Difference between revisions of "New from June"

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<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_0</p>  
 
<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_0</p>  
 
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.]
 
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.]
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(a) The surface that is expanding is two dimensional; the "center" of the balloon is in the third dimension and is not part of the surface, which has no center.<br/>
 
(a) The surface that is expanding is two dimensional; the "center" of the balloon is in the third dimension and is not part of the surface, which has no center.<br/>
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<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_1</p>  
 
<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_1</p>  
 
(into the cosmography and extended deceleration parameter) Show that \[\frac{d\dot a}{da}=-Hq.\]
 
(into the cosmography and extended deceleration parameter) Show that \[\frac{d\dot a}{da}=-Hq.\]
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\[\frac{d\dot a}{da}=\frac{d\dot a}{dt}\frac{dt}{da}=\frac{\ddot a}{\dot a}=\frac{\ddot a}{aH}\equiv-Hq.\]
 
\[\frac{d\dot a}{da}=\frac{d\dot a}{dt}\frac{dt}{da}=\frac{\ddot a}{\dot a}=\frac{\ddot a}{aH}\equiv-Hq.\]
 
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<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_2</p>  
 
<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_2</p>  
 
Give a physical interpretation of the conservation equation.
 
Give a physical interpretation of the conservation equation.
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The energy density has a time dependence determined by the conservation equation,
 
The energy density has a time dependence determined by the conservation equation,
 
\[\dot\rho=-3H\rho-3Hp.\]
 
\[\dot\rho=-3H\rho-3Hp.\]
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<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_04</p>  
 
<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_04</p>  
 
Find evolution equation for the density parameter $\Omega$ of the single-fluid FLRW models with the linear equation of state $p=w\rho$.
 
Find evolution equation for the density parameter $\Omega$ of the single-fluid FLRW models with the linear equation of state $p=w\rho$.
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\begin{align}
 
\begin{align}
 
\nonumber \Omega&=\frac\rho{3H^2},\quad (8\pi G=1),\\
 
\nonumber \Omega&=\frac\rho{3H^2},\quad (8\pi G=1),\\
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<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_05</p>  
 
<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_05</p>  
 
Solve the previous problem for the multi-component case.
 
Solve the previous problem for the multi-component case.
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The evolution equations for an $n$-fluid model use the density parameters $\Omega_1$, $\Omega_2\ldots\Omega_n$, as dynamical variables. Relation
 
The evolution equations for an $n$-fluid model use the density parameters $\Omega_1$, $\Omega_2\ldots\Omega_n$, as dynamical variables. Relation
 
\[\frac{d\Omega}{dN}=-[2q-3w-1]\Omega\]
 
\[\frac{d\Omega}{dN}=-[2q-3w-1]\Omega\]
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<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_06</p>  
 
<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_06</p>  
 
Use the conformal time to prove existence of smooth transition from the radiation-dominated era to the matter dominated one.
 
Use the conformal time to prove existence of smooth transition from the radiation-dominated era to the matter dominated one.
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Rewrite the first Friedmann equation in the form
 
Rewrite the first Friedmann equation in the form
 
\[H^2=\frac{\rho_{eq}}{3}\left[\left(\frac{a_{eq}}{a}\right)^3+\left(\frac{a_{eq}}{a}\right)^4\right],\]
 
\[H^2=\frac{\rho_{eq}}{3}\left[\left(\frac{a_{eq}}{a}\right)^3+\left(\frac{a_{eq}}{a}\right)^4\right],\]
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<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_07</p>  
 
<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_07</p>  
 
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 $q,j,s\dots$.
 
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 $q,j,s\dots$.
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\[Q=\frac{\dot H}{H^2}=\left(\frac{\ddot a}a-H^2\right)\frac1{H^2}=-(q+1);\]
 
\[Q=\frac{\dot H}{H^2}=\left(\frac{\ddot a}a-H^2\right)\frac1{H^2}=-(q+1);\]
 
\[J=\ddot H \frac{H}{\dot H^2}=\left(\frac{\dddot a}a-\frac{\dot a\ddot a}{a^2}-2H\dot H\right)\frac{H}{\dot H^2}=\frac j{Q^2}+\frac q{Q^2}-\frac2Q=\frac{j+3q+2}{(1+q)^2}.\]
 
\[J=\ddot H \frac{H}{\dot H^2}=\left(\frac{\dddot a}a-\frac{\dot a\ddot a}{a^2}-2H\dot H\right)\frac{H}{\dot H^2}=\frac j{Q^2}+\frac q{Q^2}-\frac2Q=\frac{j+3q+2}{(1+q)^2}.\]
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<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_08</p>  
 
<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_08</p>  
 
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.
 
