Difference between revisions of "Solutions of Friedman equations in the Big Bang model"

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(Problem 5.)
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[[Category:Dynamics of the Universe in the Big Bang Model|2]]
 
[[Category:Dynamics of the Universe in the Big Bang Model|2]]
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=== Problem 1. ===
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=== Problem 1: flat solutions with matter or radiation ===
 
Derive $\rho(a)$, $\rho(t)$ and $ a(t)$ for a spatially flat$^*$ ($k=0$) Universe that consists of only
 
Derive $\rho(a)$, $\rho(t)$ and $ a(t)$ for a spatially flat$^*$ ($k=0$) Universe that consists of only
  
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=== Problem 2. ===
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=== Problem 2: energy density evolution ===
 
Consider two spatially flat Universes. One is filled with radiation, the other with dust. The current energy density is the same. Compare energy densities when both of them are twice as old.
 
Consider two spatially flat Universes. One is filled with radiation, the other with dust. The current energy density is the same. Compare energy densities when both of them are twice as old.
 
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=== Problem 3. ===
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=== Problem 3: one-component-dominated Universe ===
 
Find the scale factor and density of each component as functions of time in a flat Universe which consists of dust and radiation, for the case one of the components is dominating. Present the results graphically.
 
Find the scale factor and density of each component as functions of time in a flat Universe which consists of dust and radiation, for the case one of the components is dominating. Present the results graphically.
 
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=== Problem 4: $H(t)$ ===
=== Problem 4. ===
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Derive the time dependence of the Hubble parameter for a flat Universe in which either matter or radiation is dominating.
 
Derive the time dependence of the Hubble parameter for a flat Universe in which either matter or radiation is dominating.
 
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=== Problem 5: exact solutions for matter and radiation ===
=== Problem 5. ===
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Derive the exact solutions of the Friedman equations for the Universe filled with matter and radiation. Plot the graphs of scale factor and both densities as functions of time.
 
Derive the exact solutions of the Friedman equations for the Universe filled with matter and radiation. Plot the graphs of scale factor and both densities as functions of time.
 
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=== Problem 6: equipartition time ===
=== Problem 6. ===
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At what moment after the Big Bang did matter's density exceed that of radiation for the first time?
 
At what moment after the Big Bang did matter's density exceed that of radiation for the first time?
 
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=== Problem 7. ===
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=== Problem 7: age of the Universe ===
 
Determine the age of the Universe in which either matter or radiation has always been dominating.
 
Determine the age of the Universe in which either matter or radiation has always been dominating.
 
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=== Problem 8. ===
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=== Problem 8: one component with arbitrary but constant EoS parameter ===
 
Derive the dependence  $a(t)$ for a spatially flat Universe that consists of matter with equation of state $p=w\rho$, assuming that the parameter $w$ does not change throughout the evolution.
 
Derive the dependence  $a(t)$ for a spatially flat Universe that consists of matter with equation of state $p=w\rho$, assuming that the parameter $w$ does not change throughout the evolution.
 
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=== Problem 9. ===
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=== Problem 9: $H(t)$ ===
 
Find the Hubble parameter as function of time for the previous problem.
 
Find the Hubble parameter as function of time for the previous problem.
 
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=== Problem 10. ===
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=== Problem 10: effective potential ===
 
Using the first Friedman equation, construct the effective potential $V(a)$, which governs the one-dimensional motion of a fictitious particle with coordinate $a(t)$, for the Universe filled with several non-interacting components.
 
Using the first Friedman equation, construct the effective potential $V(a)$, which governs the one-dimensional motion of a fictitious particle with coordinate $a(t)$, for the Universe filled with several non-interacting components.
 
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=== Problem 11. ===
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=== Problem 11: effective potential for radiation and dust ===
 
Construct the effective one-dimensional potential $V(a)$ (using the notation of the previous problem) for the Universe consisting of non-relativistic matter and radiation. Show that motion with $\dot{a}>0$ in such a potential can only be decelerating.
 
