Difference between revisions of "New Cosmography"

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First section of [[Cosmography]]
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First section of → [[Cosmography]]
  
 
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     <p style="text-align: left;">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 term is constant, the deceleration term $q$  is also constant and equal to $\mathrm{-}$1, as in the de Sitter and steady-state Universes. In most models of Universes the deceleration term changes in time. One can classify models of Universe on the basis of time dependence of the two parameters. All models can be characterized by whether they expand or contract, and accelerate or decelerate:</p>
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     <p style="text-align: left;">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 term is constant, the deceleration term $q$  is also constant and equal to $\mathrm{-}$1, as in the de Sitter and steady-state Universes. In most models of Universes the deceleration term changes in time. One can classify models of Universe on the basis of time dependence of the two parameters. All models can be characterized by whether they expand or contract, and accelerate or decelerate:
 +
* (a) $H>0,\; q>0$: expanding and decelerating
 +
* (b) $H>0,\; q<0$: expanding and accelerating
 +
* (c) $H<0,\; q>0$: contracting and decelerating
 +
* (d) $H<0,\; q<0$: contracting and accelerating
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* (e) $H>0,\; q=0$: expanding, zero deceleration
 +
* (f) $H<0,\; q=0$: contracting, zero deceleration
 +
* (g) $H=0,\; q=0$ : static.
 +
Of course, generally speaking, both the Hubble parameter and deceleration parameter can change their sign during the evolution. Therefore the evolving Universe can transit from one type to another. It is one of the basic tasks of cosmology to follow this evolution and clarify its causes. There is little doubt that we live in an expanding Universe, and hence only (a), (b), and (e) are possible candidates. 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>
 
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'''Problem '''
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'''Problem 7'''
<p style= "color: #999;font-size: 11px">problem id: </p>
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<p style= "color: #999;font-size: 11px">problem id: cs-7</p>
 
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Show that the deceleration parameter $q$   can be presented in the form
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$$q(x)=\frac{\dot{H}(x)}{H(x)}x-1;\,\,x=1+z$$
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'''Problem '''
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'''Problem 8'''
<p style= "color: #999;font-size: 11px">problem id: </p>
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<p style= "color: #999;font-size: 11px">problem id: cs-8</p>
 +
Show that for the deceleration parameter the following relation holds:
 +
$$q(a)=-\left(1+\frac{dH/dt}{H^2}\right)-\left(1+\frac{adH/da}{H}\right)$$
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'''Problem '''
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'''Problem 9'''
<p style= "color: #999;font-size: 11px">problem id: </p>
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<p style= "color: #999;font-size: 11px">problem id: "cs-9</p>
 
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Show that $$\frac{dq}{d\ln (1+z)}=j-q(2q+1)$$
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'''Problem '''
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'''Problem 10'''
<p style= "color: #999;font-size: 11px">problem id: </p>
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<p style= "color: #999;font-size: 11px">problem id: cs-10</p>
 
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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 $dC_{n} /d\ln (1+z)$ .
 
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     <p style="text-align: left;">\[\begin{array}{l} {\frac{dC_{n} }{d\ln (1+z)} =-a\frac{dC_{n} }{da} =-\frac{1}{H} \frac{dC_{n} }{dt} ;} \\ {\frac{dC_{n} }{d\ln (1+z)} =-\gamma _{n} \left(\frac{a^{\left(n+1\right)} }{aH^{n+1} } -\frac{a^{\left(n\right)} }{aH^{n} } -n\frac{a^{(n)} \dot{H}}{aH^{n+2} } \right)=} \\ {=-\gamma _{n} \left(\frac{1}{\gamma _{n+1} } C_{n+1} -\frac{1}{\gamma _{n} } C_{n} -\frac{1}{\gamma _{n} } nC_{n} \frac{\dot{H}}{H^{2} } \right);} \\ {\frac{dC_{n} }{d\ln (1+z)} =-\frac{\gamma _{n} }{\gamma _{n+1} } C_{n+1} +C_{n} -nC_{n} (1+q)} \end{array}\]</p>
 
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'''Problem '''
 
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'''Problem 11'''
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<p style= "color: #999;font-size: 11px">problem id: cs-11</p>
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Use result of the previous problem to find $dC_{n} /dt$.
 
