# Difference between revisions of "New from June"

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).

\paragraph{To chapter 3, section 15, if absent}

Problem 15

problem id: 150_015

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

\paragraph{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.

\paragraph{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 $\varepsilon$?

paragraph{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.

paragraph{Cardassian Model} [K. Freese and M. Lewis, Cardassian Expansion: a Model in which the Universe is Flat, Matter Dominated, and Accelerating, arXiv: 0201229] 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.

\paragraph{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} {\it Problems \ref{150_8}-\ref{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} \begin{enumerate} <div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem 39''' <p style="color: #999;font-size: 11px">problem id: 150_8</p> Consider a cosmological model in a flat Universe where the only component is a pressureless fluid with constant bulk viscosity (UNIQ-MathJax149-QINU). The pressureless fluid represent both the baryon and dark matter components. Find the dependence UNIQ-MathJax150-QINU for the considered model. <div class="NavFrame collapsed"> <div class="NavHead">solution</div> <div style="width:100%;" class="NavContent"> <p style="text-align: left;"> The conservation equation in terms of the scale factor and the first Friedmann equation are UNIQ-MathJax324-QINU UNIQ-MathJax325-QINU Here UNIQ-MathJax151-QINU is total density of the baryon and dark matter components. Having excluded the Hubble parameter and changed the independent variable from the scale factor UNIQ-MathJax152-QINU to the redshift UNIQ-MathJax153-QINU, one finds UNIQ-MathJax326-QINU The solution of this equation is: UNIQ-MathJax327-QINU </p> </div></div></div> <div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem 40''' <p style="color: #999;font-size: 11px">problem id: 150_9</p> Find UNIQ-MathJax154-QINU and UNIQ-MathJax155-QINU for the model of Universe considered in the previous problem. <div class="NavFrame collapsed"> <div class="NavHead">solution</div> <div style="width:100%;" class="NavContent"> <p style="text-align: left;"> Substitute the solution UNIQ-MathJax156-QINU obtained in the previous problem into the first Friedmann equation to obtain UNIQ-MathJax328-QINU In the considered model the bulk viscous matter is the only component of the flat Universe. Consequently, UNIQ-MathJax157-QINU and one finally obtains UNIQ-MathJax329-QINU The obtained expression allows to write the scale factor in terms of the cosmic time. Let us transform the expression for the Hubble parameter UNIQ-MathJax330-QINU to the following form UNIQ-MathJax331-QINU For UNIQ-MathJax158-QINU and UNIQ-MathJax159-QINU (UNIQ-MathJax160-QINU implies UNIQ-MathJax161-QINU and contradicts the observations) one finds UNIQ-MathJax332-QINU </p> </div></div></div> <div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem 41''' <p style="color: #999;font-size: 11px">problem id: 150_10</p> Analyze the expression for the scale factor UNIQ-MathJax162-QINU obtained in the previous problem for different types of the bulk viscosity. <div class="NavFrame collapsed"> <div class="NavHead">solution</div> <div style="width:100%;" class="NavContent"> <p style="text-align: left;"> \begin{enumerate} \item UNIQ-MathJax163-QINU When UNIQ-MathJax164-QINU then the obtained solution reproduces the de Sitter-like Universe, UNIQ-MathJax333-QINU \item UNIQ-MathJax165-QINU In this case the considered model exactly reproduces the de Sitter-like Universe, UNIQ-MathJax334-QINU The model predicts an Universe in an eternal accelerated expansion. \item UNIQ-MathJax166-QINU In this case the Universe expands forever (decelerated expansion epoch is absent). \end{enumerate}