Difference between revisions of "New Cosmography"

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     <p style="text-align: left;">It is easy to see that \textbf{$\dot{H}=-H'H/a$ }where\textbf{ $H'=dH/dx$. }Then
+
     <p style="text-align: left;">It is easy to see that $\dot{H}=-H'H/a$ where $H'=dH/dx$. Then
 
\[q=-\frac{\dot{H}}{H^{2} } -1=\frac{H'}{H} x-1\]
 
\[q=-\frac{\dot{H}}{H^{2} } -1=\frac{H'}{H} x-1\]
 
Calculating $j$, making use of $a'=-a^{2} $, we obtain
 
Calculating $j$, making use of $a'=-a^{2} $, we obtain
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<div id="cs-18"></div>
 
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'''Problem '''
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'''Problem 18'''
<p style= "color: #999;font-size: 11px">problem id: </p>
+
<p style= "color: #999;font-size: 11px">problem id: cs-18</p>
 +
Can $dH^{n} /dz^{n} $ generally be expressed in terms of the cosmographic parameters?
 +
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<div id="cs-19"></div>
 
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'''Problem '''
+
'''Problem 19'''
<p style= "color: #999;font-size: 11px">problem id: </p>
+
<p style= "color: #999;font-size: 11px">problem id: cs-19</p>
 +
Show that $$q(z)=frac 12 \frac{d\ln H^2}{d\ln(1+z)}$$
 +
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    <p style="text-align: left;"></p>
 
  </div>
 
</div></div>
 
  
 
+
<div id="cs-20"></div>
<div id=""></div>
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<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
'''Problem '''
+
'''Problem 20'''
<p style= "color: #999;font-size: 11px">problem id: </p>
+
<p style= "color: #999;font-size: 11px">problem id: cs-20</p>
 +
Show that the time derivatives of the Hubble's parameter can be expressed through the cosmographic parameters as follows:
 +
$$\dot{H}=-H^2(1+q);$$
 +
$$\ddot{H}=H^3(j+3q+2)$$
 +
$$\dddot{H}=H^4\left(s-4j-3q(q+4)-6\right)$$
 +
$$\ddddot{H}=H^5\left(l-5s+10(q+2)j+30(q+2)+24\right)$$
 +
generally be expressed in terms of the cosmographic parameters?
 +
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    <p style="text-align: left;"></p>
 
  </div>
 
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<div id="cs-21"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
'''Problem '''
+
'''Problem 21'''
<p style= "color: #999;font-size: 11px">problem id: </p>
+
<p style= "color: #999;font-size: 11px">problem id: cs-21</p>
 
+
Show that \[j=\frac{\ddot{H}}{H^{3} } +3\frac{\dot{H}}{H^{2} } +1\]
 
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     <p style="text-align: left;"></p>
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     <p style="text-align: left;">Using results of the previous problem, we find
 +
\[j=\frac{\ddot{H}}{H^{3} } -3q-2\]
 +
Substituting
 +
\[q=-\frac{\dot{H}}{H^{2} } -1\]
 +
one finally obtains
 +
\[j=\frac{\ddot{H}}{H^{3} } +3\frac{\dot{H}}{H^{2} } +1\]</p>
 
   </div>
 
   </div>
 
</div></div>
 
</div></div>
  
  
<div id=""></div>
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<div id="cs-22"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
'''Problem '''
+
'''Problem 22'''
<p style= "color: #999;font-size: 11px">problem id: </p>
+
<p style= "color: #999;font-size: 11px">problem id: cs-22</p>
 
+
Express total pressure in flat Universe through the cosmographic parameters.
 
