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(Classification of models of Universe based on the deceleration parameter)
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     <p style="text-align: left;">a) $H>0$, $q>0$: expanding and decelerating</br>
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     <p style="text-align: left;">a) $H>0$, $q>0$: expanding and decelerating<br/>
 
b) $H>0$, $q<0$: expanding and accelerating</br>
 
b) $H>0$, $q<0$: expanding and accelerating</br>
 
c) $H<0$, $q>0$: contracting and decelerating</br>
 
c) $H<0$, $q>0$: contracting and decelerating</br>

Revision as of 23:22, 29 January 2015



New from 26-12-2014


Problem 1

problem id: 2612_1

Find dimension of the Newtonian gravitational constant $G$ and electric charge $e$ in the $N$-dimensional space.


Problem 2

problem id: 2612_2

In 1881, Johnson Stoney proposed a set of fundamental units involving $c$, $G$ and $e$. Construct Stoney's natural units of length, mass and time.


Problem 3

problem id: 2612_3

Show that the fine structure constant $\alpha=e^2/(\hbar c)$ can be used to convert between Stoney and Planck units.


Problem 4

problem id: 2612_4

Gibbons [G W Gibbons, The Maximum Tension Principle in General Relativity, arXiv:0210109] formulated a hypothesis, that in General Relativity there should be a maximum value to any physically attainable force (or tension) given by \begin{equation}\label{2612_e_1} F_{max}=\frac{c^4}{4G}. \end{equation} This quantity can be constructed with help of the Planck units: \[F_{max}=\frac14M_{Pl}L_{Pl}T_{Pl}^{-2}.\] The origin of the numerical coefficient $1/4$ has no deeper meaning. It simply turns out that $1/4$ is the value that leads to the correct form of the field equations of General Relativity. The above made assumption leads to existence of a maximum power defined by \begin{equation}\label{2612_e_2} P_{max}=\frac{c^5}{G}. \end{equation} There is a hypothesis [C. Schiller, General relativity and cosmology derived from principle of maximum power or force, arXiv:0607090], that the maximum force (or power) plays the same role for general relativity as the maximum speed plays for special relativity.

The black holes are usually considered as an extremal solution of GR equations, realizing the limiting values of the physical quantities. Show that the value $c^4/(4G)$ of the force limit is the energy of a Schwarzschild black hole divided by twice its radius. The maximum power $c^5/(4G)$ is realized when such a black hole is radiated away in the time that light takes to travel along a length corresponding to twice the radius.


Problem 5

problem id: 2612_5

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 \[V_{esc}>c \quad \left(V_{esc}^2=\frac{2GM}{R}\right)/\] Therefore the condition for existence of "dark stars" in Newtonian mechanics is \[\frac{2GM}{c^2R}\ge1.\] Can this condition be satisfied in the Newtonian mechanics?


Problem 6

problem id: 2612_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 by non-relativistic matter, radiation and a component with the state equation $p=w(z)\rho$.


Problem 7

problem id: 2612_7

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


Problem 8

problem id: 2612_8

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


Problem 9

problem id:

It is of broad interest to understand better the nature of the early Universe especially the Big Bang. The discovery of the CMB in 1965 resolved a dichotomy then existing in theoretical cosmology between steady-state and Big Bang theories. The interpretation of the CMB as a relic of a Big Bang was compelling and the steady-state theory died. Actually at that time it was really a trichotomy being reduced to a dichotomy because a third theory is a cyclic cosmology model. Give a basic argument of the opponents to the latter model.


Problem 10

problem id: 2612_10

Express the present epoch value of the Ricci scalar $R$ and its first derivative in terms of $H_0$ and its derivatives (flat case).


New from 13-01-2015


Problem 1

problem id: 1301_1

Show that for a spatially flat Universe consisting of one component with equation of state $p=w\rho$ the deceleration parameter is equal to $q=(1+3w)/2$.


Problem 2

problem id: 1301_2

Find generalization of the relation $q=(1+3w)/2$ for the non-flat case.


Problem 3

problem id: 1301_3

Find deceleration parameter for multi-component non-flat Universe.


Problem 4

problem id: 1301_4

Show that the expression for deceleration parameter obtained in the previous problem can be presented in the following form \[q=\frac{\Omega_{total}}2+\frac32\sum\limits_i w_i \Omega_i.\]


Problem 5

problem id:

Let us consider the model of two-component Universe [ J. Ponce de Leon, cosmological model with variable equations of state for matter and dark energy, arXiv:1204.0589]. Such approximation is sufficient to achieve good accuracy on each stage of its evolution. At the present time the two dark components - dark matter and dark energy - are considered to be dominating. We neglect meanwhile the interaction between the components and as a result they separately satisfy the conservation equation. Let us assume that the state equation parameter for each component depends on the scale factor \begin{align} \nonumber p_{de} & = W(a) \rho_{de};\\ \nonumber p_m & = w(a)\rho_{m}. \end{align} Express the relative densities $\Omega_m$ and $\Omega_{de}$ in terms of the deceleration parameter and the state equation parameters $w(a)$ and $W(a)$.


