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Cosmo All at 00:02, 19 June 2015

2015-06-19T00:02:17Z

Cosmo All at 00:00, 19 June 2015

2015-06-19T00:00:16Z

Cosmo All at 23:53, 18 June 2015

2015-06-18T23:53:00Z

Cosmo All at 23:49, 18 June 2015

2015-06-18T23:49:04Z

Cosmo All at 23:47, 18 June 2015

2015-06-18T23:47:37Z

Cosmo All at 23:46, 18 June 2015

2015-06-18T23:46:25Z

Cosmo All at 23:39, 18 June 2015

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Cosmo All at 23:27, 18 June 2015

2015-06-18T23:27:49Z

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Cosmo All at 23:22, 18 June 2015

2015-06-18T23:22:55Z

Cosmo All at 23:22, 18 June 2015

2015-06-18T23:22:21Z

@@ Line 678: / Line 678: @@
 <div id="150_18"></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem 49''' <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 [[#150_8]].
 <div class="NavFrame collapsed">   <div class="NavHead">solution</div>   <div style="width:100%;" class="NavContent">
 <p style="text-align: left;">

@@ Line 357: / Line 357: @@
 \nonumber j&=1.
 \end{align}
 </p>  </div></div></div>
@@ Line 527: / Line 526: @@
 \nonumber \kappa(z)&\equiv\left(\frac{1-\Omega_{m0}}{\Omega_{m0}}\right)(1+z)^{-3(1-n)}.
 \end{align}
 </p>  </div></div></div>
@@ Line 540: / Line 538: @@
 \nonumber \frac{\ddot a^2}{a}&=-\frac16(\rho+3p-9\xi H).
 \end{align}
-''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''
+''Problems [[#150_8]]-[[#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''
@@ Line 596: / Line 594: @@
 <div class="NavFrame collapsed">   <div class="NavHead">solution</div>   <div style="width:100%;" class="NavContent">
 <p style="text-align: left;">
-Using the result of Problem \ref{150_9}
+Using the result of Problem [[#150_9]]
 \[a(t)=\left[\frac{3\exp\left(\frac12\bar\xi H(t-t_0)-3+\bar\xi\right)}{\bar\xi}\right]^{2/3},\]
 one finds the time of the Big Bang as $a(t_{BB})=0$:
@@ Line 637: / Line 635: @@
 <div id="150_15"></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem 46''' <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 [[#150_8]].
 <div class="NavFrame collapsed">   <div class="NavHead">solution</div>   <div style="width:100%;" class="NavContent">
 <p style="text-align: left;">

@@ Line 31: / Line 31: @@
 The two terms define the behavior of the uniform fluid which contains the energy in a dynamic Universe. The $H$ term provides the  "friction", while  the density term tracks the reduction in density due to the volume increase during expansion and  the pressure  term tracks the reduction in pressure energy during expansion.
 </p>  </div></div></div>
@@ Line 52: / Line 51: @@
 \[\frac{d\Omega}{dN}=-(1+3w)(1-\Omega)\Omega.\]
 </p>  </div></div></div>
@@ Line 81: / Line 79: @@
 We see a smooth transition between the two characteristic behaviors of radiation domination $\eta\ll\eta_{eq}$, $a\propto\eta$ and matter domination $\eta\gg\eta_{eq}$, $a\propto\eta^2$. There is no good way to rewrite this relation in terms of the cosmic times.
 </p>  </div></div></div>
@@ Line 171: / Line 168: @@
-\paragraph{To chapter 3, section 15, if absent}
+==To chapter 3==
-−
@@ Line 181: / Line 177: @@
 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 $t_0=H_0^{-1}$.
 </p>  </div></div></div>
 ==To chapter 4 The black holes==
 <div id="new2015_1"></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem 16''' <p style= "color: #999;font-size: 11px">problem id: new2015_1</p>
@@ Line 216: / Line 214: @@
 Up to a numeric factor this relation coincides with the black hole Hawking temperature $T=\hbar c^3/8\pi kGM$.
 </p>  </div></div></div>
 ==To chapter 8==
 <div id="2501_06"></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem 18''' <p style= "color: #999;font-size: 11px">problem id: 2501_06</p>
@@ Line 250: / Line 250: @@
 The number of $e$-folds depends on how fast the field is $f$-decreasing. The number of e-folds, $N$ , is inversely proportional to the square root of the slow roll parameter $\varepsilon$ or proportional to the inverse fractional change of the potential with the field, $V/(dV/d\varphi)$.
 </p>  </div></div></div>
 ==To chapter 9==
@@ Line 331: / Line 332: @@
 (place after the problem 81, chapter 9 of the book.)
 </p>  </div></div></div>
 ==A couple of problems for the SCM==

← Older revision		Revision as of 23:49, 18 June 2015
Line 1:		Line 1:
	__TOC__		__TOC__
		+
		+
		+	==To Chapter 2==

	<div id="150_0"></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem 1''' <p style= "color: #999;font-size: 11px">problem id: 150_0</p>		<div id="150_0"></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem 1''' <p style= "color: #999;font-size: 11px">problem id: 150_0</p>

← Older revision		Revision as of 23:47, 18 June 2015
Line 1:		Line 1:
		+	__TOC__
		+
	<div id="150_0"></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem 1''' <p style= "color: #999;font-size: 11px">problem id: 150_0</p>		<div id="150_0"></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem 1''' <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.]