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     <p style="text-align: left;"></p>
+
     <p style="text-align: left;">Excluding the density $\rho $ from the Friedman equations
 +
\[\begin{array}{l} {H=\frac{1}{3} \rho ,} \\ {\frac{\ddot{a}}{a} =H^{2} +\dot{H}=-\frac{1}{6} \left(\rho +3p\right)} \end{array}\]
 +
one finds
 +
\[p=-\left(3H^{2} +2\dot{H}\right)\]
 +
Using the above obtained expression $\dot{H}=-H^{2} (1+q)$ , we obtain
 +
\[p=-H^{2} \left(1-2q\right)\]</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-23"></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
'''Problem 23'''
 +
<p style= "color: #999;font-size: 11px">problem id: cs-23</p>
 +
Express time derivatives $dp/dt,d^{2} p/dt^{2} ,d^{3} p/dt^{3} ,d^{4} p/dt^{4} $ through the cosmographic parameters.
 
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     <p style="text-align: left;"></p>
+
     <p style="text-align: left;">Consequent differentiation w.r.t. time the expression $p=-H^{2} \left(1-2q\right)$ obtained in the previous problem using the expressions for the time derivatives of the cosmographic parameters
 +
\[\begin{array}{l} {\dot{q}=-H\left(j-2q^{2} -q\right),} \\ {\dot{j}=H\left[s+j(2+3q)\right],} \\ {\dot{s}=H\left[l+s(3+4q)\right],} \\ {\dot{l}=H\left[m+l(4+5q)\right]} \end{array}\]
 +
leads to
 +
$$P=-H^2(1-2q)$$
 +
$$\frac{dP}{dt}=-2H^3(j-1)$$
 +
$$\frac{d^2P}{dt^2}=-2H^4(s-j+3q+3)$$
 +
$$\frac{d^3P}{dt^3}=-2H^5\left(l-j(1+q)-3q(7+2q)-2(6+s)\right)$$</p>
 +
$$\frac{d^4P}{dt^4}=-2H^6\left(m-3l+j^2+12j(2+q)+3\left(20+s+q(48+q(27+2q)+s)\right)\right)$$</p>
 
   </div>
 
   </div>
 
</div></div>
 
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<div id=""></div>
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<div id="cs-24"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
'''Problem '''
+
'''Problem 24'''
<p style= "color: #999;font-size: 11px">problem id: </p>
+
<p style= "color: #999;font-size: 11px">problem id: cs-24</p>
 
+
Express time derivatives $d\rho /dt,d^{2} \rho /dt^{2} ,d^{3} \rho /dt^{3} ,d^{4} \rho /dt^{4} $ through the cosmographic parameters.
 
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     <p style="text-align: left;"></p>
+
     <p style="text-align: left;">Consequently differentiate the Friedman equation $\rho =3H^{2} $ and use expressions for the time derivatives of the cosmographic parameters to find[[File:image8.jpg|500px]]</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-25"></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
'''Problem 25'''
 +
<p style= "color: #999;font-size: 11px">problem id: cs-25</p>
 +
Show that the accelerated growth of expansion rate $\dot{H}>0$ takes place under the condition $q<-1$.
 
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     <p style="text-align: left;"></p>
+
     <p style="text-align: left;">\[\dot{H}=-H^{2} (1+q),\quad \dot{H}>0\to q<-1\]</p>
 
   </div>
 
   </div>
 
</div></div>
 
</div></div>
  
  
<div id=""></div>
+
<div id="cs-26"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
'''Problem '''
+
'''Problem 26'''
<p style= "color: #999;font-size: 11px">problem id: </p>
+
<p style= "color: #999;font-size: 11px">problem id: cs-26</p>
 
+
Consider the case of spatially flat Universe and express the scalar (Ricci ) curvature and its time derivatives in terms of the cosmographic parameters (-)
 
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     <p style="text-align: left;"></p>
+
     <p style="text-align: left;">Using the expression $R=-6\left(\frac{\ddot{a}}{a} +H^{2} \right)$ and definition of the deceleration parameter $q=-\frac{\ddot{a}}{aH^{2} } $ , one finds
 +
\[R=-6H^{2} \left(1-q\right)\]
 +
Using the expressions
 +
\[\begin{array}{l} {\dot{H}=-H^{2} (1+q),} \\ {\dot{q}=-H\left(j-2q^{2} -q\right)} \end{array}\]
 +
one obtains
 +
\[\dot{R}=6H^{3} \left(2-j+q\right)\]</p>
 
