Cosmography - Revision history

Cosmo All at 23:20, 18 June 2015

2015-06-18T23:20:18Z

2015-06-18T23:19:56Z

2014-04-01T02:09:57Z

2012-11-10T17:57:07Z

2012-11-10T17:53:12Z

‎Problem 24: current values of deceleration parameter

2012-11-05T20:45:27Z

‎Problem 24: current values of deceleration parameter

2012-11-05T20:29:40Z

‎Problem 23: current values of deceleration parameter

2012-11-05T20:29:23Z

2012-10-13T16:52:53Z

2012-08-10T10:27:22Z

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 <div id="150_1"></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem 50''' <p style= "color: #999;font-size: 11px">problem id: 150_1</p>

← Older revision		Revision as of 23:19, 18 June 2015
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	</div>		</div>
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		+
		+	<div id="150_1"></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem 50''' <p style= "color: #999;font-size: 11px">problem id: 150_1</p>
		+	Show that \[\frac{d\dot a}{da}=-Hq.\]
		+	<div class="NavFrame collapsed"> <div class="NavHead">solution</div> <div style="width:100%;" class="NavContent">
		+	<p style="text-align: left;">
		+	\[\frac{d\dot a}{da}=\frac{d\dot a}{dt}\frac{dt}{da}=\frac{\ddot a}{\dot a}=\frac{\ddot a}{aH}\equiv-Hq.\]
		+	</p> </div></div></div>

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    </div>
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 === Problem 24: current value of deceleration parameter ===
 Express the current values of deceleration parameter
-\[q_0  \equiv \left. { - \frac{1}{a}\frac{d^2 a}{dt^2}\left[ {\frac{1}{a}\frac{da}{dt}} \right]^{ - 2} } \right|_{t = t_9 }\]
+$q_0  \equiv \left. { - \frac{1}{a}\frac{d^2 a}{dt^2}\left[ {\frac{1}{a}\frac{da}{dt}} \right]^{ - 2} } \right|_{t = t_9 }$
- and the jerk parameter
+and the jerk parameter
-\[j \equiv \left. {\frac{1}{a}\frac{d^3 a}{dt^3}\left[ {\frac{1}{a}\frac{da}{dt}} \right]^{ - 3} } \right|_{t = t_0 }\]
+$j \equiv \left. {\frac{1}{a}\frac{d^3 a}{dt^3}\left[ {\frac{1}{a}\frac{da}{dt}} \right]^{ - 3} } \right|_{t = t_0 }$
- in terms of $N \equiv  - \ln \left( {1 + z} \right)$
+in terms of $N \equiv  - \ln \left( {1 + z} \right)$
 <div class="NavFrame collapsed">
    <div class="NavHead">solution</div>

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 <div id=""></div><div style="border: 1px solid #AAA; padding:5px;">
-=== Problem 24: current values of deceleration parameter ===
+=== Problem 24: current value of deceleration parameter ===
-Represent current values of deceleration parameter $q_0  \equiv \left. { - \frac{1}{a}\frac{d^2 a}{dt^2}\left[ {\frac{1}{a}\frac{da}{dt}} \right]^{ - 2} } \right|_{t = t_9 }$ and jerk parameter $j \equiv \left. {\frac{1}{a}\frac{d^3 a}{dt^3}\left[ {\frac{1}{a}\frac{da}{dt}} \right]^{ - 3} } \right|_{t = t_0 } $ in terms of $N \equiv  - \ln \left( {1 + z} \right)$
+Express the current values of deceleration parameter
 <div class="NavFrame collapsed">
    <div class="NavHead">solution</div>
    <div style="width:100%;" class="NavContent">
      <p style="text-align: left;">For the current time $t = t_0 $ we have  $N = 0$.
-\[
+\begin{align}
-\begin{array}{l}
+  q_0 & = \left. { - \frac{1}{H^2}\left\{ {\frac{1}{2}\frac{d\left( {H^2 } \right)}{dN} + H^2 } \right\}} \right|_{N = 0}.\\
-  q_0  = \left. { - \frac{1}{H^2}\left\{ {\frac{1}{2}\frac{d\left( {H^2 } \right)}{dN} + H^2 } \right\}} \right|_{N = 0} ,\quad  \\
+  j_0  &= \left. {\frac{1}{2H^2}\frac{d^2 H^2 }{dN^2} + \frac{3}{2H^2}\frac{dH^2}{dN} + 1} \right|_{N = 0} . \\
-  j_0  = \left. {\frac{1}{2H^2}\frac{d^2 \left( {H^2 } \right)}{dN^2} + \frac{3}{2H^2}\frac{d\left( {H^2 } \right)}{dN} + 1} \right|_{N = 0}  \\
+  \end{align}
-  \end{array}
+</p>
-\]</p>
    </div>
 </div></div>

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 \[
 \begin{array}{l}
-  q_0  = \left. { - \frac{1}{{H^2 }}\left\{ {\frac{1}{2}\frac{{d\left( {H^2 } \right)}}{{dN}} + H^2 } \right\}} \right|_{N = 0} ,\quad  \\
+  q_0  = \left. { - \frac{1}{H^2}\left\{ {\frac{1}{2}\frac{d\left( {H^2 } \right)}{dN} + H^2 } \right\}} \right|_{N = 0} ,\quad  \\
-  j_0  = \left. {\frac{1}{{2H^2 }}\frac{{d^2 \left( {H^2 } \right)}}{{dN^2 }} + \frac{3}{{2H^2 }}\frac{{d\left( {H^2 } \right)}}{{dN}} + 1} \right|_{N = 0}  \\
+  j_0  = \left. {\frac{1}{2H^2}\frac{d^2 \left( {H^2 } \right)}{dN^2} + \frac{3}{2H^2}\frac{d\left( {H^2 } \right)}{dN} + 1} \right|_{N = 0}  \\
   \end{array}
 \]</p>
    </div>
 </div></div>

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 \[z = H_0d,\]
 where $d$ is the proper distance to the object.
+<br/>
 It was Vesto Melvin Slipher who for the first time observed redshift of extragalactic objects in 1912, in the Lowell observatory in Flagstaff, Arizona. In 1922 he published the redshifts for $41$ spiral galaxies, 36 from which were positive (greater 0.006) and 5 -- negative. The Andromeda Nebula is the fastest to approach us, its redshift is $z=-0.001$.</p>
    </div>
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 </div> --></div>

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 [[Category:Dynamics of the Expanding Universe|8]]
-__NOTOC__
 ''In next problems we use an approach to the description of the evolution of the Universe's, which is called "cosmography"$^*$. It is based entirely on the cosmological principle and on some consequences of the equivalence principle. The term "cosmography" is a synonym for "cosmo-kinematics". Let us recall that kinematics represents the part of mechanics which describes motion of bodies regardless of the forces responsible for it. In this sense cosmography represents nothing else than the kinematics of cosmological expansion.''
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 $^*$See Weinberg, Gravitation and Cosmology, chapter 14.
 <div id="equ60"></div>