Difference between revisions of "Cosmological Inflation: The Canonic Theory"

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     <p style="text-align: left;">\[\begin{gathered}
 
     <p style="text-align: left;">\[\begin{gathered}
   S = {S_g} + {S_\varphi };~ {S_g} =  - \frac{1}{2}\int {{d^4}x\sqrt { - g} R;} ~
+
   S = {S_g} + {S_\varphi };\quad {S_g} =  - \frac{1}{2}\int {{d^4}x\sqrt { - g} R;} \quad
 
   R = {g^{\mu \nu }}(x){R_{\mu \nu }}(x);  \\
 
   R = {g^{\mu \nu }}(x){R_{\mu \nu }}(x);  \\
 
   {S_\varphi } = \int {{d^4}x\sqrt { - g} \left( {\frac{1}{2}{g^{\mu \nu }}{\partial _\mu }\varphi {\partial _\nu }\varphi  - V\left( \varphi  \right)} \right)} ; \\
 
   {S_\varphi } = \int {{d^4}x\sqrt { - g} \left( {\frac{1}{2}{g^{\mu \nu }}{\partial _\mu }\varphi {\partial _\nu }\varphi  - V\left( \varphi  \right)} \right)} ; \\
 
   L =  - \frac{1}{2}\sqrt { - g} R + \sqrt { - g} \left( {\frac{1}{2}{g^{\mu \nu }}{\partial _\mu }\varphi {\partial _\nu }\varphi  - V\left( \varphi  \right)} \right); \\
 
   L =  - \frac{1}{2}\sqrt { - g} R + \sqrt { - g} \left( {\frac{1}{2}{g^{\mu \nu }}{\partial _\mu }\varphi {\partial _\nu }\varphi  - V\left( \varphi  \right)} \right); \\
   R =  - 6\left( {\frac{\ddot a}{a} + \frac{{\dot a}^2}{a^2} + \frac{k}{a^2}} \right);~ \sqrt { - g}  \propto {a^3}; \\
+
   R =  - 6\left( {\frac{\ddot a}{a} + \frac{{\dot a}^2}{a^2} + \frac{k}{a^2}} \right);\quad \sqrt { - g}  \propto {a^3}; \\
 
   L = 3\left( {{a^2}\ddot a + a{{\dot a}^2} + ak} \right) + {a^3}\left( {\frac{1}{2}{{\dot \varphi }^2} - V\left( \varphi  \right)} \right); \\
 
   L = 3\left( {{a^2}\ddot a + a{{\dot a}^2} + ak} \right) + {a^3}\left( {\frac{1}{2}{{\dot \varphi }^2} - V\left( \varphi  \right)} \right); \\
 
   {a^2}\ddot a = \frac{d}{dt}\dot a{a^2} - 2a{{\dot a}^2}; \\
 
   {a^2}\ddot a = \frac{d}{dt}\dot a{a^2} - 2a{{\dot a}^2}; \\
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     <p style="text-align: left;">
Substitute into the conservation equation the following
+
After the substitution of explicit expressions for energy density and pressure
 
$$
 
$$
 
\begin{gathered}
 
\begin{gathered}
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\end{gathered}
 
\end{gathered}
 
$$
 
$$
to obtain the required equation
+
one obtains
 
$$
 
$$
 
\ddot \varphi + 3H\dot \varphi + V'(\varphi ) = 0.
 
\ddot \varphi + 3H\dot \varphi + V'(\varphi ) = 0.
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     <p style="text-align: left;">From the equation $\dot{H}=-4\pi G(p+\rho),$ obtained in the problem \ref{equ35} of Chapter 2, one finds that $\dot{H}=-4\pi G\dot{\varphi}^2$. Then one obtains
+
     <p style="text-align: left;">From the equation $\dot{H}=-4\pi G(p+\rho),$ obtained in the problem \ref{equ35} of Chapter 2, one finds that $\dot{H}=-4\pi G\dot{\varphi}^2$. Then  
 
$$V=\frac{3H^2}{8\pi G}\left(1+\frac{\dot{H}}{3H^2}\right)$$ $$\varphi = \int dt \left( -\frac{\dot{H}}{4\pi G}\right)^{1/2}.$$</p>
 
$$V=\frac{3H^2}{8\pi G}\left(1+\frac{\dot{H}}{3H^2}\right)$$ $$\varphi = \int dt \left( -\frac{\dot{H}}{4\pi G}\right)^{1/2}.$$</p>
 
   </div>
 
   </div>
 
</div></div>
 
</div></div>
 
 
 
  
 
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     <p style="text-align: left;">The first equation is precisely the first Friedman equation with inserted energy density of scalar field. To obtain the second equation differentiate with respect to time the definition of hubble parameter:
+
     <p style="text-align: left;">The first equation is precisely the first Friedman equation with the energy density of scalar field. To obtain the second equation differentiate with respect to time the definition of Hubble parameter:
 
$$\displaystyle \begin{array}{l}
 
$$\displaystyle \begin{array}{l}
 
\displaystyle H = \frac{\dot a}{a};\quad \dot H = \frac{\ddot aa - \dot a^2 }{a^2 } = \frac{\ddot a}{a} - \frac{\dot a^2 }{a^2 } = \\
 
\displaystyle H = \frac{\dot a}{a};\quad \dot H = \frac{\ddot aa - \dot a^2 }{a^2 } = \frac{\ddot a}{a} - \frac{\dot a^2 }{a^2 } = \\
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Substitute it into the first equation of the system to obtain
 
Substitute it into the first equation of the system to obtain
 
$$
 
$$
H'^2 - 12\pi GH^2 = - 32\pi ^2 G^2 V\left( \varphi \right)
+
H'^2 - 12\pi GH^2 = - 32\pi ^2 G^2 V\left( \varphi \right).
 
$$
 
$$
 
The system of first-order equations
 
The system of first-order equations
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=== Problem 12 ===
 
=== Problem 12 ===
Express the equations for the scalar field in terms of conformal time.
+
Rewrite the equations for the scalar field in terms of conformal time.
 
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$$
 
$$
  
$$
 
H^2 = \frac{8\pi G}{3}\left( {\frac{1}{2}\dot \varphi ^2 + V(\varphi )} \right),
 
$$
 
$$
 
\dot H = - 4\pi G\dot \varphi ^2
 
$$
 
 
$$
 
$$
 
\begin{array}{l}
 
\begin{array}{l}
 +
H^2 = \frac{8\pi G}{3}\left( {\frac{1}{2}\dot \varphi ^2 + V(\varphi )} \right),\\
 +
\dot H = - 4\pi G\dot \varphi ^2\\
 
\displaystyle dt = ad\eta;~\frac{d}{dt} = \frac{d\eta }{dt}\frac{1}{d\eta } = \frac{1}{a}\frac{1}{d\eta },\\
 
\displaystyle dt = ad\eta;~\frac{d}{dt} = \frac{d\eta }{dt}\frac{1}{d\eta } = \frac{1}{a}\frac{1}{d\eta },\\
 
\displaystyle \dot \varphi = \frac{1}{a}\frac{d\varphi }{d\eta } = \frac{1}{a}\varphi'. \\
 
\displaystyle \dot \varphi = \frac{1}{a}\frac{d\varphi }{d\eta } = \frac{1}{a}\varphi'. \\
 
\end{array}
 
\end{array}
 
$$
 
$$
Direct using of these expressions enables us to transform the initial system into the following
+
Then
 
$$
 
$$
 
\begin{array}{l}
 
\begin{array}{l}
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\end{array}
 
