# New on the matter creation model

As we have seen above, negative pressure is the key ingredient required to achieve the accelerated expansion. Such pressure naturally appears when the system deviates from the thermodynamic equilibrium. In particular, as was first pointed out by Zel'dovich [Ya. B. Zel'dovich, JETP Lett. 12, 307 (1970)], the negative pressure is generated in process of particle creation due to the gravitational field. Physically, one may think that the (classical) time varying gravitational field works like a 'pump' supplying energy to the quantum fields. Construction of the models including the essentially quantum process of particle creation is problematic because of the difficulties connected with its inclusion into the classical field equations of Einstein. One can however avoid those difficulties on the phenomenological level [J.Lima, arXiv 0807.3379].

We shall now consider an open thermodynamical system where the number of fluid particles $N$ is not preserved. So the particle conservation equations $\dot N_{;\mu}=0\Rightarrow \dot n + \theta n=0$ is now modified as $\dot n + \theta n=n\Gamma,$ where $N_{\mu}=nu_\mu$ is the particle flow vector, $u_\mu$ is the particle four velocity, $\theta=u^\mu_{;\mu}$ is the fluid expansion, (for the FLRW Universe, $\theta=u^\mu_{;\mu}=3H$), $\dot n = n_{,\mu}u^{mu}$ and $\Gamma$ stands for the rate of change of the particle number in a comoving volume $V$, $\Gamma>0$ indicates particle creation while $\Gamma<0$ means particle annihilation. Any non-zero $\Gamma$ will effectively behave as a bulk viscous pressure of the thermodynamical fluid and nonequilibrium thermodynamics should come into the picture.

Problem 1

problem id: 150_dp0

Express the first law of thermodynamics in terms of the specific quantities: the energy density $\rho$ and the particle number density $n$.

Problem 2

problem id: 150_dp0

In the context of particle creation, let us consider spatially flat FRLW model of the Universe as an open thermodynamical system. Obtain the Friedmann equations for the cosmic fluid having energy-momentum tensor (in this section we will use the metric $(-,+,+,+)$ $T_{\mu\nu}=(\rho+p+\Pi)u_\mu u_\nu+(p+\Pi)g_{\mu\nu}.$ For cosmological models with particle creation the cosmic fluid may be considered as a perfect fluid and the dissipative term $\Pi$ as the effective bulk viscous pressure due to particle production. In other words, the cosmic substratum is not a conventional dissipative fluid, rather a perfect fluid with varying particle number.

Problem 3

problem id: 150_dp0

Show that for the adiabatic process the effective bulk viscous pressure due to particle production is entirely determined by the particle production rate.

Problem 4

problem id: 150_dp0

An equivalent way to see the derivation of the field equations (7)-(8) (???) is to consider the energy-momentum tensor in the Einstein field equations as a total energy momentum tensor $T^{(eff)}_{\mu\nu}=T^{(\gamma)}_{\mu\nu}+T^{(c)}_{\mu\nu},$ where $T^{(\gamma)}_{\mu\nu}$ is the energy-momentum tensor for the fluid with the EoS $p = (\gamma- 1) \rho$, i.e. $T^{(\gamma)}_{\mu\nu}=(\rho+p)u_\mu u_\nu+pg_{\mu\nu}.$ and $T^{(c)}_{\mu\nu}=p_c\left(u_\mu u_\nu+g_{\mu\nu}\right)$ is the energy-momentum tensor which corresponds to the matter creation term. Show that energy-momentum tensor $T^{(eff)}_{\mu\nu}$ provides us with the matter creation pressure $\Pi=-\frac\Gamma\theta(\rho+p),$ obtained in the previous problem.

Problem 5

problem id: 150_dp0

Find the effective parameter $w_{eff}$ of the EoS for the cosmological model with particle creation.

Problem 6

problem id: 150_dp0

Find the deceleration parameter $q$ for the cosmological model with particle creation.

Problem 7

problem id: 150_dp0

Obtain the equation for the scale factor in cosmological model with particle creation.

Problem 8

problem id: 150_dp0

Calculate the derivative $dS/dt$ and find $S(t)$ for the Creation Matter Model (CMM).

Problem 9

problem id: 150_dp0

Calculate the scale factor dependence of the Universe's temperature for the Creation Matter Model (CMM).

Problem 10

problem id: 150_dp0

Solve the previous problem using the thermodynamic identity $T\left(\frac{\partial p}{\partial T}\right)_n+n\left(\frac{\partial\rho}{\partial n}_T\right)=\rho+p.$

Problem 11

problem id: 150_dp0

Solve the previous problem for cosmological substance with $\partial p/\partial\rho=w=const$.

Problem 12

problem id: 150_dp0

Show that the matter creation process can be effectively described by the dimensionless ratio $\beta=\frac\Gamma\theta=\frac\Gamma{3H}.$

Problem 13

problem id: 150_dp0

How is the evolution of $a(t)$ affected by rate of change of the particle number $\Gamma$?

Problem 14

problem id: 150_dp0

Let us assume a radiation dominated Universe ($w=1/3$, $\Gamma=\Gamma_r$). The dynamics is determined by the ratio $\Gamma_r/(3H)$ (see previous problem). The particle production must be strongly suppressed, $\Gamma_r/(3H)\ll1$, when the Universe enters the radiation phase. The simplest formula satisfying such a criterion is linear, namely: $\Gamma_r/(3H)=H/H_I$, where $H_I$ is the inflationary expansion rate associated with the initial de Sitter expansion ($H\le H_I$). Find the Hubble parameter dependence on the scale factor $H(a)$ for the radiation phase.

