New Cosmography

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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

Use result of the previous problem to find $dC_{n} /dt$.


Problem 12

problem id: cs-12

Use the general formula for $dC_{n} /dt$ obtained in the previous problem to obtain time derivatives of the cosmographic parameters $q,j,s,l$.


Problem 13

problem id: cs-13

Find derivatives of the cosmographic parameter w.r.t. the red shift.


Problem 14

problem id: cs-14

Let $1+z=1/a\equiv x$. Find $q(x)$ and $j(x)$.


Problem 15

problem id: cs-15

Express the derivatives $d^{2} H/dz^{2} $ , $d^{3} H/dz^{3} $ and $d^{4} H/dz^{4} $ in terms of the cosmographic parameters.


Problem 16

problem id: cs-16

Find decomposition of the inverse Hubble parameter $1/H$ in powers of the red shift $z$.


Problem 17

problem id: cs-17

Obtain relations for transition from the time derivatives to that w.r.t. the red shift.


Problem 18

problem id:

Can $dH^{n} /dz^{n} $ generally be expressed in terms of the cosmographic parameters?


Problem 19

problem id:

Show that $$q(z)=frac 12 \frac{d\ln H^2}{d\ln(1+z)}$$


Problem 20

problem id:

Show that the time derivatives of the Hubble's parameter can be expressed through the cosmographic parameters as follows: $$\dot{H}=-H^2(1+q);$$ $$\ddot{H}=H^3(j+3q+2)$$ $$\dddot{H}=H^4\left(s-4j-3q(q+4)-6\right)$$ $$\ddddot{H}=H^5\left(l-5s+10(q+2)j+30(q+2)+24\right)$$ generally be expressed in terms of the cosmographic parameters?


Problem 21

problem id:

Show that \[j=\frac{\ddot{H}}{H^{3} } +3\frac{\dot{H}}{H^{2} } +1\]


Problem 22

problem id:

Express total pressure in flat Universe through the cosmographic parameters.


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