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.
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\[\frac{d}{d\ln a}=\frac1H\frac{d}{dt};\]
 
\[\frac{d}{d\ln a}=\frac1H\frac{d}{dt};\]
 
\[\bar Q\equiv\frac{\dot H}{H^2},\quad \bar J\equiv\frac{\ddot H}{H^3}-\frac{\dot H^2}{H^4},\quad \bar S\equiv\frac{\dddot H}{H^4}+3\frac{\dot H^3}{H^6}-4\frac{\dot H\ddot H}{H^5}.\]
 
\[\bar Q\equiv\frac{\dot H}{H^2},\quad \bar J\equiv\frac{\ddot H}{H^3}-\frac{\dot H^2}{H^4},\quad \bar S\equiv\frac{\dddot H}{H^4}+3\frac{\dot H^3}{H^6}-4\frac{\dot H\ddot H}{H^5}.\]
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<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_09</p>  
 
<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_09</p>  
 
Express the Ricci scalar and its time derivatives in terms of the $\bar Q$, $\bar J$ and $\bar S$.
 
Express the Ricci scalar and its time derivatives in terms of the $\bar Q$, $\bar J$ and $\bar S$.
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\begin{align}
 
\begin{align}
 
\nonumber R&= -6\left[(\bar Q+2)H^2+\frac k{a^2}\right];\\
 
\nonumber R&= -6\left[(\bar Q+2)H^2+\frac k{a^2}\right];\\
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<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_3</p>  
 
<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_3</p>  
 
Show that for a perfect fluid with EoS $p=w(a)\rho$ the adiabatic sound speed can be represented in the form \[c_S^2=w(a)-\frac13\frac{d\ln(1+w)}{d\ln a}.\]
 
Show that for a perfect fluid with EoS $p=w(a)\rho$ the adiabatic sound speed can be represented in the form \[c_S^2=w(a)-\frac13\frac{d\ln(1+w)}{d\ln a}.\]
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\[c_S^2\equiv\frac{d p}{d\rho}=w(a)+\rho\frac{dw(a)}{d\rho};\]
 
\[c_S^2\equiv\frac{d p}{d\rho}=w(a)+\rho\frac{dw(a)}{d\rho};\]
 
\[\frac{dw}{d\rho}=\frac{dw}{da}\frac{da}{d\rho};\]
 
\[\frac{dw}{d\rho}=\frac{dw}{da}\frac{da}{d\rho};\]
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<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_4</p>  
 
<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_4</p>  
 
Obtain equation for $\ddot\rho(t)$, where $\rho(t)$ is energy density of an ideal fluid participating in the cosmological expansion.
 
Obtain equation for $\ddot\rho(t)$, where $\rho(t)$ is energy density of an ideal fluid participating in the cosmological expansion.
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Take time derivative of the conservation equation to obtain
 
Take time derivative of the conservation equation to obtain
 
\begin{align}\nonumber
 
\begin{align}\nonumber
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<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_5</p>  
 
<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_5</p>  
 
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 $p=w\rho$.
 
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 $p=w\rho$.
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\[\frac{d^2 a^3}{dt^2}=3\left(2H^2+\frac{\ddot a}a\right)a^3=3\left[\frac23\rho-\frac16(\rho+3p)\right]a^3=\frac32(\rho-p)\varphi.\]
 
\[\frac{d^2 a^3}{dt^2}=3\left(2H^2+\frac{\ddot a}a\right)a^3=3\left[\frac23\rho-\frac16(\rho+3p)\right]a^3=\frac32(\rho-p)\varphi.\]
 
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<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_6</p>  
 
<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_6</p>  
 
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 $p=w(z)\rho$.
 