Problem 15

problem id: 150_dp0

Using the result obtained in the previous problem, show that in the considering model the Universe evolves continuously from de Sitter stage to the standard radiation phase.

Problem 16

problem id: 150_dp0

Find the radiation energy density $\rho_r$ as function of scale factor for the rate of change of the particle number $\Gamma=\Gamma_r$, $\Gamma_r/(3H)=H/H_I$ ($H_I$ is the inflationary expansion rate).

Problem 17

problem id: 150_dp0

For cosmological model considered in the problems \ref{2016_07_p14}-\ref{2016_07_p16} find how did the cosmic temperature evolve at the very early stages?

Problem 18

problem id: 150_dp0

Solve Problem \ref{2016_07_p14} for a substance with EoS $p=w\rho=(\gamma-1)\rho$ and $\Gamma=\Gamma_0H^2$ with $\Gamma_0=3\beta/H_e$.

Problem 19

problem id: 150_dp0

Find the deceleration parameter for the model considered in previous problem.

Problem 20

problem id: 150_dp0

Let us now consider how the CMM (Creation matter model) works at later stages of the Universe's evolution in the transition from Einstein-de Sitter to a late time de Sitter stage. Due to the conservation of baryon number one have to take into account only the production rate of cold dark matter particles $\Gamma_{dm}$. What is the exact form of $\Gamma_{dm}$?

Problem 21

problem id: 150_dp0

Let us consider the dynamical system in the context of bulk viscosity. In that case, we need to replace the pressure as $p\to p-3\zeta H$, where $\zeta$ is the coefficient of bulk viscosity. At what choice of the particle production rate $\Gamma$ the creation matter model would be equivalent to the bulk viscosity model?

## Dynamical analysis of the Creation Matter Model

Problem 22

problem id: 150_dp0

Since the equation $\dot H=-\frac32\gamma H^2\left(1-\frac\Gamma{3H}\right)$ is a one-dimensional first order differential equation, hence, the dynamics is obtained from the study of its critical points (or, fixed points) in which $\dot H=0$. Show that the fixed points $H_*$ in which $H_*\ne0$ represent the de Sitter Universe.

Problem 23

problem id: 150_dp0

Find the fixed points and determine their nature for the case with the particle creation rate $\Gamma=n/H$, $n>0$.

Problem 24

problem id: 150_dp0

Find the fixed points for the case with the particle creation rate $\Gamma=-\Gamma_0+mH+n/H$, $n>0$.

Problem 25

problem id: 150_dp0

Perform dynamic analysis of the solutions, obtained in previous problem.

Problem 26

problem id: 150_dp0

Prove that the dynamics generated by the creation matter model is equivalent to that driven by the single scalar field.

Problem 27

problem id: 150_dp0

Find the scalar field $\varphi(H)$ and the potential $V(\varphi)$, satisfying the condition that the dynamics generated by the creation matter model is equivalent to the dynamics driven by a single scalar field.

Problem 28

problem id: 150_dp0

Solve the previous problem for case $\Gamma=-\Gamma_0+mH+n/H.$

Problem 29

problem id: 150_dp0

Prove equivalence of the dynamics generated by the creation matter model with the dynamics driven by a decaying vacuum energy density model.

Problem 30

problem id: 150_dp0

Show that for spatially flat Universe the inequalities $\Gamma\ll H$ and $\Lambda\ll H^2$ should be satisfied simultaneously.

Problem 31

problem id: 150_dp0

Let us consider the particular case $\Lambda(t)=const$. Show that for such choice the creation matter model corresponds to the standard LCDM.

Problem 32

problem id: 150_dp0

Calculate the cosmographic parameter $j(H)$ for the cosmological model with particle creation.

Problem 33

problem id: 150_dp0

Solve the previous problem using the cosmographical relation $j=-\frac1H\frac{dq}{dt}+q(1+2q).$

However, in connection with the particle production at the expense of gravitational field of the expanding Universe, we recall that long back ago, Zeldovich [Ya. B. Zeldovich, JETP Lett. 12, 307 (1970)] introduced some bulk viscosity mechanism which is responsible for particle production. However, later on, Lima and Germano [J. A. S. Lima and A. S. M. Germano, Phys. Lett. A 170, 373 (1992); For a more general macroscopic approach see also R. Silva, J. A. S. Lima and M. O. Calv~ao, Gen. Rel. Grav. 34, 865 (2002).] showed that although both the processes, namely the bulk viscosity mechanism by Zeldovich [Ya. B. Zeldovich, JETP Lett. 12, 307 (1970).] and the gravitational particle production produce the same dynamics of the Universe, but in principle they are completely different from a thermodynamical point of view. Since we describe the particle production, we would like also to note an analogy which exists between the models driven by the particle production and the models of Steady State Cosmology developed in [F. Hoyle and J. V. Narlikar, Proc. Roy. Soc. A 282, 191 (1964); ibdem, 290, 143 (1966); see also Action at a Distance in Physics and Cosmology, 1974 (New York, Freeman).].

Problem 34

problem id: 150_dp0

In the model with $\Gamma=-\Gamma_0+mH$, where $0\le m\le3$ and $-3H_0\le\gamma\le0$, find the time dependence of the scale factor.

Problem 35

problem id: 150_dp0

In the model considered in the previous problem, express age of the Universe in terms of the parameters $\Gamma_0$ and $m$.

