Evolution of Universe

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Problem 1

Rewrite the first Friedman equation in terms of redshift and analyze the contributions of separate components at different stages of Universe evolution.

Problem 2

Find the time dependence for the scale factor and analyze asymptotes of the dependence. Plot $a(t)$.

Problem 3

Determine the redshift value corresponding to equality of radiation and matter densities.

Problem 4

Construct effective one-dimensional potential (see problem of Chapter 3)

Problem 5

Show that the following holds: $\dot H= -4\pi G\rho_m$ and $\ddot H= 12\pi G\rho_m H$.

Problem 6

Expand the scale factor in Taylor series near the time moment $t_0$: \[\frac{a(t)}{a(t_0)}=1+\sum\limits_{n=1}^\infty\frac{A_n(t_0)}{n!}[H_0(t-t_0)]^n;\quad A_n\equiv\frac{1}{aH^n}\frac{d^na}{dt^n}\] and calculate values for few first coefficients $A_n$.

Problem 7

Show that all the coefficients $A_n$ can be expressed through elementary functions of the deceleration parameter $q$ or the density parameter \[\Omega_m=\frac23(1+q).\]

Problem 8

Consider the case of flat Universe filled by non-relativistic matter and dark energy with state equation $p_X=w\rho_X$, where the state parameter $w$ is parameterized as the following \[w=w_0+ w_a(1-a)=w_0 + w_a \frac{z}{1+z}.\] Express current values of cosmographic parameters through $w_0$ and $w_a$. % (See Cosmography of f(R) gravity)

Problem 9

Show that the results of the previous problem applied to SCM coincide with the ones obtained in the problem.

Problem 10

Photons with $z=0.1,\ 1,\ 100,\ 1000$ are registered. What was the Universe age $t_U$ in the moment of their emission? What period of time $t_{ph}$ were the photons on the way? Plot $t_U(z)$ and $t_{ph}(z)$

Problem 11

Determine the present physical distance to the object that emitted light with current redshift $z$

Problem 12

A photon was emitted at time $t$ and registered at time $t_0$ with red shift $z$. Find and plot the dependence of emission time on the redshift $t(z)$.

Problem 13

Find the time dependence for the scale factor and analyze its asymptotes. Plot $a(t)$

Problem 14

Using the explicit solution for scale factor $a(t)$, obtained in the previous problem, find the cosmic horizon $R_h$ (see Chapter 3)

Problem 15

Show that the Universe becomes $R_h$-delimited only after the cosmological constant starts to dominate.

Problem 16

Analyze the stability of Friedmann equations.

Problem 17

Determine solutions for the perturbations $\delta\rho_m$ and $\delta H$. Make sure that the solutions are stable.

Problem 18

Rewrite the first Friedman equation in terms of conformal time.

Problem 19

Find relation between the scale factor and conformal time

Problem 20

Find explicit dependence of the scale factor on the conformal time.

Problem 21

Find relative density of dark energy $10^9$ years later.

Problem 22

Find variation rate of relative density of dark energy. What are its asymptotic values? Plot its time dependence.

Problem 23

Estimate size of the cosmological horizon.

Problem 24

Find time dependence of Hubble parameter and plot it.

Problem 25

Find time dependence of dark matter density.

Problem 26

Find the asymptotic (in time) value of the Hubble parameter.

Problem 27

At present the age of the Universe $t_0\simeq13.7\cdot10^9$ years is close to the Hubble time $t_H=H_0^{-1}\simeq14\cdot10^9$ years. Does the relation $t^*\simeq t_H(t^*)=H^{-1}(t^*)$ for the age $t^*$ of the Universe hold for any moment of its evolution

Problem 28

Find current value of the deceleration parameter.

Problem 29

Find the redshift dependence of the deceleration parameter. Analyze the limiting cases.

Problem 30

Find and plot the time dependence of the deceleration parameter.

Problem 31

Find the moment of time when the dark energy started to dominate over dark matter. What redshift did it correspond to?

Problem 32

Determine the moment of time and redshift value corresponding to the transition from decelerated expansion of the Universe to the accelerated one.

Problem 33

Solve the previous problem using the derivative $d\eta/d\ln a$.

Problem 34

Is dark energy domination necessary for transition to the accelerated expansion?

Problem 35

Consider flat Universe composed of matter and dark energy in form of cosmological constant. Find the redshift value corresponding to equality of densities of the both components $\rho_m(z_{eq})=\rho_\Lambda(z_{eq})$ and the one corresponding to beginning of the accelerated expansion $q\left(z_{accel} \right) = 0$. Obtain relation between $z_{eq}$ and $z_{accel}$.

Problem 36

Show that density perturbations stop to grow after the transition from dust to $\Lambda$-dominated era.

Problem 37

What happens to the velocity fluctuations of non-relativistic matter and radiation with respect to Hubble flow in the epoch of cosmological constant domination?

Problem 38

Find the ratio of baryon to non-baryon components in the galactic halo.