Construct the effective one-dimensional potential $V(a)$ (using the notation of the previous problem) for the Universe consisting of non-relativistic matter and radiation. Show that motion with $\dot{a}>0$ in such a potential can only be decelerating.
 
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=== Problem 12. ===
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=== Problem 12: exact solutions with dust, radiation and curvature ===
 
Derive the exact solutions of the Friedman equations for the Universe with arbitrary curvature, filled with radiation and matter.
 
Derive the exact solutions of the Friedman equations for the Universe with arbitrary curvature, filled with radiation and matter.
 
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=== Problem 13. ===
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=== Problem 13: deceleration parameter for one component ===
 
Show that for a spatially flat Universe consisting of one component with equation of state $p = w\rho$ the deceleration parameter $q\equiv -\ddot{a}/(aH^2)$ is equal to $\frac{1}{2}(1 + 3w)$.
 
Show that for a spatially flat Universe consisting of one component with equation of state $p = w\rho$ the deceleration parameter $q\equiv -\ddot{a}/(aH^2)$ is equal to $\frac{1}{2}(1 + 3w)$.
 
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=== Problem 14. ===
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=== Problem 14: age of the Universe through $q$ ===
 
Express the age of the Universe through the deceleration parameter $q=-\ddot{a}/(aH^2)$ for a spatially flat Universe filled with single component with equation of state $p=w\rho$.
 
Express the age of the Universe through the deceleration parameter $q=-\ddot{a}/(aH^2)$ for a spatially flat Universe filled with single component with equation of state $p=w\rho$.
 
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=== Problem 15. ===
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=== Problem 15: $q(w)$ in non-flat case ===
 
Find the generalization of relation $q = \frac{1}{2}(1 + 3w)$ for the non-flat case.
 
Find the generalization of relation $q = \frac{1}{2}(1 + 3w)$ for the non-flat case.
 
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=== Problem 16. ===
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=== Problem 16: $q[\rho(a)]$ ===
 
Show that for a one-component Universe filled with ideal fluid of density $\rho$
 
Show that for a one-component Universe filled with ideal fluid of density $\rho$
 
\[q=-1-\frac{1}{2}\,\frac{d\ln\rho}{d\ln a}.\]
 
\[q=-1-\frac{1}{2}\,\frac{d\ln\rho}{d\ln a}.\]
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=== Problem 17. ===
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=== Problem 17: $q$ for several components ===
 
Show that for a Universe consisting of several components with equations of state $p_{i}  = w_{i} \rho_{i}$ the deceleration parameter is
 
Show that for a Universe consisting of several components with equations of state $p_{i}  = w_{i} \rho_{i}$ the deceleration parameter is
 
\[q = \frac{\Omega }{2} +
 
\[q = \frac{\Omega }{2} +
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=== Problem 18. ===
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=== Problem 18: acceleration or deceleration ===
For which values of state parameter $w$ the rate of expansion of a one-component flat Universe increases with time?
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For which values of the state parameter $w$ the rate of expansion of a one-component flat Universe increases with time?
 
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=== Problem 19. ===
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=== Problem 19: closed dusty Universe, exact solution ===
 
Show that for a spatially closed  ($k=1$) Universe that contains only non-relativistic matter the solution of the Friedman equations can be given in the form
 
Show that for a spatially closed  ($k=1$) Universe that contains only non-relativistic matter the solution of the Friedman equations can be given in the form
 
\[a(\eta)=a_{\star}(1-\cos\eta);
 
\[a(\eta)=a_{\star}(1-\cos\eta);
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=== Problem 20. ===
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=== Problem 20: size and lifetime ===
 
Find the relation between the maximum size and the total lifetime of a closed Universe filled with dust.
 
Find the relation between the maximum size and the total lifetime of a closed Universe filled with dust.
 
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=== Problem 21. ===
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=== Problem 21: future through observables ===
 
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?
 
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?
 