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     <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>
 
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'''Problem '''
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'''Problem 12'''
<p style= "color: #999;font-size: 11px">problem id: </p>
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<p style= "color: #999;font-size: 11px">problem id: cs-12</p>
 
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Use the general formula for $dC_{n} /dt$ obtained in the previous problem to obtain time derivatives of the cosmographic parameters $q,j,s,l$.
 
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     <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>
 
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'''Problem '''
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'''Problem 13'''
<p style= "color: #999;font-size: 11px">problem id: </p>
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<p style= "color: #999;font-size: 11px">problem id: cs-13</p>
 
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Find derivatives of the cosmographic parameter w.r.t. the red shift.
 
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     <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>
 
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'''Problem '''
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'''Problem 14'''
<p style= "color: #999;font-size: 11px">problem id: </p>
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<p style= "color: #999;font-size: 11px">problem id: cs-14</p>
 
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Let $1+z=1/a\equiv x$. Find $q(x)$ and $j(x)$.
 
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     <p style="text-align: left;">It is easy to see that $\dot{H}=-H'H/a$ where $H'=dH/dx$. Then
 +
\[q=-\frac{\dot{H}}{H^{2} } -1=\frac{H'}{H} x-1\]
 +
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>
 
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'''Problem '''
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'''Problem 15'''
<p style= "color: #999;font-size: 11px">problem id: </p>
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<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.
 
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     <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>
 
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'''Problem '''
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'''Problem 16'''
<p style= "color: #999;font-size: 11px">problem id: </p>
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<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  $z$.
 
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     <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>
 
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'''Problem '''
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'''Problem 17'''
<p style= "color: #999;font-size: 11px">problem id: </p>
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<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.
 
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     <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>
 
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'''Problem '''
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'''Problem 18'''
<p style= "color: #999;font-size: 11px">problem id: </p>
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<p style= "color: #999;font-size: 11px">problem id: cs-18</p>
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Can $dH^{n} /dz^{n} $ generally be expressed in terms of the cosmographic parameters?
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'''Problem '''
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'''Problem 19'''
<p style= "color: #999;font-size: 11px">problem id: </p>
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<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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'''Problem '''
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'''Problem 20'''
<p style= "color: #999;font-size: 11px">problem id: </p>
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<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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'''Problem '''
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'''Problem 21'''
<p style= "color: #999;font-size: 11px">problem id: </p>
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<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;">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>
 
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'''Problem '''
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'''Problem 22'''
<p style= "color: #999;font-size: 11px">problem id: </p>
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<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;">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>
 
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<div id="cs-23"></div>
 
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'''Problem '''
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'''Problem 23'''
<p style= "color: #999;font-size: 11px">problem id: </p>
+
<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>
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     <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>
 
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<div id="cs-24"></div>
 
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'''Problem '''
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'''Problem 24'''
<p style= "color: #999;font-size: 11px">problem id: </p>
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<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>
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     <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>
 
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+
 
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<div id="cs-25"></div>
 
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'''Problem '''
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'''Problem 25'''
<p style= "color: #999;font-size: 11px">problem id: </p>
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<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>
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     <p style="text-align: left;">\[\dot{H}=-H^{2} (1+q),\quad \dot{H}>0\to q<-1\]</p>
 
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<div id="cs-26"></div>
 
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'''Problem '''
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'''Problem 26'''
<p style= "color: #999;font-size: 11px">problem id: </p>
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<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>
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     <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>
 
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'''Problem '''
 
<p style= "color: #999;font-size: 11px">problem id: </p>
 
  
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<div id="cs-27"></div>
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<div style="border: 1px solid #AAA; padding:5px;">
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'''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>
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     <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>
 
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<div id="cs-28"></div>
 
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'''Problem '''
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'''Problem 28'''
<p style= "color: #999;font-size: 11px">problem id: </p>
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<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>
 
   <div class="NavHead">solution</div>
 
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     <p style="text-align: left;"></p>
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     <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>
 
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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-29"></div>
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<div style="border: 1px solid #AAA; padding:5px;">
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'''Problem 29'''
 +
<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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