Difference between revisions of "New Cosmography"
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| − | <p style= "color: #999;font-size: 11px">problem id: </p> | + | <p style= "color: #999;font-size: 11px">problem id: cs-2</p> |
| − | + | Using the cosmographic parameters, expand the redshift into a Taylor series in time. | |
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| − | <p style="text-align: left;"></p> | + | <p style="text-align: left;">\[\begin{array}{l} {1+z=\left[\begin{array}{cc} {} & {1+H_{0} (t-t_{0} )-\frac{1}{2} q_{0} H_{0}^{2} (t-t_{0} )^{2} +\frac{1}{3!} j_{0} H_{0}^{3} \left(t-t_{0} \right)^{3} +\frac{1}{4!} s_{0} H_{0}^{4} \left(t-t_{0} \right)^{4} } \\ {} & {+\frac{1}{5!} l_{0} H_{0}^{5} \left(t-t_{0} \right)^{5} \; +{\rm O}\left(\left(t-t_{0} \right)^{6} \right)} \\ {} & {} \end{array}\right]^{-1} ;} \\ {z=H_{0} (t_{0} -t)+\left(1+\frac{q_{0} }{2} \right)H_{0}^{2} (t-t_{0} )^{2} +\cdots .} \end{array}\]</p> |
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Revision as of 19:54, 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.
We can write the scale factor in terms of the present time cosmographic parameters: \[a(t)\sim 1+H_{0} \Delta t-\frac{1}{2} q_{0} H_{0}^{2} \Delta t^{2} +\frac{1}{6} j_{0} H_{0}^{3} \Delta t^{3} +\frac{1}{24} s_{0} H_{0}^{4} \Delta t^{4} +120l_{0} H_{0}^{5} \Delta t^{5} \] 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.
Problem 2
problem id: cs-2
Using the cosmographic parameters, expand the redshift into a Taylor series in time.
\[\begin{array}{l} {1+z=\left[\begin{array}{cc} {} & {1+H_{0} (t-t_{0} )-\frac{1}{2} q_{0} H_{0}^{2} (t-t_{0} )^{2} +\frac{1}{3!} j_{0} H_{0}^{3} \left(t-t_{0} \right)^{3} +\frac{1}{4!} s_{0} H_{0}^{4} \left(t-t_{0} \right)^{4} } \\ {} & {+\frac{1}{5!} l_{0} H_{0}^{5} \left(t-t_{0} \right)^{5} \; +{\rm O}\left(\left(t-t_{0} \right)^{6} \right)} \\ {} & {} \end{array}\right]^{-1} ;} \\ {z=H_{0} (t_{0} -t)+\left(1+\frac{q_{0} }{2} \right)H_{0}^{2} (t-t_{0} )^{2} +\cdots .} \end{array}\]
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