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=== Problem 22. ===
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=== Problem 22: age of the Universe ===
 
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.
 
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.
 
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=== Problem 23. ===
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=== Problem 23: around the Universe ===
 
Suppose in the same Universe radiation is dominating during a negligibly small fraction of total time of evolution. How many times will a photon travel around the Universe during the time from its "birth" to its "death"?
 
Suppose in the same Universe radiation is dominating during a negligibly small fraction of total time of evolution. How many times will a photon travel around the Universe during the time from its "birth" to its "death"?
 
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=== Problem 24. ===
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=== Problem 24: different volumes in dynamical Universe ===
 
In a closed Universe filled with dust the current value of the Hubble constant is $H_0$ and of the deceleration parameter $q_0$.
 
In a closed Universe filled with dust the current value of the Hubble constant is $H_0$ and of the deceleration parameter $q_0$.
  
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=== Problem 25. ===
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=== Problem 25: open dusty Universe  ===
 
Find the solution of Friedman equations for spatially open ($k=-1$) Universe filled with dust in the parametric form $a(\eta)$, $t(\eta)$.
 
Find the solution of Friedman equations for spatially open ($k=-1$) Universe filled with dust in the parametric form $a(\eta)$, $t(\eta)$.
 
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=== Problem 26. ===
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=== Problem 26: density evolution ===
 
Suppose the density of some component in a spatially flat Universe depends on scale factor as  $\rho(t) \sim a^{-n}(t)$. How much time is needed for the density of this component to change from $\rho_1$ to $\rho_2$?
 
Suppose the density of some component in a spatially flat Universe depends on scale factor as  $\rho(t) \sim a^{-n}(t)$. How much time is needed for the density of this component to change from $\rho_1$ to $\rho_2$?
 
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=== Problem 27. ===
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=== Problem 27: $q[H(t)]$ ===
 
Using the expression for $H(t)$, calculate the deceleration parameter for the cases of domination of  
 
Using the expression for $H(t)$, calculate the deceleration parameter for the cases of domination of  
  
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=== Problem 28. ===
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=== Problem 28: equation of state ===
 
Consider a Universe consisting of $n$ components, with equations of state $p_{i}=w_{i}\rho_{i}$, and find $w_{tot}$, the parameter of the equation of state $p_{tot}=w_{tot}\rho_{tot}$.
 
Consider a Universe consisting of $n$ components, with equations of state $p_{i}=w_{i}\rho_{i}$, and find $w_{tot}$, the parameter of the equation of state $p_{tot}=w_{tot}\rho_{tot}$.
 
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=== Problem 29. ===
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=== Problem 29: equations for relative densities ===
 
Derive the equations of motion for  relative densities $\Omega_{i}=\rho_{i}/\rho_{cr}$ of the two components comprising a spatially flat two-component Universe, if their equations of state are $p=w_i\rho$, $i=1,2$.
 
Derive the equations of motion for  relative densities $\Omega_{i}=\rho_{i}/\rho_{cr}$ of the two components comprising a spatially flat two-component Universe, if their equations of state are $p=w_i\rho$, $i=1,2$.
 
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=== Problem 30. ===
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=== Problem 30: EoS for non-relativistic gas ===
 
Suppose a Universe is initially filled with a gas of non-relativistic particles of mass density  $\rho_{0}$, pressure $p_{0}$, and $c_p/c_v=\gamma$. Construct the equation of state for such a system.
 
Suppose a Universe is initially filled with a gas of non-relativistic particles of mass density  $\rho_{0}$, pressure $p_{0}$, and $c_p/c_v=\gamma$. Construct the equation of state for such a system.
 
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=== Problem 31. ===
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=== Problem 31: Newtonian derivation of critical density ===
 
Derive the expression for the critical density $\rho_{cr}$ from the condition that Hubble's expansion velocity equals the second cosmic velocity (escape velocity) $v=\sqrt{2gR}$.
 