Problem 36

problem id: 150_dp0

Find deceleration parameter in the model with matter creation rate equal to $\Gamma$.

Problem 37

problem id: 150_dp0

Show that in the creation matter model with $\Gamma=-\Gamma_0+mH$ in the case $\Gamma_0=0$ the expansion of Universe is always accelerated if $m>1$ and always decelerated if $m<1$.

Problem 38

problem id: 150_dp0

Find the deceleration parameter as a function of the redshift in the creation matter model.

Problem 39

problem id: 150_dp0

Find the redshift value $z_t$ corresponding to the transition from the decelerated expansion to the accelerated one in the creation matter model.

Problem 40

problem id: 150_dp0

Express age of the Universe in terms of the $z_t$ (see the previous Problem) in the creation matter model with $m=0$.

# New on the Ricci scalar etc

Problem 1

problem id: 10_1

Show that the Ricci scalar can be written in the following form $R=6H^2[q(t)-\Omega(t)].$

Problem 2

problem id: 10_2

Show that for a one-component flat Universe filled with ideal fluid with the EoS $p=w\rho$ the Ricci scalar can be written in the following form $R=8\pi G\rho(3w-1).$

Problem 3

problem id: 10_3

Find time derivative of the scalar curvature for the spatially flat Universe filled with a barotropic fluid.

Problem 4

problem id: 10_4

Compare entropy of the Sun in the present state and right after its transformation to a black hole due to compression.

Problem 5

problem id: 10_5

Show that black holes have negative thermal capacity.

Problem 6

problem id: 10_6

What are the physical reasons for the negative thermal capacity in Newtonnian self-gravitating systems?

Problem 7

problem id: 10_7

How the negative thermal capacity of the stars (in particular the Sun) affects the course of the nuclear reactions inside them?

Problem 8

problem id: 10_8

From the three parameters (mass, charge and angular momentum) of the Kerr-Newman black hole form all possible length scales.

Problem 9

problem id: 10_9

Following [O. Luongo and H. Quevedo, Self-accelerated universe induced by repulsive e?ects as an alternative to dark energy and modi?ed gravities, arXiv: (1507.06446)] let us introduce the parameter $\lambda=-\frac{\ddot a}{a}=qH^2$, so that $\lambda<0$ when the Universe is accelerating, whereas for $\lambda>0$ the Universe decelerates. Luongo and H. Quevedo showed, that the parameter $\lambda$ can be considered as an eigenvalue of the curvature tensor defined in special way. In particular, for FLRW metric the curvature tensor $R$ can be expressed as a ($6\times6$)-matrix $R=diag(\lambda,\lambda,\lambda,r,r,r),\quad r\equiv\frac13\rho.$ The curvature eigenvalues reflect the behavior of the gravitational interaction and if gravity becomes repulsive in some regions, the eigenvalues must change accordingly; for instance, if repulsive gravity becomes dominant at a particular point, one would expect at that point a change in the sign of at least one eigenvalue. Moreover, if the gravitational field does not diverge at infinity, the eigenvalue must have an extremal at some point before it changes its sign. This means that the extremal of the eigenvalue can be interpreted as the onset of repulsion. Obtain the onset of repulsion condition in terms of cosmographic parameters.

Problem 10

problem id: 10_10

Represent result of the previous problem in terms of the Hubble parameter and its time derivatives.

# New on the deceleration parameter

Problem 1

problem id: 150_dp0

Show that the fact that in the present time $t_0$ the relation $H_0t_0\approx1$ holds unavoidably follows existance of the accelerated expansion stage of the Universe evolution.

Problem 2

problem id: 150_dp2

Suppose that $dq/dt=f(q)$. Find the Hubble parameter in terms of $q$.

Problem 3

problem id: 150_dp2

Find connection between the scalar curvature and deceleration parameter for the flat Universe.

Problem 4

problem id: 150_dp3

Show that the deceleration parameter $q$ relates the density of the Universe $\rho$ to the critical density $\rho_{cr}$ through $q=\frac12(1+3w)\frac\rho{\rho_{cr}}.$

Problem 5

problem id: 150_dp4

Solve the previous problem using the second Friedmann's equation.

# New on cosmology with power and hybrid expansion laws

Let us consider a general class of power-law cosmology described by the scale factor $a(t)=a_0\left(\frac t{t_0}\right)^\alpha$ where $\alpha$ is a dimensionless positive parameter.

Problem 6

problem id: 150_dp5

Obtain deceleration parameter in the power-law cosmology.

Problem 7

problem id: 150_dp6

Express the scale factor $a(t)$ and Hubble parameter $H(z)$ in terms of the deceleration parameter.

Problem 8

problem id: 150_dp7

Obtain the comoving distance $r(z)$ in the power-law cosmology.

Problem 9

problem id: 150_dp8

Find the comoving distance in the Milne model.

Problem 10

problem id: 150_dp9

In the power-law cosmology, find time dependence of the CMB temperature in terms of the deceleration parameter.

Let us consider now a simple generalization of power-law cosmology, called the hybrid expansion law [O. Akarsu et. al. Cosmology with hybrid expansion law: scalar field reconstruction of cosmic history and observational constraints, arXiv:gr-qc/1307.4911] $a(t)=a_0\left(\frac t{t_0}\right)^\alpha e^{\beta\left(\frac t{t_0}-1\right)}$ where $\alpha$ and $\beta$ are non-negative constants.

Problem 11

problem id: 150_dp10

Find Hubble parameter, deceleration parameter and jerk parameter for the hybrid expansion law.