   </div>
 
   </div>
 
</div></div>
 
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<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-27"></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
'''Problem 27'''
 +
<p style= "color: #999;font-size: 11px">problem id: cs-27</p>
 +
Following [O. Luongo and H. Quevedo, Self-accelerated universe induced by repulsive e$\boldsymbol{\mathrm{\textrm{?}}}$ects as an alternative to dark energy and modi?ed gravities, arXiv: (1507.06446)], introduce the parameter$\lambda \equiv -\frac{\ddot{a}}{a} =qH^{2} $ , so that $\lambda <0$  when the Universe is accelerating, whereas for $\lambda >0$ the Universe decelerates. Luongo and H. Quevedo showed, that the parameter $\lambda $ can be considered as an eigenvalue of the curvature tensor defined in special way. In particular, for FLRW metric the curvature tensor $R$ can be expressed as a (6 $\boldsymbol{\mathrm{\times}}$ 6)$\boldsymbol{\mathrm{-}}$matrix
 +
\[R=diag\left(\lambda ,\lambda ,\lambda ,\tau ,\tau ,\tau \right),\quad \tau \equiv \frac{1}{3} \rho \]
 +
The curvature eigenvalues reflect the behavior of the gravitational interaction and if gravity becomes repulsive in some regions, the eigenvalues must change accordingly; for instance, if repulsive gravity becomes dominant at a particular point, one would expect at that point a change in the sign of at least one eigenvalue. Moreover, if the gravitational field does not diverge at infinity, the eigenvalue must have an extremal at some point before it changes its sign. This means that the extremal of the eigenvalue can be interpreted as the onset of repulsion. Obtain the onset of repulsion condition in terms of cosmographic parameters.
 
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     <p style="text-align: left;"></p>
+
     <p style="text-align: left;">As mentioned above, the onset of repulsion is determined by an extremal of the eigenvalue, i.e.$\dot{\lambda }=0,$
 +
\[\lambda =qH^{2} \quad \to \quad \dot{\lambda }=\dot{q}H^{2} +2qH\dot{H}=0\]
 +
Using the result of the previous problem for $\dot{q}$ and $\dot{H}$ we find that the repulsion onset condition $\dot{\lambda }=0$reduces to
 +
\[j=-q.\]</p>
 
   </div>
 
   </div>
 
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<div id=""></div>
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<div id="cs-28"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
'''Problem '''
+
'''Problem 28'''
<p style= "color: #999;font-size: 11px">problem id: </p>
+
<p style= "color: #999;font-size: 11px">problem id: cs-28</p>
 
+
Represent results of the previous problem in terms of the Hubble parameter and its time derivatives.
 
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   <div class="NavHead">solution</div>
 
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   <div style="width:100%;" class="NavContent">
     <p style="text-align: left;"></p>
+
     <p style="text-align: left;">Solving the system of equations
 +
\[\begin{array}{l} {\dot{H}=-H^{2} (1+q),\quad } \\ {\ddot{H}=H^{3} \left(j+3q+2\right)} \end{array}\]
 +
w.r.t. the variables $q$ and $j$ one finds, that the condition $j=-q$ transforms into
 +
\[\frac{\ddot{H}}{H} =-2\dot{H}\]</p>
 
   </div>
 
   </div>
 
</div></div>
 
</div></div>
  
<div id=""></div>
+
 
 +
<div id="cs-29"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
'''Problem '''
+
'''Problem 29'''
<p style= "color: #999;font-size: 11px">problem id: </p>
+
<p style= "color: #999;font-size: 11px">problem id: cs-29</p>
 
+
Obtain the following integral relation between the Hubble's parameter and the deceleration parameter
 
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   <div class="NavHead">solution</div>
 
   <div class="NavHead">solution</div>

Latest revision as of 03:39, 24 May 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 $j(x)$.