\end{array}
 
$$
 
$$
The prime denotes differentiation with respect to conformal time.</p>
+
where the prime denotes differentiation with respect to conformal time.</p>
 
   </div>
 
   </div>
 
</div></div>
 
</div></div>
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     <p style="text-align: left;">$$
 
     <p style="text-align: left;">$$
 
\begin{array}{l}
 
\begin{array}{l}
\displaystyle H = \frac{\dot a}{a};~\dot H = \frac{\ddot aa - \dot a^2 }{a^2 } = \frac{\ddot a}{a} - H^2,\\
+
\displaystyle H = \frac{\dot a}{a};\quad \dot H = \frac{\ddot aa - \dot a^2 }{a^2 } = \frac{\ddot a}{a} - H^2,\\
\displaystyle H^2 = \frac{8\pi G}{3}\rho ;\;\frac{\ddot a}{a} = - \frac{4\pi G}{3}(\rho + 3p), \\
+
\displaystyle H^2 = \frac{8\pi G}{3}\rho ;\quad \frac{\ddot a}{a} = - \frac{4\pi G}{3}(\rho + 3p), \\
\displaystyle \dot H > 0 \to p < - \rho. \\
+
\displaystyle \dot H > 0 \Rightarrow p < - \rho. \\
 
\end{array}
 
\end{array}
 
$$
 
$$
Satisfaction of the latter condition is impossible for a scalar field with positively defined kinetic energy.
+
The latter condition is impossible to satisfy for a scalar field with positively defined kinetic energy.
 
</p>
 
</p>
 
   </div>
 
   </div>
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$$
 
$$
 
x_{\varphi}=\frac{1+w_{\varphi}}{1-w_{\varphi}},
 
x_{\varphi}=\frac{1+w_{\varphi}}{1-w_{\varphi}},
~w_{\varphi}=\frac{\dot{\varphi}^2+2V(\varphi)}{\dot{\varphi}^2-2V(\varphi)},
+
\quad w_{\varphi}=\frac{\dot{\varphi}^2+2V(\varphi)}{\dot{\varphi}^2-2V(\varphi)},
 
$$ in the system of units such that $8\pi G=1.$
 
$$ in the system of units such that $8\pi G=1.$
 
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  \frac{V_{,\varphi}}{V} =-6\frac{H}{\dot{\varphi}}\left(\frac{x_{\varphi}}{1+ x_{\varphi}}\right)\left[1 + \frac{1}{6}\frac{d\ln \left( x_{\varphi} \right)}{d\ln a} \right].
 
  \frac{V_{,\varphi}}{V} =-6\frac{H}{\dot{\varphi}}\left(\frac{x_{\varphi}}{1+ x_{\varphi}}\right)\left[1 + \frac{1}{6}\frac{d\ln \left( x_{\varphi} \right)}{d\ln a} \right].
 
$$
 
$$
As \[\frac{x_{\varphi}}{1+ x_{\varphi}} = \frac{1+ w_{\varphi}}{2}\] then
+
As \[\frac{x_{\varphi}}{1+ x_{\varphi}} = \frac{1+ w_{\varphi}}{2},\] we have
 
$$
 
$$
 
  \frac{V_{,\varphi}}{V} =-3(1+w_{\varphi})\frac{H}{\dot{\varphi}}\left[1 + \frac{1}{6}\frac{d\ln \left( x_{\varphi} \right)}{d\ln a} \right].
 
  \frac{V_{,\varphi}}{V} =-3(1+w_{\varphi})\frac{H}{\dot{\varphi}}\left[1 + \frac{1}{6}\frac{d\ln \left( x_{\varphi} \right)}{d\ln a} \right].
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Note that $\rho_\varphi=\frac{\dot{\varphi}^2}{2}+V(\varphi)= (1+x_\varphi)=\frac{2V(\varphi)}{1-w_\varphi};~ H^2=\frac 13 \rho_{cr},~\dot{\varphi} = \pm\sqrt{2Vx_\varphi}$ and therefore
 
Note that $\rho_\varphi=\frac{\dot{\varphi}^2}{2}+V(\varphi)= (1+x_\varphi)=\frac{2V(\varphi)}{1-w_\varphi};~ H^2=\frac 13 \rho_{cr},~\dot{\varphi} = \pm\sqrt{2Vx_\varphi}$ and therefore
 
$$
 
$$
\Omega_\varphi = \frac{\rho_\varphi}{\rho_{cr}} = \frac{2}{3}\frac{V}{H^2(1-w_\varphi)},
+
\begin{array}{l}
$$
+
\Omega_\varphi = \frac{\rho_\varphi}{\rho_{cr}} = \frac{2}{3}\frac{V}{H^2(1-w_\varphi)},\\
$$
+
 
\pm \frac{V_{,\varphi}}{V} = \sqrt {\frac{3(1 + w)}{\Omega _\varphi(a)}} \left[ 1 + \frac{1}{6}\frac{d\ln \left( x_{\varphi} \right)}{d\ln a} \right].
 
\pm \frac{V_{,\varphi}}{V} = \sqrt {\frac{3(1 + w)}{\Omega _\varphi(a)}} \left[ 1 + \frac{1}{6}\frac{d\ln \left( x_{\varphi} \right)}{d\ln a} \right].
 +
\end{array}
 
$$</p>
 
$$</p>
 
   </div>
 
   </div>
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=== Problem 16 ===
 
=== Problem 16 ===
The term $3H\dot{\varphi}$ in the equation for the scalar field formally acts as friction that damps the inflation evolution. Show that, nonetheless, this term does not lead to dissipative energy production.
+
The term $3H\dot{\varphi}$ in the equation for the scalar field formally acts as friction that damps the inflation. Show that, nonetheless, this term does not lead to dissipative energy production.
 
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     <p style="text-align: left;">For the proof let us obtain the equation for scalar field from the first principle of thermodynamics. Taking into account constance of entropy in the Universe
+
     <p style="text-align: left;">For the proof let us obtain the equation for scalar field from the first law of thermodynamics. Taking into account that the entropy of the Universe is conserved
 
$$
 
$$
 
dE + pdV = 0,
 
dE + pdV = 0,
 
$$
 
$$
 
one obtains
 
one obtains
$$
 
dE = Vd\rho + \rho dV,
 
$$
 
 
\begin{equation}
 
\begin{equation}
\label{enthropy}
+
\begin{array}{l}
Vd\rho + (\rho + p)dV = 0.
+
dE = Vd\rho + \rho dV,\\
 +
Vd\rho + (\rho + p)dV = 0.\label{entropy_2}
 +
\end{array}
 
\end{equation}
 
\end{equation}
Use the definition for $\rho $ and $p$to obtain
+
Use the definition for $\rho $ and $p$ to obtain
 
$$
 
$$
 
\begin{array}{l}
 
\begin{array}{l}
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\end{array}
 
\end{array}
 
$$
 
$$
Substitute it into \ref{enthropy} to obtain the equation for scalar field ïîëÿ
+
Substitute it into \ref{entropy_2} to obtain the equation for scalar field  
 
$$
 
$$
 
\ddot \varphi + 3H\dot \varphi + V'(\varphi ) = 0.
 
\ddot \varphi + 3H\dot \varphi + V'(\varphi ) = 0.
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=== Problem 18 ===
 
=== Problem 18 ===
Calculate pressure of homogeneous scalar field in the potential $V(\varphi)$ using the obtained above energy density of the field and its equation of motion.
+
Calculate the pressure of homogeneous scalar field in the potential $V(\varphi)$ using the obtained above energy density of the field and its equation of motion.
 
