New Cosmography - Revision history

Cosmo All at 00:39, 24 May 2016

2016-05-24T00:39:59Z

Cosmo All at 22:44, 23 May 2016

2016-05-23T22:44:24Z

Cosmo All at 22:43, 23 May 2016

2016-05-23T22:43:28Z

Cosmo All at 22:41, 23 May 2016

2016-05-23T22:41:58Z

Cosmo All at 21:39, 28 March 2016

2016-03-28T21:39:29Z

Cosmo All at 21:27, 28 March 2016

2016-03-28T21:27:17Z

Cosmo All at 08:21, 2 February 2016

2016-02-02T08:21:14Z

Cosmo All at 22:45, 1 February 2016

2016-02-01T22:45:45Z

Cosmo All at 22:38, 1 February 2016

2016-02-01T22:38:27Z

Cosmo All at 22:34, 1 February 2016

2016-02-01T22:34:38Z

@@ Line 323: / Line 323: @@
    <div class="NavHead">solution</div>
    <div style="width:100%;" class="NavContent">
-     <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|400px]]</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 id=""></div>
+<div id="cs-25"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
-'''Problem '''
+'''Problem 25'''
-<p style= "color: #999;font-size: 11px">problem id: </p>
+<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 $q<-1$.
 <div class="NavFrame collapsed">
    <div class="NavHead">solution</div>
    <div style="width:100%;" class="NavContent">
-     <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 id=""></div>
+<div id="cs-26"></div>
 <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 (-)
 <div class="NavFrame collapsed">
    <div class="NavHead">solution</div>
    <div style="width:100%;" class="NavContent">
-     <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
    </div>
 </div></div>
 <div id=""></div>
 <div style="border: 1px solid #AAA; padding:5px;">
-'''Problem '''
+'''Problem 27'''
-<p style= "color: #999;font-size: 11px">problem id: </p>
+<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
 <div class="NavFrame collapsed">
    <div class="NavHead">solution</div>
    <div style="width:100%;" class="NavContent">
-     <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.$\dot{\lambda }=0,$
    </div>
 </div></div>
-<div id=""></div>
+<div id="cs-28"></div>
 <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.
 <div class="NavFrame collapsed">
    <div class="NavHead">solution</div>
    <div style="width:100%;" class="NavContent">
-     <p style="text-align: left;"></p>
+     <p style="text-align: left;">Solving the system of equations
    </div>
 </div></div>
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    <div class="NavHead">solution</div>

@@ Line 323: / Line 323: @@
    <div class="NavHead">solution</div>
    <div style="width:100%;" class="NavContent">
-     <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|400px|thumb|left|alt text]]</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|400px]]</p>
    </div>
 </div></div>
 <div id=""></div>

@@ Line 323: / Line 323: @@
    <div class="NavHead">solution</div>
    <div style="width:100%;" class="NavContent">
-     <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]]</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|400px|thumb|left|alt text]]</p>
    </div>
 </div></div>

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-<div id=""></div>
+<div id="cs-24"></div>
 <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.
 <div class="NavFrame collapsed">
    <div class="NavHead">solution</div>
    <div style="width:100%;" class="NavContent">
-     <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]]</p>
    </div>
 </div></div>

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-<div id=""></div>
+<div id="cs-18"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
 '''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?
 </div>
-<div id=""></div>
+<div id="cs-19"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
 '''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 $$q(z)=frac 12 \frac{d\ln H^2}{d\ln(1+z)}$$
 </div>
-<div id=""></div>
+<div id="cs-20"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
 '''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:
 $$\dot{H}=-H^2(1+q);$$
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-<div id=""></div>
+<div id="cs-21"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
 '''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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-<div id=""></div>
+<div id="cs-22"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
 '''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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-<div id=""></div>
+<div id="cs-23"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
-'''Problem '''
+'''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.
 <div class="NavFrame collapsed">
    <div class="NavHead">solution</div>
    <div style="width:100%;" class="NavContent">
-     <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
    </div>
 </div></div>

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    <div class="NavHead">solution</div>
    <div style="width:100%;" class="NavContent">
-     <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
 \[q=-\frac{\dot{H}}{H^{2} } -1=\frac{H'}{H} x-1\]
 Calculating $j$, making use of $a'=-a^{2} $, we obtain

← Older revision		Revision as of 22:38, 1 February 2016
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−	First section of [[Cosmography]]	+	First section of → [[Cosmography]]

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