Problem 12

problem id: 150_dp11

Find asymptotes of the scale factor and of the cosmographic parameters for the hybrid expansion law at $t\to0$ and $t\to\infty$.

Problem 13

problem id: 150_dp12

Find the moment of time $t_{tr}$ when the transition from deceleration to acceleration takes place.

Problem 14

problem id: 150_dp13

Find the range of the parameter $\alpha$ variation for the hybrid expansion law.

Problem 15

problem id: 150_dp14

In 1983, Berman [M. Berman, A special law of variation for Hubble's parameter. Nuovo Cimento B 74, 182 (1983)] proposed a special law of variation of Hubble parameter in FLRW space-time, which yields a constant value of DP, $H=Da^{-n}.$ Find time dependence of the scale factor providing the constant deceleration parameter.

Problem 16

problem id: 150_dp15

For Berman's law of variation for Hubble's parameter $$H=Da^{-n}$$ find the deceleration parameter and analyze what values of the parameter $n$ correspond to accelerated expansion, and which --- to decelerated one.

Linearly varying deceleration parameter

Inspired by O. Akarsu et al. Probing kinematics and fate of the Universe with linearlytime-varying deceleration parameter, arXiv:gr-qc/1305.5190 A general approach is to expand the deceleration parameter in Taylor series is $q(x)=q_0+q_1\left(1-\frac x{x_0}\right)+q_2\left(1-\frac x{x_0}\right)^2+\ldots$ where $x$ is a some cosmological parameter as cosmic scale factor $a$, cosmic redshift $z$, cosmic time $t$ etc. As the first step one can take the following linear approximation $q(x)=q_0+q_1\left(1-\frac x{x_0}\right).$

Problem 17

problem id: 150_dp16

Consider the linearly varying deceleration parameter in terms of cosmic redshift $z$. Analyze advantages and defeats of such interpretation.

Problem 18

problem id: 150_dp17

Unlimited growth of the deceleration parameter for large $z$ in the parametrization, used in the previous problem, forces us to consider the linearly varying deceleration parameter in terms of scale factor. Make transition to such parametrization.

Problem 19

problem id: 150_dp19

Treating the Universe as a dynamical system it is useful to consider the parametrization of the deceleration parameter directly in terms of cosmic time $t$. Make transition to such parametrization.

Problem 20

problem id: 150_dp19

Consider a single-component flat Universe with a fluid described by an EoS parameter expressed as a first order Taylor expansion in cosmic time: $w=w_0+w_1(1-t),$ where $w_0$ and $w_1$ are real constants and $t$ is the normalized time. Find the corresponding parametrization for the deceleration parameter.

Problem 21

problem id: 150_dp20

In a Universe filled with a fluid characterized by the EoS parameter $w=w_0+w_1(1-t/t_0)$, find time dependence of the scale factor. [S. Kumar, Probing the matter and dark energy sources in a viable Big Rip model of the Universe, arXiv:1404.1910]

Problem 22

problem id: 150_dp21

In a Universe filled with a fluid characterized by the EoS parameter $w=w_0+w_1(1-t/t_0)$, find the Hubble parameter, deceleration parameter and the jerk parameter.

Problem 23

problem id: 150_dp22

In a Universe filled with a fluid characterized by the EoS parameter $w=w_0+w_1(1-t/t_0)$, find the pressure and energy density.

Problem 24

problem id: 150_dp23

Show that the Universe filled with a fluid characterized by the EoS parameter $w=w_0+w_1(1-t/t_0)$ achieves de Sitter phase ($q=-1$) at the end of its half life.

# New to Quantum cosmology

Problem 1

problem id: 150_dp24

Find the solutions corrected by LQC Friedmann equations for a matter dominated Universe.

# New to Observational Cosmology

Problem 1

problem id: 150_o1

Find ratio between the illuminance of the Earth surface by the Sun and other light sources in the Universe (it is a quantitative resolution of the Olbers paradox).

Problem 2

problem id: 150_o2

Consider light emitted by surface of the Sun and observed on the Earth. Determine the shift of the observed frequency compared to the analogues light frequency emitted by atoms on the Earth.

Problem 3

problem id: 150_o3

Find the distance to a distant galaxy using recessional velocity as measured by the Doppler redshift.

Problem 4

problem id: 150_o4

Show that galaxies situated outside the Hubble sphere have imaginary redshift.

Problem 5

problem id: 150_o5

Find the exact relativistic Doppler velocity-redshift relation.

Problem 6

problem id: 150_o7

Find $V_{exp}(z)$ for three cosmological models: Einstein-de Sitter, Milne and de Sitter.

Problem 7

problem id: 150_o8

Some luminous object has apparent stellar magnitude $m=20$, and the absolute one is $M=-15$. Determine distance to it.

Problem 8

problem id: 150_o9

Due to a random coincidence the Balmer series of singly ionized helium atom in a distant star overlap with the Balmer series of hydrogen in the Sun. How fast this star recedes from us?

Problem 9

problem id: 150_o10

A gas cloud rotates around a super-massive black hole with mass equal to $M=3.6\times10^6M_\odot$ (it is a possible interpretation of recent observations). Assuming that distance between these objects is of order of $60$ lightyears, determine the expected Doppler shift.

Problem 10

problem id: 150_o11

Explain why transition from the cosmological time to the redshift can be considered as a quantitative characteristic describing evolution of the Universe.