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     <p style="text-align: left;">Energy density of scalar field reads
+
     <p style="text-align: left;">Energy density of the scalar field reads
 
\begin{equation}
 
\begin{equation}
 
\label{inf:scalar_pressure_en}
 
\label{inf:scalar_pressure_en}
\rho _\varphi = \frac{1}{2}\dot \varphi ^2 + V\left( \varphi \right)
+
\rho _\varphi = \frac{1}{2}\dot \varphi ^2 + V\left( \varphi \right).
 
\end{equation}
 
\end{equation}
  
 
Differentiate it by time to obtain
 
Differentiate it by time to obtain
 
$$
 
$$
\dot \rho _\varphi = \ddot \varphi \dot \varphi + V'(\varphi )\dot \varphi
+
\dot \rho _\varphi = \ddot \varphi \dot \varphi + V'(\varphi )\dot \varphi.
 
$$
 
$$
 
Exclude the second time derivative using the equation of motion for scalar field to obtain as th result
 
Exclude the second time derivative using the equation of motion for scalar field to obtain as th result
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 +
 
=== Problem 19 ===
 
=== Problem 19 ===
 
What condition should the homogeneous scalar field $\varphi(t)$ in potential $V(\varphi)$ satisfy in order to provide accelerated expansion of the Universe?
 
What condition should the homogeneous scalar field $\varphi(t)$ in potential $V(\varphi)$ satisfy in order to provide accelerated expansion of the Universe?
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</div></div>
 
</div></div>
  
==Inflationary Introduction==
+
==Introduction to Inflation==
  
  
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=== Problem 21 ===
 
=== Problem 21 ===
What considerations led A.Guth to name his theory describing the early Universe dynamics as ''inflation theory''?
+
What considerations led A.Guth to name his theory describing the early Universe's dynamics as ''inflation theory''?
 
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<div id="inf12_new"></div>
 
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 +
 
=== Problem 22 ===
 
=== Problem 22 ===
 
A. Vilenkin in his cosmological bestseller ''Many world in one'' remembers: ''On a Wednesday afternoon, in the winter of 1980, I was sitting in a fully packed Harvard auditorium, listening to the most fascinating talk I had heard in many years. The speaker was Alan Guth, a young physicist from Stanford, and the topic was a new theory for the origin of the universe$\ldots$ The beauty of the idea was that in a single shot inflation explained why the universe is so big, why it is expanding, and why it was so hot at the beginning. A huge expanding universe was produced from almost nothing. All that was needed was a microscopic chunk of repulsive gravity material. Guth admitted he did not know where the initial chunk came from, but that detail could be worked out later. ''It's often said that you cannot get something for nothing.'' he said, ''but the universe may be the ultimate free lunch'' ''. Explain, why can this be.
 
A. Vilenkin in his cosmological bestseller ''Many world in one'' remembers: ''On a Wednesday afternoon, in the winter of 1980, I was sitting in a fully packed Harvard auditorium, listening to the most fascinating talk I had heard in many years. The speaker was Alan Guth, a young physicist from Stanford, and the topic was a new theory for the origin of the universe$\ldots$ The beauty of the idea was that in a single shot inflation explained why the universe is so big, why it is expanding, and why it was so hot at the beginning. A huge expanding universe was produced from almost nothing. All that was needed was a microscopic chunk of repulsive gravity material. Guth admitted he did not know where the initial chunk came from, but that detail could be worked out later. ''It's often said that you cannot get something for nothing.'' he said, ''but the universe may be the ultimate free lunch'' ''. Explain, why can this be.
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     <p style="text-align: left;">Let us cite A. Vilenkin one more time:
 
     <p style="text-align: left;">Let us cite A. Vilenkin one more time:
 
<br/>
 
<br/>
''Striking growth of mass during inflation can seemingly contradict the most fundamental law of nature --- the energy conservation principle. According to the famous Einstein's formula $E = m{c^2}$ energy is proportional to mass. It comes out that energy of inflated bit should increase in huge number of times, while the energy conservation law requires that it remains constant. This paradox disappears if one takes into account the contribution to energy by gravity. It has long been known that the gravity energy is always negative. it did not appear very important before, but now it acquired really cosmic importance. As the positive energy of matter grows, it is compensated by growing negative gravity energy. Total energy remains constant, as is required by the conservation law.''</p>
+
''Striking growth of mass during inflation can seemingly contradict the most fundamental law of nature --- the energy conservation principle. According to the famous Einstein's formula $E = m{c^2}$ energy is proportional to mass. It comes out that energy of inflated bit should increase in huge number of times, while the energy conservation law requires that it remains constant. This paradox disappears if one takes into account the contribution to energy by gravity. It has long been known that the gravity energy is always negative. It did not appear very important before, but now it acquired really cosmic importance. As the positive energy of matter grows, it is compensated by growing negative gravity energy. Total energy remains constant, as is required by the conservation law.''</p>
 
   </div>
 
   </div>
 
</div></div>
 
</div></div>
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 +
 
=== Problem 24 ===
 
=== Problem 24 ===
The inflation is defined as any epoch for which the scale factor of the Universe has accelerated growth, i.e. $\ddot{a}>0$. Show that the condition is equivalent to the requirement of decreasing of the comoving Hubble radius with time.
+
Inflation is defined as any epoch for which the scale factor of the Universe has accelerated growth, i.e. $\ddot{a}>0$. Show that this condition is equivalent to requirement that the comoving Hubble radius decreases with time.
 
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     <p style="text-align: left;">Though current data point out that the density ${\Omega _k}$ is close to zero, it is not obvious that $k = 0$. As
 
     <p style="text-align: left;">Though current data point out that the density ${\Omega _k}$ is close to zero, it is not obvious that $k = 0$. As
 
${\Omega _k} =  - \frac{k}{{a{H^2}}}$
 
${\Omega _k} =  - \frac{k}{{a{H^2}}}$
then current value of ${\Omega _k}$ is sensitive to current value of $a(t)$, i.e. expansion measure of the universe after the Big Bang. Considerable expansion can make the value ${\Omega _k}$ close to zero. It is the way how the inflationary models solve the flatness problem, avoiding the fine tuning problem: the value $k = 0$ is statistically unlike.</p>
+
the current value of ${\Omega _k}$ is sensitive to the current value of $a(t)$, i.e. to measure of expansion of the Universe after the Big Bang. Considerable expansion can make the value ${\Omega _k}$ close to zero. It is the way how the inflationary models solve the flatness problem, avoiding the fine tuning problem: the value $k = 0$ is statistically unlikely.</p>
 
   </div>
 
   </div>
 
</div></div>
 
</div></div>
 
 
 
 
  
 
==Inflation in the Slow-Roll Regime==
 
==Inflation in the Slow-Roll Regime==
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     <p style="text-align: left;">For the case of homogeneous and isotropic Universe with scalar field independent of coordinates one gets
+
     <p style="text-align: left;">For the case of homogeneous and isotropic Universe with scalar field one gets
 
$$
 
$$
 
\begin{array}{l}
 
\begin{array}{l}
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     <p style="text-align: left;">Soon after start of the inflation $\ddot \varphi \ll 3H\dot \varphi ;\quad \dot \varphi ^2 \ll m^2 \varphi ^2$. So
+
     <p style="text-align: left;">Soon after the start of inflation $\ddot \varphi \ll 3H\dot \varphi ;\quad \dot \varphi ^2 \ll m^2 \varphi ^2$. So
 
$$
 
$$
 
\begin{array}{l}
 
\begin{array}{l}
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Due to fast growth of the scale factor and slow variation of the field (strong friction)
 