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 $p=w(z)\rho$.
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From the definitions of redshift $1+z=1/a$ we have
 
From the definitions of redshift $1+z=1/a$ we have
 
\[\frac{dz}{dt}=-\frac{\dot a}{a^2}=-H(z)(1+z)\]
 
\[\frac{dz}{dt}=-\frac{\dot a}{a^2}=-H(z)(1+z)\]
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<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_7</p>  
 
<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_7</p>  
 
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).
 
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).
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In the case of the power law expansion with $\alpha<1$ (the decelerated expansion) the Hubble radius indeed grows faster than the expanding Universe: $R_H=H^{-1}\propto t$, while $a(t)\propto t^\alpha$. In power law situations  the Hubble radius has an expansion velocity \[\frac{d}{dt}\left(\frac1H\right)=\frac1\alpha\] greater than light speed. This behavior is true only for a  decelerating Universe composed of matter and radiation.  This is not a physical velocity, violating special relativity, but the velocity of expansion of the metric itself.
 
In the case of the power law expansion with $\alpha<1$ (the decelerated expansion) the Hubble radius indeed grows faster than the expanding Universe: $R_H=H^{-1}\propto t$, while $a(t)\propto t^\alpha$. In power law situations  the Hubble radius has an expansion velocity \[\frac{d}{dt}\left(\frac1H\right)=\frac1\alpha\] greater than light speed. This behavior is true only for a  decelerating Universe composed of matter and radiation.  This is not a physical velocity, violating special relativity, but the velocity of expansion of the metric itself.
 
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<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_015</p>  
 
<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_015</p>  
 
Show, that in the Milne Universe the age of the Universe is equal to the Hubble time.
 
Show, that in the Milne Universe the age of the Universe is equal to the Hubble time.
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In an empty (Milne) Universe since $H=\dot a/a=t^{-1}$, the age of the Universe $t_0$ is equal to the Hubble time $t_0=H_0^{-1}$.
 
In an empty (Milne) Universe since $H=\dot a/a=t^{-1}$, the age of the Universe $t_0$ is equal to the Hubble time $t_0=H_0^{-1}$.
 
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Can this condition be satisfied in the Newtonian mechanics?
 
Can this condition be satisfied in the Newtonian mechanics?
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A naive argument tells us that as we pile up more and more material of constant density $\rho_0$, the ratio $M/R$ increases:
 
A naive argument tells us that as we pile up more and more material of constant density $\rho_0$, the ratio $M/R$ increases:
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<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_017</p>  
 
<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_017</p>  
 
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.
 
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.
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The position of quanta inside a black hole can only be known within $\Delta x=2r_{Sh}=4GM/c^2$. Thus $\Delta t =2r_{Sh}/c=4GM/c^3$. According to the uncertainty principle $\Delta E\Delta t\ge\hbar$. Thus
 
The position of quanta inside a black hole can only be known within $\Delta x=2r_{Sh}=4GM/c^2$. Thus $\Delta t =2r_{Sh}/c=4GM/c^3$. According to the uncertainty principle $\Delta E\Delta t\ge\hbar$. Thus
 
\[\Delta T=\frac{\Delta E}k\approx\frac{\bar c^3}{4kGM}\]
 
\[\Delta T=\frac{\Delta E}k\approx\frac{\bar c^3}{4kGM}\]
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<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 2501_06</p>  
 
<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 2501_06</p>  
 
Why the cosmological constant cannot be used as a source for inflation in the inflation model?
 
Why the cosmological constant cannot be used as a source for inflation in the inflation model?
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The cosmological constant do provide the accelerated expansion of Universe needed to realize the inflation: $a(t)\propto e^{Ht}$, $q=-1$. However, in approximation of the cosmological constant, $H$ is constant for all time. Therefore a dynamical mechanism for the limited time of inflation is needed. The physical mechanism for the existence of an approximately constant value of $H$ which lasts for a limited time is given by a scalar field. For a large initial potential energy of scalar field the state equation parameter $w\approx-1$ and the scalar field in process of the "slow roll" imitates the cosmological constant for sufficiently long period of time to solve the flatness and causal problems. Then due to shape of the potential the scalar field exits from the slow-roll regime, oscillates about its' potential minimum decaying  into less massive particles insuring that inflation time is finite.
 
