Difference between revisions of "New Cosmography"

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Show that for the deceleration parameter the following relation holds:
 
Show that for the deceleration parameter the following relation holds:
 
$$q(a)=-\left(1+\frac{dH/dt}{H^2}\right)-\left(1+\frac{adH/da}{H}\right)$$
 
$$q(a)=-\left(1+\frac{dH/dt}{H^2}\right)-\left(1+\frac{adH/da}{H}\right)$$
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'''Problem 9'''
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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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Let \[C_{n} \equiv \gamma _{n} \frac{a^{(n)} }{aH^{n} }  ,\]
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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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Revision as of 00:24, 2 February 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


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