Derive the expression for the critical density $\rho_{cr}$ from the condition that Hubble's expansion velocity equals the second cosmic velocity (escape velocity) $v=\sqrt{2gR}$.
 
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=== Problem 32. ===
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=== Problem 32: dust plus something else ===
 
Suppose the Universe is filled with non-relativistic matter and some substance with equation of state $p_X=w\rho_X$. Find the evolution equation for the quantity $r \equiv \frac{\rho_m}{\rho _X}$.
 
Suppose the Universe is filled with non-relativistic matter and some substance with equation of state $p_X=w\rho_X$. Find the evolution equation for the quantity $r \equiv \frac{\rho_m}{\rho _X}$.
 
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=== Problem 33. ===
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=== Problem 33:deceleration parameter  ===
 
Express the deceleration parameter through the ratio $r$ for the conditions of the previous problem.
 
Express the deceleration parameter through the ratio $r$ for the conditions of the previous problem.
 
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=== Problem 34. ===
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=== Problem 34: $\rho_{X}\propto H^2$ ===
 
Let a spatially flat Universe be filled with non-relativistic dust and a substance with equation of state $p_{X}= w\rho_{X}$. Show that in case $\rho_{X}\propto H^2$, the ratio $r =\rho_{m}/\rho_{X}$ does not depend on time.
 
Let a spatially flat Universe be filled with non-relativistic dust and a substance with equation of state $p_{X}= w\rho_{X}$. Show that in case $\rho_{X}\propto H^2$, the ratio $r =\rho_{m}/\rho_{X}$ does not depend on time.
 
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=== Problem 35. ===
+
=== Problem 35: $r(q)$ ===
 
Show that for the model of the Universe described in the previous problem the parameter $r$ is related with the deceleration parameter as
 
Show that for the model of the Universe described in the previous problem the parameter $r$ is related with the deceleration parameter as
 
\[\dot{r}=-2H\frac{\Omega_{curv}}{\Omega_X}q.\]
 
\[\dot{r}=-2H\frac{\Omega_{curv}}{\Omega_X}q.\]
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=== Problem 36. ===
+
=== Problem 36: $r$ in open and closed cases ===
 
Show that in the model of the Universe of [[#dyn46nn|problem]], in case $k=+1$ and $q>0$ (decelerated expansion) $r$ increases with time, in case $k=+1$  and $q<0$ (accelerated expansion) $r$ decreases with time, and for $k=-1$ vice-versa.
 
Show that in the model of the Universe of [[#dyn46nn|problem]], in case $k=+1$ and $q>0$ (decelerated expansion) $r$ increases with time, in case $k=+1$  and $q<0$ (accelerated expansion) $r$ decreases with time, and for $k=-1$ vice-versa.
 
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{{{\Omega }_{X}}}q\]
 
{{{\Omega }_{X}}}q\]
 
and definition
 
and definition
\[{{\Omega }_{curv}}=-\frac{k}{{{a}^{2}}{{H}^{2}}}.\]</p>
+
\[{\Omega }_{curv}=-\frac{k}{{{a}^{2}}{{H}^{2}}}.\]</p>
 
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=== Problem 37. ===
+
=== Problem 37: power-law cosmologies ===
 
Let us consider a general class of power-law cosmologies described by the scale factor
 
Let us consider a general class of power-law cosmologies described by the scale factor
 
\[a(t) =a_{0}\Big(\frac{t}{t_0}\Big)^\alpha,\]
 
\[a(t) =a_{0}\Big(\frac{t}{t_0}\Big)^\alpha,\]
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=== Problem 38. ===
+
=== Problem 38: age of the Universe ===
 
In the power-law cosmology find the age of the Universe at redshift $z$.
 
In the power-law cosmology find the age of the Universe at redshift $z$.
 
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=== Problem 39. ===
+
=== Problem 39: luminosity distance ===
 
For the power-law cosmology find  the luminosity distance between the observer and the object with redshift $z$.
 
For the power-law cosmology find  the luminosity distance between the observer and the object with redshift $z$.
 