Problem 11

problem id: 150_o12

Show, that for small separations the "Hubble law" $dz=Hdr$ holds. In other words, we have replaced the velocity by the redshift.

Problem 12

problem id: 150_o13

Solve the previous problem using the Taylor series of the scale factor in terms of time.

Problem 13

problem id: 2501_02o

Why the Linear Distance-Redshift Law in Near Space?

Problem 14

problem id: 150_o1

Determine physical distance to an object emitted light with redshift $z$ in the flat expanding Universe.

Problem 15

problem id: 150_o16

Find comoving distance to a presently observed galaxy as a function of the redshift.

Problem 16

problem id: 150_o17

Solve the previous problem for the flat Universe dominated by non-relativistic matter.

Problem 17

problem id: 150_o18

Determine the recession velocity due to the cosmological expansion for an object emitted light with the redshift $z$ in the flat expanding Universe.

Problem 18

problem id: 150_o19

Obtain relations between velocity of cosmological expansion and redshift.

Problem 19

problem id: 150_o20

Find dependence of the Hubble parameter $H$ on the redshift $z$ for the case of one-component Universe composed of the non-relativistic matter.

Problem 20

problem id: 150_o21

Find dependence of relative density $\Omega$ on the redshift $z$ for the case of one-component Universe composed of the non-relativistic matter.

Problem 21

problem id: 150_o22

Construct a scheme to determine sign of acceleration of the scale factor, based on measurement of the supernovae bursts characteristics.

Problem 22

problem id: 150_o23

Show that the time derivative of the redshift for the light emitted at time $t$ and registered at time $t_0$ can be determined as $\dot z\equiv\frac{dz}{dt_0}=H(t_0)(1+z)-H(t).$

Problem 23

problem id: 150_o24

In a flat one-component Universe with the state equation $p=w\rho$ at time $t_0$ one registers a light signal with redshift $z$. What values of the EoS parameter $w$ lead to $\frac{dz}{dt_0}>0?$ Explain physical sense of the obtained result.

Problem 24

problem id: 150_o26

Show that the luminosity distance can be generally presented as $d_L=\frac{1+z}{H_0\sqrt{\Omega_{k0}}}\sinh\left(H_0\sqrt{\Omega_{k0}}\int\limits_0^z\frac{dz}{H(z)}\right),$ where $\Omega_{k0}$ is the relative contribution of the spatial curvature.

Problem 25

problem id: 150_o27

\it When galaxy formation started in the history of the Universe remains unclear. Studies of the cosmic microwave background indicate that the Universe, after initial cooling (following the Big Bang), was reheated and reionized by hot stars in newborn galaxies at a redshift in the range $6<z<14$. Here we report a spectroscopic redshift of $z=6.96$. [M. Iue et al., A galaxy at a redshift z=6.96 , arXiv:0609393]

\bf Estimate age of the Universe when the first galaxies were created.

Problem 26

problem id: 150_o30

Express probability to find two galaxies in the infinitesimally small volumes $dV_1$ and $dV_2$ in terms of the correlation function $\xi(\vec{r_1},\vec{r_2})$, if average density of the galaxies in the considered volume equals $\bar n$ and total number of the galaxies is $N$. Point out main properties of the correlation function.

Problem 27

problem id: 150_o31

\bf Find relation between the spatial $\xi(r)$ and angular $w(\theta)$ correlation functions.

Problem 28

problem id: 150_o32

Show that power decay law of the spatial correlations leads to power decay law for the angular correlations.

Problem 29

problem id: 150_o33

Find two-point correlation function for $N$ galaxies distributed on a straight line with average density $\bar n$ in non-overlapping clusters of length $a$. Density of galaxies inside the cluster is constant and equals to $n_c$. The clusters are randomly distributed.

Problem 30

problem id: 150_o34

In a flat matter-dominated Universe of age $t_0$ light from a certain galaxy exhibits a redshift $z=0.95$. How long has it taken the light signal to reach us from this galaxy?

# New from June 2015

## To Chapter 2

Problem 1

problem id: 150_0

Comprehensive explanations of the expanding Universe often use the balloon analogy. Although the balloon analogy is useful, one must guard against misconceptions that it can generate. Point out the misconceptions that appear when using this analogy [see M.O. Farooq, Observational constraints on dark energy cosmological model parameters, arXiv: 1309.3710.]

Problem 2

problem id: 150_1

(into the cosmography and extended deceleration parameter) Show that $\frac{d\dot a}{da}=-Hq.$

Problem 3

problem id: 150_2

Give a physical interpretation of the conservation equation.

Problem 4

problem id: 150_04

Find evolution equation for the density parameter $\Omega$ of the single-fluid FLRW models with the linear equation of state $p=w\rho$.

Problem 5

problem id: 150_05

Solve the previous problem for the multi-component case.

Problem 6

problem id: 150_06

Use the conformal time to prove existence of smooth transition from the radiation-dominated era to the matter dominated one.

Problem 7

problem id: 150_07

Consider a set of the cosmographic parameters built from the Hubble parameter and its time derivatives [see S. Carloni, A new approach to the analysis of the phase space of f(R)-gravity, arXiv:1505.06015) ] $Q\equiv\frac{\dot H}{H^2},\quad J\equiv\frac{\ddot H H}{\dot H^2},\quad S\equiv\frac{\dddot H H^2}{\dot H^3},\ldots$ Express them in terms of the canonic cosmographic parameters $q,j,s\dots$.