The cosmological constant do provide the accelerated expansion of Universe needed to realize the inflation: $a(t)\propto e^{Ht}$, $q=-1$. However, in approximation of the cosmological constant, $H$ is constant for all time. Therefore a dynamical mechanism for the limited time of inflation is needed. The physical mechanism for the existence of an approximately constant value of $H$ which lasts for a limited time is given by a scalar field. For a large initial potential energy of scalar field the state equation parameter $w\approx-1$ and the scalar field in process of the "slow roll" imitates the cosmological constant for sufficiently long period of time to solve the flatness and causal problems. Then due to shape of the potential the scalar field exits from the slow-roll regime, oscillates about its' potential minimum decaying  into less massive particles insuring that inflation time is finite.
 
</p>  </div></div></div>
 
</p>  </div></div></div>
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<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 2501_09</p>  
 
<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 2501_09</p>  
 
Show that inflation ends when the parameter \[\varepsilon\equiv\frac{M_P^2}{16\pi}\left(\frac{dV}{d\varphi}\frac1V\right)^2=1.\]
 
Show that inflation ends when the parameter \[\varepsilon\equiv\frac{M_P^2}{16\pi}\left(\frac{dV}{d\varphi}\frac1V\right)^2=1.\]
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Using definition of the parameter $\varepsilon$ one finds
 
Using definition of the parameter $\varepsilon$ one finds
 
\[\varepsilon=-\frac{\dot H}{H^2}.\]
 
\[\varepsilon=-\frac{\dot H}{H^2}.\]
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<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 2501_10</p>  
 
<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 2501_10</p>  
 
How does the number of e-folds $N$ depend on the slow-roll parameter $\varepsilon$?  
 
How does the number of e-folds $N$ depend on the slow-roll parameter $\varepsilon$?  
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\[N\propto\int\limits_{\varphi_{end}}^{\varphi_{initial}}d\varphi\frac{V}{dV/d\varphi} \propto\int\limits_{\varphi_{end}}^{\varphi_{initial}}d\varphi/\sqrt\varepsilon.\]
 
\[N\propto\int\limits_{\varphi_{end}}^{\varphi_{initial}}d\varphi\frac{V}{dV/d\varphi} \propto\int\limits_{\varphi_{end}}^{\varphi_{initial}}d\varphi/\sqrt\varepsilon.\]
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<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_021</p>  
 
<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_021</p>  
 
Using the by dimensional analysis for cosmological constant $\Lambda > 0$, define the set of fundamental "de Sitter units" of length, time and mass.
 
Using the by dimensional analysis for cosmological constant $\Lambda > 0$, define the set of fundamental "de Sitter units" of length, time and mass.
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\begin{align}
 
\begin{align}
 
\nonumber L_{dS}&=\sqrt{1/\Lambda}\\
 
\nonumber L_{dS}&=\sqrt{1/\Lambda}\\
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<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_022</p>  
 
<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_022</p>  
 
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)
 
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)
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The Einstein equation with cosmological constant $\Lambda$ is
 
The Einstein equation with cosmological constant $\Lambda$ is
 
  \[R_{\mu\nu}-\frac12g_{\mu\nu}R=8\pi GT_{\mu\nu}+g_{\mu\nu}\Lambda.\]
 
  \[R_{\mu\nu}-\frac12g_{\mu\nu}R=8\pi GT_{\mu\nu}+g_{\mu\nu}\Lambda.\]
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<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_023</p>  
 
<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_023</p>  
 
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.
 
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.
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From the definition of the energy density and the pressure for the scalar field it follows that
 
From the definition of the energy density and the pressure for the scalar field it follows that
 
\begin{equation}\label{150_023_e1}
 
\begin{equation}\label{150_023_e1}
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<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_024</p>  
 
<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_024</p>  
 
Find relation between the deceleration parameter and the derivative $d\varphi/dz$ for the Universe considered in the previous problem.
 
