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In the following problems we use an approach to the description of the evolution of the Universe, which is called "cosmography"$^*$. It is based entirely on the cosmological principle and on some consequences of the equivalence principle. The term "cosmography" is a synonym for "cosmo-kinematics". Let us recall that kinematics represents the part of mechanics which describes motion of bodies regardless of the forces responsible for it. In this sense cosmography represents nothing else than the kinematics of cosmological expansion.

In order to construct the key cosmological quantity $a(t)$ one needs the equations of motion (the Einstein's equation) and some assumptions on the material composition of the Universe, which enable one to obtain the energy-momentum tensor. The efficiency of cosmography lies in the ability to test cosmological models of any kind, that are compatible with the cosmological principle. Modifications of General Relativity or introduction of new components (such as dark matter or dark energy) certainly change the dependence $a(t)$, but they absolutely do not affect the kinematics of the expanding Universe.

The rate of Universe's expansion, determined by Hubble parameter $H(t)$, depends on time. The deceleration parameter $q(t)$ is used to quantify this dependence. Let us define it through the expansion of the scale factor $a(t)$ in a Taylor series in the vicinity of current time ${{t}_{0}}$: \[a(t)=a\left( {{t}_{0}} \right) +\dot{a}\left( {{t}_{0}} \right)\left[ t-{{t}_{0}} \right] +\frac{1}{2}\ddot{a}({t}_{0}) {{\left[ t-{{t}_{0}} \right]}^{2}}+\cdots\] Let us present this in the form \[\frac{a(t)}{a\left( {{t}_{0}} \right)} =1+{{H}_{0}}\left[ t-{{t}_{0}} \right] -\frac{{{q}_{0}}}{2}H_{0}^{2} {{\left[ t-{{t}_{0}} \right]}^{2}}+\cdots\] where the deceleration parameter is \[q(t)\equiv -\frac{\ddot{a}(t)a(t)}{{{{\dot{a}}}^{2}}(t)} =-\frac{\ddot{a}(t)}{a(t)}\frac{1}{{{H}^{2}}(t)}.\]

Note that the accelerated growth of scale factor takes place for $q<0$. When the sign of the deceleration parameter was originally defined, it seemed evident that gravity is the only force that governs the dynamics of Universe and it should slow down its expansion. The choice of the sign was determined then by natural wish to deal with positive quantities. This choice turned out to contradict the observable dynamics and became an example of historical curiosity.

In order to describe the kinematics of the cosmological expansion in more detail it is useful to consider the extended set of the parameters: \begin{align} H(t)\equiv &\frac{1}{a}\frac{da}{dt}\\ q(t)\equiv& -\frac{1}{a}\frac{{{d}^{2}}a}{d{{t}^{2}}}{{\left[ \frac{1}{a}\frac{da}{dt} \right]}^{-2}}\\ j(t)\equiv &\frac{1}{a}\frac{{{d}^{3}}a}{d{{t}^{3}}}{{\left[ \frac{1}{a}\frac{da}{dt} \right]}^{-3}}\\ s(t)\equiv &\frac{1}{a}\frac{{{d}^{4}}a}{d{{t}^{4}}}{{\left[ \frac{1}{a}\frac{da}{dt} \right]}^{-4}}\\ l(t)\equiv&\frac{1}{a}\frac{{{d}^{5}}a}{d{{t}^{5}}}{{\left[ \frac{1}{a}\frac{da}{dt} \right]}^{-5}} \end{align}

$^*$See Weinberg, Gravitation and Cosmology, chapter 14.

Problem 1: scale factor series

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

Problem 2: redshift series

Using these cosmographic parameters, expand the redshift into a Taylor series in time.

