: 11px">problem id: cs-13</p>
 +
Find derivatives of the cosmographic parameter w.r.t. the red shift.
 
<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 can transit from the time derivatives to the derivative w.r.t. the red shift according to the relation $\frac{d}{dz} =-\frac{1}{H(1+z)} \frac{d}{dt} $:
 +
\[\begin{array}{l} {\frac{dH}{dz} =H\frac{1+q}{1+z} ,} \\ {\frac{dq}{dz} =\frac{j-2q^{2} -q}{1+z} ,} \\ {\frac{dj}{dz} =-\frac{s+j(2+3q)}{1+z} ,} \\ {\frac{ds}{dz} =-\frac{l+s(3+4q)}{1+z} ,} \\ {\frac{dl}{dz} =-\frac{m+l\left(4+5q\right)}{1+z} } \end{array}\]</p>
 
   </div>
 
   </div>
 
</div></div>
 
</div></div>
  
  
<div id=""></div>
+
<div id="cs-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: cs-14</p>
 
+
Let $1+z=1/a\equiv x$. Find $q(x)$ and $j(x)$.
 
<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 is easy to see that \textbf{$\dot{H}=-H'H/a$ }where\textbf{ $H'=dH/dx$. }Then
 +
\[q=-\frac{\dot{H}}{H^{2} } -1=\frac{H'}{H} x-1\]
 +
Calculating $j$, making use of $a'=-a^{2} $, we obtain
 +
\[j(x)=1-2\frac{H'}{H} x+\left(\frac{H'^{2} }{H^{2} } +\frac{H''}{H} \right)x^{2} \]</p>
 
   </div>
 
   </div>
 
</div></div>
 
</div></div>
  
<div id=""></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
'''Problem '''
 
<p style= "color: #999;font-size: 11px">problem id: </p>
 
  
 +
<div id="cs-15"></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
'''Problem 15'''
 +
<p style= "color: #999;font-size: 11px">problem id: cs-15</p>
 +
Express the derivatives $d^{2} H/dz^{2} $  , $d^{3} H/dz^{3} $  and $d^{4} H/dz^{4} $ in terms of the cosmographic parameters.
 
<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 two previous problems, one finds
 +
\[\begin{array}{l} {\frac{d^{2} H}{dz^{2} } =\frac{j-q^{2} }{(1+z)^{2} } H,} \\ {\frac{d^{3} H}{dz^{3} } =\frac{H}{(1+z)^{3} } \left(3q^{2} +3q^{3} -4qi-3j-s\right)} \\ {\frac{d^{4} H}{dz^{4} } =\frac{H}{(1+z)^{4} } \left(-12q^{2} -24q^{3} -15q^{4} =32qj+25q^{2} j+7qs+12j-4j^{2} +8s+1\right)} \end{array}\]</p>
 
   </div>
 
   </div>
 
</div></div>
 
</div></div>
  
  
<div id=""></div>
+
<div id="cs-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: cs-16</p>
 
+
Find decomposition of the inverse Hubble parameter $1/H$ in powers of the red shift  $z$.
 
<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;">\[\begin{array}{l} {\frac{d}{dz} \left(\frac{1}{H} \right)=-\frac{1}{H^{2} } \frac{dH}{dz} =-\frac{1+q}{1+z} \frac{1}{H} ;} \\ {\frac{d^{2} }{dz^{2} } \left(\frac{1}{H} \right)=2\left(\frac{1+q}{1+z} \right)^{2} \frac{1}{H} -\frac{j-q^{2} }{\left(1+z\right)^{2} } \frac{1}{H} =\frac{2+4q+3q^{2} -j}{(1+z)^{2} } \frac{1}{H} ;} \\ {\frac{1}{H(z)} =\frac{1}{H_{0} } \left[1-\left(1+q_{0} \right)z+\frac{2+4q_{0} +3q_{0}^{2} -j_{0} }{6} z^{2} +\ldots \right]} \end{array}\]</p>
 
   </div>
 
   </div>
 
</div></div>
 
</div></div>
  
<div id=""></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
'''Problem '''
 
<p style= "color: #999;font-size: 11px">problem id: </p>
 
  
 +
<div id="cs-17"></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
'''Problem 17'''
 +
<p style= "color: #999;font-size: 11px">problem id: cs-17</p>
 +
Obtain relations for transition from the time derivatives to that w.r.t. the red shift.
 