Find relation between the deceleration parameter and the derivative $d\varphi/dz$ for the Universe considered in the previous problem.
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From the definition of the energy density and the pressure for the scalar field it follows that
 
From the definition of the energy density and the pressure for the scalar field it follows that
 
\begin{equation}\label{150_024_e1}
 
\begin{equation}\label{150_024_e1}
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<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: new_30</p>  
 
<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: new_30</p>  
 
Find the sound speed for the modified Chaplygin gas with the state equation \[p=B\rho-\frac A{\rho^\alpha}.\]
 
Find the sound speed for the modified Chaplygin gas with the state equation \[p=B\rho-\frac A{\rho^\alpha}.\]
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\[c_s^2=\frac{\delta p}{\delta\rho}=\frac{\dot p}{\dot\rho}=-\alpha w+(1+\alpha)B,\quad w=\frac p\rho.\]
 
\[c_s^2=\frac{\delta p}{\delta\rho}=\frac{\dot p}{\dot\rho}=-\alpha w+(1+\alpha)B,\quad w=\frac p\rho.\]
 
(place after the problem 81, chapter 9 of the book.)
 
(place after the problem 81, chapter 9 of the book.)
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<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_026</p>  
 
<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_026</p>  
 
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.
 
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.
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\begin{align}
 
\begin{align}
 
\nonumber \Omega_m&=\frac{\rho_m}{3H^2}=\frac{\Omega_{m0}\exp(-3N)}{\Omega_{m0}\exp(-3N)+\Omega_{\Lambda0}};\\
 
\nonumber \Omega_m&=\frac{\rho_m}{3H^2}=\frac{\Omega_{m0}\exp(-3N)}{\Omega_{m0}\exp(-3N)+\Omega_{\Lambda0}};\\
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<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_027</p>  
 
<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_027</p>  
 
Express the cosmographic parameters $H,q,j$ as functions of $N=\ln a/a_0$ for the SCM.
 
Express the cosmographic parameters $H,q,j$ as functions of $N=\ln a/a_0$ for the SCM.
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\begin{align}
 
\begin{align}
 
\nonumber H&=H_0\sqrt{\Omega_{m0}\exp(-3N)+\Omega_{\Lambda0}},\\
 
\nonumber H&=H_0\sqrt{\Omega_{m0}\exp(-3N)+\Omega_{\Lambda0}},\\
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<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_cardas1</p>  
 
<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_cardas1</p>  
 
Show that once the new term dominates the right hand side of the Friedmann equation, we have accelerated expansion.
 
Show that once the new term dominates the right hand side of the Friedmann equation, we have accelerated expansion.
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When the new term is so large that the ordinary first term can be neglected, we find
 
When the new term is so large that the ordinary first term can be neglected, we find
 
\[a\propto t^{\frac2{3n}}.\]
 
\[a\propto t^{\frac2{3n}}.\]
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Let us represent the Cardassian model in the form
 
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.
 
\[H^2\propto \rho+\rho_X,\quad\rho_X=\rho^n.\] 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.
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Use the result of the previous problem \[a\propto t^{2/(3n)}.\] As \[a\propto t^{\frac{2}{3(w+1)}},\] one finds that \[w=n-1\]in the considered case.
 
Use the result of the previous problem \[a\propto t^{2/(3n)}.\] As \[a\propto t^{\frac{2}{3(w+1)}},\] one finds that \[w=n-1\]in the considered case.
 
</p>  </div></div></div>
 
</p>  </div></div></div>
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<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_cardas3</p>  
 
<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_cardas3</p>  
 
Show that the result obtained in the previous problem takes place for arbitrary one-component fluid with $w_X=const$.
 
Show that the result obtained in the previous problem takes place for arbitrary one-component fluid with $w_X=const$.
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The relation
 
The relation
 
\[\rho_X(z)=\rho_{X0}\exp\left[3\int\limits_0^zdz'\frac{1+w_X(z')}{1+z'}\right]\]
 
\[\rho_X(z)=\rho_{X0}\exp\left[3\int\limits_0^zdz'\frac{1+w_X(z')}{1+z'}\right]\]
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<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_cardas4</p>  
 
<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_cardas4</p>  
 
Generalize the previous problem for the case of two-component ideal liquid (non-relativistic matter $+$ radiation) with density $\rho=\rho_m+\rho_r$.
 