Problem 3: $q$, $z$ and $H$

Obtain the following relations between the deceleration parameter and Hubble's parameter \[q(t)=\frac{d}{dt}\left( \frac{1}{H} \right)-1;\quad q(z)=\frac{1+z}{H}\frac{dH}{dz}-1;\quad q(z)=\frac{d\ln H}{dz}(1+z)-1.\]

Problem 4: $q(a)$

Show that for the deceleration parameter the following relation holds: \[q\left( a \right)=-\left( 1+ \frac{\frac{dH}{dt}}{{{H}^{2}}} \right) -\left( 1+\frac{a\frac{dH}{da}}{H} \right).\]

Problem 5: derivatives

Show that derivatives of lower cosmographic parameters can expressed through the higher ones.

Problem 6: $q(z)$

Prove that \[\frac{dq}{d\ln (1+z)}=j-q(2q+1).\]

Problem 7: $d^{n}H/dz^{n}$

Show that the derivatives $\frac{dH}{dz}$ and $\frac{{{d}^{2}}H}{d{{z}^{2}}}$ can be expressed through the parameters $q$ and $j$.

Problem 8: $d^{n}H/dt^{n}$

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

Problem 9: Ricci scalar

Consider the case of spatially flat Universe and express the scalar (Ricci) curvature and its time derivatives in terms of the cosmographic parameters $q,j,s,l$.

Problem 10: acceleration and deceleration

Show that the accelerated growth of expansion rate $\dot{H}>0$ takes place on the condition $q<-1$.

Problem 11: $H[z]$

Obtain the relation \[H(z)=-\frac{1}{1+z}\frac{dz}{dt}.\]

Problem 12: inflection point

Let $t_a$ be the moment in the history of the Universe when the decelerated expansion turned to the accelerated one, i.e. $q(t_a)=0$, and let $t_{1}<t_{a}$ and $t_{2}>t_{a}$ be two moments in the vicinity of $t_{a}$. Show that \[\Delta t\equiv t_1-t_2 =\frac{1}{H_1}-\frac{1}{H_2}.\]

Problem 13: $H[q]$

Obtain the following integral relation between the Hubble's parameter and the deceleration parameter \[H=H_0\exp\left[ \int\limits_0^z [q(z^\prime)+1] d\ln(1+z^\prime)\right].\]

Problem 14: Hubble law for close galaxies

Reformulate the Hubble's law in terms of redshift for close galaxies $(z\ll 1)$.

Problem 15: $\tfrac{d}{dt}[\tfrac{d}{dz}]$

Show that \[\frac{d}{dt}=-(1+z)H\frac{d}{dz}.\]

Problem 16: $\tfrac{d^{n}}{dt^{n}}[\tfrac{d}{dz}]$

Obtain the transformation law from the higher time derivatives to the derivatives with respect to redshift: \[\frac{d^{(i)}}{dt}\to \frac{d^{(i)}}{dz}.\]

Problem 17: $\tfrac{d^{n}H^2}{dz^n}$

Calculate the derivatives of Hubble's parameter squared with respect to redshift \[\frac{d^{(i)}H^2}{dz^{(i)}},\quad i=1,2,3,4\] and express them in terms of the cosmographic parameters.

Problem 18: $q[H(1+z)]$

Show that the deceleration parameter $q$ can be presented in the form \[q(x) = \frac{H'(x)}{H(x)}x - 1;\; x = 1 + z.\]

Problem 19: $q[H(z)]$

Show that \[q(z)=\frac{1}{2}\frac{d\ln {H^2}}{d\ln (1+z)}.\]

Problem 20: $dH/dz$

Express the derivatives $dH/dz$ and $d^2H/dz^2$ throgh the parameters $q$ and \[r \equiv \frac{\dddot a}{aH^3}.\]

Problem 21: averaged $q$

Consider the time average of the deceleration parameter \[\bar{q}\left( {{t}_{0}} \right) =\frac{1}{{{t}_{0}}}\int_{0}^{{{t}_{0}}}{q(t)dt}\] and show that it can be evaluated without integration of equation of motion for the scale factor.

Problem 22: age of the Universe via $q$

Show that the current age of the Universe is proportional to $H_{0}^{-1}$ and the proportionality coefficient is determined by the average value of the deceleration parameter.