Due to fast growth of the scale factor and slow variation of the field (strong friction)
 
$$
 
$$
a \propto e^{Ht} ;\quad H = \frac{2m\varphi }{M_{Pl}}\sqrt {\frac{\pi }{3}}
+
a \propto e^{Ht} ;\quad H = \frac{2m\varphi }{M_{Pl}}\sqrt {\frac{\pi }{3}}.
 
$$
 
$$
  
As the field decreases (slowly rolls), the viscosity falls down, and the inflation regime (the exponential growth of the scale factor) terminates.</p>
+
As the field decreases (rolls slowly), the ''viscosity'' falls down, and the inflation regime (the exponential growth of the scale factor) terminates.</p>
 
   </div>
 
   </div>
 
</div></div>
 
</div></div>
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\displaystyle 3H\dot \varphi + V'(\varphi ) \simeq 0,\quad H \equiv \frac{d\ln a}{dt} \simeq \sqrt {\frac{8\pi G}{3}V(\varphi )},\\
 
\displaystyle 3H\dot \varphi + V'(\varphi ) \simeq 0,\quad H \equiv \frac{d\ln a}{dt} \simeq \sqrt {\frac{8\pi G}{3}V(\varphi )},\\
 
\displaystyle \frac{d\ln a}{dt} = \dot \varphi \frac{d\ln a}{d\varphi} \simeq - \frac{V'\left( \varphi \right)}{3H}\frac{d\ln a}{d\varphi },\\
 
\displaystyle \frac{d\ln a}{dt} = \dot \varphi \frac{d\ln a}{d\varphi} \simeq - \frac{V'\left( \varphi \right)}{3H}\frac{d\ln a}{d\varphi },\\
\displaystyle - V'(\varphi )\frac{d\ln a}{d\varphi } \simeq 8\pi GV(\varphi )\\
+
\displaystyle - V'(\varphi )\frac{d\ln a}{d\varphi } \simeq 8\pi GV(\varphi ),
 
\end{array}$$
 
\end{array}$$
 
and therefore
 
and therefore
 
$$
 
$$
a(\varphi ) \simeq a_0 \exp \left( {8\pi G\int_\varphi ^{\varphi _0 } {\frac{V}{V'(\varphi )}d\varphi } } \right)
+
a(\varphi ) \simeq a_0 \exp \left( {8\pi G\int_\varphi ^{\varphi _0 } {\frac{V}{V'(\varphi )}d\varphi } } \right).
 
$$</p>
 
$$</p>
 
   </div>
 
   </div>
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=== Problem 31 ===
 
=== Problem 31 ===
Show that the condition $\varepsilon\ll1$ for the realization of the slow--roll limit obtained in the previous problem is also sufficient condition for the inflation.
+
Show that the condition $\varepsilon\ll1$ for the realization of the slow--roll limit obtained in the previous problem is also the sufficient condition for inflation.
 
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<div class="NavFrame collapsed">
 
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   <div class="NavHead">solution</div>
 
   <div style="width:100%;" class="NavContent">
 
   <div style="width:100%;" class="NavContent">
     <p style="text-align: left;">The most general definition of inflation reads $\ddot a > 0$
+
     <p style="text-align: left;">The most general definition of inflation reads $\ddot a > 0$;
 
$$
 
$$
\frac{\ddot a}{a} = \dot H + H^2 > 0
+
\frac{\ddot a}{a} = \dot H + H^2 > 0.
 
$$
 
$$
This condition is evidently satisfied for $\dot H > 0$. However such possibility cannot be realized for the scalar field (see problem \ref{inf17}) Therefore assume that $\dot H < 0$ and require
+
This condition is evidently satisfied for $\dot H > 0$. However such possibility cannot be realized for the scalar field (see problem \ref{inf17}). Therefore assume that $\dot H < 0$ and require
 
$$
 
$$
 
- \frac{\dot H}{H^2 } < 1
 
- \frac{\dot H}{H^2 } < 1
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to obtain
 
to obtain
 
$$
 
$$
- \frac{\dot H}{H^2 } \simeq \frac{M_{Pl}^{*2} }{2}\left( {\frac{V'}{V}} \right)^2 = \varepsilon
+
- \frac{\dot H}{H^2 } \simeq \frac{M_{Pl}^{*2} }{2}\left( {\frac{V'}{V}} \right)^2 = \varepsilon.
 
$$
 
$$
  
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and therefore
 
and therefore
 
$$
 
$$
\dot \varphi ^2 \sim \left( {\frac{\partial V}{\partial \phi }} \right)^2 \cdot \frac{1}{H^2 } \sim \left( {\frac{\partial V}{\partial \varphi }} \right)^2 \cdot \frac{M_{Pl}^2 }{V}
+
\dot \varphi ^2 \sim \left( {\frac{\partial V}{\partial \phi }} \right)^2 \cdot \frac{1}{H^2 } \sim \left( {\frac{\partial V}{\partial \varphi }} \right)^2 \cdot \frac{M_{Pl}^2 }{V}.
 
$$
 
$$
  
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$$
 
$$
 
and the considered slow-roll condition takes on the form
 
and the considered slow-roll condition takes on the form
$\varphi \gg M_{Pl} $
+
$\varphi \gg M_{Pl}.$
It easy to see that the second slow-roll condition
+
It is easy to see that the second slow-roll condition
 
$$
 
$$
 
H\dot \varphi \gg \ddot \varphi
 
H\dot \varphi \gg \ddot \varphi
 
$$
 
$$
is also satisfied for the power-law potentials with $\varphi \gg M_{Pl}$. Thus the inflation appears any time when the scalar field amplitude 9considerably) exceeds the Planck mass.</p>
+
is also satisfied for the power-law potentials with $\varphi \gg M_{Pl}$. Thus the inflation appears any time when the scalar field amplitude (considerably) exceeds the Planck mass.</p>
 
   </div>
 
   </div>
 
</div></div>
 
</div></div>
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     <p style="text-align: left;">The inflationary potential near the inflection point $\varphi _0 $, where
 
     <p style="text-align: left;">The inflationary potential near the inflection point $\varphi _0 $, where
 
$$V'(\varphi _0 ) = V''(\varphi _0 ) = 0,$$
 
$$V'(\varphi _0 ) = V''(\varphi _0 ) = 0,$$
can be approximated in the form
+
can be approximated by
 
$$
 
$$
V(\varphi ) \approx V_0 + V_3 \left( {\varphi - \varphi _0 } \right)^3
+
V(\varphi ) \approx V_0 + V_3 \left( {\varphi - \varphi _0 } \right)^3.
 
$$
 
$$
  
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=== Problem 34 ===
 
=== Problem 34 ===
Show that the inflation parameter $\varepsilon$ can be expressed through the parameter $w$ in the state equation for the scalar field.
+
Show that the inflation parameter $\varepsilon$ can be expressed through the state equation parameter $w$ for the scalar field.
 