Generalize the previous problem for the case of two-component ideal liquid (non-relativistic matter $+$ radiation) with density $\rho=\rho_m+\rho_r$.
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Use the result of the previous problem \[\frac{d\rho_X}{dz}=3\rho_X\frac{1+w_X}{1+z}.\]
 
Use the result of the previous problem \[\frac{d\rho_X}{dz}=3\rho_X\frac{1+w_X}{1+z}.\]
 
In the considered case \[\rho_X=\rho^n=\left(\rho_m+\rho_r\right)^n=\left(\rho_{m0}(1+z)^3+\rho_{r0}(1+z)^4\right)^n.\] Substitution of the latter expression into the equation for $\rho_X$ gives \[n\rho^{n-1}\left(\rho_{m0}(1+z)^2+\rho_{r0}(1+z)^3\right)=3\rho^n\frac{1+w_X}{1+z};\]
 
In the considered case \[\rho_X=\rho^n=\left(\rho_m+\rho_r\right)^n=\left(\rho_{m0}(1+z)^3+\rho_{r0}(1+z)^4\right)^n.\] Substitution of the latter expression into the equation for $\rho_X$ gives \[n\rho^{n-1}\left(\rho_{m0}(1+z)^2+\rho_{r0}(1+z)^3\right)=3\rho^n\frac{1+w_X}{1+z};\]
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<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_cardas5</p>  
 
<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_cardas5</p>  
 
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.
 
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.
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Equivalence of the equations
 
Equivalence of the equations
 
\[H^2=A\rho_m+B(\rho_m)^n\]
 
\[H^2=A\rho_m+B(\rho_m)^n\]
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<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_cardas6</p>  
 
<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_cardas6</p>  
 
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.
 
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.
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The second term starts to dominate at the redshift $z_{eq}$ when $A\rho(z_{eq})=B\rho^n(z_{eq})$, i.e., when
 
The second term starts to dominate at the redshift $z_{eq}$ when $A\rho(z_{eq})=B\rho^n(z_{eq})$, i.e., when
 
\begin{equation}\label{150_cardas6_e1}
 
\begin{equation}\label{150_cardas6_e1}
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<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_cardas7</p>  
 
<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_cardas7</p>  
 
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.
 
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.
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From
 
From
 
\[H^2=A\rho+B\rho^n\]
 
\[H^2=A\rho+B\rho^n\]
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\[H^2=A\rho\left[1+\left(\frac\rho{\rho_{car}}\right)^{n-1}\right],\]
 
\[H^2=A\rho\left[1+\left(\frac\rho{\rho_{car}}\right)^{n-1}\right],\]
 
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.
 
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.
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\[\rho(z_{eq})=\rho_m(z_{eq})+\rho_r(z_{eq})=\rho_{m0}(1+z_{eq})^3\left[1+\frac{\Omega_{r0}}{\Omega_{m0}}(1+z_{eq})\right].\]
 
\[\rho(z_{eq})=\rho_m(z_{eq})+\rho_r(z_{eq})=\rho_{m0}(1+z_{eq})^3\left[1+\frac{\Omega_{r0}}{\Omega_{m0}}(1+z_{eq})\right].\]
 
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\[H^2=\frac{8\pi G}{3}g(\rho),\]
 
\[H^2=\frac{8\pi G}{3}g(\rho),\]
 
where $\rho$ consists only of non-relativistic matter. Find the effective total pressure.
 
where $\rho$ consists only of non-relativistic matter. Find the effective total pressure.
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In the case of adiabatic expansion one has
 
In the case of adiabatic expansion one has
 
\begin{align}
 
\begin{align}
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<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_cardas10</p>  
 
<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_cardas10</p>  
 
Find the speed of sound in the Cardassian model. [P.Gandolo, K. Freese, Fluid Interpretation of Cardassian Expansion, 0209322 ]
 
Find the speed of sound in the Cardassian model. [P.Gandolo, K. Freese, Fluid Interpretation of Cardassian Expansion, 0209322 ]
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The Cardassian model
 
The Cardassian model
 
\[H^2=\frac{8\pi}{M_{Pl}^2}\rho_m+B\rho_m^n\]
 
\[H^2=\frac{8\pi}{M_{Pl}^2}\rho_m+B\rho_m^n\]
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<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_cardas11</p>  
 
<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_cardas11</p>  
 
Find the deceleration parameter for the canonic Cardassian model.
 