@@ Line 177: / Line 177: @@
 </p>  </div></div></div>
-\paragraph{To chapter 4 The black holes}
+==To chapter 4 The black holes==
 <div id="new2015_1"></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem 16''' <p style= "color: #999;font-size: 11px">problem id: new2015_1</p>
@@ Line 212: / Line 212: @@
 </p>  </div></div></div>
-\paragraph{To chapter 8}
+==To chapter 8==
 <div id="2501_06"></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem 18''' <p style= "color: #999;font-size: 11px">problem id: 2501_06</p>
@@ Line 236: / Line 236: @@
 (see Inflation and the Higgs Scalar, DANIEL GREEN (1412.2107).)
 </p>  </div></div></div>
@@ Line 247: / Line 246: @@
 </p>  </div></div></div>
-paragraph{To chapter 9}
+==To chapter 9==
@@ Line 328: / Line 327: @@
 </p>  </div></div></div>
-(a couple of problems for the SCM:)
+==A couple of problems for the SCM==
@@ Line 355: / Line 354: @@
-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.)
+==Cardassian  Model==
 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 $M_{Pl}^2\equiv1/G$.
+---------------
@@ Line 524: / Line 525: @@
 ----
-{Models with Cosmic Viscosity}
+==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
-	  	\[T_{\mu\nu}=(\rho=p-\xi\theta)u_\mu u_\nu+(p-\xi\theta)g_{\mu\nu},\]
+\[T_{\mu\nu}=(\rho=p-\xi\theta)u_\mu u_\nu+(p-\xi\theta)g_{\mu\nu},\]
 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}
@@ Line 532: / Line 533: @@
 \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}
+''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}
@@ Line 571: / Line 571: @@
 <div class="NavFrame collapsed">   <div class="NavHead">solution</div>   <div style="width:100%;" class="NavContent">
 <p style="text-align: left;">
-\begin{enumerate}
+(a) $0<\bar\xi<3$ <br/>
-\item $0<\bar\xi<3$
-−
 When $t\to\infty$ then the obtained solution reproduces the de Sitter-like Universe,
 \[a(t)\propto\exp\left[\frac{\bar\xi}3H(t-t_0)\right]\]
+(b) $\bar\xi=3$<br/>
-\item $\bar\xi=3$
-−
 In this case the considered model exactly reproduces the de Sitter-like Universe,
 \[a(t)=\exp\left[H_0(t-t_0)\right]\]
 The model predicts an Universe in an eternal accelerated expansion.
+<br/><br/>
-\item $\bar\xi>3$
+(c) $\bar\xi>3$
+<br/>
 In this case the Universe expands forever (decelerated expansion epoch is absent).
 </p>  </div></div></div>
@@ Line 654: / Line 649: @@
 <div class="NavFrame collapsed">   <div class="NavHead">solution</div>   <div style="width:100%;" class="NavContent">
 <p style="text-align: left;">
-\begin{enumerate}
+(a) For the case $\bar\xi=0$ we have $q=1/2$ that corresponds to a matter-dominated universe with null bulk viscosity.<br/>
-\item For the case $\bar\xi=0$ we have $q=1/2$ that corresponds to a matter-dominated universe with null bulk viscosity.
+(b) For the case $\bar\xi=3$ we have $q=-1$ that corresponds to the de Sitter Universe.<br/>
-\item For the case $\bar\xi=3$ we have $q=-1$ that corresponds to the de Sitter Universe.
+(c) For the case $0<\bar\xi<3$ it is always a decreasing function from $q(0)=1/2$ to $q(\infty)=-1$ with a transition from positive to negative values in the value of the scale factor \[a_t=\left[\frac{3-\bar\xi}{2\bar\xi}\right]^{3/2}.\]
-\item For the case $0<\bar\xi<3$ it is always a decreasing function from $q(0)=1/2$ to $q(\infty)=-1$ with a transition from positive to negative values in the value of the scale factor \[a_t=\left[\frac{3-\bar\xi}{2\bar\xi}\right]^{3/2}.\]
+(d) For the case $\bar\xi>3$ the Universe expands forever, and this expansion is always accelerated (there is no decelerating epoch or acceleration-deceleration transition).
-\item For the case $\bar\xi>3$ the Universe expands forever, and this expansion is always accelerated (there is no decelerating epoch or acceleration-deceleration transition).
+<br/>
-−
 When $a\to a_*\equiv(1-3/\bar\xi)^{2/3}$ $q\to-\infty$ and when $a\to\infty$ then $q\to-1$. It is a negative and increasing function but it never becomes positive, its maximum value is $-1$.
+<br/>
-\item The case $\bar\xi<3$ predicts an eternal decelerated expanding Universe. The universe begins with a Big-Bang and expands forever until to reach its maximum value $a_*=(1-3/\bar\xi)^{2/3}$ when $t\to\infty$. Deceleration parameter is a positive and increasing monotonic function where its minimum value is $1/2$.
+(e) The case $\bar\xi<3$ predicts an eternal decelerated expanding Universe. The universe begins with a Big-Bang and expands forever until to reach its maximum value $a_*=(1-3/\bar\xi)^{2/3}$ when $t\to\infty$. Deceleration parameter is a positive and increasing monotonic function where its minimum value is $1/2$.
-\end{enumerate}
 </p>  </div></div></div>

@@ Line 11: / Line 11: @@
 <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|Cosmography#150_1]] and extended deceleration parameter) Show that \[\frac{d\dot a}{da}=-Hq.\]
+(into the [[Cosmography#150_1|cosmography]] and extended deceleration parameter) Show that \[\frac{d\dot a}{da}=-Hq.\]
 <div class="NavFrame collapsed">   <div class="NavHead">solution</div>   <div style="width:100%;" class="NavContent">
 <p style="text-align: left;">