<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;">\[\begin{array}{l} {\frac{d^{2} }{dt^{2} } =(1+z)H\left[H+(1+z)\frac{dH}{dz} \right]\frac{d}{dz} +(1+z)^{2} H^{2} \frac{d^{2} }{dz^{2} } ,} \\ {\frac{d^{3} }{dt^{3} } =-(1+z)H\left\{H^{2} +(1+z)^{2} \left(\frac{dH}{dz} \right)^{2} +(1+z)H\left[4\frac{dH}{dz} +(1+z)\frac{d^{2} H}{dz^{2} } \right]\right\}\frac{d}{dz} -3(1+z)^{2} H^{2} } \\ {\times \left[H+(1+z)\frac{dH}{dz} \right]\frac{d^{2} }{dz^{2} } -(1+z)^{3} H^{3} \frac{d^{3} }{dz^{3} } ,} \\ {\frac{d^{4} }{dt^{4} } =(1+z)H\left[H^{2} +11(1+z)H^{2} \frac{dH}{dz} +11(1+z)H\frac{dH}{dz} +(1+z)^{3} \left(\frac{dH}{dz} \right)^{3} +7(1+z)^{2} H\frac{d^{2} H}{dz^{2} } \right. } \\ {+\left. 4(1+z)^{3} H\frac{dH}{dz} \frac{d^{2} H}{d^{2} z} +(1+z)^{3} H^{2} \frac{d^{3} H}{d^{3} z} \right]\frac{d}{dz} +(1+z)^{2} H^{2} \left[7H^{2} +22H\frac{dH}{dz} +7(1+z)^{2} \left(\frac{dH}{dz} \right)^{2} \right. } \\ {+\left. 4H\frac{d^{2} H}{dz^{2} } \right]\frac{d^{2} }{dz^{2} } +6(1+z)^{3} H^{3} \left[H+(1+z)\frac{dH}{dz} \right]\frac{d^{3} }{dz^{3} } +(1+z)^{4} H^{4} \frac{d^{4} }{dz^{4} } +(1+z)^{4} H^{4} \frac{d^{4} }{dz^{4} } .} \end{array}\]</p>
 
   </div>
 
   </div>
 
</div></div>
 
</div></div>

Revision as of 01:34, 2 February 2016

First section of Cosmography


Problem 1

problem id: cs-1

Using the cosmographic parameters introduced above, expand the scale factor into a Taylor series in time.


Problem 2

problem id: cs-2

Using the cosmographic parameters, expand the redshift into a Taylor series in time.


Problem 3

problem id: cs-3

What is the reason for the statement that the cosmological parameters are model-independent?


Problem 4

problem id: cs-4

Obtain the following relations between the deceleration parameter and Hubble's parameter $$q(t)=\frac{d}{dt}(\frac{1}{H})-1;\,\,q(z)=\frac{1+z}{H}\frac{dH}{dz}-1;\,\,q(z)=\frac{d\ln H}{dz}(1+z)-1.$$


Problem 5

problem id: cs-5

Show that the deceleration parameter can be defined by the relation \[q=-\frac{d\dot{a}}{Hda} \]


Problem 6

problem id: cs-6

Classify models of Universe basing on the two cosmographic parameters -- the Hubble parameter and the deceleration parameter.


Problem 7

problem id: cs-7

Show that the deceleration parameter $q$ can be presented in the form $$q(x)=\frac{\dot{H}(x)}{H(x)}x-1;\,\,x=1+z$$


Problem 8

problem id: cs-8

Show that for the deceleration parameter the following relation holds: $$q(a)=-\left(1+\frac{dH/dt}{H^2}\right)-\left(1+\frac{adH/da}{H}\right)$$


Problem 9

problem id: "cs-9

Show that $$\frac{dq}{d\ln (1+z)}=j-q(2q+1)$$


Problem 10

problem id: cs-10

Let \[C_{n} \equiv \gamma _{n} \frac{a^{(n)} }{aH^{n} } ,\] where $a^{(n)} $ is n-th time derivative of the scale factor, $n\ge 2$ , $\gamma _{2} =-1,\; \gamma _{n} =1$ äëÿ $n>2$ . Then$C_{2} =q,\; C_{3} =j,\; C_{4} =s\ldots $ Obtain $dC_{n} /d\ln (1+z)$ .


Problem 11

problem id: cs-11

Use result of the previous problem to find $dC_{n} /dt$.


Problem 12

problem id: cs-12

Use the general formula for $dC_{n} /dt$ obtained in the previous problem to obtain time derivatives of the cosmographic parameters $q,j,s,l$.


Problem 13

problem id: cs-13

Find derivatives of the cosmographic parameter w.r.t. the red shift.


Problem 14

problem id: cs-14

Let $1+z=1/a\equiv x$. Find $q(x)$ and $j(x)$.