Problem 15

problem id: cs-15

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.


Problem 16

problem id: cs-16

Find decomposition of the inverse Hubble parameter $1/H$ in powers of the red shift $z$.


Problem 17

problem id: cs-17

Obtain relations for transition from the time derivatives to that w.r.t. the red shift.


Problem 18

problem id: cs-18

Can $dH^{n} /dz^{n} $ generally be expressed in terms of the cosmographic parameters?


Problem 19

problem id: cs-19

Show that $$q(z)=frac 12 \frac{d\ln H^2}{d\ln(1+z)}$$


Problem 20

problem id: cs-20

Show that the time derivatives of the Hubble's parameter can be expressed through the cosmographic parameters as follows: $$\dot{H}=-H^2(1+q);$$ $$\ddot{H}=H^3(j+3q+2)$$ $$\dddot{H}=H^4\left(s-4j-3q(q+4)-6\right)$$ $$\ddddot{H}=H^5\left(l-5s+10(q+2)j+30(q+2)+24\right)$$ generally be expressed in terms of the cosmographic parameters?


Problem 21

problem id: cs-21

Show that \[j=\frac{\ddot{H}}{H^{3} } +3\frac{\dot{H}}{H^{2} } +1\]


Problem 22

problem id: cs-22

Express total pressure in flat Universe through the cosmographic parameters.


Problem 23

problem id: cs-23

Express time derivatives $dp/dt,d^{2} p/dt^{2} ,d^{3} p/dt^{3} ,d^{4} p/dt^{4} $ through the cosmographic parameters.


Problem 24

problem id: cs-24

Express time derivatives $d\rho /dt,d^{2} \rho /dt^{2} ,d^{3} \rho /dt^{3} ,d^{4} \rho /dt^{4} $ through the cosmographic parameters.


Problem 25

problem id: cs-25

Show that the accelerated growth of expansion rate $\dot{H}>0$ takes place under the condition $q<-1$.


Problem 26

problem id: cs-26

Consider the case of spatially flat Universe and express the scalar (Ricci ) curvature and its time derivatives in terms of the cosmographic parameters (-)


Problem 27

problem id: cs-27

Following [O. Luongo and H. Quevedo, Self-accelerated universe induced by repulsive e$\boldsymbol{\mathrm{\textrm{?}}}$ects as an alternative to dark energy and modi?ed gravities, arXiv: (1507.06446)], introduce the parameter$\lambda \equiv -\frac{\ddot{a}}{a} =qH^{2} $ , so that $\lambda <0$ when the Universe is accelerating, whereas for $\lambda >0$ the Universe decelerates. Luongo and H. Quevedo showed, that the parameter $\lambda $ can be considered as an eigenvalue of the curvature tensor defined in special way. In particular, for FLRW metric the curvature tensor $R$ can be expressed as a (6 $\boldsymbol{\mathrm{\times}}$ 6)$\boldsymbol{\mathrm{-}}$matrix \[R=diag\left(\lambda ,\lambda ,\lambda ,\tau ,\tau ,\tau \right),\quad \tau \equiv \frac{1}{3} \rho \] The curvature eigenvalues reflect the behavior of the gravitational interaction and if gravity becomes repulsive in some regions, the eigenvalues must change accordingly; for instance, if repulsive gravity becomes dominant at a particular point, one would expect at that point a change in the sign of at least one eigenvalue. Moreover, if the gravitational field does not diverge at infinity, the eigenvalue must have an extremal at some point before it changes its sign. This means that the extremal of the eigenvalue can be interpreted as the onset of repulsion. Obtain the onset of repulsion condition in terms of cosmographic parameters.


Problem 28

problem id: cs-28

Represent results of the previous problem in terms of the Hubble parameter and its time derivatives.


Problem 29

problem id: cs-29

Obtain the following integral relation between the Hubble's parameter and the deceleration parameter


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