Problem 6

problem id: 1301_6

Find the upper and lower limits on the deceleration parameter using results of the previous problem.


Problem 7

problem id: 1301_7

Find the relation between the total pressure and the deceleration parameter for the flat one-component Universe


Problem 8

problem id: 1301_8

The expansion of pressure by the cosmic time is given by \[p(t)=\left.\sum\limits_{k=0}^\infty\frac1{k!}\frac{d^kp}{dt^k}\right|_{t=t_0}(t-t_0)^k.\] Using cosmography parameters , evaluate the derivatives up to the fourth order.


Problem 9

problem id: 1301_9

Show that for one-component flat Universe filled with ideal fluid of density $\rho$.


Problem 10

problem id: 1301_10

For what values of the state parameter $w$ the rate of expansion of a one-component flat Universe increases with time?


Problem 11

problem id: 1301_11

Express the age of the spatially flat Universe filled with a single component with equation of state $p=w\rho$ through the deceleration parameter.


Problem 12

problem id: 1301_12

Suppose we know the current values of the Hubble constant $H_0$ and the deceleration parameter $q_0$ for a closed Universe filled with dust only. How many times larger will it ever become? Find lifetime of such a Universe.


Problem 13

problem id: 1301_13

In a closed Universe filled with non-relativistic matter the current values of the Hubble constant is $H_0$, the deceleration parameter is $q_0$. Find the current age of this Universe.


Problem 14

problem id: 1301_14

In a closed Universe filled with dust the current value of the Hubble constant is $H_0$ and of the deceleration parameter $q_0$.

a) What is the total proper volume of the Universe at present time?

b) What is the total current proper volume of space occupied by matter which we are presently observing?

c) What is the total proper volume of space which we are directly observing?


Problem 15

problem id: 1301_15

For the closed ($k=+1$) model of Universe, filled with non-relativistic matter, show that solutions of the Friedmann equations can be represented in terms of the two parameters $H_0$ and $q_0$. [Y.Shtanov, Lecture Notes on theoretical cosmology, 2010 ]


Problem 16

problem id: 1301_16

Do the same as in the previous problem for the case of open ($k=-1$) model of Universe.


Classification of models of Universe based on the deceleration parameter

Problem 17

problem id: 1301_17

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 parameter is constant, the deceleration parameter $q$ is also constant and equal to $-1$, as in the de Sitter model. All models can be characterized by whether they expand or contract, and accelerate or decelerate. Build a classification of such type using the signature of the Hubble parameter and the deceleration parameter.


Problem 18

problem id: 1301_18

Can the Universe variate its type of evolutions in frames of the classification given in the previous problem?


Problem 19

problem id: 1301_19

Point out possible regime of expansion of Universe in the case of constant deceleration parameter.


Problem 20

problem id: 1301_20

Having fixed the material content we can classify the model of Universe using the connection between the deceleration parameter and the spatial geometry. Perform this procedure in the case of Universe filled with non-relativistic matter.


Problem 21

problem id: 1301_21

Models of the Universe can be classified basing on the relation between the deceleration parameter and age of the Universe. Build such a classification in the case of Universe filled with non-relativistic matter (the Einstein-de Sitter Universe)


Problem 22

problem id: 1301_22

Show that sign of the deceleration parameter determines the difference between the actual age of the Universe and the Hubble time.


Problem 23

problem id: 1301_23

Suppose the flat Universe is filled with non-relativistic matter with density $\rho_m$ and some substance with equation of state $p_X=w\rho_X$. Express the deceleration parameter through the ratio $r\equiv\rho_m/\rho_X$.


Problem 24

problem id: 1301_23_1

Obtain Friedmann equations for the case of spatially flat $n$-dimensional Universe. (see Shouxin Chen, Gary W. Gibbons, Friedmann's Equations in All Dimensions and Chebyshev's Theorem, arXiv: 1409.3352)


Problem 25

problem id: 1301_23_2

Analyze exact solutions of the Friedmann equations obtained in the previous problem in the case of flat ($k=0$) $n$-dimensional Universe filled with a barotropic liquid with the state equation \begin{equation} \label{10} p_m=w \rho_m. \end{equation} and obtain corresponding explicit expressions for the deceleration parameter.

Deceleration as a cosmographic parameter

Problem 26

problem id: 1301_24

Make transition from the derivatives w.r.t. cosmological time to that w.r.t. conformal time in definitions of the Hubble parameter and the deceleration parameter.


Problem 27

problem id: 1301_25

Make Taylor expansion of the scale factor in time using the cosmographic parameters.


Problem 28

problem id: 1301_26

Make Taylor expansion of the redshift in time using the cosmographic parameters.