<div class="NavFrame collapsed">
 
<div class="NavFrame collapsed">
 
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   <div class="NavHead">solution</div>
 
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   <div style="width:100%;" class="NavContent">
 
     <p style="text-align: left;">$$
 
     <p style="text-align: left;">$$
\varepsilon = \frac{3}{2}(w + 1)
+
\varepsilon = \frac{3}{2}(w + 1);
 
$$
 
$$
  
 
$$
 
$$
\varepsilon = \frac{3}{2}(w + 1) = \frac{3}{2}\left( {\frac{p}{\rho } + 1} \right) = \frac{3}{2}\frac{\dot \varphi ^2 }{\rho }
+
\varepsilon = \frac{3}{2}(w + 1) = \frac{3}{2}\left( {\frac{p}{\rho } + 1} \right) = \frac{3}{2}\frac{\dot \varphi ^2 }{\rho }.
 
$$
 
$$
  
 
In the inflation regime $3H\dot \varphi + V'(\varphi ) \simeq 0$ and $\rho \sim V$, $H^2 \sim \frac{1}{3M_{Pl} ^{*2} }V$. So we recover the initial definition of the inflation parameter
 
In the inflation regime $3H\dot \varphi + V'(\varphi ) \simeq 0$ and $\rho \sim V$, $H^2 \sim \frac{1}{3M_{Pl} ^{*2} }V$. So we recover the initial definition of the inflation parameter
 
$$
 
$$
\varepsilon = \frac{M_{Pl}^{*2} }{2}\left( {\frac{V'}{V}} \right)^2
+
\varepsilon = \frac{M_{Pl}^{*2} }{2}\left( {\frac{V'}{V}} \right)^2.
 
$$</p>
 
$$</p>
 
   </div>
 
   </div>
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$$\frac{\ddot a}{a} = H^2 (1 - \varepsilon ).$$
 
$$\frac{\ddot a}{a} = H^2 (1 - \varepsilon ).$$
  
The Hamilton-Jakobi equation (see problem \ref{inf15})
+
The Hamilton-Jacobi equation (see problem \ref{inf15})
 
$$
 
$$
 
H'^2 (\phi ) - \frac{3}{2M^* _{Pl}{}^2}H^2 (\phi ) = - \frac{1}{2M^* _{Pl}{}^4 }V(\phi )
 
H'^2 (\phi ) - \frac{3}{2M^* _{Pl}{}^2}H^2 (\phi ) = - \frac{1}{2M^* _{Pl}{}^4 }V(\phi )
 
$$
 
$$
enables to consider $H(\phi )$, (and not $V(\phi )$) as a fundamental quantity. In terms of this function the inflation is described in more natural way. As soon $H(\phi )$ is determined, on immediately finds $V(\phi )$. Introduce the inflation parameters $\varepsilon _H ,\eta _H$ in terms of $H(\phi )$
+
enables one to consider $H(\phi )$ (instead of $V(\phi )$) as a fundamental quantity. In terms of this function the inflation is described in more natural way. As soon as $H(\phi )$ is determined, one immediately finds $V(\phi )$. The inflation parameters $\varepsilon _H ,\eta _H$ in terms of $H(\phi )$ are
 
$$
 
$$
\varepsilon _H = 2M_{Pl}^{*2} \left( {\frac{H'(\phi )}{H(\phi )}} \right)^2 ;\quad \eta _H = 2M_{Pl}^{*2} \frac{H''(\phi )}{H(\phi )}
+
\varepsilon _H = 2M_{Pl}^{*2} \left( {\frac{H'(\phi )}{H(\phi )}} \right)^2 ;\quad \eta _H = 2M_{Pl}^{*2} \frac{H''(\phi )}{H(\phi )}.
 
$$</p>
 
$$</p>
 
   </div>
 
   </div>
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\displaystyle - \frac{d\ln H}{d\ln a} = - \frac{d\ln H}{Hdt} = - \frac{\dot H}{H^2 } = - \frac{H'}{H^2 }\dot \varphi,\\
 
\displaystyle - \frac{d\ln H}{d\ln a} = - \frac{d\ln H}{Hdt} = - \frac{\dot H}{H^2 } = - \frac{H'}{H^2 }\dot \varphi,\\
 
\displaystyle \dot \varphi = - \frac{1}{4\pi G}H' = - 2M_{Pl}^{*2} H'; \\
 
\displaystyle \dot \varphi = - \frac{1}{4\pi G}H' = - 2M_{Pl}^{*2} H'; \\
- \frac{d\ln H}{d\ln a} = 2M_{Pl}^{*2} \left( \frac{H'}{H} \right)^2 = \varepsilon _H \\
+
- \frac{d\ln H}{d\ln a} = 2M_{Pl}^{*2} \left( \frac{H'}{H} \right)^2 = \varepsilon _H.
\\
+
 
\end{array}
 
\end{array}
 
$$
 
$$
Line 1,012: Line 1,002:
  
 
=== Problem 38 ===
 
=== Problem 38 ===
Prove that the definition of inflation as the regime for which $\ddot a>0$ is equivalent to condition $\varepsilon_H<1$.
+
Prove that condition $\ddot a>0$, which defines inflation, is equivalent to $\varepsilon_H<1$.
 
<!-- <div class="NavFrame collapsed">
 
<!-- <div class="NavFrame collapsed">
 
   <div class="NavHead">solution</div>
 
   <div class="NavHead">solution</div>
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<div id="inf34"></div>
 
<div id="inf34"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
 +
 
=== Problem 39 ===
 
=== Problem 39 ===
 
Show that inflation appears every time when the scalar field's value exceeds the Planck mass.
 
Show that inflation appears every time when the scalar field's value exceeds the Planck mass.
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   <div class="NavHead">solution</div>
 
   <div class="NavHead">solution</div>
 
   <div style="width:100%;" class="NavContent">
 
   <div style="width:100%;" class="NavContent">
     <p style="text-align: left;">For homogeneous scalar field in the potential $V\left( \varphi \right)$ the non-zero components of energy-momentum tensor in the local Lorentz system equal to
+
     <p style="text-align: left;">For homogeneous scalar field in potential $V\left( \varphi \right)$ the non-zero components of energy-momentum tensor in the local Lorentz frame equal to
 
$$
 
$$
T_{00} = \frac{1}{2}\dot \varphi ^2 + V\left( \varphi \right) = \rho _\varphi ;\;T_{ij} = \left( {\frac{1}{2}\dot \varphi ^2 - V\left( \varphi \right)} \right)\delta _{ij} = p_\varphi \delta _{ij} \quad
+
T_{00} = \frac{1}{2}\dot \varphi ^2 + V\left( \varphi \right) = \rho _\varphi ;\;T_{ij} = \left( {\frac{1}{2}\dot \varphi ^2 - V\left( \varphi \right)} \right)\delta _{ij} = p_\varphi \delta _{ij}.
 
$$
 
$$
  
Line 1,052: Line 1,043:
 
   </div>
 
   </div>
 
</div></div>
 
</div></div>
 
 
  
 
==Solution of the Hot Big Bang Theory Problems==
 
==Solution of the Hot Big Bang Theory Problems==
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   <div class="NavHead">solution</div>
 
   <div class="NavHead">solution</div>
 
   <div style="width:100%;" class="NavContent">
 
   <div style="width:100%;" class="NavContent">
     <p style="text-align: left;">Assume that the inflation started at some initial moment $t_i $ and continued to final moment$t_f $. During the inflation $a \sim e^{Ht} $ with $H = const$. So in the inflation phase
+
     <p style="text-align: left;">Assume that the inflation started at some initial moment $t_i $ and continued to final moment $t_f$. During inflation $a \sim e^{Ht} $ with $H = const$. So in the inflation phase
 
$$
 
$$
\Omega - 1 = \frac{k}{\dot a^2 } \sim e^{ - 2H\,t}
+
\Omega - 1 = \frac{k}{\dot a^2 } \sim e^{ - 2H\,t}.
 
$$
 
$$
This result means that during the inflation the relative density $\Omega $ exponentially tends to unity.
+
This result means that during inflation the relative density $\Omega $ exponentially tends to unity.
 