Problem 23: proper distance through $q_0$

Show that the proper distance to an object with redshift $z$ is related to the current deceleration parameter $q_0$ as \[R=\frac{c}{H_{0}q_{0}^{2}}\,\frac{1}{1+z}\, \Big[q_{0}z +(q_{0}-1)\big(\sqrt{1+2q_0}-1\big)\Big].\]

Problem 24: current value of deceleration parameter

Express the current values of deceleration parameter $q_0 \equiv \left. { - \frac{1}{a}\frac{d^2 a}{dt^2}\left[ {\frac{1}{a}\frac{da}{dt}} \right]^{ - 2} } \right|_{t = t_9 }$ and the jerk parameter $j \equiv \left. {\frac{1}{a}\frac{d^3 a}{dt^3}\left[ {\frac{1}{a}\frac{da}{dt}} \right]^{ - 3} } \right|_{t = t_0 }$ in terms of $N \equiv - \ln \left( {1 + z} \right)$

Problem 25

problem id:

Using $d_l (z)=a_0 (1+z)f(\chi)$, where \[\chi=\frac1{a_0}\int\limits_0^z\frac{du}{H(u)},\quad f(\chi)=\left\{\begin{array}{rcl} \chi & - & flat\ case\\ \sinh(\chi) & - & open\ case\\ \sin(\chi) & - & closed\ case. \end{array}\right.\] find the standard luminosity distance-versus-redshift relation up to the second order in $z$: \[d_L=\frac z{H_0}\left[1+\left(\frac{1-q_0}2\right)z+O(z^2)\right].\]

Problem 26

problem id:

Many supernovae give data in the $z>1$ redshift range. Why is this a problem for the above formula for the $z$-redshift? (Problems 2) - 10) are all from the Visser convergence article)

Problem 27

problem id:

Give physical reasons for the above divergence at $z=-1$.

Problem 28

problem id:

Sometimes a "pivot" is used: \[z=z_{pivot}+\Delta z,\] \[\frac1{1+z_{pivot}+\Delta z}=\frac{a(t)}{a_0}=1+H_0(t-t_0)-\frac1{2!}{q_0 H_0^2}(t-t_0)^2+\frac1{3!}{j_0 H_0^3}(t-t_0)^3+O(|t-t_0|^4).\] What is the convergence radius now?

Problem 29

problem id:

The most commonly used definition of redshift is \[z=\frac{\lambda_0-\lambda_e}\lambda_e=\frac{\Delta\lambda}\lambda_e.\] Let's introduce a new redshift: \[z=\frac{\lambda_0-\lambda_e}\lambda_0=\frac{\Delta\lambda}\lambda_0.\]

Problem 30

problem id:

Argue why, on physical grounds, we can't extrapolate beyond $y=1$.

Problem 31

problem id: cg_7

The most commonly used distance is the luminosity distance, and it is related to the distance modulus in the following way: \[\mu_D=5\log_10[d_L/(10\ pc)]=5\log_10[d_L/(1\ Mpc)]+25.\] However, alternative distances are also used (for a variety of mathematical purposes):
1) The "photon flux distance": \[d_F=\frac{d_L}{(1+z)^{1/2}}.\]
2) The "photon count distance": \[d_P=\frac{d_L}{(1+z)}.\]
3) The "deceleration distance": \[d_Q=\frac{d_L}{(1+z)^{3/2}}.\]
4) The "angular diameter distance": \[d_A=\frac{d_L}{(1+z)^{2}}.\]

Obtain the Hubble law for these distances (in terms of $z$-redshift, up to the second power by z)

Problem 32

problem id:

Do the same for the distance modulus directly.

Problem 33

problem id:

Do problem #cg_7 but for $y$-redshift.

Problem 34

problem id: cg_10

Obtain the THIRD-order luminosity distance expansion in terms of $z$-redshift

Problem 35

problem id:

Alternative (if less physically evident) redshifts are also viable. One promising redshift is the $y_4$ redshift: $y_4 = \arctan(y)$. Obtain the second order redshift formula for $d_L$ in terms of $y_4$.