Problem 29

problem id: 1301_27

Show that \[q(t)=\frac{d}{dt}\left(\frac1H\right)-1.\]


Problem 30

problem id: 1301_28

Show that the deceleration parameter as a function of the red shift satisfies the following relations \begin{align} \nonumber q(z)=&\frac{1+z}{H}\frac{d H}{dz}-1;\\ \nonumber q(z)=&\frac12(1+z)\frac1{H^2}\frac{d H^2}{dz}-1;\\ \nonumber q(z)=&\frac12\frac{d\ln H^2}{d\ln(1+z)}-1;\\ \nonumber q(z)=&\frac{d\ln H}{dz}(1+z)-1. \end{align}


Problem 31

problem id: 1301_29

Show that the deceleration parameter as a function of the scale factor satisfies the following relations \begin{align} \nonumber q(a)=&-\left(1+\frac{a\frac{dH}{da}}{H}\right);\\ \nonumber q(a)=&\frac{d\ln(aH)}{d\ln a}. \end{align}


Problem 32

problem id: 1301_30

Show that the derivatives $dH/dz$, $d^2H/dz^2$, $d^3H/dz^3$ and $d^4H/dz^4$ can be expressed through the deceleration parameter $q$ and other cosmological parameters.


Problem 33

problem id: 1301_31

Use results of the previous problem to make Taylor expansion of the Hubble parameter in redshift.


Problem 34

problem id: 1301_32

Express derivatives of the Hubble parameter squared w.r.t. the redshift $d^iH^2/dz^i$, $i=1,2,3,4$ in terms of the cosmographic parameters.


Problem 35

problem id: 1301_33

Express the current values of deceleration and jerk parameters in terms of $N=-\ln(1+z)$.


Problem 36

problem id: 1301_34

Express time derivatives of the Hubble parameter in terms of the cosmographic parameters.


Problem 37

problem id: 1301_35

Express the deceleration parameter as power series in redshift $z$ or $y$-redshift $z/(1+z)$.


Problem 38

problem id: 1301_36

Show that the Hubble parameter is connected to the deceleration parameter by the integral relation \[H=H_0\exp\left[\int\limits_0^z[q(z')+1]d\ln (1+z')\right].\]


Problem 39

problem id: 1301_37

Show that derivatives of the lower order parameters can be expressed through the higher ones, for instance \[\frac{dq}{d\ln(1+z)}=j-q(2q+1).\]


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Exactly Integrable n-dimensional Universes

Exactly Integrable n-dimensional Universes

Problem 1

problem id: gnd_1

Derive Friedmann equations for the spatially n-dimensional Universe.


Problem 2

problem id: gnd_2

Obtain the energy conservation law for the case of n-dimensional Universe.


Problem 3

problem id: gnd_3

Obtain relation between the energy density and scale factor in the case of a two-component n-dimensional Universe dominated by the cosmological constant and a barotropic fluid.


Problem 4

problem id: gnd_4

Obtain equation of motion for the scale factor for the previous problem.


To integrate (\ref{14}), we recall Chebyshev's theorem:

For rational numbers $p,q,r$ ($r\neq0$) and nonzero real numbers $\alpha,\beta$, the integral $\int x^p(\alpha+\beta x^r)^q\,d x$ is elementary if and only if at least one of the quantities \begin{equation}\label{cd} \frac{p+1}r,\quad q,\quad \frac{p+1}r+q, \end{equation} is an integer.

Another way to see the validity of the Chebyshev theorem is to represent the integral of concern by a hypergeometric function such that when a quantity in (\ref{cd}) is an integer the hypergeometric function is reduced into an elementary function. Consequently, when $k=0$ or $\Lambda=0$, and $w$ is rational, the Chebyshev theorem enables us to know that, for exactly what values of $n$ and $w$, the equation (\ref{14}) may be integrated.


Problem 5

problem id: gnd_4_0

Obtain analytic solutions for the equation of motion for the scale factor of the previous problem for spatially flat ($k=0$) Universe.


Problem 6

problem id: gnd_5

Use the analytic solutions obtained in the previous problem to study cosmology with $w>-1$ and $a(0)=0$.


Problem 7

problem id: gnd_6

Use results of the previous problem to calculate the deceleration $q=-a\ddot a/\dot a^2$ parameter.


Problem 8

problem id: gnd_4_1

Obtain the exact solvability conditions for the case $\Lambda=0$ in equation (\ref{14}) (see problem #gnd_4 ).


Problem 9

problem id: gnd_4_2

Obtain the explicit solutions for the case $n=3$ and $w=-5/9$ in the previous problem.


Problem 10

problem id: gnd_10

Rewrite equation of motion for the scale factor (\ref{14}) in terms of the conformal time.


Problem 11

problem id: gnd_11

Find the rational values of the equation of state parameter $w$ which provide exact integrability of the equation (\ref{48}) with $k=0$.


Problem 12

problem id: gnd_12

Obtain explicit solutions for the case $w=\frac1n-1$ in the previous problem.


Problem 13

problem id: gnd_13

Obtain exact solutions for the equation (\ref{48}) with $\Lambda=0$.


Problem 14

problem id: gnd_14

Obtain inflationary solutions using the results of the previous problem.


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.