</p>
 
</p>
 
   </div>
 
   </div>
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   <div class="NavHead">solution</div>
 
   <div class="NavHead">solution</div>
 
   <div style="width:100%;" class="NavContent">
 
   <div style="width:100%;" class="NavContent">
     <p style="text-align: left;">Let $R_0$ is current size of Universe, and $R_{eq}$ is its size at the moment $t_{eq}$ of equality between energy densities of matter and radiation ($t_{eq} \simeq 50\,000~\mbox{ years}$). Then
+
     <p style="text-align: left;">Let $R_0$ be the current size of Universe, and $R_{eq}$ is its size at the moment $t_{eq}$ of equality between energy densities of matter and radiation ($t_{eq} \simeq 50\,000~\mbox{ years}$). Then
 
$$
 
$$
 
R_{eq} \approx R_0 \left( {\frac{t_0 }{t_{eq} }} \right)^{ - 2/3}.
 
R_{eq} \approx R_0 \left( {\frac{t_0 }{t_{eq} }} \right)^{ - 2/3}.
 
$$
 
$$
If the inflation stopped at the time moment $t_{inf} $ ($t_{inf} \approx 10^{ - 36}~\mbox{sec}$), then
+
If the inflation stopped at time $t_{inf} $ ($t_{inf} \approx 10^{ - 36}~\mbox{sec}$), then
 
$$
 
$$
 
\frac{R_{eq} }{R_{inf} } = \left( {\frac{t_{eq} }{t_{inf} }} \right)^{1/2}
 
\frac{R_{eq} }{R_{inf} } = \left( {\frac{t_{eq} }{t_{inf} }} \right)^{1/2}
Line 1,110: Line 1,099:
 
and therefore
 
and therefore
 
$$
 
$$
R_{inf} = R_0 \left( {\frac{t_0 }{t_{eq} }} \right)^{ - 2/3} \left( \frac{t_{eq}}{t_{inf}} \right)^{-{1/2}} \simeq R_0 \times 10^{ - 28}
+
R_{inf} = R_0 \left( {\frac{t_0 }{t_{eq} }} \right)^{ - 2/3} \left( \frac{t_{eq}}{t_{inf}} \right)^{-{1/2}} \simeq R_0 \times 10^{ - 28}.
 
$$</p>
 
$$</p>
 
   </div>
 
   </div>
Line 1,122: Line 1,111:
  
 
=== Problem 44 ===
 
=== Problem 44 ===
Find the number $N_e$ of $e$-foldings of the scale factor in the inflation epoch.
+
Find the number $N_e$ of $e$-foldings of the scale factor during the inflation epoch.
 
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   <div class="NavHead">solution</div>
 
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   <div style="width:100%;" class="NavContent">
     <p style="text-align: left;">Let during the inflation the scale factor increased in $e^N $ times
+
     <p style="text-align: left;">Suppose during inflation the scale factor increased $e^N $ times
 
$$
 
$$
 
\frac{a_i }{a_e } = e^N.
 
\frac{a_i }{a_e } = e^N.
 
$$
 
$$
The indices $i,e$ refer to the start and the end of inflation respectively. From the Hubble parameter definition $H \equiv \frac{d}{dt}\ln a$ it follows that
+
The indices $i,e$ refer to the start and end of inflation respectively. From the Hubble parameter definition $H \equiv \frac{d}{dt}\ln a$ it follows that
 
$$
 
$$
N = \int_{t_i }^{t_e } {Hdt} = \int_{\varphi _i }^{\varphi _e } {H\frac{dt}{d\varphi}} d\varphi = \int_{\varphi _i }^{\varphi _e } {H\frac{1}{\dot \varphi }} d\varphi
+
N = \int_{t_i }^{t_e } {Hdt} = \int_{\varphi _i }^{\varphi _e } {H\frac{dt}{d\varphi}} d\varphi = \int_{\varphi _i }^{\varphi _e } {H\frac{1}{\dot \varphi }} d\varphi.
 
$$
 
$$
 
Using that in the inflation regime
 
Using that in the inflation regime
Line 1,138: Line 1,127:
 
\begin{array}{l}
 
\begin{array}{l}
 
\displaystyle 3H\dot \varphi + V'(\varphi ) = 0,\\
 
\displaystyle 3H\dot \varphi + V'(\varphi ) = 0,\\
\displaystyle H^2 = \frac{8\pi }{3M_{Pl}^2 }V\left( \varphi \right).\\
+
\displaystyle H^2 = \frac{8\pi }{3M_{Pl}^2 }V\left( \varphi \right),
 
\end{array}
 
\end{array}
 
$$
 
$$
one obtains (omitting the coefficients of order of unity)
+
one obtains (omitting the coefficients of the order of unity)
 
\[
 
\[
N \sim \int_{\varphi _e }^{\varphi _i } {d\varphi \frac{V(\varphi )}{M_{Pl}^2 V'(\varphi )}}
+
N \sim \int_{\varphi _e }^{\varphi _i } {d\varphi \frac{V(\varphi )}{M_{Pl}^2 V'(\varphi )}}.
 
\]</p>
 
\]</p>
 
   </div>
 
   </div>
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   <div style="width:100%;" class="NavContent">
 
   <div style="width:100%;" class="NavContent">
 
     <p style="text-align: left;">$$
 
     <p style="text-align: left;">$$
N = \frac{1}{{M_{Pl}^2 }}\int_{\varphi _i }^\varphi {d\varphi \frac{V}{V'}} \approx \frac{1}{3M_{Pl}^2 }\frac{V_0 }{V_3 }\frac{1}{\varphi _0 - \varphi }
+
N = \frac{1}{{M_{Pl}^2 }}\int_{\varphi _i }^\varphi {d\varphi \frac{V}{V'}} \approx \frac{1}{3M_{Pl}^2 }\frac{V_0 }{V_3 }\frac{1}{\varphi _0 - \varphi }.
 
$$</p>
 
$$</p>
 
   </div>
 
   </div>
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   <div class="NavHead">solution</div>
 
   <div class="NavHead">solution</div>
 
   <div style="width:100%;" class="NavContent">
 
   <div style="width:100%;" class="NavContent">
     <p style="text-align: left;">Transform the first Friedman equation o the following
+
     <p style="text-align: left;">Transform the first Friedman equation to the following
 
$$
 
$$
 
\begin{array}{l}
 
\begin{array}{l}
Line 1,191: Line 1,180:
 
and write down the derivative of $x$ to obtain
 
and write down the derivative of $x$ to obtain
 
$$
 
$$
x={const\over \rho a^2} \Rightarrow {dx\over dN}={1\over H}{dx\over dt}=\left(1+3w\right)x.
+
x={const\over \rho a^2}\quad\Rightarrow\quad {dx\over dN}={1\over H}{dx\over dt}=\left(1+3w\right)x.
 
$$
 
$$
 
This equation has a fixed point at $x=0$. Its type depends on the parameter $w$:
 
This equation has a fixed point at $x=0$. Its type depends on the parameter $w$:
 
$$
 
$$
\begin{array}{l}
+
\begin{align}
\mbox{for matter}~w=0~\Rightarrow x=e^{N},\\
+
& \mbox{for matter} & w & = 0 & &\Rightarrow x=e^{N},\\
\mbox{for radiation}~w=1/3~\Rightarrow x=e^{2N},\\
+
& \mbox{for radiation} & w & = 1/3 & &\Rightarrow x=e^{2N},\\
\end{array}
+
\end{align}
 
$$
 
$$
I.e. the fixed point is unstable both for matter-dominated and the radiation-dominated case---any deviation of $\Omega$ from $1$ grows with time. From the other hand during the inflation $w\simeq -1$, therefore
+
i.e. the fixed point is unstable both for matter-dominated and the radiation-dominated case---any deviation of $\Omega$ from $1$ grows with time. On the other hand during inflation $w\simeq -1$, therefore
 
$$
 
$$
 
x=e^{-2N}
 
x=e^{-2N}
 
$$
 
$$
and the fixed point is stable. It means that any deviation of density from the critical one decrease with time.</p>
+
and the fixed point is stable. It means that any deviation of density from the critical one decreases with time.</p>
 
   </div>
 
   </div>
 
</div></div>
 
</div></div>
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<div id="inf24"></div>
 
<div id="inf24"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
 +
 
=== Problem 47 ===
 
=== Problem 47 ===
 
Find the particle horizon in the inflationary regime, assuming $H\approx const$.
 
