Difference between revisions of "New Cosmography"

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'''Problem 1'''
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<p style= "color: #999;font-size: 11px">problem id: cs-1</p>
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Using the cosmographic parameters introduced above, expand the scale factor into a Taylor series in time.
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    <p style="text-align: left;">We can write the scale factor in terms of the present time cosmographic parameters:
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\[a(t)\sim 1+H_{0} \Delta t-\frac{1}{2} q_{0} H_{0}^{2} \Delta t^{2} +\frac{1}{6} j_{0} H_{0}^{3} \Delta t^{3} +\frac{1}{24} s_{0} H_{0}^{4} \Delta t^{4} +120l_{0} H_{0}^{5} \Delta t^{5} \]
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This decomposition describes evolution of the Universe on the time interval $\Delta t$ directly through the measurable cosmographic parameters. Each of them describes certain characteristic of the evolution. In particular, the sign of deceleration parameter $q$  indicates whether the dynamics is accelerated or decelerated. In other words, a positive\textbf{ }acceleration parameter indicates that standard gravity predominates over the other species, whereas a negative sign\textbf{ }provides a repulsive e\textbf{ff}ect which overcomes the standard attraction due to gravity. Evolution of the deceleration parameter is described by the jerk parameter $j$. In particular, a positive jerk parameter would\textbf{ }indicate that there exists a transition time when the Universe modifies its expansion. In the vicinity of this transition the modulus of deceleration parameters tends to zero and then changes its sign\textbf{. }The two terms, i.e., $q$ and $j$ fix the local dynamics, but they may be not sufficient to remove the degeneration between different cosmological models and one will need higher terms of the decomposition.</p>
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</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 class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </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 class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </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 class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </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 class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </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 class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </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 class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </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 class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </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 class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </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 class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </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 class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </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 class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </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 class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </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 class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </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 class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </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 class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </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 class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </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 class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </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 class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </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 class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </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 class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </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 class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </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 class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </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 class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </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 class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </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 class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </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 class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </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 class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </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 class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </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 class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </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 class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </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 class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </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 class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </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 class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </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 class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </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 class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </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 class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </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 class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </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 class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </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 class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </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 class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </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 class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </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 class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </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 class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </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 class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </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 class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </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 class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
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Revision as of 22:52, 1 February 2016


Problem 1

problem id: cs-1

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


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