Problem 39

Imagine that in the Universe described by SCM the dark energy was instantly switched off. Analyze further dynamics of the Universe.

Problem 40

Estimate density of dark energy in form of cosmological constant using the Hubble diagram for the neighborhood of the Local group.

Problem 41

Estimate the Local group mass by methods used in the previous problem.

Problem 42

Find "weak" points in the argumentation of the two preceding problems.

Problem 43

The product of the age of the Universe and current Hubble parameter (the Hubble's constant) is a very important test (Sandage consistency test) of internal consistency for any model of Universe. Analyze the parameters on which the product $H_0 t_0$ depends in the Big Bang model and in the SCM.

Problem 44

Show that for a fixed source of radiation the luminosity distance for high redshift values in flat Universe is greater for the dark energy dominated case compared to the non-relativistic matter dominated one.

Problem 45

In the observations that discovered the accelerated expansion of the Universe the researchers in particular detected two $Ia$ type supernovae: $1992P$, $z=0.026$, $m=16.08$ and $1997ap$, $z=0.83$, $m=24.32$. Show that these observed parameters are in accordance with the SCM.

Problem 46

Find the redshift value, at which a source of linear dimension $d$ has minimum visible angular size.

Problem 47

Compare the observed value of the dark energy density with the one expected from the dimensionality considerations (the cosmological constant problem).

Problem 48

Determine the density of vacuum energy using the Planck scale as cutoff parameter.

Problem 49

Identifying the vacuum fluctuations density with the observable dark energy value in SCM, find the required frequency cutoff magnitude in the fluctuation spectrum.

Problem 50

What purely cosmological problem originates from the divergence of the zero-point energy density?

Problem 51

With exact supersymmetry, the bosonic contribution to cosmological constant is canceled by its fermionic counterpart. However, we know that our world looks like not supersymmetric. Supersymmetry, if exists, has to be broken above or around $100GeV$ scale. Compare the observed value of the dark energy density with the one expected from broken supersymmetry.

Problem 52

Determine duration of the inflation period.

Problem 53

Plot the dependence of luminosity distance $d_L$ (in units of $H_0^{-1}$) on the redshift $z$ for the two-component flat Universe with non-relativistic liquid ($w=0$) and cosmological constant ($w=-1$). Consider the following cases:
a) $\Omega_\Lambda^0=0$;
b) $\Omega_\Lambda^0=0.3$;
c) $\Omega_\Lambda^0=0.7$;
d) $\Omega_\Lambda^0=1$.

Problem 54

For a Universe filled by dark energy with state equation $p_{DE}=w_{DE}\rho_{DE}$ and non-relativistic matter obtain the Taylor series for $d_L$ in terms of $z$ near the observation point $z_0=0$. Explain the obtained result.

Problem 55

Determine position of the first acoustic peak in the CMB power spectrum produced by baryon oscillations on the surface of the last scattering.

Problem 56

\bf Compare the asymptotes of time dependence of the scale factor $a(t)$ for the SCM and de Sitter models. Explain physical reasons of their distinction.

Problem 57

Redshift for any object slowly changes due to the acceleration (or deceleration) of the Universe expansion. Estimate change of velocity in one year.

Problem 58

Determine the lower limit of ratio of the total volume of the Universe to the observed one?

Problem 59

What is the difference between the inflationary expansion in the early Universe and the present accelerated expansion?

Problem 60

Compare the values of Hubble parameter at the beginning of the inflation period and at the beginning of the present accelerated expansion of Universe.

Problem 61

Plot the dependencies $h(x)=H(x)/H_0(x)$, where $x=1+z$, in the SCM, for quintessence and for phantom energy cases.

Problem 62

Find the constraints imposed by the Weak Energy Condition (WEC) for the dark energy on the redshift-dependent Hubble parameter. (Inspired by A.Sen, R.Scherrer)

Problem 63

Using the statefinders, show that the power law cosmology mimics SCM model at (see Chapters 3 and 9)

Problem 64

Find expressions for $H(z)$ for a homogeneous and isotropic FRW Universe.
a) For a flat universe with generic equation of state parameter for dark energy;
b) for a non-flat Universe, equation of state parameter for dark energy given by the $w_0, w_a$ parameterization;
c) for a flat $\Lambda$CDM model.
For more detail see article The expansion rate of the intermediate Universe in light of Planck

Problem 65

Consider the model of the universe with a cosmological time-dependent "constant": \begin{equation} (\frac{\dot {a}}{a})^{2} = \frac{8\pi G}{3}\rho + \frac{\Lambda(t) }{3} - \frac{k}{R_0^2a^{2}} \label{EoS1} \end{equation} \begin{equation} \frac{\ddot {a}}{a} = -\frac{4\pi G}{3}(\rho + 3p ) + \frac{\Lambda(t) }{3}. \label{EoS2} \end{equation} where $\rho$ - density of matter (baryons and dark matter). Find dependence of the density of matter on the scale factor $\rho(a)$.