Find the deceleration parameter for the canonic Cardassian model.
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\begin{align}
 
\begin{align}
 
\nonumber q(z)&=\frac12\frac{(1+z)}{E^2(z)}\frac{dE^2(z)}{dz}-1,\\
 
\nonumber q(z)&=\frac12\frac{(1+z)}{E^2(z)}\frac{dE^2(z)}{dz}-1,\\
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<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_8</p>  
 
<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <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 ($\xi=const$). The pressureless fluid represent both the baryon and dark matter components. Find the dependence $\rho_m(z)$ for the considered model.
 
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.
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The conservation equation in terms of the scale factor and the first Friedmann equation are
 
The conservation equation in terms of the scale factor and the first Friedmann equation are
 
\[a\frac{d\rho_m}{da}-3(3H\xi-\rho_m)=0,\]
 
\[a\frac{d\rho_m}{da}-3(3H\xi-\rho_m)=0,\]
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<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_9</p>  
 
<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_9</p>  
 
Find $H(z)$ and $a(t)$ for the model of Universe considered in the previous problem.
 
Find $H(z)$ and $a(t)$ for the model of Universe considered in the previous problem.
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Substitute the solution $\rho_m(z)$ obtained in the previous problem into the first Friedmann equation to obtain \[H(z)=H_0\left[\frac{\bar\gamma}3+\left(\Omega_{m0}^{1/2}-\frac{\bar\gamma}3\right)(1+z)^{3/2}\right],\quad \bar\xi\equiv\frac{24\pi G}{H_0}\xi.\]
 
Substitute the solution $\rho_m(z)$ obtained in the previous problem into the first Friedmann equation to obtain \[H(z)=H_0\left[\frac{\bar\gamma}3+\left(\Omega_{m0}^{1/2}-\frac{\bar\gamma}3\right)(1+z)^{3/2}\right],\quad \bar\xi\equiv\frac{24\pi G}{H_0}\xi.\]
 
In the considered model the bulk viscous matter is the only component of the flat Universe. Consequently, $\Omega_{m0}=1$ and one finally obtains
 
In the considered model the bulk viscous matter is the only component of the flat Universe. Consequently, $\Omega_{m0}=1$ and one finally obtains
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<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_10</p>  
 
<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_10</p>  
 
Analyze the expression for the scale factor $a(t)$ obtained in the previous problem for different types of the bulk viscosity.
 
Analyze the expression for the scale factor $a(t)$ obtained in the previous problem for different types of the bulk viscosity.
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\begin{enumerate}
 
\begin{enumerate}
 
\item $0<\bar\xi<3$
 
\item $0<\bar\xi<3$
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<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_11</p>  
 
<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_11</p>  
 
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.
 
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.
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Using the result of Problem \ref{150_9}
 
Using the result of Problem \ref{150_9}
 
\[a(t)=\left[\frac{3\exp\left(\frac12\bar\xi H(t-t_0)-3+\bar\xi\right)}{\bar\xi}\right]^{2/3},\]
 
\[a(t)=\left[\frac{3\exp\left(\frac12\bar\xi H(t-t_0)-3+\bar\xi\right)}{\bar\xi}\right]^{2/3},\]
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<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_12</p>  
 
<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_12</p>  
 
Show that the result of the previous Problem for zero bulk viscosity ($\xi=0$) correctly reproduces the lifetime of the matter-dominated Universe.
 
Show that the result of the previous Problem for zero bulk viscosity ($\xi=0$) correctly reproduces the lifetime of the matter-dominated Universe.
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For $\xi\to0$ \[H_0(t_0-t_{BB})=\frac23.\]
 
For $\xi\to0$ \[H_0(t_0-t_{BB})=\frac23.\]
 
</p>  </div></div></div>
 
</p>  </div></div></div>
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<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_13</p>  
 
<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_13</p>  
 
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.
 
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.
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Using
 
Using
 
\[H(a)=\frac13H_0\frac{\bar\xi a^{3/2}+3-\bar\xi}{a^{3/2}},\]
 
\[H(a)=\frac13H_0\frac{\bar\xi a^{3/2}+3-\bar\xi}{a^{3/2}},\]
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<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_14</p>  
 
<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_14</p>  
 
Analyze the dependence \[a_t=\left(\frac{3-\bar\xi}{2\bar\xi}\right)^{2/3},\] obtained in the previous problem.
 