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Revision as of 18:09, 13 October 2012

Contents

Problem 1: flat solutions with matter or radiation

Derive $\rho(a)$, $\rho(t)$ and $ a(t)$ for a spatially flat$^*$ ($k=0$) Universe that consists of only

a) radiation,

b) non-relativistic matter$^{**}$.

$^*$ The "spatial" part is often omitted when there cannot be any confusion (and even when there can be), generally and hereatfer.

$^{**}$ In this context and below also quite often called just "matter" or dust.


Problem 2: energy density evolution

Consider two spatially flat Universes. One is filled with radiation, the other with dust. The current energy density is the same. Compare energy densities when both of them are twice as old.


Problem 3: one-component-dominated Universe

Find the scale factor and density of each component as functions of time in a flat Universe which consists of dust and radiation, for the case one of the components is dominating. Present the results graphically.


Problem 4: $H(t)$

Derive the time dependence of the Hubble parameter for a flat Universe in which either matter or radiation is dominating.



Problem 5: exact solutions for matter and radiation

Derive the exact solutions of the Friedman equations for the Universe filled with matter and radiation. Plot the graphs of scale factor and both densities as functions of time.


Problem 6: equipartition time

At what moment after the Big Bang did matter's density exceed that of radiation for the first time?


Problem 7: age of the Universe

Determine the age of the Universe in which either matter or radiation has always been dominating.


Problem 8: one component with arbitrary but constant EoS parameter

Derive the dependence $a(t)$ for a spatially flat Universe that consists of matter with equation of state $p=w\rho$, assuming that the parameter $w$ does not change throughout the evolution.


Problem 9: $H(t)$

Find the Hubble parameter as function of time for the previous problem.


Problem 10: effective potential

Using the first Friedman equation, construct the effective potential $V(a)$, which governs the one-dimensional motion of a fictitious particle with coordinate $a(t)$, for the Universe filled with several non-interacting components.


Problem 11: effective potential for radiation and dust

Construct the effective one-dimensional potential $V(a)$ (using the notation of the previous problem) for the Universe consisting of non-relativistic matter and radiation. Show that motion with $\dot{a}>0$ in such a potential can only be decelerating.


Problem 12: exact solutions with dust, radiation and curvature

Derive the exact solutions of the Friedman equations for the Universe with arbitrary curvature, filled with radiation and matter.


Problem 13: deceleration parameter for one component

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


Problem 14: age of the Universe through $q$

Express the age of the Universe through the deceleration parameter $q=-\ddot{a}/(aH^2)$ for a spatially flat Universe filled with single component with equation of state $p=w\rho$.


Problem 15: $q(w)$ in non-flat case

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


Problem 16: $q[\rho(a)]$

Show that for a one-component Universe filled with ideal fluid of density $\rho$ \[q=-1-\frac{1}{2}\,\frac{d\ln\rho}{d\ln a}.\]


Problem 17: $q$ for several components

Show that for a Universe consisting of several components with equations of state $p_{i} = w_{i} \rho_{i}$ the deceleration parameter is \[q = \frac{\Omega }{2} + \frac{3}{2}\sum\limits_i {w_i \Omega_i },\] where $\Omega$ is the total relative density.


Problem 18: acceleration or deceleration

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


Problem 19: closed dusty Universe, exact solution

Show that for a spatially closed ($k=1$) Universe that contains only non-relativistic matter the solution of the Friedman equations can be given in the form \[a(\eta)=a_{\star}(1-\cos\eta); \qquad t(\eta)=a_{\star}(\eta -\sin\eta); \qquad a_{\star}=\frac{4\pi G\rho_0}{3}; \quad 0<\eta<2\pi.\]


Problem 20: size and lifetime

Find the relation between the maximum size and the total lifetime of a closed Universe filled with dust.


Problem 21: future through observables

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?