Find the particle horizon in the inflationary regime, assuming $H\approx const$.
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\begin{array}{l}
 
\begin{array}{l}
 
\displaystyle L_p (t) = a(t)\int_0^t {\frac{dt'}{a(t')}},\\
 
\displaystyle L_p (t) = a(t)\int_0^t {\frac{dt'}{a(t')}},\\
\displaystyle a(t) \propto e^{Ht} \to L_p (t) = \frac{1}{H}\left( {e^{Ht} - 1} \right).\\
+
\displaystyle a(t) \propto e^{Ht} \quad\Rightarrow\quad L_p (t) = \frac{1}{H}\left( {e^{Ht} - 1} \right).\\
 
\end{array}
 
\end{array}
 
$$</p>
 
$$</p>
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   <div class="NavHead">solution</div>
 
   <div class="NavHead">solution</div>
 
   <div style="width:100%;" class="NavContent">
 
   <div style="width:100%;" class="NavContent">
     <p style="text-align: left;">Assume that the inflation started at $t_i = 10^{ - 38}~\mbox{sec}$ and stopped at $t_f = 10^{ - 36}~\mbox{sec}$. Before start of the inflation the universe was dominated by radiation and the particle horizon at that time was equal to
+
     <p style="text-align: left;">Assume that the inflation started at $t_i = 10^{ - 38}~\mbox{sec}$ and stopped at $t_f = 10^{ - 36}~\mbox{sec}$. Before the start of inflation the Universe was dominated by radiation and the particle horizon at that time was equal to
 
$$
 
$$
 
L_p = 2t_i = 2 \cdot 3 \cdot 10^{10} \mbox{cm}\,c^{ - 1} \cdot 10^{ - 38} c = 6 \cdot 10^{ - 28}\mbox{ cm}.
 
L_p = 2t_i = 2 \cdot 3 \cdot 10^{10} \mbox{cm}\,c^{ - 1} \cdot 10^{ - 38} c = 6 \cdot 10^{ - 28}\mbox{ cm}.
 
$$
 
$$
It is the maximum size of the region where the thermal equilibrium could be established before the start of inflation. During the inflation this region increased in $e^N $ times, and then endured further expansion in radiation-dominated and matter-dominated epochs. At present this region has the size
+
It is the maximum size of the region where the thermal equilibrium could be established before the start of inflation. During inflation this region increased $e^N $ times, and then endured further expansion in radiation-dominated and matter-dominated epochs. At present this region has the size
 
$$
 
$$
 
l_0 \approx L_p (t_i )e^N \left( {\frac{t_{eq} }{t_f }} \right)^{1/2} \left( {\frac{t_0 }{t_{eq}}} \right)^{1/2}.
 
l_0 \approx L_p (t_i )e^N \left( {\frac{t_{eq} }{t_f }} \right)^{1/2} \left( {\frac{t_0 }{t_{eq}}} \right)^{1/2}.
 
$$
 
$$
 
Choosing value $N \approx 100$ (sufficiently ''modest'' estimate) and using the above cited values $t_{eq}$
 
Choosing value $N \approx 100$ (sufficiently ''modest'' estimate) and using the above cited values $t_{eq}$
and $t_0$, one obtins
+
and $t_0$, one obtains
 
$$
 
$$
l_0 \approx 10^{40} \mbox{cm}
+
l_0 \approx 10^{40} \mbox{cm}.
 
$$
 
$$
This value considerable exceeds the size of presently observed Universe $l \approx 10^{28} \mbox{ cm}$ and therefore the horizon problem is solved.</p>
+
This value considerably exceeds the size of presently observed Universe $l \approx 10^{28} \mbox{ cm}$ and therefore the horizon problem is solved.</p>
 
   </div>
 
   </div>
 
</div></div>
 
</div></div>
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<div id="inf56"></div>
 
<div id="inf56"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
 +
 
=== Problem 49 ===
 
=== Problem 49 ===
 
Did entropy change during the inflation period? If yes, then estimate what its change was.
 
Did entropy change during the inflation period? If yes, then estimate what its change was.
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<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
 
=== Problem 50 ===
 
=== Problem 50 ===
Does the inflation theory explain the modern value of entropy?
+
Does the inflation theory explain the modern value of entropy of the Universe?
 
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<!-- <div class="NavFrame collapsed">
 
   <div class="NavHead">solution</div>
 
   <div class="NavHead">solution</div>
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<div id="inf45"></div>
 
<div id="inf45"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
 +
 
=== Problem 51 ===
 
=== Problem 51 ===
Find the solution of the monopole problem in frame of inflation theory.
+
Find the solution of the monopole problem in the frame of inflation theory.
 
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<!-- <div class="NavFrame collapsed">
 
   <div class="NavHead">solution</div>
 
   <div class="NavHead">solution</div>

Latest revision as of 10:06, 27 August 2013


Inflation hasn't won the race,
But so far it's the only horse
Andrei Linde.



Scalar Field In Cosmology

Problem 1

A scalar field $\varphi(\vec r,t)$ in a potential $V(\varphi)$ on flat background is described by Lagrangian \[L=\frac12\left(\dot\varphi^2-\nabla\varphi\cdot\nabla\varphi\right)-V(\varphi)\] Obtain the equation of motion (evolution) for this field from the least action principle.


Problem 2

Rewrite the action for free scalar field minimally coupled to gravitation \[S_\varphi=\int d^4x\sqrt{-g}\left(\frac12 g^{\mu\nu}\partial_\mu\varphi\partial_\nu\varphi\right)\] for the case of FLRW metric.


Problem 3

Using the action obtained in the previous problem, obtain the evolution equation for the scalar field in the expanding Universe.


Problem 4

Calculate the density and pressure of homogeneous scalar field $\varphi(t)$ in potential $V(\varphi)$ in the FLRW metric.


Problem 5

Starting from the scalar field's action in the form \[ S = \int {d^4 x\sqrt { - g} \left[ {{1 \over 2}(\nabla \varphi )^2 - V(\varphi )} \right]} \] obtain the equation of motion for this field for the case of FLRW metric.



Problem 6

Construct the Lagrange function describing the dynamics of the Universe filled with a scalar field in potential $V(\varphi)$. Using the obtained Lagrangian, obtain the Friedman equations and the Klein--Gordon equation.



Problem 7

Obtain the equation of motion for a homogeneous scalar field $\varphi(t)$ in potential $V(\varphi)$ starting from the conservation equation \[\dot\rho+3\frac{\dot a}{a}(\rho+p)=0.\]



Problem 8

Obtain the equation of motion for the homogeneous scalar field $\varphi(t)$ in potential $V(\varphi)$ using the analogy with Newtonian dynamics.