Problem 8

problem id: 150_08

Consider another set of the cosmographic parameters [see S. Carloni, A new approach to the analysis of the phase space of f(R)-gravity, arXiv:1505.06015) ] $\bar Q\equiv\frac{H_{,N}}{H},\quad \bar J\equiv\frac{H_{,NN}}{H},\quad \bar S\equiv\frac{H_{,NNN}}{H},\ldots,$ where $H_{,N}\equiv \frac{dH}{d\ln a}.$ Express them in terms of the Hubble parameter and its time derivatives.

Problem 9

problem id: 150_09

Express the Ricci scalar and its time derivatives in terms of the $\bar Q$, $\bar J$ and $\bar S$.

Problem 10

problem id: 150_3

Show that for a perfect fluid with EoS $p=w(a)\rho$ the adiabatic sound speed can be represented in the form $c_S^2=w(a)-\frac13\frac{d\ln(1+w)}{d\ln a}.$

Problem 11

problem id: 150_4

Obtain equation for $\ddot\rho(t)$, where $\rho(t)$ is energy density of an ideal fluid participating in the cosmological expansion.

Problem 12

problem id: 150_5

Show that in the case of the flat Friedmann metric, the third power of the scale factor $\varphi\equiv a^3$ satisfies the equation $\frac{d^2\varphi}{dt^2}=\frac32(\rho-p)\varphi,\quad 8\pi G=1.$ Check validity of this equation for different cosmological components: non-relativistic matter, cosmological constant and a component with EoS $p=w\rho$.

Problem 13

problem id: 150_6

The lookback time is defined as the difference between the present day age of the Universe and its age at redshift $z$, i.e. the difference between the age of the Universe at observation $t_0$ and the age of the Universe, $t$, when the photons were emitted. Find the lookback time for the Universe filled with non-relativistic matter, radiation and a component with the EoS $p=w(z)\rho$.

Problem 14

problem id: 150_7

Show that the Hubble radius grows faster than the expanding Universe in the case of power law expansion $a(t)\propto t^\alpha$ with $\alpha<1$ (the decelerated expansion).

## To chapter 3

Problem 15

problem id: 150_015

Show, that in the Milne Universe the age of the Universe is equal to the Hubble time.

## To chapter 4 The black holes

Problem 16

problem id: new2015_1

see E. Berti, A Black-Hole Primer: Particles, Waves, Critical Phenomena and Superradiant instabilities (arXiv:1410.4481[gr-qc])

A Newtonian analog of the black hole concept is a so-called "dark star". If we consider light as a corpuscle traveling at speed $c$, light cannot escape to infinity whenever $V_{esc}>c$, where $V_{esc}^2=\frac{2GN}R.$ Therefore the condition for existence of "dark stars" in Newtonian mechanics is $\frac{2GN}{c^2R}\ge1.$

Can this condition be satisfied in the Newtonian mechanics?

Problem 17

problem id: 150_017

Derive the relation $T\propto M^{-1}$ from the Heisenberg uncertainty principle and the fact that the size of a black hole is given by the Schwarzschild radius.

## To chapter 8

Problem 18

problem id: 2501_06

Why the cosmological constant cannot be used as a source for inflation in the inflation model?

Problem 19

problem id: 2501_09

Show that inflation ends when the parameter $\varepsilon\equiv\frac{M_P^2}{16\pi}\left(\frac{dV}{d\varphi}\frac1V\right)^2=1.$

Problem 20

problem id: 2501_10

How does the number of e-folds $N$ depend on the slow-roll parameter $\varepsilon$?

## To chapter 9

Problem 21

problem id: 150_021

Using the by dimensional analysis for cosmological constant $\Lambda > 0$, define the set of fundamental "de Sitter units" of length, time and mass.

Problem 22

problem id: 150_022

In the Newtonian approximation, find the force acting on the point unit mass in the Universe filled by non-relativistic matter and cosmological constant. (see Chiu Man Ho and Stephen D. H. Hsu, The Dark Force: Astrophysical Repulsion from Dark Energy, arXiv: 1501.05592)

Problem 23

problem id: 150_023

Consider a spatially flat FLRW Universe, which consists of two components: the non-relativistic matter and the scalar field $\varphi$ in the potential $V(\varphi)$. Find relation between the scalar field potential and the deceleration parameter.

Problem 24

problem id: 150_024

Find relation between the deceleration parameter and the derivative $d\varphi/dz$ for the Universe considered in the previous problem.

Problem 25

problem id: new_30

Find the sound speed for the modified Chaplygin gas with the state equation $p=B\rho-\frac A{\rho^\alpha}.$

## A couple of problems for the SCM

Problem 26

problem id: 150_026

Let $N=\ln(a/a_0)$, where $a_0=a(t_0)$ and $t_0$ is some chosen reference time. Usually the reference time is the present time and in that case $\tau=-\ln(1+z)$. Find $\Omega_m(N)$ and $\Omega_\Lambda(N)$ for the SCM.

Problem 27

problem id: 150_027

Express the cosmographic parameters $H,q,j$ as functions of $N=\ln a/a_0$ for the SCM.