Problem 22: age of the Universe

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 23: around the Universe

Suppose in the same Universe radiation is dominating during a negligibly small fraction of total time of evolution. How many times will a photon travel around the Universe during the time from its "birth" to its "death"?


Problem 24: different volumes in dynamical Universe

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 25: open dusty Universe

Find the solution of Friedman equations for spatially open ($k=-1$) Universe filled with dust in the parametric form $a(\eta)$, $t(\eta)$.


Problem 26: density evolution

Suppose the density of some component in a spatially flat Universe depends on scale factor as $\rho(t) \sim a^{-n}(t)$. How much time is needed for the density of this component to change from $\rho_1$ to $\rho_2$?


Problem 27: $q[H(t)]$

Using the expression for $H(t)$, calculate the deceleration parameter for the cases of domination of

a) radiation,

b) matter.


Problem 28: equation of state

Consider a Universe consisting of $n$ components, with equations of state $p_{i}=w_{i}\rho_{i}$, and find $w_{tot}$, the parameter of the equation of state $p_{tot}=w_{tot}\rho_{tot}$.


Problem 29: equations for relative densities

Derive the equations of motion for relative densities $\Omega_{i}=\rho_{i}/\rho_{cr}$ of the two components comprising a spatially flat two-component Universe, if their equations of state are $p=w_i\rho$, $i=1,2$.


Problem 30: EoS for non-relativistic gas

Suppose a Universe is initially filled with a gas of non-relativistic particles of mass density $\rho_{0}$, pressure $p_{0}$, and $c_p/c_v=\gamma$. Construct the equation of state for such a system.


Problem 31: Newtonian derivation of critical density

Derive the expression for the critical density $\rho_{cr}$ from the condition that Hubble's expansion velocity equals the second cosmic velocity (escape velocity) $v=\sqrt{2gR}$.


Problem 32: dust plus something else

Suppose the Universe is filled with non-relativistic matter and some substance with equation of state $p_X=w\rho_X$. Find the evolution equation for the quantity $r \equiv \frac{\rho_m}{\rho _X}$.


Problem 33:deceleration parameter

Express the deceleration parameter through the ratio $r$ for the conditions of the previous problem.


Problem 34: $\rho_{X}\propto H^2$

Let a spatially flat Universe be filled with non-relativistic dust and a substance with equation of state $p_{X}= w\rho_{X}$. Show that in case $\rho_{X}\propto H^2$, the ratio $r =\rho_{m}/\rho_{X}$ does not depend on time.


Problem 35: $r(q)$

Show that for the model of the Universe described in the previous problem the parameter $r$ is related with the deceleration parameter as \[\dot{r}=-2H\frac{\Omega_{curv}}{\Omega_X}q.\]


Problem 36: $r$ in open and closed cases

Show that in the model of the Universe of problem, in case $k=+1$ and $q>0$ (decelerated expansion) $r$ increases with time, in case $k=+1$ and $q<0$ (accelerated expansion) $r$ decreases with time, and for $k=-1$ vice-versa.


The following three problems on power-law cosmology are inspired by Kumar [1].


Problem 37: power-law cosmologies

Let us consider a general class of power-law cosmologies described by the scale factor \[a(t) =a_{0}\Big(\frac{t}{t_0}\Big)^\alpha,\] where $t_0$ is the present age of theUniverse and $\alpha$ is a dimensionless positive parameter. Show that:

1) the scale factor in terms of the deceleration parameter may be written as \[a(t) =a_{0}\Big(\frac{t}{t_0}\Big)^{1/1 + q}, \quad\text{i.e.}\quad \alpha=\frac{1}{1+q}.\]

2) the expansion of the Universe is described by Hubble parameter \[H=\frac{1}{(1+q)\;t}\] or in terms of redshift \[H(z)=H_{0}(1+z)^{1+q}.\]


Problem 38: age of the Universe

In the power-law cosmology find the age of the Universe at redshift $z$.


Problem 39: luminosity distance

For the power-law cosmology find the luminosity distance between the observer and the object with redshift $z$.