Problem 36

problem id:

$H_0a_0/c\gg1$ is a generic prediction of inflationary cosmology. Why is this an obstacle in proving/measuring the curvature of pace based on cosmographic methods?

Problem 37

problem id:

When looking at the various series formulas, you might be tempted to just take the highest-power formulas you can get and work with them. Why is this not a good idea?

Problem 38

problem id:

Using the definition of the redshift, find the ratio \[\frac{\Delta z}{\Delta t_{obs}},\] called "redshift drift" through the Hubble Parameter for a fixed/commoving observer and emitter: \[\int\limits_{t_s}^{t_o}\frac{dt}{a(t)}=\int\limits_{t_s+\Delta t_s}^{t_o+\Delta t_o}\frac{dt}{a(t)}.\]

Problem 39

problem id:

A photon's physical distance traveled is \[D=c\int dt=c(t_0-t_s).\] Using the definition of the redshift, construct a power series for $z$-redshift based on this physical distance.

Problem 40

problem id:

Invert result of the previous problem (up to $z^3$).

Problem 41

problem id:

Obtain a power series (in $z$) breakdown of redshift drift up to $z^3$ (hint: use $(dH)/(dz)$ formulas)

Problem 42

problem id:

Using the Friedmann equations, continuity equations, and the standard definitions for heat capacity(that is, \[C_V=\frac{\partial U}{\partial T},\quad C_P=\frac{\partial h}{\partial T},\] where $U=V_0\rho_ta^3$, $h=V_0(\rho_t+P)a^3$, show that \[C_P=\frac{V_0}{4\pi G}\frac{H^2}{T'}\frac{j-1}{(1+z)^4},\] \[C_V=\frac{V_0}{8\pi G}\frac{H^2}{T'}\frac{2q-1}{(1+z)^4}.\]

Problem 43

problem id:

What signs of $C_p$ and $C_v$ are predicted by the $\Lambda-CDM$ model? ($q_{0\Lambda CDM}=-1+\frac32\Omega_m$, $j_{0\Lambda CDM}=1$ and experimentally, $\Omega_m=0.274\pm0.015$)

Problem 44

problem id: cg_20

In cosmology, the scale factor is sometimes presented as a power series \[a(t)=c_0|t-t_\odot|^{\eta_0}+c_1|t-t_\odot|^{\eta_1}+c_2|t-t_\odot|^{\eta_2}+c_3|t-t_\odot|^{\eta_3}+\ldots\]. Get analogous series for $H$ and $q$.

Problem 45

problem id:

What values of the powers in the power series are required for the following singularities?
a) Big bang/crunch (scale factor $a=0$)
b) Big Rip (scale factor is infinite)
c) Sudden singularity ($n$th derivative of the scale factor is infinite)
d) Extremality event (derivative of scale factor $= 1$) \end{description}

Problem 46

problem id:

Analyze the possible behavior of the Hubble parameter around the cosmological milestones (hint: use your solution of problem #cg_20).

Problem 47

problem id:

Going outside of the bounds of regular cosmography, let's assume the validity of the Friedmann equations. It is sometimes useful to expand the Equation of State as a series (like $p=p_0+\kappa_0(\rho-\rho_0)+O[(\rho-\rho_0)^2],$) and describe the EoS parameter ($w9t)=p/\rho$) at arbitrary times through the cosmographic parameters.

Problem 48

problem id:

Analogously to the previous problem, analyze the slope parameter \[\kappa=\frac{dp}{d\rho}.\] Specifically, we are interested in the slope parameter at the present time, since that is the value that is seen in the series expansion.

Problem 49

problem id:

Continuing the previous problem, let's look at the third order term - $d^2 p/d\rho^2$.

Problem 50

problem id: 150_1

Show that \[\frac{d\dot a}{da}=-Hq.\]