Problem 9

Express $V(\varphi)$ and $\varphi$ through the Hubble parameter $H$ and its derivative $\dot{H}$ for the Universe filled with quintessence.

Problem 10

Show that Friedman equations for the scalar field $\varphi(t)$ in potential $V(\varphi)$ can be presented in the form \[H^2=\frac{8\pi G}{3}\left(\frac12 \dot\varphi^2+V(\varphi)\right),\] \[\dot H=-4\pi G\dot\varphi^2.\]



Problem 11

Provided that the scalar field $\varphi(t)$ is a single--valued function of time, transform the second order equation for the scalar field into a system of first order equations.



Problem 12

Rewrite the equations for the scalar field in terms of conformal time.



Problem 13

Show that condition $\dot H>0$ cannot be realized for the scalar field with positively defined kinetic energy.



Problem 14

Show that the Klein--Gordon equation could be rewritten in dimensionless form $$ \varphi '' + \left( {2 - q} \right)\varphi ' = \chi ;\quad \chi \equiv - \frac{1}{H^2 }\frac{dV}{d\varphi }, $$ where prime denotes the derivative by $\ln a$, and $q = - {{a\ddot a} / {\dot a^2 }}$ is the deceleration parameter.



Problem 15

Represent the equation of motion for the scalar field in the form $$ \pm \frac{V_{,\varphi}}{V} = \sqrt {\frac{3(1 + w)}{\Omega _\varphi(a)}} \left[ 1 + \frac{1}{6}\frac{d\ln \left( x_{\varphi} \right)}{d\ln a} \right], $$ where $$ x_{\varphi}=\frac{1+w_{\varphi}}{1-w_{\varphi}}, \quad w_{\varphi}=\frac{\dot{\varphi}^2+2V(\varphi)}{\dot{\varphi}^2-2V(\varphi)}, $$ in the system of units such that $8\pi G=1.$


Problem 16

The term $3H\dot{\varphi}$ in the equation for the scalar field formally acts as friction that damps the inflation. Show that, nonetheless, this term does not lead to dissipative energy production.


Problem 17

Obtain the system of equations describing the scalar field dynamics in the expanding Universe containing radiation and matter in the conformal time.


Problem 18

Calculate the pressure of homogeneous scalar field in the potential $V(\varphi)$ using the obtained above energy density of the field and its equation of motion.


Problem 19

What condition should the homogeneous scalar field $\varphi(t)$ in potential $V(\varphi)$ satisfy in order to provide accelerated expansion of the Universe?


Problem 20

What conditions should the scalar field satisfy in order to provide expansion of the Universe close to the exponential one?

Introduction to Inflation

Problem 21

What considerations led A.Guth to name his theory describing the early Universe's dynamics as inflation theory?


Problem 22

A. Vilenkin in his cosmological bestseller Many world in one remembers: On a Wednesday afternoon, in the winter of 1980, I was sitting in a fully packed Harvard auditorium, listening to the most fascinating talk I had heard in many years. The speaker was Alan Guth, a young physicist from Stanford, and the topic was a new theory for the origin of the universe$\ldots$ The beauty of the idea was that in a single shot inflation explained why the universe is so big, why it is expanding, and why it was so hot at the beginning. A huge expanding universe was produced from almost nothing. All that was needed was a microscopic chunk of repulsive gravity material. Guth admitted he did not know where the initial chunk came from, but that detail could be worked out later. It's often said that you cannot get something for nothing. he said, but the universe may be the ultimate free lunch . Explain, why can this be.


Problem 23

Is energy conservation violated during the inflation?


Problem 24

Inflation is defined as any epoch for which the scale factor of the Universe has accelerated growth, i.e. $\ddot{a}>0$. Show that this condition is equivalent to requirement that the comoving Hubble radius decreases with time.


Problem 25

Show that in the process of inflation the curvature term in the Friedman equation becomes negligible. Even if that condition was not initially satisfied, the inflation quickly realizes it.


Problem 26

It is sometimes said, that the choice of $k = 0$ is motivated by observations: the density of curvature is close to zero. Is this claim correct?

Inflation in the Slow-Roll Regime

Problem 27

Obtain the evolution equations for the scalar field in expanding Universe in the inflationary slow-roll regime.


Problem 28

Find the time dependence of scale factor in the slow--roll regime for the case $V(\varphi)={m^2 \varphi^2 / 2}$.


Problem 29

Find the dependence of scale factor on the scalar field in the slow-roll regime.


Problem 30

Show that the conditions for realization of the slow--roll limit can be presented in the form: \[\varepsilon(\varphi)\equiv\frac{M^{*2}_{Pl}}{2}\left(\frac{V^\prime}{V}\right)^2\ll1; \ |\eta(\varphi)|\equiv\left|M^{*2}_{Pl}\frac{V^{\prime\prime}}{V}\right|\ll1; \ M^*_{Pl}\equiv(8\pi G)^{-1/2}.\]


Problem 31

Show that the condition $\varepsilon\ll1$ for the realization of the slow--roll limit obtained in the previous problem is also the sufficient condition for inflation.


Problem 32

Find the slow--roll condition for power law potentials.


Problem 33

Show that the condition $\varepsilon \ll\eta$ is satisfied in the vicinity of inflection point of the inflationary potential $V(\varphi)$.


Problem 34

Show that the inflation parameter $\varepsilon$ can be expressed through the state equation parameter $w$ for the scalar field.



Problem 35

Show that the second Friedman equation \[\frac{\ddot{a}}{a}=-\frac{4\pi G}{3}(\rho+3p)\] can be presented in the form \[\frac{\ddot{a}}{a}=H^2(1-\varepsilon).\]



Problem 36

Show that in the slow--roll regime $\varepsilon_H\rightarrow\varepsilon$ and $\eta_H\rightarrow\eta-\varepsilon$.



Problem 37

Show that the inflation parameters $\varepsilon_H,\ \eta_H$ can be presented in the following symmetric form \[\varepsilon_H=-\frac{d\ln H}{d\ln a};\ \eta_H=-\frac{d\ln H'}{d\ln a}.\]



Problem 38

Prove that condition $\ddot a>0$, which defines inflation, is equivalent to $\varepsilon_H<1$.



Problem 39

Show that inflation appears every time when the scalar field's value exceeds the Planck mass.



Problem 40

Find the energy momentum tensor for homogeneous scalar field in the slow--roll regime.

Solution of the Hot Big Bang Theory Problems

Problem 41

Show that in the inflation epoch the relative density $\Omega$ exponentially tends to unity.



Problem 42

Estimate the temperature of the Universe at the end of inflation.



Problem 43

Estimate the size of the Universe at the end of inflation.



Problem 44

Find the number $N_e$ of $e$-foldings of the scale factor during the inflation epoch.



Problem 45

Find the number $N_e$ of $e$-foldings of the scale factor for the inflation process near the inflection point.



Problem 46

Show that inflation transforms the unstable fixed point $x=0$ for the quantity \[x\equiv\frac{\Omega-1}{\Omega}\] into the stable one, therefore solving the problem of the flatness of the Universe.



Problem 47

Find the particle horizon in the inflationary regime, assuming $H\approx const$.



Problem 48

Find the solution of the horizon problem in the framework of inflation theory.



Problem 49

Did entropy change during the inflation period? If yes, then estimate what its change was.



Problem 50

Does the inflation theory explain the modern value of entropy of the Universe?



Problem 51

Find the solution of the monopole problem in the frame of inflation theory.