## Cardassian Model

[K. Freese and M. Lewis, Cardassian Expansion: a Model in which the Universe is Flat, Matter Dominated, and Accelerating, arXiv: 0201229] Cardassian Model is a modification to the Friedmann equation in which the Universe is flat, matter dominated, and accelerating. An additional term, which contains only matter or radiation (no vacuum contribution), becomes the dominant driver of expansion at a late epoch of the universe. During the epoch when the new term dominates, the universe accelerates. The authors named this period of acceleration by the Cardassian era. (The name Cardassian refers to a humanoid race in Star Trek whose goal is to take over the universe, i.e., accelerated expansion. This race looks foreign to us and yet is made entirely of matter.) Pure matter (or radiation) alone can drive an accelerated expansion if the first Friedmann equation is modified by the addition of a new term on the right hand side as follows: $H^2=A\rho+B\rho^n,$ where the energy density $\rho$ contains only ordinary matter and radiation, and $n<2/3$. In the usual Friedmann equation $B=0$. To be consistent with the usual result, we take $A=\frac{8\pi}{3M_{Pl}^2},$ where $M_{Pl}^2\equiv1/G$.

Problem 28

problem id: 150_cardas1

Show that once the new term dominates the right hand side of the Friedmann equation, we have accelerated expansion.

Problem 29

problem id: 150_cardas2

Let us represent the Cardassian model in the form $H^2\propto \rho+\rho_X,\quad\rho_X=\rho^n.$ Find the parameter $w_X$ of the EoS $p_X=w_X\rho_X$, assuming that the Universe is filled exclusively by the non-relativistic matter.

Problem 30

problem id: 150_cardas3

Show that the result obtained in the previous problem takes place for arbitrary one-component fluid with $w_X=const$.

Problem 31

problem id: 150_cardas4

Generalize the previous problem for the case of two-component ideal liquid (non-relativistic matter $+$ radiation) with density $\rho=\rho_m+\rho_r$.

Problem 32

problem id: 150_cardas5

Show that we can interpret the Cardassian empirical term in the modified Friedmann equation as the superposition of a quintessential fluid with $w=n-1$ and a background of dust.

Problem 33

problem id: 150_cardas6

We have two parameters in the original Cardassian model: $B$ and $n$. Make the transition $\{B,n\}\to\{z_{eq},n\}$, where $z_{eq}$ is the redshift value at which the second term $B\rho^n$ starts to dominate.

Problem 34

problem id: 150_cardas7

What is the current energy density of the Universe in the Cardassian model? Show that the corresponding energy density is much smaller than in the standard Friedmann cosmology, so that only matter can be sufficient to provide a flat geometry.

Problem 35

problem id: 150_cardas8

Let us represent the basic relation of Cardassian model in the following way $H^2=A\rho\left[1+\left(\frac\rho{\rho_{car}}\right)^{n-1}\right],$ where $\rho_{car}=\rho(z_{eq})$ is the energy density at which the two terms are equal. Find the function $\rho(z_{eq})$ under assumption that the Universe is filled with non-relativistic matter and radiation.

Problem 36

problem id: 150_cardas9

Let Friedmann equation is modified to be $H^2=\frac{8\pi G}{3}g(\rho),$ where $\rho$ consists only of non-relativistic matter. Find the effective total pressure.

Problem 37

problem id: 150_cardas10

Find the speed of sound in the Cardassian model. [P.Gandolo, K. Freese, Fluid Interpretation of Cardassian Expansion, 0209322 ]

Problem 38

problem id: 150_cardas11

Find the deceleration parameter for the canonic Cardassian model.

## Models with Cosmic Viscosity

A Universe filled with a perfect fluid represents quite a simple which seems to be in good agreement with cosmological observations. But, on a more physical and realistic basis we can replace the energy-momentum tensor for the simplest perfect fluid by introducing cosmic viscosity. The energy momentum tensor with bulk viscosity is given by $T_{\mu\nu}=(\rho=p-\xi\theta)u_\mu u_\nu+(p-\xi\theta)g_{\mu\nu},$ where $\xi$ is bulk viscosity, and $\theta\equiv3H$ is the expansion scalar. This modifies the equation of state of the cosmic fluid. The Friedmann equations with inclusion of the bulk viscosity, i.e. using the energy-momentum tensor $T_{\mu\nu}$, read \begin{align} \nonumber \frac{\dot a^2}{a^2}&=\frac13\rho,\quad \rho=\rho_m+\rho_\Lambda,\quad 8\pi G=1;\\ \nonumber \frac{\ddot a^2}{a}&=-\frac16(\rho+3p-9\xi H). \end{align} Problems #150_8-#150_14 are inspired by A. Avelino and U. Nucamendi, Can a matter-dominated model with constant bulk viscosity drive the accelerated expansion of the universe? arXiv:0811.3253

Problem 39

problem id: 150_8

Consider a cosmological model in a flat Universe where the only component is a pressureless fluid with constant bulk viscosity ($\xi=const$). The pressureless fluid represent both the baryon and dark matter components. Find the dependence $\rho_m(z)$ for the considered model.

Problem 40

problem id: 150_9

Find $H(z)$ and $a(t)$ for the model of Universe considered in the previous problem.

Problem 41

problem id: 150_10

Analyze the expression for the scale factor $a(t)$ obtained in the previous problem for different types of the bulk viscosity.

Problem 42

problem id: 150_11

Show that the Universe in the considered model with $\xi=const$ had the Big Bang in the past for all values of the bulk viscosity in the interval $0<\bar\xi<3$ and determine how far in the past (in terms of the cosmic time) it happened.

Problem 43

problem id: 150_12

Show that the result of the previous Problem for zero bulk viscosity ($\xi=0$) correctly reproduces the lifetime of the matter-dominated Universe.