Analyze the dependence \[a_t=\left(\frac{3-\bar\xi}{2\bar\xi}\right)^{2/3},\] obtained in the previous problem.
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\begin{enumerate}
 
\begin{enumerate}
 
\item For $0<\bar\xi<1$ the transition between the decelerated epoch to the accelerated one takes place in the future $a_t>1$.
 
\item For $0<\bar\xi<1$ the transition between the decelerated epoch to the accelerated one takes place in the future $a_t>1$.
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<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_15</p>  
 
<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_15</p>  
 
Find the deceleration parameter $q(a,\bar\xi)$ for the cosmological model presented in the Problem \ref{150_8}.
 
Find the deceleration parameter $q(a,\bar\xi)$ for the cosmological model presented in the Problem \ref{150_8}.
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\[q(a)=-\frac{\ddot a}{aH^2}.\]
 
\[q(a)=-\frac{\ddot a}{aH^2}.\]
 
The term $\ddot a/a$ can be calculated from the second Friedmann equation, that for a matter-dominated universe with bulk viscosity reads:
 
The term $\ddot a/a$ can be calculated from the second Friedmann equation, that for a matter-dominated universe with bulk viscosity reads:
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<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_16</p>  
 
<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_16</p>  
 
Analyze behavior of the deceleration parameter $q(a,\bar\xi)$ obtained in the previous problem for different values of the bulk viscosity $\bar\xi(\xi)$.
 
Analyze behavior of the deceleration parameter $q(a,\bar\xi)$ obtained in the previous problem for different values of the bulk viscosity $\bar\xi(\xi)$.
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\begin{enumerate}
 
\begin{enumerate}
 
\item For the case $\bar\xi=0$ we have $q=1/2$ that corresponds to a matter-dominated universe with null bulk viscosity.
 
\item For the case $\bar\xi=0$ we have $q=1/2$ that corresponds to a matter-dominated universe with null bulk viscosity.
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<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_17</p>  
 
<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_17</p>  
 
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.
 
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.
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From the expression
 
From the expression
 
\[a_t=\left[\frac{3-\bar\xi}{2\bar\xi}\right]^{3/2}.\]
 
\[a_t=\left[\frac{3-\bar\xi}{2\bar\xi}\right]^{3/2}.\]
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<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_18</p>  
 
<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_18</p>  
 
Find the curvature scalar $R(a,\xi)$ for the cosmological model presented in the Problem \ref{150_8}.
 
Find the curvature scalar $R(a,\xi)$ for the cosmological model presented in the Problem \ref{150_8}.
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For a flat Universe
 
For a flat Universe
 
\[R=-6\left(\frac{\ddot a}a +H^2\right).\]
 
\[R=-6\left(\frac{\ddot a}a +H^2\right).\]
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<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_19</p>  
 
<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_19</p>  
 
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.
 
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.
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The Friedmann equations for the considered model read
 
The Friedmann equations for the considered model read
 
\[H^2=\frac{8\pi G}{3}\rho;\]
 
\[H^2=\frac{8\pi G}{3}\rho;\]
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<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_0013</p>  
 
<div id=""></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem ''' <p style= "color: #999;font-size: 11px">problem id: 150_0013</p>  
 
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].
 
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].
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Use the first law of thermodynamics
 
Use the first law of thermodynamics
 
\[d(\rho a^3)+pd(a^3)=TdS\]
 
\[d(\rho a^3)+pd(a^3)=TdS\]

Revision as of 02:17, 19 June 2015

Problem

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

problem id: 150_1

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


Problem

problem id: 150_2

Give a physical interpretation of the conservation equation.


Problem

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

problem id: 150_05

Solve the previous problem for the multi-component case.


Problem

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

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

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

problem id: 150_09

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


Problem

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

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

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

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

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

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

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

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

problem id: 2501_06

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


Problem

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

problem id: 2501_10

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

paragraph{To chapter 9}


Problem

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

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

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

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

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

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

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

problem id: 150_cardas1

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


Problem

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

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

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

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

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

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

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

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

problem id: 150_cardas10

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


Problem

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 ''' <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 ''' <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 ''' <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}