# New Cosmography

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.

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.

\[\begin{array}{l} {1+z=\left[\begin{array}{cc} {} & {1+H_{0} (t-t_{0} )-\frac{1}{2} q_{0} H_{0}^{2} (t-t_{0} )^{2} +\frac{1}{3!} j_{0} H_{0}^{3} \left(t-t_{0} \right)^{3} +\frac{1}{4!} s_{0} H_{0}^{4} \left(t-t_{0} \right)^{4} } \\ {} & {+\frac{1}{5!} l_{0} H_{0}^{5} \left(t-t_{0} \right)^{5} \; +{\rm O}\left(\left(t-t_{0} \right)^{6} \right)} \\ {} & {} \end{array}\right]^{-1} ;} \\ {z=H_{0} (t_{0} -t)+\left(1+\frac{q_{0} }{2} \right)H_{0}^{2} (t-t_{0} )^{2} +\cdots .} \end{array}\]

**Problem 3**

problem id: cs-3

What is the reason for the statement that the cosmological parameters are model-independent?

The cosmographic parameters are model-independent quantities for the simple reason: these parameters are not functions of the EoS parameters $w$ or $w_{i} $ of the cosmic fluid filling the Universe in a concrete model.

**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 \[q=-\frac{d\dot{a}}{Hda} \]

\[q=-\frac{d\dot{a}}{Hda} =-\frac{\ddot{a}dt}{Hda} =-\frac{\ddot{a}}{aH^{2} } .\] It corresponds to the standard definition of the deceleration parameter \[q-\frac{\ddot{a}}{aH^{2} } .\]

**Problem 6**

problem id: cs-6

Classify models of Universe basing on the two cosmographic parameters -- the Hubble parameter and the deceleration parameter.

When the rate of expansion never changes, and $\dot{a}$ is constant, the scaling factor is proportional to time $t$ , and the deceleration term is zero. When the Hubble term is constant, the deceleration term $q$ is also constant and equal to $\mathrm{-}$1, as in the de Sitter and steady-state Universes. In most models of Universes the deceleration term changes in time. One can classify models of Universe on the basis of time dependence of the two parameters. All models can be characterized by whether they expand or contract, and accelerate or decelerate:

- (a) $H>0,\; q>0$: expanding and decelerating
- (b) $H>0,\; q<0$: expanding and accelerating
- (c) $H<0,\; q>0$: contracting and decelerating
- (d) $H<0,\; q<0$: contracting and accelerating
- (e) $H>0,\; q=0$: expanding, zero deceleration
- (f) $H<0,\; q=0$: contracting, zero deceleration
- (g) $H=0,\; q=0$ : static.

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