Problem 44

problem id: 150_13

As we have seen in the previous problems, in the interval $0<\bar\xi<3$ the Universe begins with a Big-Bang followed by an eternal expansion and this expansion begins with a decelerated epoch followed by an eternal accelerated one. The transition between the decelerated-accelerated expansion epochs depends on the value of $\bar\xi$. Find the value of the scale factor where the transition happens.

Problem 45

problem id: 150_14

Analyze the dependence $a_t=\left(\frac{3-\bar\xi}{2\bar\xi}\right)^{2/3},$ obtained in the previous problem.

Problem 46

problem id: 150_15

Find the deceleration parameter $q(a,\bar\xi)$ for the cosmological model presented in the Problem #150_8.

Problem 47

problem id: 150_16

Analyze behavior of the deceleration parameter $q(a,\bar\xi)$ obtained in the previous problem for different values of the bulk viscosity $\bar\xi(\xi)$.

Problem 48

problem id: 150_17

Use result of the problem \ref{150_15} to find the current value of the deceleration parameter and make sure that for $\bar\xi=1$ the transition from the decelerated to accelerated epochs of the Universe takes place today.

Problem 49

problem id: 150_18

Find the curvature scalar $R(a,\xi)$ for the cosmological model presented in the Problem #150_8.

Problem 50

problem id: 150_19

Let us consider a flat homogeneous and isotropic Universe filled by a fluid with bulk viscosity. We shall assume that the EoS for the fluid is $p=w\rho$, $w=const$ and that the viscosity coefficient $\xi(\rho)=\xi_0\rho^\nu$. Find the dependence $\rho(a)$ for the considered model.

Problem 51

problem id: 150_0013

Usually the inflationary models of the early Universe contain two distinct phases. During the first phase entropy of the Universe remains constant. The second phase is essentially non-adiabatic, particles are produced through the damping of the coherent oscillations of the inflaton field by coupling to other fields and by its subsequent decay. Find relation between the bulk viscosity and the entropy production [J. A. S. Lima, R. Portugal, I. Waga, Bulk viscosity and deflationary universes, arXiv:0708.3280].

# UNSORTED NEW Problems

The history of what happens in any chosen sample region is the same as the history of what happens everywhere. Therefore it seems very tempting to limit ourselves with the formulation of Cosmology for the single sample region. But any region is influenced by other regions near and far. If we are to pay undivided attention to a single region, ignoring all other regions, we must in some way allow for their influence. E. Harrison in his book Cosmology, Cambridge University Press, 1981 suggests a simple model to realize this idea. The model has acquired the name of "Cosmic box" and it consists in the following.

Imaginary partitions, comoving and perfectly reflecting, are used to divide the Universe into numerous separate cells. Each cell encloses a representative sample and is sufficiently large to contain galaxies and clusters of galaxies. Each cell is larger than the largest scale of irregularity in the Universe, and the contents of all cells are in identical states. A partitioned Universe behaves exactly as a Universe without partitions. We assume that the partitions have no mass and hence their insertion cannot alter the dynamical behavior of the Universe. The contents of all cells are in similar states, and in the same state as when there were no partitions. Light rays that normally come from very distant galaxies come instead from local galaxies of long ago and travel similar distances by multiple re?ections. What normally passes out of a region is reflected back and copies what normally enters a region.

Let us assume further that the comoving walls of the cosmic box move apart at a velocity given by the Hubble law. If the box is a cube with sides of length $L$, then opposite walls move apart at relative velocity $HL$.Let us assume that the size of the box $L$ is small compared to the Hubble radius $L_{H}$ , the walls have a recession velocity that is small compared to the velocity of light. Inside a relatively small cosmic box we use ordinary everyday physics and are thus able to determine easily the consequences of expansion. We can even use Newtonian mechanics to determine the expansion if we embed a spherical cosmic box in Euclidean space.

Problem 1

problem id:

As we have shown before (see Chapter 3): $p(t)\propto a(t)^{-1}$ , so all freely moving particles, including galaxies (when not bound in clusters), slowly lose their peculiar motion and ultimately become stationary in expanding space. Try to understand what happens by considering a moving particle inside an expanding cosmic box

Problem 2

problem id:

Show that at redshift $z=1$ , when the Universe is half its present size, the kinetic energy of a freely moving nonrelativistic particle is four times its present value, and the energy of a relativistic particle is twice its present value.

Problem 3

problem id:

Let the cosmic box is filled with non-relativistic gas. Find out how the gas temperature varies in the expanding cosmic box.

Problem 4

problem id:

Show that entropy of the cosmic box is conserved during its expansion.

Problem 5

problem id:

Consider a (cosmic) box of volume V, having perfectly reflecting walls and containing radiation of mass density $\rho$. The mass of the radiation in the box is $M=\rho V$ . We now weigh the box and find that its mass, because of the enclosed radiation, has increased not by M but by an amount 2M. Why?

Problem 6

problem id:

Show that the jerk parameter is $j(t)=q+2q^{2} -\frac{\dot{q}}{H}$

Problem 7

problem id:

We consider FLRW spatially flat Universe with the general Friedmann equations $\begin{array}{l} {H^{2} =\frac{1}{3} \rho +f(t),} \\ {\frac{\ddot{a}}{a} =-\frac{1}{6} \left(\rho +3p\right)+g(t)} \end{array}$ Obtain the general conservation equation.

Problem 8

problem id:

Show that for extra driving terms in the form of the cosmological constant the general conservation equation (see previous problem) transforms in the standard conservation equation.

Problem 9

problem id:

Show that case $f(t)=g(t)=\Lambda /3$ corresponds to $\Lambda (t)CDM$ model.